>Instructions
Complete the tab quity using the Capital Asset Pricing Model (CAPM)
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Complete the tab re-evaluation table
CF Payback timeline
Complete the tab venue increases % annually
Cost of Capital Find the cost of Equity using the Capital Asset Pricing Model (CAPM) End . Find the cost of equity using CAPM.
. Find the cost of debt.
WACC Information from Largo Global ,3 3,341,000.1 Payback .0%
million Interest Expense 1
PV NPV IRR NPV $900.0 $1,300.0 $1,300.0 NPV IRR | | | | | | | | 3 $0.00 After-Tax Cash Flow Re-evlauation and Payback Timelines Instructions Budget Projections Complete the budget projections for years – using the following information
s assume the Acutal figure as the base for the budget and forecast.
2024 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Discussion
Answer the following questions: 1. Discuss the concepts that were most challenging for you in the readings and review material. How did the practice exercises help clarify these? 2. What did you learn that will help you determine the most appropriate way to finance the investments you previously recommended for LGI? Lesson 5 Project 5: Cost of Capital, Risk/Return, and Capital Budgeting PeopleImages / E+ / Getty Images You have developed an in-depth understanding of LGI’s operating efficiency related to costing and how that impacts the bottom line. You feel confident that your investment choices will positively boost LGI’s productivity and improve the company’s operations. Thanks to your efforts, the company will have a plan for financing its investments appropriately. LGI will finally be on a path of a sustainable future. Answer the questions in the Project 5 Questions – Report Template document. Prepare your analysis report including recommendations for how the company can improve its financial situation.
Complete the analysis report: · Download the Project 5 Questions – Report Template
· Read the instructions. · Answer all the questions. · Include your recommendations. · Submit the analysis report (Word document) and analysis calculation (Excel file) to the dropbox as your final deliverable at the end of Week 10. Label your files as follows: · P5_Final_lastname_Report_date x · P5_Final_lastname_Calculation_date.xlsx
Check Your Evaluation Criteria
Before you submit your assignment, review the competencies below, which your instructor will use to evaluate your work. A good practice would be to use each competency as a self-check to confirm you have incorporated all of them. To view the complete grading rubric, click My Tools, select Assignments from the drop-down menu, and then click the project title. · 3.1: Identify numerical or mathematical information that is relevant in a problem or situation. · 3.2: Employ mathematical or statistical operations and data analysis techniques to arrive at a correct or optimal solution. · 3.3: Analyze mathematical or statistical information, or the results of quantitative inquiry and manipulation of data. · 10.4: Make strategic managerial decisions for obtaining capital required for achieving organizational goals. Project 5: Cost of Capital, Risk/Return, and Capital Budgeting Print Project Transcript Scenario Your team’s work with Largo Global Inc. (LGI) is nearly complete. In your consulting role, you have recommended steps for improving the company’s financial health. You have offered advice on a revenue target, recommended steps for optimizing operations, and suggested investments that will improve LGI’s competitive position. In your final project, you will continue working on capital budgeting with a focus on the best way for LGI to finance the investments you recommend. Your Project 5 business report will focus on ensuring LGI’s capital structure is sound and that the company is on a financially sustainable path. You will recommend a plan for financing investments that does not expose LGI to unnecessary risk. By the end of this project, the company’s financial statements should demonstrate that it has returned to a competitive position.
Competencies
Your work will be evaluated using the competencies listed below. · 3.1: Identify numerical or mathematical information that is relevant in a problem or situation. Project 5: Cost of Capital, Risk/Return, and Capital Budgeting For the next two weeks, you will focus on the concepts of risk and return, the cost of capital, and capital budgeting. You will ensure that the financing plan you recommend supports LGI’s long-term financial position. Log into O’Reilly by following these instructions and complete the required reading.
Required Reading
Parrino, R., Kidwell, D. S., & Bates, T. W. (2012). Fundamentals of Corporate Finance (2nd ed.) Wiley. Chapter 7: Risk and Return · Section 7.1 to 7.7 Chapter 10: The Fundamentals of Capital Budgeting · Sections 10.1 to 10.6 Chapter 13: The Cost of Capital · Sections 13.1 to 13.4 Project 5: Cost of Capital, Risk/Return, and Capital Budgeting Using the Project 5 Review and Practice Guide , review the options for financing investments available to LGI. Then, apply what you have learned by completing the exercises and problems referenced in the Project 5 Review and Practice Guide. You must review the guide and do the practice exercises and problems so that you · are prepared to have informed discussions with your team about capital structure and appropriate leverage, · understand risk and return, · can make recommendations for financing LGI’s investments, and · can complete your mission, turning around LGI. Complete this review and practice by the end of Week 9.
Review and Practice
You must complete the review and practice content to participate in the discussion in Step 3.
Project 5 Review and Practice Guide
Project 5: Cost of Capital, Risk/Return, and Capital Budgeting PeopleImages / E+ / Getty Images You have finished reviewing the material and performing the exercises, but you have some questions. Participate in the Project 5 class discussion. Respond to the two questions below by posting in the discussion; then, respond to two of your classmates’ discussion posts by the end of the week.
Discussion
Answer the following questions: 1. Discuss the concepts that were most challenging for you in the readings and review material. How did the practice exercises help clarify these? 2. What did you learn that will help you determine the most appropriate way to finance the investments you previously recommended for LGI? Before you participate in the discussion activity, see
MBA discussion guidelines
. Project 5: Cost of Capital, Risk/Return, and Capital Budgeting Your team has provided you with an Excel workbook containing LGI’s financials. You will use the
Project 5 Excel workbook
to perform advanced capital budgeting techniques to assess the viability of the investment you made in the previous project.
Complete the analysis calculation for the project:
· Download the
Project 5 Excel Workbook
, click the Instructions tab, and read the instructions. · Calculate cost of debt and equity as well as weighted average cost of capital (WACC). · Apply the capital asset pricing model (CAPM). · Develop a capital budget. If you would like instructor feedback on this step, submit your Excel file to the Assignments folder as a milestone by the end of Week 9. This is optional. If you choose to submit the milestone, you will receive instructor feedback you may use to make corrections before submitting your final Project 5. To distinguish the milestone submission from the file you will submit in Step 5, label your file as follows: P5_milestone_lastname_Calculation_date · You should use the comments your instructor made about your optional milestone submission to revise your calculations as needed. Next, proceed to Step 5, where you will answer questions about your analysis, make recommendations for LGI, and compile and submit your final report. Project 5: Cost of Capital, Risk/Return, and Capital Budgeting Lesson 5 reading © David Young-Wolff/PhotoEdit BUILDING INTUITION MORE RISK MEANS A HIGHER EXPECTED RETURN total holding period return where P0 is the price paid for the asset at time zero and P1 is the price at a later point in time. where CF1 is the cash flow from the dividend. You can download actual realized investment returns for a large number of stock market indexes at the Callan Associates Web site, What return would you have earned if Dell had paid a $1 dividend and today’s price was $12.90? With the $1 dividend and a correspondingly lower price, the total return is the same: You can see from this example that a dollar of capital appreciation is worth the same as a dollar of income. Calculating the Return on an Investment PROBLEM: You purchased a beat-up 1974 Datsun 240Z sports car a year ago for $1,500. Datsun is what Nissan, the Japanese car company, was called in the 1970s. The 240Z was the first in a series of cars that led to the Nissan 370Z that is being sold today. Recognizing that a mint-condition 240Z is a much sought-after car, you invested $7,000 and a lot of your time fixing up the car. Last week, you sold it to a collector for $18,000. Not counting the value of the time you spent restoring the car, what is the total return you earned on this investment over the one-year holding period? Expected Returns What is the expected value of your bonus? If you have taken a statistics course, you might recall that an expected value represents the sum of the products of the possible outcomes and the probabilities that those outcomes will be realized. In our example the expected value of the bonus can be calculated using the following formula: where E(Bonus) is your expected bonus, p Notice that the expected bonus of $530,000 is not equal to either of the two possible payoffs. Neither is it equal to the simple average of the two possible payoffs. This is because the expected bonus takes into account the probability of each event occurring. If the probability of each event had been 50 percent, then the expected bonus would have equaled the simple average of the two payoffs: However, since it is more likely that you will not get a hit (a 67.5 percent chance) than that you will get a hit (a 32.5 percent chance), and the payoff is lower if you do not get a hit, the expected bonus is less than the simple average. The key point here is that the expected value reflects the relative likelihoods of the possible outcomes. expected return is mathematical shorthand indicating that n values are added together. In Equation 7.2, each of the n possible returns is multiplied by the probability that it will be realized, and these products are then added together to calculate the expected return. To see how we calculate the expected return on an asset, suppose you are considering purchasing Dell, Inc. stock for $13.90 per share. You plan to sell the stock in one year. You estimate that there is a 30 percent chance that Dell stock will sell for $13.40 at the end of one year, a 30 percent chance that it will sell for $14.90, a 30 percent that it will sell for $15.40, and a 10 percent chance that it will sell for $16.00. If Dell pays no dividends on its shares, what is the return that you expect from this stock in the next year? Therefore, we can calculate the return from owning Dell stock under each of the four possible outcomes using the approach we used for the similar Dell problem we solved earlier in the chapter. These returns are calculated as follows: Applying Equation 7.2, the expected return on Dell stock over the next year is therefore 5.83 percent, calculated as follows: Notice that the negative return is entered into the formula just like any other. Also notice that the sum of the pi Calculating Expected Returns PROBLEM: You have just purchased 100 railroad cars that you plan to lease to a large railroad company. Demand for shipping goods by rail has recently increased dramatically due to the rising price of oil. You expect oil prices, which are currently at $98.81 per barrel, to reach $115.00 per barrel in the next year. If this happens, railroad shipping prices will increase, thereby driving up the value of your railroad cars as increases in demand outpace the rate at which new cars are being produced. you must recognize that ΔP/P0 is the capital appreciation under each outcome and that CF1/P0 equals the 10 percent that you expect to receive from leasing the rail cars. The expected returns for the three outcomes are: You can then use Equation 7.2 to calculate the expected return for your rail car investment: Alternatively, since there is a 100 percent probability that the return from leasing the railroad cars is 10 percent, you could have simply calculated the expected increase in value of the railroad cars: and added the 10 percent to arrive at the answer of 33.5 percent. Of course, this simpler approach only works if the return from leasing is known with certainty. SITUATION: You are deciding whether you should advertise your pizza business on the radio or on billboards placed on local taxicabs. For $1,000 per month, you can either buy 20 one-minute ads on the radio or place your ad on 40 taxicabs. This means that you expect 20 one-minute ads to bring in 20 × 48 = 960 new customers. Placing ads on 40 taxicabs is therefore expected to bring in 40 35 1,400 new customers. Unfortunately, using this calculation to obtain a measure of risk presents two problems. First, since one difference is positive and the other difference is negative, one difference partially cancels the other. As a result, you are not getting an accurate measure of total risk. Second, this calculation does not take into account the number of potential outcomes or the probability of each outcome. Note that the square of the Greek symbol sigma, σ2, is generally used to represent the variance. standard deviation (σ) Equation 7.3 simply extends the calculation illustrated above to the situation where there are n possible outcomes. Like the expected return calculation (Equation 7.2), Equation 7.3 can be simplified if all of the possible outcomes are equally likely. In this case it becomes: In both the general case and the case where all possible outcomes are equally likely, the standard deviation is simply the square root of the variance . EXHIBIT 7.1 Normal Distribution Since the returns on many assets are approximately normally distributed, the standard deviation provides a convenient way of computing the probability that the return on an asset will fall within a particular range. In these applications, the expected return on an asset equals the mean of the distribution, and the standard deviation is a measure of the uncertainty associated with the return. EXHIBIT 7.2 Standard Deviation and Width of the Normal Distribution Understanding the Standard Deviation PROBLEM: You are considering investing in a share of Google Inc., stock and want to evaluate how risky this potential investment is. You know that stock returns tend to be normally distributed, and you have calculated the expected return on Google stock to be 4.67 percent and the standard deviation of the annual return to be 23 percent. Based on these statistics, within what range would you expect the return on this stock to fall during the next year? Calculate this range for a 90 percent level of confidence (that is, 90 percent of the time, the returns will fall within the specified range). EXHIBIT 7.3 Distributions of Annual Total Returns for U.S. Stocks and Bonds from 1926 to 2009 EXHIBIT 7.4 Monthly Returns for Apple Inc. stock and the S&P 500 Index from September 2005 through September 2010 EXHIBIT 7.5 Cumulative Value of $1 Invested in 1926 With this information, we can calculate the expected returns for AMD and Intel by using Equation 7.2: and Similarly, we can calculate the standard deviations of the returns for AMD and Intel in the same way that we calculated the standard deviation for our baseball bonus example in Section 7.2: and Having calculated the expected returns and standard deviations for the expected returns on AMD and Intel stock, the natural question to ask is which provides the highest risk- adjusted return. Before we answer this question, let’s return to the example at the beginning of Section 7.1. Recall that, in this example, we proposed choosing among three stocks: A, B, and C. We stated that investors would prefer the investment that provides the highest expected return for a given level of risk or the lowest risk for a given expected return. This made it fairly easy to choose between Stocks A and B, which had the same return but different risk levels, and between Stocks B and C, which had the same risk but different returns. We were stuck when trying to choose between Stocks A and C, however, because they differed in both risk and return. Now, armed with tools for quantifying expected returns and risk, we can at least take a first pass at comparing stocks such as these. In this equation, CV is a measure of the risk associated with an investment for each 1 percent of expected return. Since these values are equal, the coefficient of variation measure suggests that these two investments are equally attractive on a risk-adjusted basis. where CVi* is a modified coefficient of variation that is computed by subtracting the risk-free rate from the expected return. We can see that the modified coefficient of variation for AMD is smaller than the modified coefficient of variation for Intel. This tells us that an investment in AMD stock is expected to have less risk for each 1 percent of return. Since investors prefer less risk for a given level of return, the AMD stock is a more attractive investment. Sharpe Ratio You can read more about the Sharpe Ratio and other ratios that are used to measure risk-adjusted returns for investments at the following Web site: PROBLEM: You are trying to choose between two investments. The first investment, a painting by Picasso, has an expected return of 14 percent with a standard deviation of 30 percent over the next year. The second investment, a pair of blue suede shoes once worn by Elvis, has an expected return of 20 percent with a standard deviation of return of 40 percent. The risk-free rate of interest is 3 percent. What is the Sharpe Ratio for each of these investments, and what do these ratios tell us? The Sharpe Ratio for Elvis’s blue suede shoes is larger than the Sharpe Ratio for the painting. This indicates that the return for each 1 standard deviation of risk is greater for Elvis’s shoes than for the painting. Notice that this formula is just like the expected return formula for an individual stock. However, in this case, instead of multiplying outcomes by their associated probabilities, we are multiplying expected returns for individual stocks by the fraction of the total portfolio value that each of these stocks represents. In other words, the formula for the expected return for a two-stock portfolio is: where x This equation is just like Equation 7.2, except that (1) the returns are expected returns for individual assets and (2) instead of multiplying by the probability of an outcome, we are multiplying by the fraction of the portfolio invested in each asset. Note that this equation can be used only if you have already calculated the expected return for each stock. Therefore, the expected return on the portfolio is: Risk of a Portfolio with more than one Asset PROBLEM: You have become concerned that you have too much of your money invested in your pizza restaurant and have decided to diversify your personal portfolio. Right now the pizza restaurant is your only investment. To diversify, you plan to sell 45 percent of your restaurant and invest the proceeds from the sale, in equal proportions, into a stock market index fund and a bond market index fund. Over the next year, you expect to earn a return of 15 percent on your remaining investment in the pizza restaurant, 12 percent on your investment in the stock market index fund, and 8 percent on your investment in the bond market index fund. What return will you expect from your diversified portfolio over the next year? At 12.75 percent, the expected return is an average of the returns on the individual assets in your portfolio, weighted by the fraction of your portfolio that is invested in each. Exhibit 7.6 where x covariance of returns EXHIBIT 7.6 Monthly Returns for Southwest Airlines and Netflix Stock from September 2005 through September 2010 You can see that the covariance calculation is very similar to the variance calculation. The difference is that, instead of squaring the difference between the return from each outcome and the expected return for an individual asset, we calculate the product of this difference for two different assets. The correlation between the returns on two assets will always have a value between −1 and + 1. This makes the interpretation of this variable straightforward. A negative correlation means that the returns tend to have opposite signs. For example, when the return on one asset is positive, the return on the other asset tends to be negative. If the correlation is exactly − 1, the returns on the two assets are perfectly negatively correlated. In other words, when the return on one asset is positive, the return on the other asset will always be negative. A positive correlation means that when the return on one asset is positive, the return on the other asset also tends to be positive. If the correlation is exactly equal to + 1, then the returns of the two assets are said to be perfectly positively correlated. The return on one asset will always be positive when the return on the other asset is positive. Finally, a correlation of 0 means that the returns on the assets are not correlated. In this case, the fact that the return on one asset is positive or negative tells you nothing about how likely it is that the return on the other asset will be positive or negative. You can see that this portfolio variance is smaller than the variance of either the Southwest Airlines or Netflix stock on its own. Exhibit 7.7 EXHIBIT 7.7 Monthly Returns for Southwest Airlines and Netflix Stock and for a Portfolio with 50 Percent of the Value in Each of these Two Stocks from September 2005 through September 2010 The positive correlation tells us that the prices of Southwest Airlines and Netflix stock tend to move in the same direction. However, the correlation of less than one tells us that they do not always do so. The fact that the prices of these two shares do not always move together is the reason that the returns on a portfolio of the two stocks have less variation than the returns on the individual company shares. This example illustrates the benefit of diversification—how holding more than one asset with different risk characteristics can reduce the risk of a portfolio. Note that if the correlation of the returns between Southwest Airlines and Netflix stock equaled one, holding these two stocks would not reduce risk because their prices would always move up or down together. Calculating the Variance of a Two-Asset Portfolio PROBLEM: You are still planning to sell 45 percent of your pizza restaurant in order to diversify your personal portfolio. However, you have now decided to invest all of the proceeds in the stock market index fund. After you diversify, you will have 55 percent of your wealth invested in the restaurant and 45 percent invested in the stock market index fund. You have estimated the variances of the returns for these two investments and the covariance between their returns to be as follows: What will be the variance and standard deviation of returns in your portfolio after you have sold the ownership interest in your restaurant and invested in the stock market index fund? and the standard deviation is (0.0394)1/2 = 0.1985, or 19.85 percent. EXHIBIT 7.8 Total Risk in a Portfolio as the Number of Assets Increases EXHIBIT 7.9 Plot of Monthly General Electric Company Stock and S&P 500 Index Returns: October 2005 through September 2010 EXHIBIT 7.10 Slope of Relation Between General Electric Company Monthly Stock Returns and S&P 500 Index Returns: October 2005 through September 2010 Exhibit 7.10 Although we have to be careful about drawing conclusions when we have only two data points, we might interpret the slope of 2 to indicate that new information that causes the market return to increase by 1 percent will tend to cause the return on Nike stock to increase by 2 percent. Of course, the reverse might also be true. That is, new information that causes the market return to decrease by 1 percent may also cause the return on Nike stock to go down by 2 percent. To the extent that the same information is driving the changes in returns on Nike stock and on the market, it would not be possible for an investor in Nike stock to diversify this risk away. It is nondiversifiable, or systematic, risk. Now you might ask yourself what happened to the unsystematic risk of GE or Nike stock. This is best illustrated by the GE example, where we have more than two observations. As you can see in where Rrf is the return on a security with a risk-free rate of return, which analysts typically estimate by looking at returns on government securities. The compensation for taking risk, which varies with the risk of the asset, is added to the risk-free rate of return to get an estimate of the expected rate of return for an asset. If we recognize that the compensation for taking risk varies with asset risk and that systematic risk is what matters, we can write the preceding equation as follows: where units of systematic risk where β where E (Rm) is the expected return on the market. The term E (Rm) − Rrf is called the market risk premium. Consequently, we can now write the equation for expected return as: 7.7 THE CAPITAL ASSET PRICING MODEL Note that we must have three pieces of information in order to use Equation 7.10: (1) the risk-free rate, (2) beta, and (3) either the market risk premium or the expected return on the market. Recall that the market risk premium is the difference between the expected return on the market and the risk-free rate [E(Rm) − Rrf], which is 6 percent in the above example. Security Market Line (SML) Exhibit 7.11 EXHIBIT 7.11 The Security Market Line where P0 is the price that the asset is currently selling for. If an asset’s price implies that the expected return is greater than that predicted by the CAPM, that asset will plot above the SML in PROBLEM: You are considering buying 100 shares of General Electric stock. Value Line (a financial reporting service) reports that the beta for General Electric is 1.61. The risk-free rate is 4 percent, and the market risk premium is 6 percent. What is the expected rate of return on General Electric stock according to the CAPM? The Capital Asset Pricing Model and Portfolio Returns Of course, this should not be surprising since investing in a portfolio is simply an alternative to investing in a single asset. If we substitute Equation 7.10 into Equation 7.6 for each of the n assets and rearrange the equation, we find that the beta for a portfolio is simply a weighted average of the betas for the individual assets in the portfolio. In other words: where x Your portfolio has 1.25 times as much systematic risk as the market. Based on Equation 7.10, you would, therefore, expect to earn a return of 11.5 percent, calculated as follows: Up to this point, we have focused on calculating the expected rate of return for an investment in any asset from the perspective of an investor, such as a stockholder. A natural question that might arise is how these concepts relate to the rate of return that should be used within a firm to evaluate a project. The short answer is that they are the same. The rate of return used to discount the cash flows for a project with a particular level of systematic risk is exactly the same as the rate of return that an investor would expect to receive from an investment in any asset having the same level of systematic risk. In Portfolio Risk and Expected Return PROBLEM: You have recently become very interested in real estate. To gain some experience as a real estate investor, you have decided to get together with nine of your friends to buy three small cottages near campus. If you and your friends pool your money, you will have just enough to buy the three properties. Since each investment requires the same amount of money and you will have a 10 percent interest in each, you will effectively have one-third of your portfolio invested in each cottage. Therefore, from Equation 7.6, the expected return on the portfolio is: Using the second approach, from Equation 7.11, the beta of the portfolio is: and from Equation 7.10, the expected return is: EXAMPLE 7.2 DECISION MAKING SITUATION: You are trying to decide whether to invest in one or both of two different stocks. Stock 1 has a beta of 0.8 and an expected return of 7.0 percent. Stock 2 has a beta of 1.2 and an expected return of 9.5 percent. You remember learning about the CAPM in school and believe that it does a good job of telling you what the appropriate expected return should be for a given level of risk. Since the risk-free rate is 4 percent and the market risk premium is 6 percent, the CAPM tells you that the appropriate expected rate of return for an asset with a beta of 0.8 is 8.8 percent. The corresponding value for an asset with a beta of 1.2 is 11.2 percent. Should you invest in either or both of these stocks? Explain the relation between risk and return. Self-Study Problems and solve for σ. Doing this with either equation yields: · 7.4 A comparison of the Sharpe Ratios for the two stocks will tell you which has the highest expected return per unit of total risk. Stock B has the highest expected return per unit of risk. · 7.1 Returns: Describe the difference between a total holding period return and an expected return. · 7.13 Expected returns: Jose is thinking about purchasing a softdrink machine and placing it in a business office. He knows that there is a 5 percent probability that someone who walks by the machine will make a purchase from the machine, and he knows that the profit on each softdrink sold is $0.10. If Jose expects a thousand people per day to pass by the machine and requires a complete return of his investment in one year, then what is the maximum price that he should be willing to pay for the softdrink machine? Assume 250 working days in a year and ignore taxes and the time value of money. · 7.17 Calculating the variance and standard deviation: Ben would like to invest in gold and is aware that the returns on such an investment can be quite volatile. Use the following table of states, probabilities, and returns to determine the expected return and the standard deviation of the return on Ben’s gold investment. · 7.18 Single-asset portfolios: Using the information from Problems 7.15, 7.16, and 7.17, calculate the coefficient of variation for each of the investments in those problems. · 7.21 Compensation for bearing systematic risk: You have constructed a diversified portfolio of stocks such that there is no unsystematic risk. Explain why the expected return of that portfolio should be greater than the expected return of a risk-free security. · 7.27 David is going to purchase two stocks to form the initial holdings in his portfolio. Iron stock has an expected return of 15 percent, while Copper stock has an expected return of 20 percent. If David plans to invest 30 percent of his funds in Iron and the remainder in Copper, what will be the expected return from his portfolio? What if David invests 70 percent of his funds in Iron stock? Assume the CAPM and SML are true and fill in the missing values in the table. Would you invest in the stock of any of the three firms? If so, which one(s) and why? 1 2 3 4 5 6 AFP/Getty Images, Inc. capital budgeting Reason Description Renewal: Over time, equipment must be repaired, overhauled, rebuilt, or retrofitted with new technology to keep the firm’s manufacturing or service operations going. For example, a company that has a fleet of delivery trucks may decide to overhaul the trucks and their engines rather than purchase new trucks. Renewal decisions typically do not require an elaborate analysis and are made on a routine basis. Replacement: At some point, an asset will have to be replaced rather than repaired or overhauled. The major decision is whether to replace the asset with a similar piece of equipment or purchase equipment that would require a change in the production process. Sometimes, replacement decisions involve equipment that is operating satisfactorily but has become obsolete. The new or retrofitted equipment may provide cost savings with respect to labor or material usage and/or may improve product quality. These decisions typically originate at the plant level. Expansion: Strategically, the most important motive for capital expenditures is to expand the level of operating output. One type of expansion decision involves increasing the output of existing products. This may mean new equipment to produce more products or expansion of the firm’s distribution system. These types of decisions typically require a more complex analysis than a renewal or replacement decision. Another type of expansion decision involves producing a new product or entering a new market. This type of expansion often involves large dollar amounts and significant business risk and requires the approval of the firm’s board of directors. Regulatory: Some capital expenditures are required by federal and state regulations. These mandatory expenditures usually involve meeting workplace safety standards and environmental standards. Other: This category includes items such as parking facilities, office buildings, and executive aircraft. Many of these capital expenditures are hard to analyze because it is difficult to estimate their cash inflows. Ultimately, the decisions can be more subjective than analytical. Sources of Information mutually exclusive projects Cost of Capital cost of capital If a capital project has a positive NPV, the value of the cash flows the project is expected to generate exceeds the project’s cost. Thus, a positive NPV project increases the value of the firm and, hence, stockholders’ wealth. If a capital project has a negative NPV, the value of the cash flows from the project is less than its cost. If accepted, a negative NPV project will decrease the value of the firm and stockholders’ wealth. The $475,000 price paid for the pizza parlor exceeds the cost ($155,000) by $320,000. You have created $320,000 in value. How did you do this? You did it by improving the food, customer service, and dining ambiance while keeping prices competitive. Your management skills and knowledge of the pizza business resulted in significant growth in the current year’s cash flows and the prospect of even larger cash flows in the future. EXHIBIT 10.2 Sample Worksheet for Net Present Value Analysis where: Next, we will work an example to see how the NPV is calculated for a capital project. Suppose you are the president of a small regional firm located in Chicago that manufactures frozen pizzas, which are sold to grocery stores and to firms in the hospitality and food service industry. Your market research group has developed an idea for a “pocket” pizza that can be used as an entrée with a meal or as an “on the go” snack. The sales manager believes that, with an aggressive advertising campaign, sales of the product will be about $300,000 per year. The cost to modify the existing production line will also be $300,000, according to the plant manager. The marketing and plant managers estimate that the cost to produce the pocket pizzas, to market and advertise them, and to deliver them to customers will be about $220,000 per year. The product’s life is estimated to be five years, and the specialized equipment necessary for the project has an estimated salvage value of $30,000. The appropriate cost of capital is 15 percent. EXHIBIT 10.3 Pocket Pizza Project Time Line and Cash Flows ($ thousands) The NPV for the pocket pizza project is therefore $16,910. The PV of the future cash flows is − $283.09. With that information, we can compute the NPV using Equation 10.1 as follows: APPLICATION 10.1 LEARNING BY DOING The Dough’s Up: The Self-Rising Pizza Project PROBLEM: Let’s continue our frozen pizza example. Suppose the head of the research and development (R&D) group announces that R&D engineers have developed a breakthrough technology—self-rising frozen pizza dough that, when baked, rises and tastes exactly like fresh-baked dough. 5. The project is economically viable. The NPV for the self-rising pizza dough project is $118,567. Because the NPV is positive, management should accept the project. The project is estimated to increase the value of the firm by $118,567. EXHIBIT 10.4 Self-Rising Pizza Dough Project Time Line and Cash Flows ($ thousands) Mutually Exclusive Projects and NPV SITUATION: Suppose you are the manager of the information systems (IS) department of the frozen pizza manufacturer we have been discussing. Your department has identified four possible capital projects with the following NPVs: (1) $4,500, (2) $3,000, (3) $0.0, and (4) $1,000. What should you decide about each project if the projects are independent? What should you decide if the projects are mutually exclusive? > BEFORE YOU GO ON EXHIBIT 10.5 Payback Period Cash Flows and Calculations Exhibit 10.5 Let’s look at this calculation in more detail. Note in A Payback Calculation PROBLEM: A firm has two capital projects, A and B, which are under review for funding. Both projects cost $500, and the projects have the following cash flows: What is the payback period for each project? If the projects are independent, which project should management select? If the projects are mutually exclusive, which project should management accept? The firm’s payback cutoff point is two years. Whether the projects are independent or mutually exclusive, management should accept only project B since project A’s payback period exceeds the two-year cutoff point. Projects D and E: Projects D and E dramatically illustrate the problem when a capital budgeting evaluation tool fails to consider cash flows after the payback period. Project D has a payback period of one year, suggesting that it should be accepted, and project E has a payback period of 2.5 years, suggesting that it should be rejected. However, a simple look at the future cash flows suggests otherwise. It is clear that project D, with a negative $5,000 cash flow in year 4, is a disaster and should be rejected, while project E, with a positive $5,000 cash flow in year 4, should be accepted. Indeed, the NPV analysis confirms these conclusions: project D has a negative NPV of $2,924, and project E has a positive NPV of $2,815. In both instances, the payback rule led to the wrong economic decision. These examples illustrate that a rapid payback does not necessarily mean a good investment. EXHIBIT 10.7 Discounted Payback Period Cash Flows and Calculations accounting rate of return (ARR) where: where BV0 is the original cost of the asset. This means that we can also describe the IRR as the discount rate that causes the NPV to equal zero. This relation can be written in a general form as follows: Because of their close relation, it may seem that the IRR and the NPV are interchangeable—that is, either should tell you to accept or reject the same capital projects. After all, both methods are based on whether the project’s return exceeds the cost of capital and, hence, whether the project will add value to the firm. In many circumstances, the IRR and NPV methods do give us the same answer. As you will see later, however, some of the properties of the IRR equation can lead to incorrect decisions concerning whether to accept or reject a particular capital project. Trial-and-Error Method Recall that the result we are looking for is zero. Because our result is $16.44, the discount rate of 12 percent is too low, and we must try a higher rate. Let’s try 13 percent: We are very close; let’s try 14 percent: Because our result is now a negative number, we know the correct rate is between 13 and 14 percent, and looking at the magnitude of the numbers, we know that the answer is closer to 14 percent. Let’s try 13.7 percent. EXHIBIT 10.8 Time Line and Expected Net Cash Flows for the Ford Project ($ thousands) Calculating the IRR at Larry’s PROBLEM: Larry’s Ice Cream in the DuPont Circle area of Washington, D.C., is famous for its gourmet ice cream. However, some customers have asked for a health- oriented, low-cal, softyogurt. The machine that makes this confection is manufactured in Italy and costs $5,000 plus $1,750 for installation. Larry estimates that the machine will generate a net cash flow of $2,000 a year (the shop closes November through March of each year). Larry also estimates the machine’s life to be 10 years and that it will have a $400 salvage value. His cost of capital is 15 percent. Larry thinks the machine is overpriced and it’s a bum deal. Is he right? The hand trial-and-error calculations are shown below. The first calculation uses 15 percent, the cost of capital, our recommended starting point, and the answer is $3,386.41 (which is also the project’s NPV). Because the value is a positive number, we need to use a larger discount rate than 15 percent. Our guess is 27.08 percent. At that value the NPV equals zero; thus, the IRR for the yogurt machine is 27.08 percent. Because the project’s future cash flow pattern resembles that for a bond, we can also solve for the IRR on a financial calculator, just as we would solve for the yield to maturity of a bond. Just enter the data directly into the corresponding keys on the calculator and press the interest key and we have our answer: 27.08 percent. As with present value calculations, for projects with unequal cash flows, you should consult your financial calculator’s manual. When the IRR and NPV Methods Agree Exhibit 10.9 Discount Rate NPV ($ thousands) 0% $160 5 94 10 37 15 −12 20 −54 25 −92 30 −124 EXHIBIT 10.9 NPV Profile for the Ford Project Once again, the best way to understand the effect of these cash flows is to look at an NPV profile. Shown here are NPV calculations we made at various discount rates to generate the data necessary to plot the NPV profile shown in Discount Rate NPV ($ millions) 0% −$5.00 10 −1.28 20 0.56 30 1.21 40 1.12 50 0.56 60 −0.31 70 1.37 Looking at the data in the table, you can probably spot a problem. The NPV is initially negative (− $5.00); then, at a discount rate of 20 percent, switches to positive ($0.56); and then, at a discount rate of 60 percent, switches back to negative (− $0.31). EXHIBIT 10.10 NPV Profile for Gold-Mining Operation Showing Multiple IRR Solutions The IRR is 20.7 percent for project A and 19.0 percent for project B. Because the two projects are mutually exclusive, only one project can be accepted. If you were following the IRR decision rule, you would accept project A. However, as you will see, it turns out that project B might be the better choice. Notice that the project with the higher NPV depends on what rate of return is used to discount the cash flows. Our example shows a conflict in ranking order between the IRR and NPV methods at discount rates between 0 and 13 percent. In this range, project B has the lower IRR, but it has the higher NPV and should be the project selected. If the discount rate is above 15 percent, however, project A has the higher NPV as well as the higher IRR. In this range there is no conflict between the two evaluation methods. EXHIBIT 10.11 NPV Profiles for Two Mutually Exclusive Projects SITUATION: Suppose you work for an entrepreneur who owns a number of small businesses in Fresno, California, as well as a small piece of property near California State University at Fresno, which he believes would be an ideal site for a student-oriented convenience store. His 12-year-old son, who happens to be in the office after school, says he has a better idea: his father should open a lemonade stand. Your boss tells you to find the NPV and IRR for both projects, assuming a 10 percent discount rate. After collecting data, you present the following analysis: Assuming the projects are mutually exclusive, which should be selected? To compute the MIRR, we have to make two preliminary calculations. First, we calculate the value of PVCost, which is the present value of the cash outflows that make up the investment cost of the project. Since for most capital projects, the investment cost cash flows are incurred at the beginning of the project, t = 0, there is often no need to calculate a present value. If investment costs are incurred over time (t > 0), then the cash flows must be discounted at the cost of capital. where: Once we have computed the values of PVCost and TV, we use Equation 10.5 to compute the MIRR. Note that by combining intermediate cash flows into a single terminal value, MIRR has the added advantage of always yielding a conventional cash flow. The terminal value of $809.86 equals the sum of the $240 in year 1 compounded at 12 percent for two years plus the $240 in year 2 compounded at 12 percent for 1 year plus the $240 in year 3. Mathematically, this calculation is: With the information that the cost of the project is $560 and the TV is $809.86, we can calculate the MIRR using Equation 10.5: At 13.09 percent, the MIRR is higher than Ford’s cost of capital of 12 percent, so the project should be accepted. > BEFORE YOU GO ON EXHIBIT 10.12 Capital Budgeting Techniques Used by Business Firms Sources: Stanley, Marjorie T. and Stanley B. Block, “A Survey of Multinational Capital Budgeting” The Financial Review, March 1984. Graham, John R. and Campbell R. Harvey, “The Theory and Practice of Corporate Finance,” Journal of Financial Economics, May/June 2001. Discuss why capital budgeting decisions are the most important investment decisions made by a firm’s management. Self-Study Problems · 10.2 Rutledge, Inc., has invested $100,000 in a project that will produce cash flows of $45,000, $37,500, and $42,950 over the next three years. Find the payback period for the project. Which project will be chosen if the company’s payback criterion is three years? What if the company accepts all projects as long as the payback period is less than five years? The company will accept all projects that provide an accounting rate of return (ARR) of at least 45 percent. Should the company accept this project? NPV for Craigmore Forklifts: Premium should purchase the Otis forklift since it has a larger NPV. · 10.3 Payback periods for Perryman projects A and B: Payback Period for Project A: Payback Period for Project B: If the payback period is three years, project B will be chosen. If the payback criterion is five years, both A and B will be chosen. The company should accept the project. Try a higher rate. At k 17 percent: Try a higher rate. At k = 17.5 percent: Thus, the IRR for Otis is less than 17.5 percent. Using a financial calculator, you can find the exact rate to be 17.43 percent. Try a lower rate. At k = 13 percent: Try a higher rate. At k = 13.1 percent: Thus, the IRR for Craigmore is less than 13.1 percent. The exact rate is 13.06 percent. Based on the IRR, we would still choose the Otis system over the Craigmore system. The decision is the same as that indicated by NPV. Questions and Problems · 10.1 Net present value: Riggs Corp. management is planning to spend $650,000 on a new marketing campaign. They believe that this action will result in additional cash flows of $325,000 over the next three years. If the discount rate is 17.5 percent, what is the NPV on this project? Year Cash Flow 0 −$3,300,000 1 875,123 2 966,222 3 1,145,000 4 1,250,399 5 1,504,445 · 10.4 Net present value: Franklin Mints, a confectioner, is considering purchasing a new jelly bean-making machine at a cost of $312,500. The company’s management projects that the cash flows from this investment will be $121,450 for the next seven years. If the appropriate discount rate is 14 percent, what is the NPV for the project? · 10.6 Payback: Refer to Problem 10.5. What are the payback periods for production systems 1 and 2? If the systems are mutually exclusive and the firm always chooses projects with the lowest payback period, in which system should the firm invest? · 10.13 Net present value: Champlain Corp. is investigating two computer systems. The Alpha 8300 costs $3,122,300 and will generate cost savings of $1,345,500 in each of the next five years. The Beta 2100 system costs $3,750,000 and will produce cost savings of $1,125,000 in the first three years and then $2 million for the next two years. If the company’s discount rate for similar projects is 14 percent, what is the NPV for the two systems? Which one should be chosen based on the NPV? Year Cash Flow 1 $512,496 2 −242,637 3 814,558 4 887,225 5 712,642 · 10.15 Net present value: Cranjet Industries is expanding its product line and its production capacity. The costs and expected cash flows of the two independent projects are given in the following table. The firm uses a discount rate of 16.4 percent for such projects. 1. 10.16 Net present value: Emporia Mills management is evaluating two alternative heating systems. Costs and projected energy savings are given in the following table. The firm uses 11.5 percent to discount such project cash flows. Which system should be chosen? 1. 10.17 Payback: Creative Solutions, Inc., has just invested $4,615,300 in new equipment. The firm uses payback period criteria of not accepting any project that takes more than four years to recover its costs. The company anticipates cash flows of $644,386, $812,178, $943,279, $1,364,997, $2,616,300, and $2,225,375 over the next six years. Does this investment meet the firm’s payback criteria? 1. 10.19 Payback: Regent Corp. is evaluating three competing types of equipment. Costs and cash flow projections for all three are given in the following table. Which would be the best choice based on payback period? 1. 10.20 Discounted payback: Nugent Communication Corp. is investing $9,365,000 in new technologies. The company expects significant benefits in the first three years after installation (as can be seen by the following cash flows), and smaller constant benefits in each of the next four years. What is the discounted payback period for the project assuming a discount rate of 10 percent? 1. 10.21 Modified internal rate of return (MIRR): Morningside Bakeries has recently purchased equipment at a cost of $650,000. The firm expects to generate cash flows of $275,000 in each of the next four years. The cost of capital is 14 percent. What is the MIRR for this project? Years Cash Flow 1–4 $23,500,000 5–7 72,000,000 8–10 80,000,000 1. 10.24 Internal rate of return: Refer to Problem 10.5. Compute the IRR for both production system 1 and production system 2. Which has the higher IRR? Which production system has the higher NPV? Explain why the IRR and NPV rankings of systems 1 and 2 are different. 1. 10.26 Internal rate of return: Compute the IRR on the following cash flow streams: 1. 10.28 Draconian Measures, Inc., is evaluating two independent projects. The company uses a 13.8 percent discount rate for such projects. The costs and cash flows for the projects are shown in the following table. What are their NPVs? 1. 10.29 Refer to Problem 10.28. 1. 10.31 Intrepid, Inc., is considering investing in three independent projects. The costs and the cash flows are given in the following table. The appropriate cost of capital is 14.5 percent. Compute the project IRRs and identify the projects that should be accepted. 1. 10.32 Jekyll & Hyde Corp. is evaluating two mutually exclusive projects. The cost of capital is 15 percent. Costs and cash flows are given in the following table. Which project should be accepted? 1. 10.33 Larsen Automotive, a manufacturer of auto parts, is considering investing in two projects. The company typically compares project returns to a cost of funds of 17 percent. Compute the IRRs based on the cash flows in the following table. Which project(s) will be accepted? 1. 10.34 Compute the IRR for each of the following projects: 1. 10.35 Primus Corp. is planning to convert an existing warehouse into a new plant that will increase its production capacity by 45 percent. The cost of this project will be $7,125,000. It will result in additional cash flows of $1,875,000 for the next eight years. The discount rate is 12 percent. 38. Calculate the projects’ NPV. Years CF 1–2 0 3–5 $ 845,000 6–9 $1,450,000 39. Compute the payback and discounted payback periods for the project. 1. 10.40 Given the following cash flows for a capital project, calculate the NPV and IRR. The required rate of return is 8 percent. NPV IRR a. $1,905 10.9% b. $1,905 26.0% c. $3,379 10.9% d. $3,379 26.0% 1. 10.41 Given the following cash flows for a capital project, calculate its payback period and discounted payback period. The required rate of return is 8 percent. The discounted payback period is 1 Jeff Greenberg/Alamy Chapter 7 where E (R EXHIBIT 13.1 The Finance Balance Sheet If the cash flows that the apartment building is expected to produce are worth $4,000,000, then investors would be willing to pay $3,700,000 for the equity in the firm. This is the value of the cash flows that they would expect to receive after making the interest and principal payments on the mortgage. Furthermore, since, by definition, the mortgage is worth $300,000, the value of the debt plus the value of the equity is $300,000 + $3,700,000 = $4,000,000—which is exactly equal to the market value of the firm’s assets. where β to estimate the expected return on the firm’s assets, which is also the firm’s cost of capital. Unfortunately, because analysts are not typically able to estimate betas for individual projects, they generally cannot use this approach. In Equation 13.2, k This tells us that the value of the debt claims equals 7.5 percent of the value of the firm and that the value of the equity claims equals the remaining 92.5 percent of the value of the firm. If the cost of the debt for this business is 6 percent and the cost of the equity is 10 percent, the cost of capital for the firm can be calculated as a weighted average of the costs of the debt and equity: Notice that we have used Equation 13.2 to calculate a weighted average cost of capital (WACC) for the firm in this example. In fact, this is what people typically call the firm’s cost of capital, k Calculating the Cost of Capital for a Firm PROBLEM: You are considering purchasing a rug cleaning company that will cost $2,000,000. You plan to finance the purchase with a $1,500,000 loan from Bank of America (BofA) that has a 6.5 percent interest rate, a $300,000 loan from the seller of the company that has an 8 percent interest rate, and $200,000 of your own money. You will own all of the equity (stock) in the firm. You estimate that the opportunity cost of your $200,000 investment—that is, what you could earn on an investment of similar risk in the capital market—is 12 percent with that much debt. What is the cost of capital for this investment? We can then calculate the WACC using Equation 13.2: On average, you would be paying 7.3 percent per year on every dollar you invested in the firm. This is the opportunity cost of capital for the firm. It is the rate that you would use to discount the cash flows associated with the rug cleaning business in an NPV analysis. By trial and error or with a financial calculator, we solve for i and find: If this bond was sold at par, it paid 7 percent when it was issued five years ago. Someone who buys it today will expect to earn only 6.1 percent per year. This is the annual rate of return required by the market on this bond, which is known as the effective annual yield. Converting the adjusted semiannual rate to an EAR, we see that the actual annual cost of this debt financing is: This after-tax cost of debt is the cost that firms actually use to calculate the WACC. The reason is simply that investors care only about the after-tax cost of capital—just as they care only about after-tax cash flows. Managers are concerned about what they actually have to pay for capital, and the actual cost is reduced if the government subsidizes debt by providing a tax break. Suppose that your taxable income before the interest deduction is $75,000 and, for simplicity, that both your average and marginal tax rates are 20 percent. Without the interest deduction, you would pay taxes totaling $15,000 ($75,000 × 0.20 = $15,000). However, because the interest payments reduce your taxable income, your taxes with the interest deduction will be only $12,600 [($75,000 − $12,000) × 0.20 = $12,600]. The ability to deduct the interest payments you made saved you $2,400 ($15,000 − $12,600 = $2,400)! This savings is exactly equal to the interest payment you make times your marginal tax rate: $12,000 × 0.20 = $2,400. Since you are saving $2,400, the after-tax cost of your interest payments is $9,600 ($12,000 − $2,400 = $9,600), which means that the after-tax cost of this debt is 4.8 percent ($9,600/$200,000 = 0.048, or 4.8 percent). This is exactly what Equation 13.3 tells us. With k Estimating the Cost of Debt for a Firm The pretax cost of the bank loan that you took out today is simply the 6 percent rate that the bank is charging you, assuming that the bank is charging you the market rate. The weighted average pretax cost of debt is: The after-tax cost of debt is therefore: APPLICATION 13.2 LEARNING BY DOING Calculating the After-Tax Cost of Debt for a Firm PROBLEM: You have just successfully completed a leveraged buyout of the firm that you have been working for. To finance this $35 million transaction, you and three partners put up a total of $10 million in equity capital, and you borrowed $25 million from banks and other investors. The bank debt consists of $10 million of secured debt borrowed at a rate of 6 percent from Bank of America and $7 million of senior unsecured debt borrowed at a rate of 7 percent from JPMorgan Chase. The remaining $8 million was borrowed from an investment group managed by a private equity firm. The rate on this subordinated (junior) unsecured debt is 9.5 percent. What is the overall after-tax cost of the debt financing used to buy the firm if you expect the firm’s average and marginal tax rates to both be 25 percent? (2) The weighted average pretax cost of debt is: (3) The after-tax cost of debt is therefore: EXAMPLE 13.1 DECISION MAKING SITUATION: Your pizza parlor business has developed such a strong reputation that you have decided to take advantage of the restaurant’s name recognition by selling frozen pizzas through grocery stores. In order to do this, you will have to build a manufacturing facility. You estimate that this will cost you $10 million. Since your business currently has only $2 million in the bank, you will have to borrow the remaining $8 million. You have spoken with two bankers about possible loan packages. The banker from Easy Money Financial Services offered you a loan for $6 million with a 6 percent rate and $2 million with a 7.5 percent rate. You calculate the pretax cost of debt for this package to be: Your local banker offered you a single $8 million loan for 6.350 percent. Which financing should you choose if all terms on all of the loans, other than the interest rates, are the same? In this equation, the expected return on an asset is a linear function of the systematic risk associated with that asset. Equation 13.4 is just another way of writing Equation 7.10. It tells us that the cost of common stock equals the risk-free rate of return plus compensation for the systematic risk associated with the common stock. You already saw some examples of how to use this equation to calculate the cost of equity in the discussion of the Capital Asset Pricing Model (CAPM) in This example illustrates how Equation 13.4 is used to estimate the cost of common stock for a company. How would the analysis differ for a private company? The only difference is that we would not be able to estimate the beta directly. We would have to estimate the beta using betas for similar public companies. PROBLEM: You have decided to estimate the cost of the common equity in your pizza business on November 19, 2010. As noted earlier, the risk-free rate and the market risk premium on that day were 3.95 percent and 6.01 percent, respectively. Since you have already decided that Domino’s Pizza is a reasonably comparable company, you obtain Domino’s beta from the Yahoo! finance Web site ( k where D1 is the dividend expected to be paid one period from today, R is the required rate of return, and g is the annual rate at which the dividends are expected to grow in perpetuity. While Equation 13.5 is just a variation of Equation 9.4, it is important enough to identify as a separate equation because it provides a direct way of estimating the cost of equity under certain circumstances. If we can estimate the dividend that stockholders will receive next period, D1, and we can estimate the rate at which the market expects dividends to grow over the long run, g, then we can use today’s market price, P0, in Equation 13.5 to tell us what rate of return investors in the firm’s common stock are expecting to earn. This approach can be useful for a firm that pays dividends when it is reasonable to assume dividends will grow at a constant rate and when the analyst has a good idea what that growth rate will be. An electric utility firm is an example of this type of firm. Some electric utility firms pay relatively high and predictable dividends that increase at a fairly consistent rate. In contrast, this approach would not be appropriate for use by a high-tech firm that pays no dividends or that pays a small dividend that is likely to increase at a high rate in the short run. Equation 13.5, like any other equation, should be used only if it is appropriate for the particular stock. where D In this equation, we have replaced the R in Equation 9.6 with k All this equation does is add the present values of the dividends that are expected in each of the next three years and the present value of a growing perpetuity that begins in the fourth year. EXHIBIT 13.2 The Three-Stage Dividend Growth Equation From the information given in the problem statement, we know the following: Substituting these values into the above equation gives us the following, which we solve for k As mentioned earlier, we can solve this equation for k Which Method Should We Use? as: where Pps is the present value of the expected dividends (the current preferred stock price), Dps is the annual preferred stock dividend, and k Plugging the information from our example into Equation 13.6, we see that k This is the rate of return at which the present value of the annual $5 cash flows equals the market price of $85. Therefore, 5.9 percent is the rate that investors currently require for investing in this preferred stock. APPLICATION 13.4 LEARNING BY DOING PROBLEM: You work in the Treasury Department at Wells Fargo & Company, and your manager has asked you to estimate the cost of each of the different types of stock that Wells Fargo has outstanding. One of these issues is a 8 percent non-cumulative preferred stock that has a stated value of $1,000 and is currently selling for $927.90. Although this preferred stock is publicly traded, it does not trade very often. This means that you cannot use the CAPM to estimate k You may recall from the discussion in where x Note that because the $4 million and $3 million loans have rates that equal what it would cost to refinance them today, their market values equal the amount that is owed. Since the $10 million market value of the bond issue is below the $11 million face value, the rate that firm is actually paying must be lower than the 7 percent rate you estimated to reflect the current cost of this debt. Recall that as interest rates increase, the market value of a bond decreases. This is the negative relation that we referred to earlier in this chapter. From Equation 13.4, we calculate the cost of the common equity to be: You can see real-world applications of the WACC calculation at the New Zealand Web site for Pricewaterhouse-Coopers, the international accounting and consulting firm, at APPLICATION 13.5 LEARNING BY DOING Calculating the WACC with Equation 13.7 PROBLEM: After calculating the cost of the common equity in your pizza business to be 12.1 percent (see Learning by Doing Application 13.3), you have decided to estimate the WACC. You recently hired a business appraiser to estimate the value of your stock, which includes all of the outstanding common equity. His report indicates that it is worth $500 million. Your company has no other long-term debt or any preferred stock outstanding. Both the marginal and average tax rates for your company are 20 percent. What is the WACC for your pizza business? and the WACC is: EXAMPLE 13.2 DECISION MAKING SITUATION: You are a financial analyst for the company whose WACC of 7.3 percent we just calculated in the main text. One day, your manager walks in to your office and tells you that she is thinking about selling $23 million of common stock and using the proceeds from the sale to pay back both of the firm’s loans and to repurchase all of the outstanding bonds and preferred stock. She tells you that this is a smart move because if she does this, the beta of the firm’s common stock will decline to 0.70 and the overall k What do you tell your manager? Chapter 11 EXHIBIT 13.3 Potential Errors When Using the WACC to Evaluate Projects This is an attractive project because it returns more than the investors’ opportunity cost of capital. As you can see, when the WACC is used to discount the cash flows, the firm will end up rejecting a positive NPV project. The firm will be passing up an opportunity to create value for its stockholders. As an exercise, you might try constructing a numerical example in which a firm accepts a negative NPV project. This equation provides a measure of the firm’s cost of capital that reflects both how the firm’s assets have been financed—that is, the mix of debt and preferred and common stock that was used to acquire those assets—and the current cost of each type of financing. In other words, the WACC reflects both the x’s and the k’s associated with the firm’s financing. Why is this important? Because the costs of the different types of capital depend on the fraction of the total firm financing that each represents. If the firm uses more or less debt, the cost of debt will be higher or lower. In turn, the cost of both preferred stock and common stock will be affected. This means that even if the underlying business risk of the project is the same as that for the firm as a whole, if the project is financed differently than the firm, the appropriate discount rate for the project analysis will be different from that for the firm as a whole. EXHIBIT 13.4 Potential Errors When Using Multiple Discount Rates to Evaluate Projects Explain what the weighted average cost of capital for a firm is and why it is often used as a discount rate to evaluate projects. Self-Study Problems · 13.5 The pretax debt contribution to the cost of capital is x · 13.1 Finance balance sheet: KneeMan Markup Company has total debt obligations with book and market values equal to $30 million and $28 million, respectively. It also has total equity with book and market values equal to $20 million and $70 million, respectively. If you were going to buy all of the assets of KneeMan Markup today, how much should you be willing to pay? · 13.13 Finance balance sheet: Explain why the total value of all of the securities used to finance a firm must be equal to the value of the firm. · 13.27 You are analyzing the cost of capital for MacroSwift Corporation, which develops software operating systems for computers. The firm’s dividend growth rate has been a very constant 3 percent per year for the past 15 years. Competition for the firm’s current products is expected to develop in the next year, and MacroSwift is currently expanding its revenue stream into the multimedia industry. Evaluate the appropriateness of continuing to use a 3 percent growth rate in dividends for MacroSwift in your cost of capital model. the beta of the assets of a firm also equals a weighted average of the betas for the debt, preferred stock, and common stock of a firm: Why must this be true? 1. 13.38 The cost of equity is equal to the: 1 2 3 4 5 6 7 8 9 Lesson 5 reading © David Young-Wolff/PhotoEdit BUILDING INTUITION MORE RISK MEANS A HIGHER EXPECTED RETURN total holding period return where P0 is the price paid for the asset at time zero and P1 is the price at a later point in time. where CF1 is the cash flow from the dividend. You can download actual realized investment returns for a large number of stock market indexes at the Callan Associates Web site, What return would you have earned if Dell had paid a $1 dividend and today’s price was $12.90? With the $1 dividend and a correspondingly lower price, the total return is the same: You can see from this example that a dollar of capital appreciation is worth the same as a dollar of income. Calculating the Return on an Investment PROBLEM: You purchased a beat-up 1974 Datsun 240Z sports car a year ago for $1,500. Datsun is what Nissan, the Japanese car company, was called in the 1970s. The 240Z was the first in a series of cars that led to the Nissan 370Z that is being sold today. Recognizing that a mint-condition 240Z is a much sought-after car, you invested $7,000 and a lot of your time fixing up the car. Last week, you sold it to a collector for $18,000. Not counting the value of the time you spent restoring the car, what is the total return you earned on this investment over the one-year holding period? Expected Returns What is the expected value of your bonus? If you have taken a statistics course, you might recall that an expected value represents the sum of the products of the possible outcomes and the probabilities that those outcomes will be realized. In our example the expected value of the bonus can be calculated using the following formula: where E(Bonus) is your expected bonus, p Notice that the expected bonus of $530,000 is not equal to either of the two possible payoffs. Neither is it equal to the simple average of the two possible payoffs. This is because the expected bonus takes into account the probability of each event occurring. If the probability of each event had been 50 percent, then the expected bonus would have equaled the simple average of the two payoffs: However, since it is more likely that you will not get a hit (a 67.5 percent chance) than that you will get a hit (a 32.5 percent chance), and the payoff is lower if you do not get a hit, the expected bonus is less than the simple average. The key point here is that the expected value reflects the relative likelihoods of the possible outcomes. expected return is mathematical shorthand indicating that n values are added together. In Equation 7.2, each of the n possible returns is multiplied by the probability that it will be realized, and these products are then added together to calculate the expected return. To see how we calculate the expected return on an asset, suppose you are considering purchasing Dell, Inc. stock for $13.90 per share. You plan to sell the stock in one year. You estimate that there is a 30 percent chance that Dell stock will sell for $13.40 at the end of one year, a 30 percent chance that it will sell for $14.90, a 30 percent that it will sell for $15.40, and a 10 percent chance that it will sell for $16.00. If Dell pays no dividends on its shares, what is the return that you expect from this stock in the next year? Therefore, we can calculate the return from owning Dell stock under each of the four possible outcomes using the approach we used for the similar Dell problem we solved earlier in the chapter. These returns are calculated as follows: Applying Equation 7.2, the expected return on Dell stock over the next year is therefore 5.83 percent, calculated as follows: Notice that the negative return is entered into the formula just like any other. Also notice that the sum of the pi Calculating Expected Returns PROBLEM: You have just purchased 100 railroad cars that you plan to lease to a large railroad company. Demand for shipping goods by rail has recently increased dramatically due to the rising price of oil. You expect oil prices, which are currently at $98.81 per barrel, to reach $115.00 per barrel in the next year. If this happens, railroad shipping prices will increase, thereby driving up the value of your railroad cars as increases in demand outpace the rate at which new cars are being produced. you must recognize that ΔP/P0 is the capital appreciation under each outcome and that CF1/P0 equals the 10 percent that you expect to receive from leasing the rail cars. The expected returns for the three outcomes are: You can then use Equation 7.2 to calculate the expected return for your rail car investment: Alternatively, since there is a 100 percent probability that the return from leasing the railroad cars is 10 percent, you could have simply calculated the expected increase in value of the railroad cars: and added the 10 percent to arrive at the answer of 33.5 percent. Of course, this simpler approach only works if the return from leasing is known with certainty. SITUATION: You are deciding whether you should advertise your pizza business on the radio or on billboards placed on local taxicabs. For $1,000 per month, you can either buy 20 one-minute ads on the radio or place your ad on 40 taxicabs. This means that you expect 20 one-minute ads to bring in 20 × 48 = 960 new customers. Placing ads on 40 taxicabs is therefore expected to bring in 40 35 1,400 new customers. Unfortunately, using this calculation to obtain a measure of risk presents two problems. First, since one difference is positive and the other difference is negative, one difference partially cancels the other. As a result, you are not getting an accurate measure of total risk. Second, this calculation does not take into account the number of potential outcomes or the probability of each outcome. Note that the square of the Greek symbol sigma, σ2, is generally used to represent the variance. standard deviation (σ) Equation 7.3 simply extends the calculation illustrated above to the situation where there are n possible outcomes. Like the expected return calculation (Equation 7.2), Equation 7.3 can be simplified if all of the possible outcomes are equally likely. In this case it becomes: In both the general case and the case where all possible outcomes are equally likely, the standard deviation is simply the square root of the variance . EXHIBIT 7.1 Normal Distribution Since the returns on many assets are approximately normally distributed, the standard deviation provides a convenient way of computing the probability that the return on an asset will fall within a particular range. In these applications, the expected return on an asset equals the mean of the distribution, and the standard deviation is a measure of the uncertainty associated with the return. EXHIBIT 7.2 Standard Deviation and Width of the Normal Distribution Understanding the Standard Deviation PROBLEM: You are considering investing in a share of Google Inc., stock and want to evaluate how risky this potential investment is. You know that stock returns tend to be normally distributed, and you have calculated the expected return on Google stock to be 4.67 percent and the standard deviation of the annual return to be 23 percent. Based on these statistics, within what range would you expect the return on this stock to fall during the next year? Calculate this range for a 90 percent level of confidence (that is, 90 percent of the time, the returns will fall within the specified range). EXHIBIT 7.3 Distributions of Annual Total Returns for U.S. Stocks and Bonds from 1926 to 2009 EXHIBIT 7.4 Monthly Returns for Apple Inc. stock and the S&P 500 Index from September 2005 through September 2010 EXHIBIT 7.5 Cumulative Value of $1 Invested in 1926 With this information, we can calculate the expected returns for AMD and Intel by using Equation 7.2: and Similarly, we can calculate the standard deviations of the returns for AMD and Intel in the same way that we calculated the standard deviation for our baseball bonus example in Section 7.2: and Having calculated the expected returns and standard deviations for the expected returns on AMD and Intel stock, the natural question to ask is which provides the highest risk- adjusted return. Before we answer this question, let’s return to the example at the beginning of Section 7.1. Recall that, in this example, we proposed choosing among three stocks: A, B, and C. We stated that investors would prefer the investment that provides the highest expected return for a given level of risk or the lowest risk for a given expected return. This made it fairly easy to choose between Stocks A and B, which had the same return but different risk levels, and between Stocks B and C, which had the same risk but different returns. We were stuck when trying to choose between Stocks A and C, however, because they differed in both risk and return. Now, armed with tools for quantifying expected returns and risk, we can at least take a first pass at comparing stocks such as these. In this equation, CV is a measure of the risk associated with an investment for each 1 percent of expected return. Since these values are equal, the coefficient of variation measure suggests that these two investments are equally attractive on a risk-adjusted basis. where CVi* is a modified coefficient of variation that is computed by subtracting the risk-free rate from the expected return. We can see that the modified coefficient of variation for AMD is smaller than the modified coefficient of variation for Intel. This tells us that an investment in AMD stock is expected to have less risk for each 1 percent of return. Since investors prefer less risk for a given level of return, the AMD stock is a more attractive investment. Sharpe Ratio You can read more about the Sharpe Ratio and other ratios that are used to measure risk-adjusted returns for investments at the following Web site: PROBLEM: You are trying to choose between two investments. The first investment, a painting by Picasso, has an expected return of 14 percent with a standard deviation of 30 percent over the next year. The second investment, a pair of blue suede shoes once worn by Elvis, has an expected return of 20 percent with a standard deviation of return of 40 percent. The risk-free rate of interest is 3 percent. What is the Sharpe Ratio for each of these investments, and what do these ratios tell us? The Sharpe Ratio for Elvis’s blue suede shoes is larger than the Sharpe Ratio for the painting. This indicates that the return for each 1 standard deviation of risk is greater for Elvis’s shoes than for the painting. Notice that this formula is just like the expected return formula for an individual stock. However, in this case, instead of multiplying outcomes by their associated probabilities, we are multiplying expected returns for individual stocks by the fraction of the total portfolio value that each of these stocks represents. In other words, the formula for the expected return for a two-stock portfolio is: where x This equation is just like Equation 7.2, except that (1) the returns are expected returns for individual assets and (2) instead of multiplying by the probability of an outcome, we are multiplying by the fraction of the portfolio invested in each asset. Note that this equation can be used only if you have already calculated the expected return for each stock. Therefore, the expected return on the portfolio is: Risk of a Portfolio with more than one Asset PROBLEM: You have become concerned that you have too much of your money invested in your pizza restaurant and have decided to diversify your personal portfolio. Right now the pizza restaurant is your only investment. To diversify, you plan to sell 45 percent of your restaurant and invest the proceeds from the sale, in equal proportions, into a stock market index fund and a bond market index fund. Over the next year, you expect to earn a return of 15 percent on your remaining investment in the pizza restaurant, 12 percent on your investment in the stock market index fund, and 8 percent on your investment in the bond market index fund. What return will you expect from your diversified portfolio over the next year? At 12.75 percent, the expected return is an average of the returns on the individual assets in your portfolio, weighted by the fraction of your portfolio that is invested in each. Exhibit 7.6 where x covariance of returns EXHIBIT 7.6 Monthly Returns for Southwest Airlines and Netflix Stock from September 2005 through September 2010 You can see that the covariance calculation is very similar to the variance calculation. The difference is that, instead of squaring the difference between the return from each outcome and the expected return for an individual asset, we calculate the product of this difference for two different assets. The correlation between the returns on two assets will always have a value between −1 and + 1. This makes the interpretation of this variable straightforward. A negative correlation means that the returns tend to have opposite signs. For example, when the return on one asset is positive, the return on the other asset tends to be negative. If the correlation is exactly − 1, the returns on the two assets are perfectly negatively correlated. In other words, when the return on one asset is positive, the return on the other asset will always be negative. A positive correlation means that when the return on one asset is positive, the return on the other asset also tends to be positive. If the correlation is exactly equal to + 1, then the returns of the two assets are said to be perfectly positively correlated. The return on one asset will always be positive when the return on the other asset is positive. Finally, a correlation of 0 means that the returns on the assets are not correlated. In this case, the fact that the return on one asset is positive or negative tells you nothing about how likely it is that the return on the other asset will be positive or negative. You can see that this portfolio variance is smaller than the variance of either the Southwest Airlines or Netflix stock on its own. Exhibit 7.7 EXHIBIT 7.7 Monthly Returns for Southwest Airlines and Netflix Stock and for a Portfolio with 50 Percent of the Value in Each of these Two Stocks from September 2005 through September 2010 The positive correlation tells us that the prices of Southwest Airlines and Netflix stock tend to move in the same direction. However, the correlation of less than one tells us that they do not always do so. The fact that the prices of these two shares do not always move together is the reason that the returns on a portfolio of the two stocks have less variation than the returns on the individual company shares. This example illustrates the benefit of diversification—how holding more than one asset with different risk characteristics can reduce the risk of a portfolio. Note that if the correlation of the returns between Southwest Airlines and Netflix stock equaled one, holding these two stocks would not reduce risk because their prices would always move up or down together. Calculating the Variance of a Two-Asset Portfolio PROBLEM: You are still planning to sell 45 percent of your pizza restaurant in order to diversify your personal portfolio. However, you have now decided to invest all of the proceeds in the stock market index fund. After you diversify, you will have 55 percent of your wealth invested in the restaurant and 45 percent invested in the stock market index fund. You have estimated the variances of the returns for these two investments and the covariance between their returns to be as follows: What will be the variance and standard deviation of returns in your portfolio after you have sold the ownership interest in your restaurant and invested in the stock market index fund? and the standard deviation is (0.0394)1/2 = 0.1985, or 19.85 percent. EXHIBIT 7.8 Total Risk in a Portfolio as the Number of Assets Increases EXHIBIT 7.9 Plot of Monthly General Electric Company Stock and S&P 500 Index Returns: October 2005 through September 2010 EXHIBIT 7.10 Slope of Relation Between General Electric Company Monthly Stock Returns and S&P 500 Index Returns: October 2005 through September 2010 Exhibit 7.10 Although we have to be careful about drawing conclusions when we have only two data points, we might interpret the slope of 2 to indicate that new information that causes the market return to increase by 1 percent will tend to cause the return on Nike stock to increase by 2 percent. Of course, the reverse might also be true. That is, new information that causes the market return to decrease by 1 percent may also cause the return on Nike stock to go down by 2 percent. To the extent that the same information is driving the changes in returns on Nike stock and on the market, it would not be possible for an investor in Nike stock to diversify this risk away. It is nondiversifiable, or systematic, risk. Now you might ask yourself what happened to the unsystematic risk of GE or Nike stock. This is best illustrated by the GE example, where we have more than two observations. As you can see in where Rrf is the return on a security with a risk-free rate of return, which analysts typically estimate by looking at returns on government securities. The compensation for taking risk, which varies with the risk of the asset, is added to the risk-free rate of return to get an estimate of the expected rate of return for an asset. If we recognize that the compensation for taking risk varies with asset risk and that systematic risk is what matters, we can write the preceding equation as follows: where units of systematic risk where β where E (Rm) is the expected return on the market. The term E (Rm) − Rrf is called the market risk premium. Consequently, we can now write the equation for expected return as: 7.7 THE CAPITAL ASSET PRICING MODEL Note that we must have three pieces of information in order to use Equation 7.10: (1) the risk-free rate, (2) beta, and (3) either the market risk premium or the expected return on the market. Recall that the market risk premium is the difference between the expected return on the market and the risk-free rate [E(Rm) − Rrf], which is 6 percent in the above example. Security Market Line (SML) Exhibit 7.11 EXHIBIT 7.11 The Security Market Line where P0 is the price that the asset is currently selling for. If an asset’s price implies that the expected return is greater than that predicted by the CAPM, that asset will plot above the SML in PROBLEM: You are considering buying 100 shares of General Electric stock. Value Line (a financial reporting service) reports that the beta for General Electric is 1.61. The risk-free rate is 4 percent, and the market risk premium is 6 percent. What is the expected rate of return on General Electric stock according to the CAPM? The Capital Asset Pricing Model and Portfolio Returns Of course, this should not be surprising since investing in a portfolio is simply an alternative to investing in a single asset. If we substitute Equation 7.10 into Equation 7.6 for each of the n assets and rearrange the equation, we find that the beta for a portfolio is simply a weighted average of the betas for the individual assets in the portfolio. In other words: where x Your portfolio has 1.25 times as much systematic risk as the market. Based on Equation 7.10, you would, therefore, expect to earn a return of 11.5 percent, calculated as follows: Up to this point, we have focused on calculating the expected rate of return for an investment in any asset from the perspective of an investor, such as a stockholder. A natural question that might arise is how these concepts relate to the rate of return that should be used within a firm to evaluate a project. The short answer is that they are the same. The rate of return used to discount the cash flows for a project with a particular level of systematic risk is exactly the same as the rate of return that an investor would expect to receive from an investment in any asset having the same level of systematic risk. In Portfolio Risk and Expected Return PROBLEM: You have recently become very interested in real estate. To gain some experience as a real estate investor, you have decided to get together with nine of your friends to buy three small cottages near campus. If you and your friends pool your money, you will have just enough to buy the three properties. Since each investment requires the same amount of money and you will have a 10 percent interest in each, you will effectively have one-third of your portfolio invested in each cottage. Therefore, from Equation 7.6, the expected return on the portfolio is: Using the second approach, from Equation 7.11, the beta of the portfolio is: and from Equation 7.10, the expected return is: EXAMPLE 7.2 DECISION MAKING SITUATION: You are trying to decide whether to invest in one or both of two different stocks. Stock 1 has a beta of 0.8 and an expected return of 7.0 percent. Stock 2 has a beta of 1.2 and an expected return of 9.5 percent. You remember learning about the CAPM in school and believe that it does a good job of telling you what the appropriate expected return should be for a given level of risk. Since the risk-free rate is 4 percent and the market risk premium is 6 percent, the CAPM tells you that the appropriate expected rate of return for an asset with a beta of 0.8 is 8.8 percent. The corresponding value for an asset with a beta of 1.2 is 11.2 percent. Should you invest in either or both of these stocks? Explain the relation between risk and return. Self-Study Problems and solve for σ. Doing this with either equation yields: · 7.4 A comparison of the Sharpe Ratios for the two stocks will tell you which has the highest expected return per unit of total risk. Stock B has the highest expected return per unit of risk. · 7.1 Returns: Describe the difference between a total holding period return and an expected return. · 7.13 Expected returns: Jose is thinking about purchasing a softdrink machine and placing it in a business office. He knows that there is a 5 percent probability that someone who walks by the machine will make a purchase from the machine, and he knows that the profit on each softdrink sold is $0.10. If Jose expects a thousand people per day to pass by the machine and requires a complete return of his investment in one year, then what is the maximum price that he should be willing to pay for the softdrink machine? Assume 250 working days in a year and ignore taxes and the time value of money. · 7.17 Calculating the variance and standard deviation: Ben would like to invest in gold and is aware that the returns on such an investment can be quite volatile. Use the following table of states, probabilities, and returns to determine the expected return and the standard deviation of the return on Ben’s gold investment. · 7.18 Single-asset portfolios: Using the information from Problems 7.15, 7.16, and 7.17, calculate the coefficient of variation for each of the investments in those problems. · 7.21 Compensation for bearing systematic risk: You have constructed a diversified portfolio of stocks such that there is no unsystematic risk. Explain why the expected return of that portfolio should be greater than the expected return of a risk-free security. · 7.27 David is going to purchase two stocks to form the initial holdings in his portfolio. Iron stock has an expected return of 15 percent, while Copper stock has an expected return of 20 percent. If David plans to invest 30 percent of his funds in Iron and the remainder in Copper, what will be the expected return from his portfolio? What if David invests 70 percent of his funds in Iron stock? Assume the CAPM and SML are true and fill in the missing values in the table. Would you invest in the stock of any of the three firms? If so, which one(s) and why? 1 2 3 4 5 6 AFP/Getty Images, Inc. capital budgeting Reason Description Renewal: Over time, equipment must be repaired, overhauled, rebuilt, or retrofitted with new technology to keep the firm’s manufacturing or service operations going. For example, a company that has a fleet of delivery trucks may decide to overhaul the trucks and their engines rather than purchase new trucks. Renewal decisions typically do not require an elaborate analysis and are made on a routine basis. Replacement: At some point, an asset will have to be replaced rather than repaired or overhauled. The major decision is whether to replace the asset with a similar piece of equipment or purchase equipment that would require a change in the production process. Sometimes, replacement decisions involve equipment that is operating satisfactorily but has become obsolete. The new or retrofitted equipment may provide cost savings with respect to labor or material usage and/or may improve product quality. These decisions typically originate at the plant level. Expansion: Strategically, the most important motive for capital expenditures is to expand the level of operating output. One type of expansion decision involves increasing the output of existing products. This may mean new equipment to produce more products or expansion of the firm’s distribution system. These types of decisions typically require a more complex analysis than a renewal or replacement decision. Another type of expansion decision involves producing a new product or entering a new market. This type of expansion often involves large dollar amounts and significant business risk and requires the approval of the firm’s board of directors. Regulatory: Some capital expenditures are required by federal and state regulations. These mandatory expenditures usually involve meeting workplace safety standards and environmental standards. Other: This category includes items such as parking facilities, office buildings, and executive aircraft. Many of these capital expenditures are hard to analyze because it is difficult to estimate their cash inflows. Ultimately, the decisions can be more subjective than analytical. Sources of Information mutually exclusive projects Cost of Capital cost of capital If a capital project has a positive NPV, the value of the cash flows the project is expected to generate exceeds the project’s cost. Thus, a positive NPV project increases the value of the firm and, hence, stockholders’ wealth. If a capital project has a negative NPV, the value of the cash flows from the project is less than its cost. If accepted, a negative NPV project will decrease the value of the firm and stockholders’ wealth. The $475,000 price paid for the pizza parlor exceeds the cost ($155,000) by $320,000. You have created $320,000 in value. How did you do this? You did it by improving the food, customer service, and dining ambiance while keeping prices competitive. Your management skills and knowledge of the pizza business resulted in significant growth in the current year’s cash flows and the prospect of even larger cash flows in the future. EXHIBIT 10.2 Sample Worksheet for Net Present Value Analysis where: Next, we will work an example to see how the NPV is calculated for a capital project. Suppose you are the president of a small regional firm located in Chicago that manufactures frozen pizzas, which are sold to grocery stores and to firms in the hospitality and food service industry. Your market research group has developed an idea for a “pocket” pizza that can be used as an entrée with a meal or as an “on the go” snack. The sales manager believes that, with an aggressive advertising campaign, sales of the product will be about $300,000 per year. The cost to modify the existing production line will also be $300,000, according to the plant manager. The marketing and plant managers estimate that the cost to produce the pocket pizzas, to market and advertise them, and to deliver them to customers will be about $220,000 per year. The product’s life is estimated to be five years, and the specialized equipment necessary for the project has an estimated salvage value of $30,000. The appropriate cost of capital is 15 percent. EXHIBIT 10.3 Pocket Pizza Project Time Line and Cash Flows ($ thousands) The NPV for the pocket pizza project is therefore $16,910. The PV of the future cash flows is − $283.09. With that information, we can compute the NPV using Equation 10.1 as follows: APPLICATION 10.1 LEARNING BY DOING The Dough’s Up: The Self-Rising Pizza Project PROBLEM: Let’s continue our frozen pizza example. Suppose the head of the research and development (R&D) group announces that R&D engineers have developed a breakthrough technology—self-rising frozen pizza dough that, when baked, rises and tastes exactly like fresh-baked dough. 5. The project is economically viable. The NPV for the self-rising pizza dough project is $118,567. Because the NPV is positive, management should accept the project. The project is estimated to increase the value of the firm by $118,567. EXHIBIT 10.4 Self-Rising Pizza Dough Project Time Line and Cash Flows ($ thousands) Mutually Exclusive Projects and NPV SITUATION: Suppose you are the manager of the information systems (IS) department of the frozen pizza manufacturer we have been discussing. Your department has identified four possible capital projects with the following NPVs: (1) $4,500, (2) $3,000, (3) $0.0, and (4) $1,000. What should you decide about each project if the projects are independent? What should you decide if the projects are mutually exclusive? > BEFORE YOU GO ON EXHIBIT 10.5 Payback Period Cash Flows and Calculations Exhibit 10.5 Let’s look at this calculation in more detail. Note in A Payback Calculation PROBLEM: A firm has two capital projects, A and B, which are under review for funding. Both projects cost $500, and the projects have the following cash flows: What is the payback period for each project? If the projects are independent, which project should management select? If the projects are mutually exclusive, which project should management accept? The firm’s payback cutoff point is two years. Whether the projects are independent or mutually exclusive, management should accept only project B since project A’s payback period exceeds the two-year cutoff point. Projects D and E: Projects D and E dramatically illustrate the problem when a capital budgeting evaluation tool fails to consider cash flows after the payback period. Project D has a payback period of one year, suggesting that it should be accepted, and project E has a payback period of 2.5 years, suggesting that it should be rejected. However, a simple look at the future cash flows suggests otherwise. It is clear that project D, with a negative $5,000 cash flow in year 4, is a disaster and should be rejected, while project E, with a positive $5,000 cash flow in year 4, should be accepted. Indeed, the NPV analysis confirms these conclusions: project D has a negative NPV of $2,924, and project E has a positive NPV of $2,815. In both instances, the payback rule led to the wrong economic decision. These examples illustrate that a rapid payback does not necessarily mean a good investment. EXHIBIT 10.7 Discounted Payback Period Cash Flows and Calculations accounting rate of return (ARR) where: where BV0 is the original cost of the asset. This means that we can also describe the IRR as the discount rate that causes the NPV to equal zero. This relation can be written in a general form as follows: Because of their close relation, it may seem that the IRR and the NPV are interchangeable—that is, either should tell you to accept or reject the same capital projects. After all, both methods are based on whether the project’s return exceeds the cost of capital and, hence, whether the project will add value to the firm. In many circumstances, the IRR and NPV methods do give us the same answer. As you will see later, however, some of the properties of the IRR equation can lead to incorrect decisions concerning whether to accept or reject a particular capital project. Trial-and-Error Method Recall that the result we are looking for is zero. Because our result is $16.44, the discount rate of 12 percent is too low, and we must try a higher rate. Let’s try 13 percent: We are very close; let’s try 14 percent: Because our result is now a negative number, we know the correct rate is between 13 and 14 percent, and looking at the magnitude of the numbers, we know that the answer is closer to 14 percent. Let’s try 13.7 percent. EXHIBIT 10.8 Time Line and Expected Net Cash Flows for the Ford Project ($ thousands) Calculating the IRR at Larry’s PROBLEM: Larry’s Ice Cream in the DuPont Circle area of Washington, D.C., is famous for its gourmet ice cream. However, some customers have asked for a health- oriented, low-cal, softyogurt. The machine that makes this confection is manufactured in Italy and costs $5,000 plus $1,750 for installation. Larry estimates that the machine will generate a net cash flow of $2,000 a year (the shop closes November through March of each year). Larry also estimates the machine’s life to be 10 years and that it will have a $400 salvage value. His cost of capital is 15 percent. Larry thinks the machine is overpriced and it’s a bum deal. Is he right? The hand trial-and-error calculations are shown below. The first calculation uses 15 percent, the cost of capital, our recommended starting point, and the answer is $3,386.41 (which is also the project’s NPV). Because the value is a positive number, we need to use a larger discount rate than 15 percent. Our guess is 27.08 percent. At that value the NPV equals zero; thus, the IRR for the yogurt machine is 27.08 percent. Because the project’s future cash flow pattern resembles that for a bond, we can also solve for the IRR on a financial calculator, just as we would solve for the yield to maturity of a bond. Just enter the data directly into the corresponding keys on the calculator and press the interest key and we have our answer: 27.08 percent. As with present value calculations, for projects with unequal cash flows, you should consult your financial calculator’s manual. When the IRR and NPV Methods Agree Exhibit 10.9 Discount Rate NPV ($ thousands) 0% $160 5 94 10 37 15 −12 20 −54 25 −92 30 −124 EXHIBIT 10.9 NPV Profile for the Ford Project Once again, the best way to understand the effect of these cash flows is to look at an NPV profile. Shown here are NPV calculations we made at various discount rates to generate the data necessary to plot the NPV profile shown in Discount Rate NPV ($ millions) 0% −$5.00 10 −1.28 20 0.56 30 1.21 40 1.12 50 0.56 60 −0.31 70 1.37 Looking at the data in the table, you can probably spot a problem. The NPV is initially negative (− $5.00); then, at a discount rate of 20 percent, switches to positive ($0.56); and then, at a discount rate of 60 percent, switches back to negative (− $0.31). EXHIBIT 10.10 NPV Profile for Gold-Mining Operation Showing Multiple IRR Solutions The IRR is 20.7 percent for project A and 19.0 percent for project B. Because the two projects are mutually exclusive, only one project can be accepted. If you were following the IRR decision rule, you would accept project A. However, as you will see, it turns out that project B might be the better choice. Notice that the project with the higher NPV depends on what rate of return is used to discount the cash flows. Our example shows a conflict in ranking order between the IRR and NPV methods at discount rates between 0 and 13 percent. In this range, project B has the lower IRR, but it has the higher NPV and should be the project selected. If the discount rate is above 15 percent, however, project A has the higher NPV as well as the higher IRR. In this range there is no conflict between the two evaluation methods. EXHIBIT 10.11 NPV Profiles for Two Mutually Exclusive Projects SITUATION: Suppose you work for an entrepreneur who owns a number of small businesses in Fresno, California, as well as a small piece of property near California State University at Fresno, which he believes would be an ideal site for a student-oriented convenience store. His 12-year-old son, who happens to be in the office after school, says he has a better idea: his father should open a lemonade stand. Your boss tells you to find the NPV and IRR for both projects, assuming a 10 percent discount rate. After collecting data, you present the following analysis: Assuming the projects are mutually exclusive, which should be selected? To compute the MIRR, we have to make two preliminary calculations. First, we calculate the value of PVCost, which is the present value of the cash outflows that make up the investment cost of the project. Since for most capital projects, the investment cost cash flows are incurred at the beginning of the project, t = 0, there is often no need to calculate a present value. If investment costs are incurred over time (t > 0), then the cash flows must be discounted at the cost of capital. where: Once we have computed the values of PVCost and TV, we use Equation 10.5 to compute the MIRR. Note that by combining intermediate cash flows into a single terminal value, MIRR has the added advantage of always yielding a conventional cash flow. The terminal value of $809.86 equals the sum of the $240 in year 1 compounded at 12 percent for two years plus the $240 in year 2 compounded at 12 percent for 1 year plus the $240 in year 3. Mathematically, this calculation is: With the information that the cost of the project is $560 and the TV is $809.86, we can calculate the MIRR using Equation 10.5: At 13.09 percent, the MIRR is higher than Ford’s cost of capital of 12 percent, so the project should be accepted. > BEFORE YOU GO ON EXHIBIT 10.12 Capital Budgeting Techniques Used by Business Firms Sources: Stanley, Marjorie T. and Stanley B. Block, “A Survey of Multinational Capital Budgeting” The Financial Review, March 1984. Graham, John R. and Campbell R. Harvey, “The Theory and Practice of Corporate Finance,” Journal of Financial Economics, May/June 2001. Discuss why capital budgeting decisions are the most important investment decisions made by a firm’s management. Self-Study Problems · 10.2 Rutledge, Inc., has invested $100,000 in a project that will produce cash flows of $45,000, $37,500, and $42,950 over the next three years. Find the payback period for the project. Which project will be chosen if the company’s payback criterion is three years? What if the company accepts all projects as long as the payback period is less than five years? The company will accept all projects that provide an accounting rate of return (ARR) of at least 45 percent. Should the company accept this project? NPV for Craigmore Forklifts: Premium should purchase the Otis forklift since it has a larger NPV. · 10.3 Payback periods for Perryman projects A and B: Payback Period for Project A: Payback Period for Project B: If the payback period is three years, project B will be chosen. If the payback criterion is five years, both A and B will be chosen. The company should accept the project. Try a higher rate. At k 17 percent: Try a higher rate. At k = 17.5 percent: Thus, the IRR for Otis is less than 17.5 percent. Using a financial calculator, you can find the exact rate to be 17.43 percent. Try a lower rate. At k = 13 percent: Try a higher rate. At k = 13.1 percent: Thus, the IRR for Craigmore is less than 13.1 percent. The exact rate is 13.06 percent. Based on the IRR, we would still choose the Otis system over the Craigmore system. The decision is the same as that indicated by NPV. Questions and Problems · 10.1 Net present value: Riggs Corp. management is planning to spend $650,000 on a new marketing campaign. They believe that this action will result in additional cash flows of $325,000 over the next three years. If the discount rate is 17.5 percent, what is the NPV on this project? Year Cash Flow 0 −$3,300,000 1 875,123 2 966,222 3 1,145,000 4 1,250,399 5 1,504,445 · 10.4 Net present value: Franklin Mints, a confectioner, is considering purchasing a new jelly bean-making machine at a cost of $312,500. The company’s management projects that the cash flows from this investment will be $121,450 for the next seven years. If the appropriate discount rate is 14 percent, what is the NPV for the project? · 10.6 Payback: Refer to Problem 10.5. What are the payback periods for production systems 1 and 2? If the systems are mutually exclusive and the firm always chooses projects with the lowest payback period, in which system should the firm invest? · 10.13 Net present value: Champlain Corp. is investigating two computer systems. The Alpha 8300 costs $3,122,300 and will generate cost savings of $1,345,500 in each of the next five years. The Beta 2100 system costs $3,750,000 and will produce cost savings of $1,125,000 in the first three years and then $2 million for the next two years. If the company’s discount rate for similar projects is 14 percent, what is the NPV for the two systems? Which one should be chosen based on the NPV? Year Cash Flow 1 $512,496 2 −242,637 3 814,558 4 887,225 5 712,642 · 10.15 Net present value: Cranjet Industries is expanding its product line and its production capacity. The costs and expected cash flows of the two independent projects are given in the following table. The firm uses a discount rate of 16.4 percent for such projects. 1. 10.16 Net present value: Emporia Mills management is evaluating two alternative heating systems. Costs and projected energy savings are given in the following table. The firm uses 11.5 percent to discount such project cash flows. Which system should be chosen? 1. 10.17 Payback: Creative Solutions, Inc., has just invested $4,615,300 in new equipment. The firm uses payback period criteria of not accepting any project that takes more than four years to recover its costs. The company anticipates cash flows of $644,386, $812,178, $943,279, $1,364,997, $2,616,300, and $2,225,375 over the next six years. Does this investment meet the firm’s payback criteria? 1. 10.19 Payback: Regent Corp. is evaluating three competing types of equipment. Costs and cash flow projections for all three are given in the following table. Which would be the best choice based on payback period? 1. 10.20 Discounted payback: Nugent Communication Corp. is investing $9,365,000 in new technologies. The company expects significant benefits in the first three years after installation (as can be seen by the following cash flows), and smaller constant benefits in each of the next four years. What is the discounted payback period for the project assuming a discount rate of 10 percent? 1. 10.21 Modified internal rate of return (MIRR): Morningside Bakeries has recently purchased equipment at a cost of $650,000. The firm expects to generate cash flows of $275,000 in each of the next four years. The cost of capital is 14 percent. What is the MIRR for this project? Years Cash Flow 1–4 $23,500,000 5–7 72,000,000 8–10 80,000,000 1. 10.24 Internal rate of return: Refer to Problem 10.5. Compute the IRR for both production system 1 and production system 2. Which has the higher IRR? Which production system has the higher NPV? Explain why the IRR and NPV rankings of systems 1 and 2 are different. 1. 10.26 Internal rate of return: Compute the IRR on the following cash flow streams: 1. 10.28 Draconian Measures, Inc., is evaluating two independent projects. The company uses a 13.8 percent discount rate for such projects. The costs and cash flows for the projects are shown in the following table. What are their NPVs? 1. 10.29 Refer to Problem 10.28. 1. 10.31 Intrepid, Inc., is considering investing in three independent projects. The costs and the cash flows are given in the following table. The appropriate cost of capital is 14.5 percent. Compute the project IRRs and identify the projects that should be accepted. 1. 10.32 Jekyll & Hyde Corp. is evaluating two mutually exclusive projects. The cost of capital is 15 percent. Costs and cash flows are given in the following table. Which project should be accepted? 1. 10.33 Larsen Automotive, a manufacturer of auto parts, is considering investing in two projects. The company typically compares project returns to a cost of funds of 17 percent. Compute the IRRs based on the cash flows in the following table. Which project(s) will be accepted? 1. 10.34 Compute the IRR for each of the following projects: 1. 10.35 Primus Corp. is planning to convert an existing warehouse into a new plant that will increase its production capacity by 45 percent. The cost of this project will be $7,125,000. It will result in additional cash flows of $1,875,000 for the next eight years. The discount rate is 12 percent. 38. Calculate the projects’ NPV. Years CF 1–2 0 3–5 $ 845,000 6–9 $1,450,000 39. Compute the payback and discounted payback periods for the project. 1. 10.40 Given the following cash flows for a capital project, calculate the NPV and IRR. The required rate of return is 8 percent. NPV IRR a. $1,905 10.9% b. $1,905 26.0% c. $3,379 10.9% d. $3,379 26.0% 1. 10.41 Given the following cash flows for a capital project, calculate its payback period and discounted payback period. The required rate of return is 8 percent. The discounted payback period is 1 Jeff Greenberg/Alamy Chapter 7 where E (R EXHIBIT 13.1 The Finance Balance Sheet If the cash flows that the apartment building is expected to produce are worth $4,000,000, then investors would be willing to pay $3,700,000 for the equity in the firm. This is the value of the cash flows that they would expect to receive after making the interest and principal payments on the mortgage. Furthermore, since, by definition, the mortgage is worth $300,000, the value of the debt plus the value of the equity is $300,000 + $3,700,000 = $4,000,000—which is exactly equal to the market value of the firm’s assets. where β to estimate the expected return on the firm’s assets, which is also the firm’s cost of capital. Unfortunately, because analysts are not typically able to estimate betas for individual projects, they generally cannot use this approach. In Equation 13.2, k This tells us that the value of the debt claims equals 7.5 percent of the value of the firm and that the value of the equity claims equals the remaining 92.5 percent of the value of the firm. If the cost of the debt for this business is 6 percent and the cost of the equity is 10 percent, the cost of capital for the firm can be calculated as a weighted average of the costs of the debt and equity: Notice that we have used Equation 13.2 to calculate a weighted average cost of capital (WACC) for the firm in this example. In fact, this is what people typically call the firm’s cost of capital, k Calculating the Cost of Capital for a Firm PROBLEM: You are considering purchasing a rug cleaning company that will cost $2,000,000. You plan to finance the purchase with a $1,500,000 loan from Bank of America (BofA) that has a 6.5 percent interest rate, a $300,000 loan from the seller of the company that has an 8 percent interest rate, and $200,000 of your own money. You will own all of the equity (stock) in the firm. You estimate that the opportunity cost of your $200,000 investment—that is, what you could earn on an investment of similar risk in the capital market—is 12 percent with that much debt. What is the cost of capital for this investment? We can then calculate the WACC using Equation 13.2: On average, you would be paying 7.3 percent per year on every dollar you invested in the firm. This is the opportunity cost of capital for the firm. It is the rate that you would use to discount the cash flows associated with the rug cleaning business in an NPV analysis. By trial and error or with a financial calculator, we solve for i and find: If this bond was sold at par, it paid 7 percent when it was issued five years ago. Someone who buys it today will expect to earn only 6.1 percent per year. This is the annual rate of return required by the market on this bond, which is known as the effective annual yield. Converting the adjusted semiannual rate to an EAR, we see that the actual annual cost of this debt financing is: This after-tax cost of debt is the cost that firms actually use to calculate the WACC. The reason is simply that investors care only about the after-tax cost of capital—just as they care only about after-tax cash flows. Managers are concerned about what they actually have to pay for capital, and the actual cost is reduced if the government subsidizes debt by providing a tax break. Suppose that your taxable income before the interest deduction is $75,000 and, for simplicity, that both your average and marginal tax rates are 20 percent. Without the interest deduction, you would pay taxes totaling $15,000 ($75,000 × 0.20 = $15,000). However, because the interest payments reduce your taxable income, your taxes with the interest deduction will be only $12,600 [($75,000 − $12,000) × 0.20 = $12,600]. The ability to deduct the interest payments you made saved you $2,400 ($15,000 − $12,600 = $2,400)! This savings is exactly equal to the interest payment you make times your marginal tax rate: $12,000 × 0.20 = $2,400. Since you are saving $2,400, the after-tax cost of your interest payments is $9,600 ($12,000 − $2,400 = $9,600), which means that the after-tax cost of this debt is 4.8 percent ($9,600/$200,000 = 0.048, or 4.8 percent). This is exactly what Equation 13.3 tells us. With k Estimating the Cost of Debt for a Firm The pretax cost of the bank loan that you took out today is simply the 6 percent rate that the bank is charging you, assuming that the bank is charging you the market rate. The weighted average pretax cost of debt is: The after-tax cost of debt is therefore: APPLICATION 13.2 LEARNING BY DOING Calculating the After-Tax Cost of Debt for a Firm PROBLEM: You have just successfully completed a leveraged buyout of the firm that you have been working for. To finance this $35 million transaction, you and three partners put up a total of $10 million in equity capital, and you borrowed $25 million from banks and other investors. The bank debt consists of $10 million of secured debt borrowed at a rate of 6 percent from Bank of America and $7 million of senior unsecured debt borrowed at a rate of 7 percent from JPMorgan Chase. The remaining $8 million was borrowed from an investment group managed by a private equity firm. The rate on this subordinated (junior) unsecured debt is 9.5 percent. What is the overall after-tax cost of the debt financing used to buy the firm if you expect the firm’s average and marginal tax rates to both be 25 percent? (2) The weighted average pretax cost of debt is: (3) The after-tax cost of debt is therefore: EXAMPLE 13.1 DECISION MAKING SITUATION: Your pizza parlor business has developed such a strong reputation that you have decided to take advantage of the restaurant’s name recognition by selling frozen pizzas through grocery stores. In order to do this, you will have to build a manufacturing facility. You estimate that this will cost you $10 million. Since your business currently has only $2 million in the bank, you will have to borrow the remaining $8 million. You have spoken with two bankers about possible loan packages. The banker from Easy Money Financial Services offered you a loan for $6 million with a 6 percent rate and $2 million with a 7.5 percent rate. You calculate the pretax cost of debt for this package to be: Your local banker offered you a single $8 million loan for 6.350 percent. Which financing should you choose if all terms on all of the loans, other than the interest rates, are the same? In this equation, the expected return on an asset is a linear function of the systematic risk associated with that asset. Equation 13.4 is just another way of writing Equation 7.10. It tells us that the cost of common stock equals the risk-free rate of return plus compensation for the systematic risk associated with the common stock. You already saw some examples of how to use this equation to calculate the cost of equity in the discussion of the Capital Asset Pricing Model (CAPM) in This example illustrates how Equation 13.4 is used to estimate the cost of common stock for a company. How would the analysis differ for a private company? The only difference is that we would not be able to estimate the beta directly. We would have to estimate the beta using betas for similar public companies. PROBLEM: You have decided to estimate the cost of the common equity in your pizza business on November 19, 2010. As noted earlier, the risk-free rate and the market risk premium on that day were 3.95 percent and 6.01 percent, respectively. Since you have already decided that Domino’s Pizza is a reasonably comparable company, you obtain Domino’s beta from the Yahoo! finance Web site ( k where D1 is the dividend expected to be paid one period from today, R is the required rate of return, and g is the annual rate at which the dividends are expected to grow in perpetuity. While Equation 13.5 is just a variation of Equation 9.4, it is important enough to identify as a separate equation because it provides a direct way of estimating the cost of equity under certain circumstances. If we can estimate the dividend that stockholders will receive next period, D1, and we can estimate the rate at which the market expects dividends to grow over the long run, g, then we can use today’s market price, P0, in Equation 13.5 to tell us what rate of return investors in the firm’s common stock are expecting to earn. This approach can be useful for a firm that pays dividends when it is reasonable to assume dividends will grow at a constant rate and when the analyst has a good idea what that growth rate will be. An electric utility firm is an example of this type of firm. Some electric utility firms pay relatively high and predictable dividends that increase at a fairly consistent rate. In contrast, this approach would not be appropriate for use by a high-tech firm that pays no dividends or that pays a small dividend that is likely to increase at a high rate in the short run. Equation 13.5, like any other equation, should be used only if it is appropriate for the particular stock. where D In this equation, we have replaced the R in Equation 9.6 with k All this equation does is add the present values of the dividends that are expected in each of the next three years and the present value of a growing perpetuity that begins in the fourth year. EXHIBIT 13.2 The Three-Stage Dividend Growth Equation From the information given in the problem statement, we know the following: Substituting these values into the above equation gives us the following, which we solve for k As mentioned earlier, we can solve this equation for k Which Method Should We Use? as: where Pps is the present value of the expected dividends (the current preferred stock price), Dps is the annual preferred stock dividend, and k Plugging the information from our example into Equation 13.6, we see that k This is the rate of return at which the present value of the annual $5 cash flows equals the market price of $85. Therefore, 5.9 percent is the rate that investors currently require for investing in this preferred stock. APPLICATION 13.4 LEARNING BY DOING PROBLEM: You work in the Treasury Department at Wells Fargo & Company, and your manager has asked you to estimate the cost of each of the different types of stock that Wells Fargo has outstanding. One of these issues is a 8 percent non-cumulative preferred stock that has a stated value of $1,000 and is currently selling for $927.90. Although this preferred stock is publicly traded, it does not trade very often. This means that you cannot use the CAPM to estimate k You may recall from the discussion in where x Note that because the $4 million and $3 million loans have rates that equal what it would cost to refinance them today, their market values equal the amount that is owed. Since the $10 million market value of the bond issue is below the $11 million face value, the rate that firm is actually paying must be lower than the 7 percent rate you estimated to reflect the current cost of this debt. Recall that as interest rates increase, the market value of a bond decreases. This is the negative relation that we referred to earlier in this chapter. From Equation 13.4, we calculate the cost of the common equity to be: You can see real-world applications of the WACC calculation at the New Zealand Web site for Pricewaterhouse-Coopers, the international accounting and consulting firm, at APPLICATION 13.5 LEARNING BY DOING Calculating the WACC with Equation 13.7 PROBLEM: After calculating the cost of the common equity in your pizza business to be 12.1 percent (see Learning by Doing Application 13.3), you have decided to estimate the WACC. You recently hired a business appraiser to estimate the value of your stock, which includes all of the outstanding common equity. His report indicates that it is worth $500 million. Your company has no other long-term debt or any preferred stock outstanding. Both the marginal and average tax rates for your company are 20 percent. What is the WACC for your pizza business? and the WACC is: EXAMPLE 13.2 DECISION MAKING SITUATION: You are a financial analyst for the company whose WACC of 7.3 percent we just calculated in the main text. One day, your manager walks in to your office and tells you that she is thinking about selling $23 million of common stock and using the proceeds from the sale to pay back both of the firm’s loans and to repurchase all of the outstanding bonds and preferred stock. She tells you that this is a smart move because if she does this, the beta of the firm’s common stock will decline to 0.70 and the overall k What do you tell your manager? Chapter 11 EXHIBIT 13.3 Potential Errors When Using the WACC to Evaluate Projects This is an attractive project because it returns more than the investors’ opportunity cost of capital. As you can see, when the WACC is used to discount the cash flows, the firm will end up rejecting a positive NPV project. The firm will be passing up an opportunity to create value for its stockholders. As an exercise, you might try constructing a numerical example in which a firm accepts a negative NPV project. This equation provides a measure of the firm’s cost of capital that reflects both how the firm’s assets have been financed—that is, the mix of debt and preferred and common stock that was used to acquire those assets—and the current cost of each type of financing. In other words, the WACC reflects both the x’s and the k’s associated with the firm’s financing. Why is this important? Because the costs of the different types of capital depend on the fraction of the total firm financing that each represents. If the firm uses more or less debt, the cost of debt will be higher or lower. In turn, the cost of both preferred stock and common stock will be affected. This means that even if the underlying business risk of the project is the same as that for the firm as a whole, if the project is financed differently than the firm, the appropriate discount rate for the project analysis will be different from that for the firm as a whole. EXHIBIT 13.4 Potential Errors When Using Multiple Discount Rates to Evaluate Projects Explain what the weighted average cost of capital for a firm is and why it is often used as a discount rate to evaluate projects. Self-Study Problems · 13.5 The pretax debt contribution to the cost of capital is x · 13.1 Finance balance sheet: KneeMan Markup Company has total debt obligations with book and market values equal to $30 million and $28 million, respectively. It also has total equity with book and market values equal to $20 million and $70 million, respectively. If you were going to buy all of the assets of KneeMan Markup today, how much should you be willing to pay? · 13.13 Finance balance sheet: Explain why the total value of all of the securities used to finance a firm must be equal to the value of the firm. · 13.27 You are analyzing the cost of capital for MacroSwift Corporation, which develops software operating systems for computers. The firm’s dividend growth rate has been a very constant 3 percent per year for the past 15 years. Competition for the firm’s current products is expected to develop in the next year, and MacroSwift is currently expanding its revenue stream into the multimedia industry. Evaluate the appropriateness of continuing to use a 3 percent growth rate in dividends for MacroSwift in your cost of capital model. the beta of the assets of a firm also equals a weighted average of the betas for the debt, preferred stock, and common stock of a firm: Why must this be true? 1. 13.38 The cost of equity is equal to the: 1 2 3 4 5 6 7 8 9 Word doc 5 Project 5 Report Instructions Answer the five questions below. They focus entirely on the financing, risk/return, Cost of Capital and Budget Forecasting of Largo Global Inc. (LGI) based on the investing activities that took place in project 4. Base your analysis on the data provided and calculated in the Excel workbook. Provide support for your reasoning from the readings in Project 5, Step 1, and the discussion in Project 5, Step 3. Be sure to cite your sources. Provide a detailed response below each question. Use 12-point font and double spacing. Maintain the existing margins in this document. Your final Word document, including the questions, should not exceed 5 pages. Include a title page in addition to the five pages. Any tables and graphs you choose to include are also excluded from the five-page limit. Name your document as follows: P5_Final_lastname_Report_date. You must address all five questions and make full use of the information on all tabs of project 5 as well as data in other Excel workbooks (e.g. from project 1: ratio, common-size, and cash flow analysis). You are strongly encouraged to exceed the requirements by refining your analysis. Consider other tools and techniques that were discussed in the required and recommended reading for Project 5. This means adding an in-depth explanation of what happened in that year for which data was provided to make precise recommendations to LGI. Title Page Name Course and section number Faculty name Submission date Questions
1. How would you assess the evolution of the capital structure of LGI? Reflecting on your work in Project 1, would you consider the risk exposure under control? If not, what are your recommendations? [insert your answer here]
2. What kind of information do you find valuable in CAPM to guide you in assessing the risk of LGI compared to other firms and the market in general?
[insert your answer here]
3. Identify and differentiate the stakeholders of LGI and explain how each one should perceive and weigh the risk and/or return of the firm. [insert your answer here]
4. Would you consider the investment made in Project 4 optimally financed considering the proportion of debt that is bearable by LGI? How did the current WACC in Project 5 depart from the state of the firm in Project 1? [insert your answer here]
5. If you had to advise a potential investor interested in having a minority stake in LGI, what kind of information would you provide to help the investor make a decision? Would you be bullish or have reservations? Support your answer with facts and data from all MBA 620 projects as well as your budget projections. [insert your answer here] Word doc 5 Project 5 Report Instructions Answer the five questions below. They focus entirely on the financing, risk/return, Cost of Capital and Budget Forecasting of Largo Global Inc. (LGI) based on the investing activities that took place in project 4. Base your analysis on the data provided and calculated in the Excel workbook. Provide support for your reasoning from the readings in Project 5, Step 1, and the discussion in Project 5, Step 3. Be sure to cite your sources. Provide a detailed response below each question. Use 12-point font and double spacing. Maintain the existing margins in this document. Your final Word document, including the questions, should not exceed 5 pages. Include a title page in addition to the five pages. Any tables and graphs you choose to include are also excluded from the five-page limit. Name your document as follows: P5_Final_lastname_Report_date. You must address all five questions and make full use of the information on all tabs of project 5 as well as data in other Excel workbooks (e.g. from project 1: ratio, common-size, and cash flow analysis). You are strongly encouraged to exceed the requirements by refining your analysis. Consider other tools and techniques that were discussed in the required and recommended reading for Project 5. This means adding an in-depth explanation of what happened in that year for which data was provided to make precise recommendations to LGI. Title Page Name Course and section number Faculty name Submission date Questions
1. How would you assess the evolution of the capital structure of LGI? Reflecting on your work in Project 1, would you consider the risk exposure under control? If not, what are your recommendations? [insert your answer here]
2. What kind of information do you find valuable in CAPM to guide you in assessing the risk of LGI compared to other firms and the market in general?
[insert your answer here]
3. Identify and differentiate the stakeholders of LGI and explain how each one should perceive and weigh the risk and/or return of the firm. [insert your answer here]
4. Would you consider the investment made in Project 4 optimally financed considering the proportion of debt that is bearable by LGI? How did the current WACC in Project 5 depart from the state of the firm in Project 1? [insert your answer here]
5. If you had to advise a potential investor interested in having a minority stake in LGI, what kind of information would you provide to help the investor make a decision? Would you be bullish or have reservations? Support your answer with facts and data from all MBA 620 projects as well as your budget projections. [insert your answer here] Turnitin Report Formatting Title Page Citation Outline
2
INSTRUCTIONS
Cost of Capital
o
Find the cost of
E
o Find the Weighted Average Cost of Capital (
WACC
Payback
o Complete the After-tax
Cash Flow
o Complete the
D
o Complete the questions on the tab
Budget Projections
o
Re
5
o Expense increases 2% annually
Instructions:
1
2
Find the Weighted Average Cost of Equity (WACC)
1
RF
ꞵ
RM
= CAPM
————————————–
2
E
D
Total Capital (V)
$ –
0
Last Fiscal
Year
Interest Expense
Tax Rate (TC)
1. Find the weight of equity = E / (E + D).
2. Find the weight of debt = D / (E + D).
Re
3
Rd
4
WACC
5. Find the weighted average cost of capital.
a. As of today, Largo Global market capitalization (E) is $
6
7
b. Largo Global’s book value of debt is $539,500,000.2
c. Cost of Equity = CAPM from question 1
d. Cost of Debt = Last Fiscal Year End Interest Expense3 / Book Value of Debt (D).
e. Use the tax rates given in Project 4 Tab 3.
_________
1 Market value of equity (E), also known as market cap, is calculated using the following equation:
Market Cap = Share Price x Shares Outstanding from Project 1
2 Book value of debt (D) is calculated as follows: Book Value of Debt = Last 2-Year Avg of Notes Payable + Last 2 Year Avg of Long-Term Debt from Project 1.
3 From Project 1
Payback Table View
Table 1 – Data
Cost of new equipment (at year 0)
191.10
million
Corporate income tax rate – Federal
26.0%
Corporate income tax rate – State of Maryland
8
Discount rate for the project using WACC
Loan Amount
Loan Interest rate (Prime + 2)
5.25%
Table 2 – After-tax Cash Flow Table
(all figures in $ millions)
Year
Projected Cash Inflows from Operations
Projected Cash Outflows from Operations
Depreciation Expense
Projected Taxable Income
Projected Federal Income Taxes
Projected State Income Taxes
Projected After-tax Cash Flows
PV
NPV
IRR
NPV2
Excel function to use :
SLN
IPMT
0
1
$850.0
$840.0
$23.89
$0.00
($13.89)
($3.61)
($1.11)
$14.72
2
$900.0
$810.0
3
$990.0
$870.0
4
$1,005.0
5
$1,200.0
$1,100.0
6
$1,300.0
$1,150.0
7
$1,350.0
8
$1,320.0
PV
NPV1 – calculated NPV including interest expense
NPV2 – calculated NPV from Project 4 at the lower discount rate of 4.95%
Payback Timeline View Example of Actual Cash Flows
0 1 2 3 4 5 6 7 8
|
Cash Flow $0.00 $14.72 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $14.72
Cummulative Cash Flow
$0.00 $14.72 $14.72 $14.72 $14.72 $14.72 $14.72 $14.72 $14.72
Payback Period
years
ERROR:#DIV/0!
months
0 1 2 3 4 5 6 7 8 PV
| | | | | | | | | $0.00
Discounted Cash Flow (DCF)
$0.00
Cummulative DCF
Payback Period years months
ANSWER THESE QUESTIONS:
1. What is the total depreciation for tax purposes?
2. What is the total PV of the Cash Flows using the WACC rate?
3. What is the NPV using the WACC rate?
4. What is the NPV using the alternative rate?
5. What is the IRR?
6. What is the payback period using the DCF?
7. Should the project be accepted? Why?
Technologically advanced distribution equipment proposal re-evaluation
The CFO has asked you to re-evaluate the cash flow projections associated with the equipment purchase proposal due to the proposed loan agreement, and recommend whether the purchase should go forward. Table 1 shows the data and Table 2 shows projections of the cash inflows and outflows that would occur during the first eight years using the new equipment.
Keep the following in mind: Row 34 has a suggested Excel function to use. Complete all the blank cells within the tables.
I. In the Data Table:
A. Use the WACC calulated on the Cost of Capital tab
B. Calulate the loan amount with a 10% down payment
II. In the After-tax Cash Flow:
C. Complete the Depreciation Expense from Project 4 (straight line, $0 Salvage)
D. Complete the interest expense using the WACC from Cost of Capital tab
E. Complete the After-tax Cash Flow Table including the interest expense
F. Compute the PV, NPV1, IRR, and adjusted NPV2
III. In the Payback Timeline View:
G. Complete the discounted cash flow Payback Timeline View of Discounted Cash Flows
i) complete the timeline amounts based on the DCF (DCF is the same as PV)
ii) complete the timeline amountss for the Cummulative DCF
iii) calulate the payback period in years and months
IV. Answer the following questions:
1. What is the total depreciation for tax purposes?
2. What is the total PV of the Cash Flows using the WACC rate?
3. What is the NPV using the WACC rate?
4. What is the NPV using the alternative rate?
5. What is the IRR?
6. What is the payback period using the DCF?
7. Should the project be accepted? Why?
INSTRUCTIONS:
1).
2021
2024
Revenue increases 5% annually
Expense increases 2½% annually
For Depreciation and
Interest expense
2020
2). Answer the question below the forecast.
1).
Largo Global Income Statement of December 31, 2020 (millions)
ACTUAL
BUDGET
FORECAST
2020 2021
2022
2023
Sales (net sales)
$2,013
Cost of goods sold
1400
Gross profit
613
Selling, general, and administrative expenses
125
Earnings before Interest, taxes, depreciation, and amortization (EBITDA)
488
Depreciation and amortization
174
Earning before interest and taxes (EBIT) Operating income (loss)
314
Interest expense
141
Earnings before taxes (EBT)
173
Taxes (34%)
59
Net earnings (loss)/Net Income
$ 114
2). Based on the changes suggested throughout the 5 projects, is Largo Global in a better financial position?
Step 5: Prepare the Analysis Report for Project 5
Start Here
· 3.2: Employ mathematical or statistical operations and data analysis techniques to arrive at a correct or optimal solution.
· 3.3: Analyze mathematical or statistical information, or the results of quantitative inquiry and manipulation of data.
· 10.4: Make strategic managerial decisions for obtaining capital required for achieving organizational goals.
Step 1: Prepare for the Project
Step 2: Review and Practice
Step 3: Participate in the Required Project 5 Discussion
Step 4: Complete the Analysis Calculation for Project 5
Step 5: Prepare the Analysis Report for Project 5
PeopleImages / E+ / Getty Images
You have developed an in-depth understanding of LGI’s operating efficiency related to costing and how that impacts the bottom line. You feel confident that your investment choices will positively boost LGI’s productivity and improve the company’s operations. Thanks to your efforts, the company will have a plan for financing its investments appropriately. LGI will finally be on a path of a sustainable future. Answer the questions in the Project 5 Questions – Report Template document. Prepare your analysis report including recommendations for how the company can improve its financial situation.
Complete the analysis report:
· Download the Project 5 Questions – Report Template
· Read the instructions.
· Answer all the questions.
· Include your recommendations.
· Submit the analysis report (Word document) and analysis calculation (Excel file) to the dropbox as your final deliverable at the end of Week 10. Label your files as follows:
· P5_Final_lastname_Report_date x
· P5_Final_lastname_Calculation_date.xlsx
Check Your Evaluation Criteria
Before you submit your assignment, review the competencies below, which your instructor will use to evaluate your work. A good practice would be to use each competency as a self-check to confirm you have incorporated all of them. To view the complete grading rubric, click My Tools, select Assignments from the drop-down menu, and then click the project title.
· 3.1: Identify numerical or mathematical information that is relevant in a problem or situation.
· 3.2: Employ mathematical or statistical operations and data analysis techniques to arrive at a correct or optimal solution.
· 3.3: Analyze mathematical or statistical information, or the results of quantitative inquiry and manipulation of data.
· 10.4: Make strategic managerial decisions for obtaining capital required for achieving organizational goals.
Risk and Return
Learning Objectives
Explain the relation between risk and return.
Describe the two components of a total holding period return, and calculate this return for an asset.
Explain what an expected return is and calculate the expected return for an asset.
Explain what the standard deviation of returns is and why it is very useful in finance, and calculate it for an asset.
Explain the concept of diversification.
Discuss which type of risk matters to investors and why.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate whether the expected return of an asset is sufficient to compensate an investor for the risks associated with that asset.
When Blockbuster Inc. filed for bankruptcy protection on Thursday, September 23, 2010, its days as the dominant video rental firm were long gone. Netflix had become the most successful competitor in the video rental market through its strategy of renting videos exclusively online and avoiding the high costs associated with operating video rental stores.
The bankruptcy filing passed control of Blockbuster to a group of bondholders, including the famous billionaire investor Carl Icahn, and the shares owned by the old stockholders became virtually worthless. The bondholders planned to reorganize the company and restructure its financing so that it had a chance of competing more effectively with Netflix in the future.
Over the previous five years, Blockbuster stockholders had watched the value of their shares steadily decline as, year after year, the company failed to respond effectively to the threat posed by Netflix. From September 23, 2005 to September 23, 2010, the price of Blockbuster shares fell from $4.50 to $0.04. In contrast, the price of Netflix shares rose from $24.17 to $160.47 over the same period. While the Blockbuster stockholders were losing almost 100 percent of their investments, Netflix stockholders were earning an average return of 46 percent per year!
This chapter discusses risk, return, and the relation between them. The difference in the returns earned by Blockbuster and Netflix stockholders from 2005 to 2010 illustrates a challenge faced by all investors. The shares of both of these companies were viewed as risky investments in 2005, and yet an investor who put all of his or her money in Blockbuster lost virtually everything, while an investor who put all of his or her money in Netflix earned a very high return. How should have investors viewed the risks of investing in these companys’ shares in 2005? How is risk related to the returns that investors might expect to earn? How does diversification reduce the overall risk of an investor’s portfolio? These are among the topics that we discuss in this chapter.
CHAPTER PREVIEW
Up to this point, we have often mentioned the rate of return that we use to discount cash flows, but we have not explained how that rate is determined. We have now reached the point where it is time to examine key concepts underlying the discount rate. This chapter introduces a quantitative framework for measuring risk and return. This framework will help you develop an intuitive understanding of how risk and return are related and what risks matter to investors. The relation between risk and return has implications for the rate we use to discount cash flows because the time value of money that we discussed in
Chapters 5
and
6
is directly related to the returns that investors require. We must understand these concepts in order to determine the correct present value for a series of cash flows and to be able to make investment decisions that create value for stockholders.
We begin this chapter with a discussion of the general relation between risk and return to introduce the idea that investors require a higher rate of return from riskier assets. This is one of the most fundamental relations in finance. We next develop the statistical concepts required to quantify holding period returns, expected returns, and risk. We then apply these concepts to portfolios with a single asset and with more than one asset to illustrate the benefit of diversification. From this discussion, you will see how investing in more than one asset enables an investor to reduce the total risk associated with his or her investment portfolio, and you will learn how to quantify this benefit.
Once we have discussed the concept of diversification, we examine what it means for the relation between risk and return. We find that the total risk associated with an investment consists of two components: (1) unsystematic risk and (2) systematic risk. Diversification enables investors to eliminate the unsystematic risk associated with an individual asset. Investors do not require higher returns for the unsystematic risk that they can eliminate through diversification. Only systematic risk—risk that cannot be diversified away—affects expected returns on an investment. The distinction between unsystematic and systematic risk and the recognition that unsystematic risk can be diversified away are extremely important in finance. After reading this chapter, you will understand precisely what the term risk means in finance and how it is related to the rates of return that investors require.
7.1 RISK AND RETURN
·
The rate of return that investors require for an investment depends on the risk associated with that investment. The greater the risk, the larger the return investors require as compensation for bearing that risk. This is one of the most fundamental relations in finance. The rate of return is what you earn on an investment, stated in percentage terms. We will be more specific later, but for now you might think of risk as a measure of how certain you are that you will receive a particular return. Higher risk means you are less certain.
To get a better understanding of how risk and return are related, consider an example. You are trying to select the best investment from among the following three stocks:
The greater the risk associated with an investment, the greater the return investors expect from it. A corollary to this idea is that investors want the highest return for a given level of risk or the lowest risk for a given level of return. When choosing between two investments that have the same level of risk, investors prefer the investment with the higher return. Alternatively, if two investments have the same expected return, investors prefer the less risky alternative.
Which would you choose? If you were comparing only Stocks A and B, you should choose Stock A. Both stocks have the same expected return, but Stock A has less risk. It does not make sense to invest in the riskier stock if the expected return is the same. Similarly, you can see that Stock C is clearly superior to Stock B. Stocks B and C have the same level of risk, but Stock C has a higher expected return. It would not make sense to accept a lower return for taking on the same level of risk.
But what about the choice between Stocks A and C? This choice is less obvious. Making it requires understanding the concepts that we discuss in the rest of this chapter.
7.2 QUANTITATIVE MEASURES OF RETURN
·
Before we begin a detailed discussion of the relation between risk and return, we should define more precisely what these terms mean. We begin with measures of return.
Holding Period Returns
the total return on an asset over a specific period of time or holding period
When people refer to the return from an investment, they are generally referring to the total return over some investment period, or holding period. The total holding period return consists of two components: (1) capital appreciation and (2) income. The capital appreciation component of a return, RCA, arises from a change in the price of the asset over the investment or holding period and is calculated as follows:
The income component of a return arises from income that an investor receives from the asset while he or she owns it. For example, when a firm pays a cash dividend on its stock, the income component of the return on that stock, RI, is calculated as follows:
The total holding period return, RT, is simply the sum of the capital appreciation and income components of return:
http://www.callan.com/research/periodic
/.
Let’s consider an example of calculating the total holding period return on an investment. One year ago today, you purchased a share of Dell Inc. stock for $12.50. Today it is worth $13.90. Dell paid no dividend on its stock. What total return did you earn on this stock over the past year?
If Dell paid no dividend and you received no other income from holding the stock, the total return for the year equals the return from the capital appreciation. The total return is calculated as follows:
APPLICATION 7.1 LEARNING BY DOING
APPROACH: Use Equation 7.1 to calculate the total holding period return. To calculate RT using Equation 7.1, you must know P0, P1, and CF1. In this problem, you can assume that the $7,000 was spent at the time you bought the car to purchase parts and materials. Therefore, your initial investment, P0, was $1,500 $7,000 $8,500. Since there were no other cash inflows or outflows between the time that you bought the car and the time that you sold it, CF1 equals $0.
SOLUTION: The total holding period return is:
·
Suppose that you are a senior who plays college baseball and that your team is in the College World Series. Furthermore, suppose that you have been drafted by the Washington Nationals and are coming up for what you expect to be your last at-bat as a college player. The fact that you expect this to be your last at-bat is important because you just signed a very unusual contract with the Nationals. Your signing bonus will be determined solely by whether you get a hit in your final collegiate at-bat. If you get a hit, then your signing bonus will be $800,000. Otherwise, it will be $400,000. This past season, you got a hit 32.5 percent of the times you were at bat (you did not get a hit 67.5 percent of the time), and you believe this percentage reflects the likelihood that you will get a hit in your last collegiate at-bat.
1
H is the probability of a hit, p
NH is the probability of no hit, BH is the bonus you receive if you get a hit, and BNH is the bonus you receive if you get no hit. Since p
H equals 0.325, p
NH equals 0.675, BH equals $800,000, and BNH equals $400,000, the expected value of your bonus is:
What would your expected payoff be if you got a hit 99 percent of the time? We intuitively know that the expected bonus should be much closer to $800,000 in this case. In fact, it is:
We calculate an expected return in finance in the same way that we calculate any expected value. The expected return is a weighted average of the possible returns from an investment, where each of these returns is weighted by the probability that it will occur. In general terms, the expected return on an asset, E (RAsset), is calculated as follows:
an average of the possible returns from an investment, where each return is weighted by the probability that it will occur
where R
i
is possible return i and p
i
is the probability that you will actually earn R
i
. The summation symbol in this equation
It is important to make sure that the sum of the n individual probabilities, the pi
‘s, always equals 1, or 100 percent, when you calculate an expected value. The sum of the probabilities cannot be less than 100 percent because you must account for all possible outcomes in the calculation. On the other hand, as you may recall from statistics, the sum of the probabilities of all possible outcomes cannot exceed 100 percent. For example, notice that the sum of the pi
‘s equals 1 in each of the expected bonus calculations that we discussed earlier (0.325 0.625 in the first calculation, 0.5 0.5 in the second, and 0.99 0.01 in the third).
The expected return on an asset reflects the return that you can expect to receive from investing in that asset over the period that you plan to own it. It is your best estimate of this return, given the possible outcomes and their associated probabilities.
Note that if each of the possible outcomes is equally likely (that is, p
1 = p
2 = p
3 = … = pn
= p = 1/n), this formula reduces to the formula for a simple (equally weighted) average of the possible returns:
Since Dell pays no dividends, the total return on its stock equals the return from capital appreciation:
‘s equals 1.
APPLICATION 7.2 LEARNING BY DOING
Given your oil price prediction, you estimate that there is a 30 percent chance that the value of your railroad cars will increase by 15 percent, a 40 percent chance that their value will increase by 25 percent, and a 30 percent chance that their value will increase by 30 percent in the next year. In addition to appreciation in the value of your cars, you expect to earn 10 percent on your investment over the next year (after expenses) from leasing the railroad cars. What total return do you expect to earn on your railroad car investment over the next year?
APPROACH: Use Equation 7.1 first to calculate the total return that you would earn under each of the three possible outcomes. Next use these total return values, along with the associated probabilities, in Equation 7.2 to calculate the expected total return.
SOLUTION: To calculate the total returns using Equation 7.1,
EXAMPLE 7.1 DECISION MAKING
Using Expected Values in Decision Making
There is some uncertainty regarding how many new customers will visit your restaurant after hearing one of your radio ads. You estimate that there is a 30 percent chance that 35 people will visit, a 45 percent chance that 50 people will visit, and a 25 percent chance that 60 people will visit. Therefore, you expect the following number of new customers to visit your restaurant in response to each radio ad:
Similarly, you estimate that there is a 20 percent chance you will get 20 new customers in response to an ad placed on a taxi, a 30 percent chance you will get 30 new customers, a 30 percent chance that you will get 40 new customers, and a 20 percent chance that you will get 50 new customers. Therefore, you expect the following number of new customers in response to each ad that you place on a taxi:
Which of these two advertising options is more attractive? Is it cost effective?
DECISION: You should advertise on taxicabs. For a monthly cost of $1,000, you expect to attract 1,400 new customers with taxicab advertisements but only 960 new customers if you advertise on the radio.
The answer to the question of whether advertising on taxicabs is cost effective depends on how much the gross profits (profits after variable costs) of your business are increased by those 1,400 customers. Monthly gross profits will have to increase by $1,000, or average 72 cents per new customer ($1,000/1,400 $0.72) to cover the cost of the advertising campaign.
> BEFORE YOU GO ON
1. What are the two components of a total holding period return?
2. How is the expected return on an investment calculated?
7.3 THE VARIANCE AND STANDARD DEVIATION AS MEASURES OF RISK
·
We turn next to a discussion of the two most basic measures of risk used in finance—the variance and the standard deviation. These are the same variance and standard deviation measures that you studied if you took a course in statistics.
Calculating the Variance and Standard Deviation
Let’s begin by returning to our College World Series example. Recall that you will receive a bonus of $800,000 if you get a hit in your final collegiate at-bat and a bonus of $400,000 if you do not. The expected value of your bonus is $530,000. Suppose you want to measure the risk, or uncertainty, associated with the bonus. How can you do this? One approach would be to compute a measure of how much, on average, the bonus payoffs deviate from the expected value. The underlying intuition here is that the greater the difference between the actual bonus and the expected value, the greater the risk. For example, you might calculate the difference between each possible bonus payment and the expected value, and sum these differences. If you do this, you will get the following result:
variance (σ2)
a measure of the uncertainty associated with an outcome
A better approach would be to square the differences (squaring the differences makes all the numbers positive) and multiply each squared difference by its associated probability before summing them up. This calculation yields the variance (σ2) of the possible outcomes. The variance does not suffer from the two problems mentioned earlier and provides a measure of risk that has a consistent interpretation across different situations or assets. For the original bonus arrangement, the variance is:
Because it is somewhat awkward to work with units of squared dollars, in a calculation such as this we would typically take the square root of the variance. The square root gives us the standard deviation (σ) of the possible outcomes. For our example, the standard deviation is:
the square root of the variance
As you will see when we discuss the normal distribution, the standard deviation has a natural interpretation that is very useful for assessing investment risks.
The general formula for calculating the variance of returns can be written as follows:
Interpreting the Variance and Standard Deviation
The variance and standard deviation are especially useful measures of risk for variables that are normally distributed—those that can be represented by a normal distribution. The normal distribution is a symmetric frequency distribution that is completely described by its mean (average) and standard deviation.
Exhibit 7.1
illustrates what this distribution looks like. Even if you have never taken a statistics course, you have already encountered the normal distribution. It is the “bell curve” on which instructors often base their grade distributions. SAT scores and IQ scores are also based on normal distributions.
normal distribution
a symmetric frequency distribution that is completely described by its mean and standard deviation; also known as a bell curve due to its shape
This distribution is very useful in finance because the returns for many assets are approximately normally distributed. This makes the variance and standard deviation practical measures of the uncertainty associated with investment returns. Since the standard deviation is more easily interpreted than the variance, we will focus on the standard deviation as we discuss the normal distribution and its application in finance.
In
Exhibit 7.1
, you can see that the normal distribution is symmetric: the left and right sides are mirror images of each other. The mean falls directly in the center of the distribution, and the probability that an outcome is less than or greater than a particular distance from the mean is the same whether the outcome is on the left or the right side of the distribution. For example, if the mean is 0, the probability that a particular outcome is 3 or less is the same as the probability that it is + 3 or more (both are 3 or more units from the mean). This enables us to use a single measure of risk for the normal distribution. That measure is the standard deviation.
The normal distribution is a symmetric distribution that is completely described by its mean and standard deviation. The mean is the value that defines the center of the distribution, and the standard deviation, s, describes the dispersion of the values centered around the mean.
The standard deviation tells us everything we need to know about the width of the normal distribution or, in other words, the variation in the individual values. This variation is what we mean when we talk about risk in finance. In general terms, risk is a measure of the range of potential outcomes. The standard deviation is an especially useful measure of risk because it tells us the probability that an outcome will fall a particular distance from the mean, or within a particular range. You can see this in the following table, which shows the fraction of all observations in a normal distribution that are within the indicated number of standard deviations from the mean.
For example, if the expected return for a real estate investment in Miami, Florida, is 10 percent with a standard deviation of 2 percent, there is a 90 percent chance that the actual return will be within 3.29 percent of 10 percent. How do we know this? As shown in the table, 90 percent of all outcomes in a normal distribution have a value that is within 1.645 standard deviations of the mean value, and 1.645 × 2 percent = 3.29 percent. This tells us that there is a 90 percent chance that the realized return on the investment in Miami will be between 6.71 percent (10 percent − 3.29 percent = 6.71 percent) and 13.29 percent (10 percent + 3.29 percent = 13.29 percent), a range of 6.58 percent (13.29 percent − 6.71 percent = 6.58 percent).
You may be wondering what is standard about the standard deviation. The answer is that this statistic is standard in the sense that it can be used to directly compare the uncertainties (risks) associated with the returns on different investments. For instance, suppose you are comparing the real estate investment in Miami with a real estate investment in Fresno, California. Assume that the expected return on the Fresno investment is also 10 percent. If the standard deviation for the returns on the Fresno investment is 3 percent, there is a 90 percent chance that the actual return is within 4.935 percent (1.645 × 3 percent = 4.935 percent) of 10 percent. In other words, 90 percent of the time, the return will be between 5.065 percent (10 percent − 4.935 percent = 5.065 percent) and 14.935 percent (10 percent + 4.935 percent = 14.935 percent), a range of 9.87 percent (14.935 percent − 5.065 percent = 9.87 percent).
This range is exactly 9.87 percent/6.58 percent = 1.5 times as large as the range for the Miami investment opportunity. Notice that the ratio of the two standard deviations also equals 1.5 (3 percent/2 percent = 1.5). This is not a coincidence. We could have used the standard deviations to directly compute the relative uncertainty associated with the Fresno and Miami investment returns. The relation between the standard deviation of returns and the width of a normal distribution (the uncertainty) is illustrated in
Exhibit 7.2
.
Let’s consider another example of how the standard deviation is interpreted. Suppose customers at your pizza restaurant have complained that there is no consistency in the number of slices of pepperoni that your cooks are putting on large pepperoni pizzas. One night you decide to work in the area where the pizzas are made so that you can count the number of pepperoni slices on the large pizzas to get a better idea of just how much variation there is. After counting the slices of pepperoni on 50 pizzas, you estimate that, on average, your pies have 18 slices of pepperoni and that the standard deviation is 3 slices.
With this information, you estimate that 95 percent of the large pepperoni pizzas sold in your restaurant have between 12.12 and 23.88 slices. You are able to estimate this range because you know that 95 percent of the observations in a normal distribution fall within 1.96 standard deviations of the mean. With a standard deviation of three slices, this implies that the number of pepperoni slices on 95 percent of your pizzas is within 5.88 slices of the mean (3 slices × 1.96 = 5.88 slices). This, in turn, indicates a range of 12.12 (18 − 5.88 = 12.12) to 23.88 (18 + 5.88 = 23.88) slices.
Since you put only whole slices of pepperoni on your pizzas, 95 percent of the time the number of slices is somewhere between 12 and 24. No wonder your customers are up in arms! In response to this information, you decide to implement a standard policy regarding the number of pepperoni slices that go on each type of pizza.
The larger standard deviation for the return on the Fresno investment means that the Fresno investment is riskier than the Miami investment. The actual return for the Fresno investment is more likely to be further from its expected return.
APPLICATION 7.3 LEARNING BY DOING
APPROACH: Use the values in the previous table or
Exhibit 7.1
to compute the range within which Google’s stock return will fall 90 percent of the time. First, find the number of standard deviations associated with a 90 percent level of confidence in the table or
Exhibit 7.1
and then multiply this number by the standard deviation of the annual return for Google’s stock. Then subtract the resulting value from the expected return (mean) to obtain the lower end of the range and add it to the expected return to obtain the upper end.
SOLUTION: From the table, you can see that we would expect the return over the next year to be within 1.645 standard deviations of the mean 90 percent of the time. Multiplying this value by the standard deviation of Google’s stock (23 percent) yields 23 percent × 1.645 = 37.835 percent. This means that there is a 90 percent chance that the return will be between −33.165 percent (4.67 percent − 37.835 percent = −33.165 percent) and 42.505 percent (4.67 percent + 37.835 percent = 42.505 percent).
While the expected return of 4.67 percent is relatively low, the returns on Google stock vary considerably, and there is a reasonable chance that the stock return in the next year could be quite high or quite low (even negative). As you will see shortly, this wide range of possible returns is similar to the range we observe for typical shares in the U.S. stock market.
Historical Market Performance
Now that we have discussed how returns and risks can be measured, we are ready to examine the characteristics of the historical returns earned by securities such as stocks and bonds.
Exhibit 7.3
illustrates the distributions of historical returns for some securities in the United States and shows the average and standard deviations of these annual returns for the period from 1926 to 2009.
Note that the statistics reported in
Exhibit 7.3
are for indexes that represent total average returns for the indicated types of securities, not total returns on individual securities. We generally use indexes to represent the performance of the stock or bond markets. For instance, when news services report on the performance of the stock market, they often report that the Dow Jones Industrial Average (an index based on 30 large stocks), the S&P 500 Index (an index based on 500 large stocks), or the NASDAQ Composite Index (an index based on all stocks that are traded on NASDAQ) went up or down on a particular day. These and other indexes are discussed in
Chapter 9
.
The plots in
Exhibit 7.3
are arranged in order of decreasing risk, which is indicated by the decreasing standard deviation of the annual returns. The top plot shows returns for a small-stock index that represents the 10 percent of U.S. firms that have the lowest total equity value (number of shares multiplied by price per share). The second plot shows returns for the S&P 500 Index, representing large U.S. stocks. The remaining plots show three different types of government debt: Long-term government bonds that mature in 20 years, intermediate-term government bonds that mature in five years, and U.S. Treasury bills, which are short-term debts of the U.S. government, that mature in 30 days.
Higher standard deviations of returns have historically been associated with higher returns. For example, between 1926 and 2009, the standard deviation of the annual returns for small stocks was higher than the standard deviations of the returns earned by other types of securities, and the average return that investors earned from small stocks was also higher. At the other end of the spectrum, the returns on Treasury bills had the smallest standard deviation, and Treasury bills earned the smallest average return.
Source: Data from Morningstar, 2010 SBBI Yearbook
The key point to note in
Exhibit 7.3
is that, on average, annual returns have been higher for riskier securities. Small stocks, which have the largest standard deviation of total returns, at 32.79 percent, also have the largest average annual return, 16.57 percent. On the other end of the spectrum, Treasury bills have the smallest standard deviation, 3.08 percent, and the smallest average annual return, 3.71 percent. Returns for small stocks in any particular year may have been higher or lower than returns for the other types of securities, but on average, they were higher. This is evidence that investors require higher returns for investments with greater risks.
The statistics in
Exhibit 7.3
describe actual investment returns, as opposed to expected returns. In other words, they represent what has happened in the past. Financial analysts often use historical numbers such as these to estimate the returns that might be expected in the future. That is exactly what we did in the baseball example earlier in this chapter. We used the percentage of at-bats in which you got a hit this past season to estimate the likelihood that you would get a hit in your last collegiate at-bat. We assumed that your past performance was a reasonable indicator of your future performance.
To see how historical numbers are used in finance, let’s suppose that you are considering investing in a fund that mimics the S&P 500 Index (this is what we call an index fund) and that you want to estimate what the returns on the S&P 500 Index are likely to be in the future. If you believe that the 1926 to 2009 period provides a reasonable indication of what we can expect in the future, then the average historical return on the S&P 500 Index of 11.84 percent provides a perfectly reasonable estimate of the return you can expect from your investment in the S&P 500 Index fund. In
Chapter 13
we will explore in detail how historical data can be used in this way to estimate the discount rate used to evaluate projects in the capital budgeting process.
Comparing the historical returns for an individual stock with the historical returns for an index can also be instructive.
Exhibit 7.4
shows such a comparison for Apple Inc. and the S&P 500 Index using monthly returns for the period from September 2005 to September 2010. Notice in the exhibit that the returns on Apple stock are much more volatile than the average returns on the firms represented in the S&P 500 Index. In other words, the standard deviation of returns for Apple stock is higher than that for the S&P 500 Index. This is not a coincidence; we will discuss shortly why returns on individual stocks tend to be riskier than returns on indexes.
One last point is worth noting while we are examining historical returns: the value of a $1.00 investment in 1926 would have varied greatly by 2009, depending on where that dollar was invested.
Exhibit 7.5
shows that $1.00 invested in U.S. Treasury bills in 1926 would have been worth $20.53 by 2009. In contrast, that same $1.00 invested in small stocks would have been worth $12,231.13 by 2009!
2
Over a long period of time, earning higher rates of return can have a dramatic impact on the value of an investment. This huge difference reflects the impact of compounding of returns (returns earned on returns), much like the compounding of interest we discussed in
Chapter 5
.
The returns on shares of individual stocks tend to be much more volatile than the returns on portfolios of stocks, such as the S&P 500.
The value of a $1 investment in stocks, small or large, grew much more rapidly than the value of a $1 investment in bonds or Treasury bills over the 1926 to 2009 period. This graph illustrates how earning a higher rate of return over a long period of time can affect the value of an investment portfolio. Although annual stock returns were less certain between 1926 and 2009, the returns on stock investments were much greater.
Source: Data from Morningstar, 2010 SBBI Yearbook
> BEFORE YOU GO ON
1. What is the relation between the variance and the standard deviation?
2. What relation do we generally observe between risk and return when we examine historical returns?
3. How would we expect the standard deviation of the return on an individual stock to compare with the standard deviation of the return on a stock index?
7.4 RISK AND DIVERSIFICATION
·
It does not generally make sense to invest all of your money in a single asset. The reason is directly related to the fact that returns on individual stocks tend to be riskier than returns on indexes. By investing in two or more assets whose values do not always move in the same direction at the same time, an investor can reduce the risk of his or her collection of investments, or portfolio. This is the idea behind the concept of diversification.
portfolio
the collection of assets an investor owns
diversification
Reducing risk by investing in two or more assets whose values do not always move in the same direction at the same time
This section develops the tools necessary to evaluate the benefits of diversification. We begin with a discussion of how to quantify risk and return for a single-asset portfolio, and then we discuss more realistic and complicated portfolios that have two or more assets. Although our discussion focuses on stock portfolios, it is important to recognize that the concepts discussed apply equally well to portfolios that include a range of assets, such as stocks, bonds, gold, art, and real estate, among others.
Single-Asset Portfolios
Returns for individual stocks from one day to the next have been found to be largely independent of each other and approximately normally distributed. In other words, the return for a stock on one day is largely independent of the return on that same stock the next day, two days later, three days later, and so on. Each daily return can be viewed as having been randomly drawn from a normal distribution where the probability associated with the return depends on how far it is from the expected value. If we know what the expected value and standard deviation are for the distribution of returns for a stock, it is possible to quantify the risks and expected returns that an investment in the stock might yield in the future.
To see how we might do this, assume that you are considering investing in one of two stocks for the next year: Advanced Micro Devices (AMD) or Intel. Also, to keep things simple, assume that there are only three possible economic conditions (outcomes) a year from now and that the returns on AMD and Intel under each of these outcomes are as follows:
The coefficient of variation (CV) is a measure that can help us in making comparisons such as that between Stocks A and C. The coefficient of variation for stock i is calculated as:
coefficient of variation (CV)
a measure of the risk associated with an investment for each one percent of expected return
Recall that Stock A has an expected return of 12 percent and a risk level of 12 percent, while Stock C has an expected return of 16 percent and a risk level of 16 percent. If we assume that the risk level given for each stock is equal to the standard deviation of its return, we can find the coefficients of variation for the stocks as follows:
While this analysis appears to make sense, there is a conceptual problem with using the coefficient of variation to compute the amount of risk an investor can expect to realize for each 1 percent of expected return. This problem arises because investors expect to earn a positive return even when assets are completely risk free. For example, as shown in
Exhibit 7.3
, from 1926 to 2009 investors earned an average return of 3.71 percent each year on 30-day Treasury bills, which are considered to be risk free.
3
If investors can earn a positive risk-free rate without bearing any risk, then it really only makes sense to compare the risk of the investment, s
Ri
, with the return that investors expect to earn over and above the risk-free rate. As we will discuss in detail in Section 7.6, the expected return over and above the risk-free rate is a measure of the return that investors expect to earn for bearing risk.
This suggests that we should use the difference between the expected return, E (R
i
), and the risk-free rate, Rrf, instead of E (R
i
) alone in the coefficient of variation calculation. With this change, Equation 7.4 would be written as:
Let’s compute this modified coefficient of variation for the AMD and Intel example. If the risk-free rate equals 0.03, or 3 percent, the modified coefficients of variation for the two stocks are:
A popular version of this modified coefficient of variation calculation is known as the Sharpe Ratio. This ratio is named after 1990 Nobel Prize Laureate William Sharpe who developed the concept and was one of the originators of the capital asset pricing model which is discussed in Section 7.7. The Sharpe Ratio is simply the inverse of the modified coefficient of variation:
A measure of the return per unit of risk for an investment
For the stocks of AMD and Intel, the Sharpe Ratios are:
http://en.wikipedia.org/wiki/sharpe-ratio
.
This tells us that investors in AMD stock can expect to earn 0.524 percent for each one standard deviation of return while investors in Intel stock can expect to earn 0.458 percent for each one standard deviation of return. Many people find the Sharpe Ratio to be a more intuitive measure than the coefficient of variation because they find it easier to think about the return per unit of risk than risk per unit of return.
APPLICATION 7.4 LEARNING BY DOING
Calculating and Interpreting the Sharpe Ratio
APPROACH: Use Equation 7.5 to compute the Sharpe Ratios for the two investments.
SOLUTION: The Sharpe Ratios are:
Portfolios with More Than One Asset
It may seem like a good idea to evaluate investments by calculating a measure of risk for each 1 percent of expected return or the expected return per unit of risk. However, the coefficient of variation and the Sharpe Ratio have a critical shortcoming that is not quite evident when we are considering only a single asset. In order to explain this shortcoming, we must discuss the more realistic setting in which an investor has constructed a portfolio with more than one asset.
Expected Return on a Portfolio with more than one Asset
Suppose that you own a portfolio that consists of $500 of AMD stock and $500 of Intel stock and that over the next year you expect to earn returns on the AMD and Intel shares of 9.9 percent and 8.1 percent, respectively. How would you calculate the expected return for the overall portfolio?
Let’s try to answer this question using our intuition. If half of your funds are invested in each stock, it would seem reasonable that the expected return for this portfolio should be a 50-50 mixture of the expected returns from the two stocks, or:
i
represents the fraction of the portfolio invested in asset i. The corresponding equation for a portfolio with n assets is:
To see how Equation 7.6 is used to calculate the expected return on a portfolio with more than two assets, consider an example. Suppose that you were recently awarded a $500,000 grant from a national foundation to pursue your interest in advancing the art of noodling—a popular pastime in some parts of the country in which people catch 40- to 50-pound catfish by putting their hands into catfish holes and wiggling their fingers like noodles to attract the fish.
4
Since your grant is intended to support your activities for five years, you kept $100,000 to cover your expenses for the next year and invested the remaining $400,000 in U.S. Treasury bonds and stocks. Specifically, you invested $100,000 in Treasury bonds (TB) that yield 4.5 percent; $150,000 in Procter & Gamble stock (P&G), which has an expected return of 7.5 percent; and $150,000 in Exxon Mobil Corporation stock (EMC), which has an expected return of 9.0 percent. What is the expected return on this $400,000 portfolio?
In order to use Equation 7.6, we must first calculate xi, the fraction of the portfolio invested in asset i, for each investment. These fractions are as follows:
Now that we have calculated the expected return on a portfolio with more than one asset, the next question is how to quantify the risk of such a portfolio. Before we discuss the mechanics of how to do this, it is important to have some intuitive understanding of how volatility in the returns for different assets interact to determine the volatility of the overall portfolio.
APPLICATION 7.5 LEARNING BY DOING
Calculating the Expected Return on a Portfolio
APPROACH: First, calculate the fraction of your portfolio that will be invested in each type of asset after you have diversified. Then use Equation 7.6 to calculate the expected return on the portfolio.
SOLUTION: After you have diversified, 55 percent (100 percent − 45 percent = 55 percent) of your portfolio will be invested in your restaurant, 22.5 percent (45 percent × 0.50 = 22.5 percent) will be invested in the stock market index fund, and 22.5 percent (45 percent × 0.50 = 22.5 percent) will be invested in the bond market index fund. Therefore, from Equation 7.6, we know that the expected return for your portfolio is:
The prices of two stocks in a portfolio will rarely, if ever, change by the same amount and in the same direction at the same time. Normally, the price of one stock will change by more than the price of the other. In fact, the prices of two stocks will frequently move in different directions. These differences in price movements affect the total volatility of the returns for a portfolio.
shows monthly returns for the stock of Southwest Airlines and Netflix over the period from September 2005 through September 2010. Notice that the returns on these shares are generally different and that the prices of the shares can move in different directions in a given month (one stock has a positive return when the other has a negative return). When the stock prices move in opposite directions, the change in the price of one stock off sets at least some of the change in the price of the other stock. As a result, the level of risk for a portfolio of the two stocks is less than the average of the risks associated with the individual shares.
This means that we cannot calculate the variance of a portfolio containing two assets simply by calculating the weighted average of the variances of the individual stocks. We have to account for the fact that the returns on different shares in a portfolio tend to partially offset each other. For a two-asset portfolio, we calculate the variance of the returns using the following formula:
i
represents the fraction of the portfolio invested in stock i, s2R i
is the variance of the return of stock i, and sR1,2 is the covariance of the returns between stocks 1 and 2. The covariance of returns is a measure of how the returns on two assets covary, or move together. The third term in Equation 7.7 accounts for the fact that returns from the two assets will offset each other to some extent. The covariance of returns is calculated using the following formula:
a measure of how the returns on two assets covary, or move together
The returns on two stocks are generally different. In some periods, the return on one stock is positive, while the return on the other is negative. Even when the returns on both are positive or negative, they are rarely exactly the same.
where i represents outcomes rather than assets. Compare this equation with Equation 7.3, reproduced here:
Just as it is difficult to directly interpret the variance of the returns for an asset—recall that the variance is in units of squared dollars—it is difficult to directly interpret the covariance of returns between two assets. We get around this problem by dividing the covariance of returns by the product of the standard deviations of the returns for the two assets. This gives us the correlation, r, between the returns on those assets:
Let’s work an example to see how Equation 7.7 is used to calculate the variance of the returns on a portfolio that consists of 50 percent Southwest Airlines stock and 50 percent Netflix stock. Using the data plotted in
Exhibit 7.6
, we can calculate the variance of the annual returns for the Southwest Airlines and Netflix stocks, , to be 0.1065 and 0.2021, respectively. The covariance between the annual returns on these two stocks is 0.0070. We do not show the calculations for the variances and the covariance because each of these numbers was calculated using 60 different monthly returns. These calculations are too cumbersome to illustrate. Rest assured, however, that they were calculated using Equations 7.3 and 7.8.
5
With these values, we can calculate the variance of a portfolio that consists of 50 percent Southwest Airlines (SW) stock and 50 percent Netflix stock as:
If we calculate the standard deviations by taking the square roots of the variances, we find that the standard deviations for Southwest Airlines stock, Netflix stock, and the portfolio consisting of those two stocks are 0.326 (32.6 percent), 0.450 (45.0 percent), and 0.284 (28.4 percent), respectively.
illustrates the monthly returns for the portfolio of Southwest Airlines and Netflix stock, along with the monthly returns for the individual stocks. You can see in this exhibit that, while the returns on the portfolio vary quite a bit, this variation is less than that for the individual company shares.
The variation in the returns from a portfolio that consists of Southwest Airlines and Netflix stock in equal proportions is less than the variation in the returns from either of those stocks alone.
Using Equation 7.9, we can calculate the correlation of the returns between Southwest Airlines and Netflix stock as:
As we add more and more assets to a portfolio, calculating the variance using the approach illustrated in Equation 7.7 becomes increasingly complex. The reason for this is that we have to account for the covariance between each pair of assets. These more extensive calculations are beyond the scope of this book, but they are conceptually the same as those for a portfolio with two assets.
APPLICATION 7.6 LEARNING BY DOING
APPROACH: Use Equation 7.7 to calculate the variance of the portfolio returns and then take the square root of this value to obtain the standard deviation.
SOLUTION: The variance of the portfolio returns is:
Comparing the variance of the portfolio returns of 0.0394 with the variances of the restaurant returns, 0.0625, and the stock market index fund returns, 0.0400, shows once again that a portfolio with two or more assets can have a smaller variance of returns (and thus a smaller standard deviation of returns) than any of the individual assets in the portfolio.
The Limits of Diversification
BUILDING INTUITION DIVERSIFIED PORTFOLIOS ARE LESS RISKY
Diversified portfolios generally have less risk for a given level of return than the individual risky assets in the portfolio. This is because the values of individual assets rarely change by the same amount and in the same direction at the same time. As a result, some of the variation in an asset’s value can be diversified away by owning another asset at the same time. This is important because it tells us that investors can eliminate some of the risk associated with individual investments by holding them in a diversified portfolio.
In the sample calculations for the portfolio containing Southwest Airlines and Netflix stock, we saw that the standard deviation of the returns for a portfolio consisting of equal investments in those two stocks was 28.4 percent from September 2005 through September 2010 and that this figure was lower than the standard deviation of returns for either of the individual stocks (32.6 percent and 45.0 percent). You might wonder how the standard deviation for the portfolio is likely to change if we increase the number of assets in the portfolio. The answer is simple. If the returns on the individual stocks added to our portfolio do not all change in the same way, then increasing the number of stocks in the portfolio will reduce the standard deviation of the portfolio returns even further.
Let’s consider a simple example to illustrate this point. Suppose that all assets have a standard deviation of returns that is equal to 40 percent and that the covariance between the returns for each pair of assets is 0.048. If we form a portfolio in which we have an equal investment in two assets, the standard deviation of returns for the portfolio will be 32.25 percent. If we add a third asset, the portfolio standard deviation of returns will decrease to 29.21 percent. It will be even lower, at 27.57 percent, for a four-asset portfolio.
Exhibit 7.8
illustrates how the standard deviation for the portfolio declines as more stocks are added.
In addition to showing how increasing the number of assets decreases the overall risk of a portfolio,
Exhibit 7.8
illustrates three other very important points. First, the decrease in the standard deviation for the portfolio gets smaller and smaller as more assets are added. You can see this effect by looking at the distance between the straight horizontal line and the plot of the standard deviation of the portfolio returns.
The second important point is that, as the number of assets becomes very large, the portfolio standard deviation does not approach zero. It decreases only so far. In the example in
Exhibit 7.8
, it approaches 21.9 percent. The standard deviation does not approach zero because we are assuming that the variations in the asset returns do not completely cancel each other out. This is a realistic assumption because in practice investors can rarely diversify away all risk. They can diversify away risk that is unique to the individual assets, but they cannot diversify away risk that is common to all assets. The risk that can be diversified away is called unsystematic or diversifiable risk, and the risk that cannot be diversified away is called systematic or nondiversifiable risk. In the next section, we will discuss systematic risk in detail.
unsystematic or diversifiable risk
risk that can be eliminated through diversification
systematic or nondiversifiable risk
risk that cannot be eliminated through diversification
The total risk of a portfolio decreases as the number of assets increases. This is because the amount of unsystematic risk in the portfolio decreases. The diversification benefit from adding another asset declines as the total number of assets in the portfolio increases and the unsystematic risk approaches zero. Most of the diversification benefit can often be achieved with as few as 15 or 20 assets.
The third key point illustrated in
Exhibit 7.8
is that most of the risk-reduction benefits from diversification can be achieved in a portfolio with 15 to 20 assets. Of course, the number of assets required to achieve a high level of diversification depends on the covariances between the assets in the portfolio. However, in general, it is not necessary to invest in a very large number of different assets.
> BEFORE YOU GO ON
1. What does the coefficient of variation tell us, and how is it related to the Sharpe Ratio?
2. What are the two components of total risk?
3. Why does the total risk of a portfolio not approach zero as the number of assets in a portfolio becomes very large?
7.5 SYSTEMATIC RISK
·
The objective of diversification is to eliminate variation in returns that is unique to individual assets. We diversify our investments across a number of different assets in the hope that these unique variations will cancel each other out. With complete diversification, all of the unsystematic risk is eliminated from the portfolio. An investor with a diversified portfolio still faces systematic risk, however, and we now turn our attention to that form of risk.
Why Systematic Risk Is All That Matters
The idea that unsystematic risk can be diversified away has direct implications for the relation between risk and return. If the transaction costs associated with constructing a diversified portfolio are relatively low, then rational, informed investors, such as the students who are taking this class, will prefer to hold diversified portfolios.
Diversified investors face only systematic risk, whereas investors whose portfolios are not well diversified face systematic risk plus unsystematic risk. Because they face less risk, the diversified investors will be willing to pay higher prices for individual assets than the other investors. Therefore, expected returns on individual assets will be lower than the total risk (systematic plus unsystematic risk) of those assets suggests they should be.
To illustrate, consider two individual investors, Emily and Jane. Each of them is trying to decide if she should purchase stock in your pizza restaurant. Emily holds a diversified portfolio and Jane does not. Assume your restaurant’s stock has five units of systematic risk and nine units of total risk. You can see that Emily faces less risk than Jane and will require a lower expected rate of return. Consequently, Emily will be willing to pay a higher price than Jane.
BUILDING INTUITION SYSTEMATIC RISK IS THE RISK THAT MATTERS
The required rate of return on an asset depends only on the systematic risk associated with that asset. Because unsystematic risk can be diversified away, investors can and will eliminate their exposure to this risk. Competition among diversified investors will drive the prices of assets to the point where the expected returns will compensate investors for only the systematic risk that they bear.
If the market includes a large number of diversified investors such as Emily, competition among these investors will drive the price of your restaurant’s shares up further. This competition will ultimately push the price up to the point where the expected return just compensates all investors for the systematic risk associated with your stock. The bottom line is that, because of competition among diversified investors, all investors are only rewarded for bearing systematic risk in asset markets. For this reason, we are concerned only about systematic risk when we think about the relation between risk and return in finance.
Measuring Systematic Risk
If systematic risk is all that matters when we think about expected returns, then we cannot use the standard deviation as a measure of risk.
6
The standard deviation is a measure of total risk. We need a way of quantifying the systematic risk of individual assets.
A natural starting point for doing this is to recognize that the most diversified portfolio possible will come closest to eliminating all unsystematic risk. Such a portfolio provides a natural benchmark against which we can measure the systematic risk of an individual asset. What is the most diversified portfolio possible? The answer is simple. It is the portfolio that consists of all assets, including stocks, bonds, real estate, precious metals, commodities, art, baseball cards, and so forth from all over the world. In finance, we call this the market portfolio.
market portfolio
the portfolio of all assets
Unfortunately, we do not have very good data for most of these assets for most of the world, so we use the next best thing: the U.S. public stock market. A large number of stocks from a broad range of industries trade in this market. The companies that issue these stocks own a wide range of assets all over the world. These characteristics, combined with the facts that the U.S. market has been operating for a very long time and that we have very reliable and detailed information on prices for U.S. stocks, make the U.S. stock market a natural benchmark for estimating systematic risk.
Since systematic risk is, by definition, risk that cannot be diversified away, the systematic risk of an individual asset is really just a measure of the relation between the returns on the individual asset and the returns on the market. In fact, systematic risk is often referred to as market risk. To see how we might use data from the U.S. public stock market to estimate the systematic risk of an individual asset, look at
Exhibit 7.9
, which plots 60 historical monthly returns for General Electric Company (GE) against the corresponding monthly returns for the S&P 500 index (a proxy for the U.S. stock market). In this plot, you can see that returns on GE stock tend to be higher when returns on the S&P 500 tend to be higher. The measure of systematic risk that we use in finance is a statistical measure of this relation.
market risk
a term commonly used to refer to nondiversifiable, or systematic, risk
We quantify the relation between the returns on GE stock and the market by finding the slope of the line that best represents the relation illustrated in
Exhibit 7.9
. Specifically, we estimate the slope of the line of best fit. We do this using the statistical technique called regression analysis. If you are not familiar with regression analysis, don’t worry; the details are beyond the scope of this course. All you have to know is that this technique gives us the line that fits the data best.
The monthly returns on General Electric stock are positively related to the returns on the S&P 500 index. In other words, the return on General Electric’s stock tends to be higher when the return on the S&P 500 Index is higher and lower when the return on the S&P 500 index is lower.
The line shown In the exhibit best represents the relation between the monthly returns on General Electric stock and the returns on the S&P 500 index. The slope of this line, which equals 1.61, indicates that the return on General Electric stock tends to equal about 1.61 times the return on the S&P 500 index.
illustrates the line that was estimated for the data in
EXHIBIT 7.9
using regression analysis. Note that the slope of this line is 1.61. Recall from your math classes that the slope of a line equals the ratio of the rise (vertical distance) divided by the corresponding run (horizontal distance). In this case, the slope is the change in the return on GE stock divided by the change in the return on the U.S. stock market. A slope of 1.61 therefore means that, on average, the change in the return on GE stock was 1.61 times as large as the change in the return on the S&P 500 index. Thus, if the S&P 500 index goes up 1 percent, the average increase in GE’s stock is 1.61 percent. This is a measure of systematic risk because it tells us that the volatility of the returns on GE stock is 1.61 times as large as that for the S&P 500 as a whole.
To explore this idea more completely, let’s consider another, simpler example. Suppose that you have data for Nike stock and for the U.S. stock market (S&P 500 index) for each of the past two years. In the first year, the return on the market was 10 percent, and the return on Nike stock was 15 percent. In the second year, the return on the market was 12 percent, and the return on Nike stock was 19 percent. From this information, we know that the return on Nike stock increased by 4 percent while the return on the market increased 2 percent. If we plotted the returns for Nike stock and for the market for each of the last two periods, as we did for GE stock and the market in
Exhibits 7.9
and
7.10
, and estimated the line that best fit the data, it would be a line that connected the dots for the two periods. The slope of this line would equal 2, calculated as follows:
In finance, we call the slope of the line of best fit beta. Often we simply use the corresponding Greek letter, β, to refer to this measure of systematic risk. As shown below, a beta of 1 tells us that an asset has just as much systematic risk as the market. A beta higher than or lower than 1 tells us that the asset has more or less systematic risk than the market, respectively. A beta of 0 indicates a risk-free security, such as a U.S. Treasury bill.
beta (β)
a measure of nondiversifiable, systematic, or market, risk
Exhibit 7.10
, the line of best fit does not go through each data point. That is because some of the change in GE’s stock price each month reflected information that did not affect the S&P 500 as a whole. That information is the unsystematic component of the risk of GE’s stock. The vertical distance between each data point and the line of best fit represents variation in GE’s stock return that can be attributed to this unsystematic risk.
A convenient place to find betas for individual companies is MSN Money Central at
http://moneycentral.msn.com
. Just enter the stock symbol in the “Get Quote” box near the top of the page and hit “Enter” on your computer (try the railroad company CSX, for example). You will get prices, an estimate of the beta, and other financial information.
The positive slope (β) of the regression line in
Exhibit 7.10
tells us that returns for the S&P 500 and for GE stock will tend to move in the same direction. The return on the S&P 500 and the return on GE’s stock will not always change in the same direction, however, because the unsystematic risk associated with GE stock can more than offset the effect of the market in any particular period. In the next section, we will discuss the implications of beta for the level (as opposed to the change) in the expected return for a stock such as GE.
> BEFORE YOU GO ON
1. Why are returns on the stock market used as a benchmark in measuring systematic risk?
2. How is beta estimated?
3. How would you interpret a beta of 1.5 for an asset? A beta of 0.75?
7.6 COMPENSATION FOR BEARING SYSTEMATIC RISK
Now that we have identified the measure of the risk that diversified investors care about—systematic risk—we are in a position to examine how this measure relates to expected returns. Earlier, in our discussion of the coefficient of variation and the Sharpe Ratio, we asserted that the expected return over and above the risk-free rate is the return that investors expect to earn for bearing risk. To see why this must be true, think about the rate of return that you would require for an investment. First, you would want to make sure that you were compensated for inflation. It would not make sense to invest if you expected the investment to return an amount that did not at least allow you to have the same purchasing power that the money you invested had when you made the investment. Second, you would want some compensation for the fact that you are giving up the use of your money for a period of time. This compensation may be very small if you are forgoing the use of your money for only a short time, such as when you invest in a 30-day Treasury bill, but it might be relatively large if you are investing for several years. Finally, you would also require compensation for the systematic risk associated with the investment.
When you invest in a U.S. government security such as a Treasury bill, note, or bond, you are investing in a security that has no risk of default. After all, the U.S. government can always increase taxes or print more money to pay you back. Changes in economic conditions and other factors that affect the returns on other assets do not affect the default risk of U.S. government securities. As a result, these securities do not have systematic risk, and their returns can be viewed as risk free. In other words, returns on government bonds reflect the compensation required by investors to account for the impact of inflation on purchasing power and for their inability to use the money during the life of the investment.
It follows that the difference between required returns on government securities and required returns for risky investments represents the compensation investors require for taking risk. Recognizing this allows us to write the expected return for an asset i as:
i
is the number of units of systematic risk associated with asset i. Finally, if beta, β, is the appropriate measure for the number of units of systematic risk, we can also define compensation for taking risk as follows:
i
is the beta for asset i.
Remember that beta is a measure of systematic risk that is directly related to the risk of the market as a whole. If the beta for an asset is 2, that asset has twice as much systematic risk as the market. If the beta for an asset is 0.5, then the asset has half as much systematic risk as the market. Recognizing this natural interpretation of beta suggests that the appropriate “unit of systematic risk” is the level of risk in the market as a whole and that the appropriate “compensation per unit of systematic risk” is the expected return required for the level of systematic risk in the market as a whole. The required rate of return on the market, over and above that of the risk-free return, represents compensation required by investors for bearing a market (systematic) risk. This suggests that:
·
In deriving Equation 7.10, we intuitively arrived at the Capital Asset Pricing Model (CAPM). Equation 7.10 is the CAPM, a model that describes the relation between risk and expected return. We will discuss the predictions of the CAPM in more detail shortly, but first let’s look more closely at how it works.
Capital Asset Pricing Model (CAPM)
a model that describes the relation between risk and expected return
Suppose that you want to estimate the expected return for a stock that has a beta of 1.5 and that the expected return on the market and risk-free rate are 10 percent and 4 percent, respectively. We can use Equation 7.10 (the CAPM) to find the expected return for this stock:
While the expected return on the market is known in the above example, we actually cannot observe it in practice. For this reason, financial analysts estimate the market risk premium using historical data. We discuss how they do this in
Chapter 13
.
The Security Market Line
a plot of the relation between expected return and systematic risk
displays a plot of Equation 7.10 to illustrate how the expected return on an asset varies with systematic risk. This plot shows that the relation between the expected return on an asset and beta is positive and linear. In other words, it is a straight line with a positive slope. The line in
Exhibit 7.11
is known as the Security Market Line (SML).
The Security Market Line (SML) is the line that shows the relation between expected return and systematic risk, as measured by beta. When beta equals zero and there is no systematic risk, the expected return equals the risk-free rate. As systematic risk (beta) increases, the expected return increases. This is an illustration of the positive relation between risk and return. The SML shows that it is systematic risk that matters to investors.
In
Exhibit 7.11
you can see that the expected rate of return equals the risk-free rate when beta equals 0. This makes sense because when investors do not face systematic risk, they will only require a return that reflects the expected rate of inflation and the fact that they are giving up the use of their money for a period of time.
Exhibit 7.11
also shows that the expected return on an asset equals the expected return on the market when beta equals 1. This is not surprising given that both the asset and the market would have the same level of systematic risk if this were the case.
It is important to recognize that the SML illustrates what the CAPM predicts the expected total return should be for various values of beta. The actual expected total return depends on the price of the asset. You can see this from Equation 7.1:
Exhibit 7.11
. This means that the asset’s price is lower than the CAPM suggests it should be. Conversely, if the expected return on an asset plots below the SML, this implies that the asset’s price is higher than the CAPM suggests it should be. The point at which a particular asset plots relative to the SML, then, tells us something about whether the price of that asset might be low or high. Recognizing this fact can be helpful in evaluating the attractiveness of an investment such as the General Electric stock in Learning by Doing Application 7.7.
APPLICATION 7.7 LEARNING BY DOING
Expected Returns and Systematic Risk
APPROACH: Use Equation 7.10 to calculate the expected return on General Electric stock.
SOLUTION: The expected return is:
The expected return for a portfolio can also be predicted using the CAPM. The expected return on a portfolio with n assets is calculated using the relation:
The fact that the SML is a straight line turns out to be rather convenient if we want to estimate the beta for a portfolio. Recall that the equation for the expected return for a portfolio with n assets was given by Equation 7.6:
i
is the proportion of the portfolio value that is invested in asset i, β i is the beta of asset i, and n is the number of assets in the portfolio. This formula makes it simple to calculate the beta of any portfolio of assets once you know the betas of the individual assets. As an exercise, you might prove this to yourself by using Equations 7.6 and 7.10 to derive Equation 7.11.
Let’s consider an example to see how Equation 7.11 is used. Suppose that you invested 25 percent of your wealth in a fully diversified market fund, 25 percent in risk-free Treasury bills, and 50 percent in a house with twice as much systematic risk as the market. What is the beta of your overall portfolio? What rate of return would you expect to earn from this portfolio if the risk-free rate is 4 percent and the market risk premium is 6 percent?
We know that the beta for the market must equal 1 by definition and that the beta for a risk-free asset equals 0. The beta for your home must be 2 since it has twice the systematic risk of the market. Therefore, the beta of your portfolio is:
Chapter 13
we will explore the relation between the expected return and the rate used to discount project cash flows in much more detail. By the time we finish that discussion, you will understand thoroughly how businesses determine the rate that they use to discount the cash flows from their investments.
APPLICATION 7.8 LEARNING BY DOING
While the cottages cost the same, they are different distances from campus and in different neighborhoods. You believe that this causes them to have different levels of systematic risk, and you estimate that the betas for the individual cottages are 1.2, 1.3, and 1.5. If the risk-free rate is 4 percent and the market risk premium is 6 percent, what will be the expected return on your real estate portfolio after you make all three investments?
APPROACH: There are two approaches that you can use to solve this problem. First, you can estimate the expected return for each cottage using Equation 7.10 and then calculate the expected return on the portfolio using Equation 7.6. Alternatively, you can calculate the beta for the portfolio using Equation 7.11 and then use Equation 7.10 to calculate the expected return.
SOLUTION: Using the first approach, we find that Equation 7.10 gives us the following expected returns:
Choosing between Two Investments
DECISION: You should not invest in either stock. The expected returns for both of them are below the values predicted by the CAPM for investments with the same level of risk. In other words, both would plot below the line in
Exhibit 7.11
. This implies that they are both overpriced.
> BEFORE YOU GO ON
1. How is the expected return on an asset related to its systematic risk?
2. What name is given to the relation between risk and expected return implied by the CAPM?
3. If an asset’s expected return does not plot on the line in question 2 above, what does that imply about its price?
SUMMARY OF Learning Objectives
Investors require greater returns for taking greater risk. They prefer the investment with the highest possible return for a given level of risk or the investment with the lowest risk for a given level of return.
Describe the two components of a total holding period return, and calculate this return for an asset.
The total holding period return on an investment consists of a capital appreciation component and an income component. This return is calculated using Equation 7.1. It is important to recognize that investors do not care whether they receive a dollar of return through capital appreciation or as a cash dividend. Investors value both sources of return equally.
Explain what an expected return is and calculate the expected return for an asset.
The expected return is a weighted average of the possible returns from an investment, where each of these returns is weighted by the probability that it will occur. It is calculated using Equation 7.2.
Explain what the standard deviation of returns is and why it is very useful in finance, and calculate it for an asset.
The standard deviation of returns is a measure of the total risk associated with the returns from an asset. It is useful in evaluating returns in finance because the returns on many assets tend to be normally distributed. The standard deviation of returns provides a convenient measure of the dispersion of returns. In other words, it tells us about the probability that a return will fall within a particular distance from the expected value or within a particular range. To calculate the standard deviation, the variance is first calculated using Equation 7.3. The standard deviation of returns is then calculated by taking the square root of the variance.
Explain the concept of diversification.
Diversification is reducing risk by investing in two or more assets whose values do not always move in the same direction at the same time. Investing in a portfolio containing assets whose prices do not always move together reduces risk because some of the changes in the prices of individual assets offset each other. This can cause the overall volatility in the value of an investor’s portfolio to be lower than if it consisted of only a single asset.
Discuss which type of risk matters to investors and why.
Investors care about only systematic risk. This is because they can eliminate unsystematic risk by holding a diversified portfolio. Diversified investors will bid up prices for assets to the point at which they are just being compensated for the systematic risks they must bear.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate whether the expected return of an asset is sufficient to compensate an investor for the risks associated with that asset.
The CAPM tells us that the relation between systematic risk and return is linear and that the risk-free rate of return is the appropriate return for an asset with no systematic risk. From the CAPM we know what rate of return investors will require for an investment with a particular amount of systematic risk (beta). This means that we can use the expected return predicted by the CAPM as a benchmark for evaluating whether expected returns for individual assets are sufficient. If the expected return for an asset is less than that predicted by the CAPM, then the asset is an unattractive investment because its return is lower than the CAPM indicates it should be. By the same token, if the expected return for an asset is greater than that predicted by the CAPM, then the asset is an attractive investment because its return is higher than it should be.
SUMMARY OF Key Equations
· 7.1 Kaaran made a friendly wager with a colleague that involves the result from flipping a coin. If heads comes up, Kaaran must pay her colleague $15; otherwise, her colleague will pay Kaaran $15. What is Kaaran’s expected cash flow, and what is the variance of that cash flow if the coin has an equal probability of coming up heads or tails? Suppose Kaaran’s colleague is willing to handicap the bet by paying her $20 if the coin toss results in tails. If everything else remains the same, what are Kaaran’s expected cash flow and the variance of that cash flow?
· 7.2 You know that the price of CFI, Inc., stock will be $12 exactly one year from today. Today the price of the stock is $11. Describe what must happen to the price of CFI, Inc., today in order for an investor to generate a 20 percent return over the next year. Assume that CFI does not pay dividends.
· 7.3 The expected value of a normal distribution of prices for a stock is $50. If you are 90 percent sure that the price of the stock will be between $40 and $60, then what is the variance of the stock price?
· 7.4 You must choose between investing in stock A or stock B. You have already used CAPM to calculate the rate of return you should expect to receive for each stock given their systematic risk and decided that the expected return for both exceeds that predicted by CAPM by the same amount. In other words, both are equally attractive investments for a diversified investor. However, since you are still in school and do not have a lot of money, your investment portfolio is not diversified. You have decided to invest in the stock that has the highest expected return per unit of total risk. If the expected return and standard deviation of returns for stock A are 10 percent and 25 percent, respectively, and the expected return and standard deviation of returns for stock B are 15 percent and 40 percent, respectively, which should you choose? Assume that the risk-free rate is 5 percent.
· 7.5 CSB, Inc., has a beta of 1.35. If the expected market return is 14.5 percent and the risk-free rate is 5.5 percent, what is the appropriate required return of CSB (using the CAPM)?
Solutions to Self-Study Problems
· 7.1
· 7.2 The expected return for CFI based on today’s stock price is ($12 − $11)/$11 = 9.09 percent, which is lower than 20 percent. Since the stock price one year from today is fixed, the only way that you will generate a 20 percent return is if the price of the stock drops today. Consequently, the price of the stock today must drop to $10. It is found by solving the following: 0.2 = ($12 − x)/x, or x = $10.
· 7.3 Since you know that 1.645 standard deviations around the expected return captures 90 percent of the distribution, you can set up either of the following equations:
· 7.5 E(RCSB = Rrf + β
CSB[E(RM) − Rrf] = 0.055 + [1.35 × (0.145 − 0.055)] = 0.1765 or, 17.65%
Critical Thinking Questions
· 7.1 Given that you know the risk as well as the expected return for two stocks, discuss what process you might utilize to determine which of the two stocks is a better buy. You may assume that the two stocks will be the only assets held in your portfolio.
· 7.2 What is the difference between the expected rate of return and the required rate of return? What does it mean if they are different for a particular asset at a particular point in time?
· 7.3 Suppose that the standard deviation of the returns on the shares of stock at two different companies is exactly the same. Does this mean that the required rate of return will be the same for these two stocks? How might the required rate of return on the stock of a third company be greater than the required rates of return on the stocks of the first two companies even if the standard deviation of the returns of the third company’s stock is lower?
· 7.4 The correlation between stocks A and B is 0.50, while the correlation between stocks A and C is −0.5. You already own stock A and are thinking of buying either stock B or stock C. If you want your portfolio to have the lowest possible risk, would you buy stock B or C? Would you expect the stock you choose to affect the return that you earn on your portfolio?
· 7.5 The idea that we can know the return on a security for each possible outcome is overly simplistic. However, even though we cannot possibly predict all possible outcomes, this fact has little bearing on the risk-free return. Explain why.
· 7.6 Which investment category has shown the greatest degree of risk in the United States since 1926? Explain why that makes sense in a world where the value of an asset in this investment category is likely to be more adversely affected by a particular negative event than the price of a corporate bond. Use the same type of explanation to help explain other investment choices since 1926.
· 7.7 You are concerned about one of the investments in your fully diversified portfolio. You just have an uneasy feeling about the CFO, I am Shifty, of that particular firm. You do believe, however, that the firm makes a good product and that it is appropriately priced by the market. Should you be concerned about the effect on your portfolio if Shifty embezzles a portion of the firm’s cash?
· 7.8 The CAPM is used to price the risk in any asset. Our examples have focused on stocks, but we could also price the expected rate of return for bonds. Explain how debt securities are also subject to systematic risk.
· 7.9 In recent years, investors have correctly agreed that the market portfolio consists of more than just a group of U.S. stocks and bonds. If you are an investor who invests in only U.S. stocks, describe the effects on the risk in your portfolio.
· 7.10 You may have heard the statement that you should not include your home as an asset in your investment portfolio. Assume that your house will comprise up to 75 percent of your assets in the early part of your investment life. Evaluate the implications of omitting it from your portfolio when calculating the risk of your overall investment portfolio.
Questions and Problems
· 7.2 Expected returns: John is watching an old game show rerun on television called Let’s Make a Deal in which the contestant chooses a prize behind one of two curtains. Behind one of the curtains is a gag prize worth $150, and behind the other is a round-the-world trip worth ,200. The game show has placed a subliminal message on the curtain containing the gag prize, which makes the probability of choosing the gag prize equal to 75 percent. What is the expected value of the selection, and what is the standard deviation of that selection?
· 7.3 Expected returns: You have chosen biology as your college major because you would like to be a medical doctor. However, you find that the probability of being accepted to medical school is about 10 percent. If you are accepted to medical school, then your starting salary when you graduate will be $300,000 per year. However, if you are not accepted, then you would choose to work in a zoo, where you will earn $40,000 per year. Without considering the additional educational years or the time value of money, what is your expected starting salary as well as the standard deviation of that starting salary?
· 7.4 Historical market: Describe the general relation between risk and return that we observe in the historical bond and stock market data.
· 7.5 Single-asset portfolios: Stocks A, B, and C have expected returns of 15 percent, 15 percent, and 12 percent, respectively, while their standard deviations are 45 percent, 30 percent, and 30 percent, respectively. If you were considering the purchase of each of these stocks as the only holding in your portfolio and the risk-free rate is 0 percent, which stock should you choose?
· 7.6 Diversification: Describe how investing in more than one asset can reduce risk through diversification.
· 7.7 Systematic risk: Define systematic risk.
· 7.8 Measuring systematic risk: Susan is expecting the returns on the market portfolio to be negative in the near term. Since she is managing a stock mutual fund, she must remain invested in a portfolio of stocks. However, she is allowed to adjust the beta of her portfolio. What kind of beta would you recommend for Susan’s portfolio?
· 7.9 Measuring systematic risk: Describe and justify what the value of the beta of a U.S. Treasury bill should be.
· 7.10 Measuring systematic risk: If the expected rate of return for the market is not much greater than the risk-free rate of return, what is the general level of compensation for bearing systematic risk?
· 7.11 CAPM: Describe the Capital Asset Pricing Model (CAPM) and what it tells us.
· 7.12 The Security market line: If the expected return on the market is 10 percent and the risk-free rate is 4 percent, what is the expected return for a stock with a beta equal to 1.5? What is the market risk premium for the set of circumstances described?
· 7.14 Interpreting the variance and standard deviation: The distribution of grades in an introductory finance class is normally distributed, with an expected grade of 75. If the standard deviation of grades is 7, in what range would you expect 95 percent of the grades to fall?
· 7.15 Calculating the variance and standard deviation: Kate recently invested in real estate with the intention of selling the property one year from today. She has modeled the returns on that investment based on three economic scenarios. She believes that if the economy stays healthy, then her investment will generate a 30 percent return. However, if the economy softens, as predicted, the return will be 10 percent, while the return will be 25 percent if the economy slips into a recession. If the probabilities of the healthy, soft, and recessionary states are 0.4, 0.5, and 0.1, respectively, then what are the expected return and the standard deviation of the return on Kate’s investment?
· 7.16 Calculating the variance and standard deviation: Barbara is considering investing in a stock and is aware that the return on that investment is particularly sensitive to how the economy is performing. Her analysis suggests that four states of the economy can affect the return on the investment. Using the table of returns and probabilities below, find the expected return and the standard deviation of the return on Barbara’s investment.
· 7.19 Portfolios with more than one asset: Emmy is analyzing a two-stock portfolio that consists of a Utility stock and a Commodity stock. She knows that the return on the Utility stock has a standard deviation of 40 percent and the return on the Commodity stock has a standard deviation of 30 percent. However, she does not know the exact covariance in the returns of the two stocks. Emmy would like to plot the variance of the portfolio for each of three cases—covariance of 0.12, 0, and 0.12—in order to understand how the variance of such a portfolio would react. Do the calculation for all three cases (0.12, 0, and 0.12), assuming an equal proportion of each stock in the portfolio.
· 7.20 Portfolios with more than one asset: Given the returns and probabilities for the three possible states listed below, calculate the covariance between the returns of Stock A and Stock B. For convenience, assume that the expected returns of Stock A and Stock B are 11.75 percent and 18 percent, respectively.
· 7.22 Compensation for bearing systematic risk: Write out the equation for the covariance in the returns of two assets, Asset 1 and Asset 2. Using that equation, explain the easiest way for the two asset returns to have a covariance of zero.
· 7.23 Compensation for bearing systematic risk: Evaluate the following statement: By fully diversifying a portfolio, such as by buying every asset in the market, we can completely eliminate all types of risk, thereby creating a synthetic Treasury bill.
· 7.24 CAPM: Damien knows that the beta of his portfolio is equal to 1, but he does not know the risk-free rate of return or the market risk premium. He also knows that the expected return on the market is 8 percent. What is the expected return on Damien’s portfolio?
· 7.25 CAPM: In February 2011 the risk-free rate was 4.75 percent, the market risk premium was 6 percent, and the beta for Dell stock was 1.31. What is the expected return that was consistent with the systematic risk associated with the returns on Dell stock?
· 7.26 CAPM: The market risk premium is 6 percent, and the risk-free rate is 5 percent. If the expected return on a bond is 6.5 percent, what is its beta?
· 7.28 Sumeet knows that the covariance in the return on two assets is 0.0025. Without knowing the expected return of the two assets, explain what that covariance means.
· 7.29 In order to fund her retirement, Glenda requires a portfolio with an expected return of 12 percent per year over the next 30 years. She has decided to invest in Stocks 1, 2, and 3, with 25 percent in Stock 1, 50 percent in Stock 2, and 25 percent in Stock 3. If Stocks 1 and 2 have expected returns of 9 percent and 10 percent per year, respectively, then what is the minimum expected annual return for Stock 3 that will enable Glenda to achieve her investment requirement?
· 7.30 Tonalli is putting together a portfolio of 10 stocks in equal proportions. What is the relative importance of the variance for each stock versus the covariance for the pairs of stocks in her portfolio? For this exercise, ignore the actual values of the variance and covariance terms and explain their importance conceptually.
· 7.31 Explain why investors who have diversified their portfolios will determine the price and, consequently, the expected return on an asset.
· 7.32 Brad is about to purchase an additional asset for his well-diversified portfolio. He notices that when he plots the historical returns of the asset against those of the market portfolio, the line of best fit tends to have a large amount of prediction error for each data point (the scatter plot is not very tight around the line of best fit). Do you think that this will have a large or a small impact on the beta of the asset? Explain your opinion.
· 7.33 The beta of an asset is equal to 0. Discuss what the asset must be.
· 7.34 The expected return on the market portfolio is 15 percent, and the return on the risk-free security is 5 percent. What is the expected return on a portfolio with a beta equal to 0.5?
· 7.35 Draw the Security Market Line (SML) for the case where the market risk premium is 5 percent and the risk-free rate is 7 percent. Now suppose an asset has a beta of 1.0 and an expected return of 4 percent. Plot it on your graph. Is the security properly priced? If not, explain what we might expect to happen to the price of this security in the market. Next, suppose another asset has a beta of 3.0 and an expected return of 20 percent. Plot it on the graph. Is this security properly priced? If not, explain what we might expect to happen to the price of this security in the market.
· 7.36 If the CAPM describes the relation between systematic risk and expected returns, can both an individual asset and the market portfolio of all risky assets have negative expected real rates of return? Why or why not?
· 7.37 You have been provided the following data on the securities of three firms and the market:
Sample Test Problems
· 7.1 Friendly Airlines stock is selling at a current price of $37.50 per share. If the stock does not pay a dividend and has a 12 percent expected return, what is the expected price of the stock one year from today?
· 7.2 Stefan’s parents are about to invest their nest egg in a stock that he has estimated to have an expected return of 9 percent over the next year. If the return on the stock is normally distributed with a 3 percent standard deviation, in what range will the stock return fall 95 percent of the time?
· 7.3 Elaine has narrowed her investment alternatives to two stocks (at this time she is not worried about diversifying): Stock M, which has a 23 percent expected return, and Stock Y, which has an 8 percent expected return. If Elaine requires a 16 percent return on her total investment, then what proportion of her portfolio will she invest in each stock?
· 7.4 You have just prepared a graph similar to
Exhibit 7.9
, comparing historical data for Pear Computer Corp. and the general market. When you plot the line of best fit for these data, you find that the slope of that line is 2.5. If you know that the market generated a return of 12 percent and that the risk-free rate is 5 percent, then what would your best estimate be for the return of Pear Computer during that same time period?
· 7.5 The CAPM predicts that the return of MoonBucks Tea Corp. is 23.6 percent. If the risk-free rate of return is 8 percent and the expected return on the market is 20 percent, then what is MoonBucks’ beta?
For simplicity, let’s ignore the possibility of your hitting a sacrifice fly and other such outcomes.
From a practical standpoint, it would not really have been possible to grow $1.00 to $12,231.13 by investing in small U.S. stocks because this increase assumes that an investor is able to rebalance the stock portfolio by buying and selling shares as necessary at no cost. Since buying and selling shares is costly, the final wealth would have been lower. Nevertheless, even after transaction costs, it would have been much more profitable to invest in small stocks than in U.S. Treasury bills.
On August 5, 2011, Standard and Poor’s, the credit rating agency, lowered its rating on U.S. Treasury securities from AAA to AA+, indicating that it considered these securities to have a very small amount of default risk. The other two large credit rating agencies, Moody’s and Fitch, decided not to lower their ratings of U.S. Treasury securities at that time, suggesting that if there was any default risk, it remained extremely small.
For more information on noodling, see the April 21, 2006, New York Times article titled “In the Jaws of a Catfish,” by Ethan Todras-Whitehill and the May 16, 2011, Wall Street Journal article titled “Long Arm of the Law Penalizes Texans Who Nab Catfish by Hand,” by Ana Campoy.
The only adjustment that we had to make was to account for the fact that our calculations used monthly returns rather than annual returns. This adjustment simply required us to multiply each number we calculated by 12 because there are 12 months in a year.
This statement is true in the context of how expected returns are determined. However, the standard deviation is still a very useful measure of the risk faced by an individual investor who does not hold a diversified portfolio. For example, the owners of most small businesses have much of their personal wealth tied up in their businesses. They are certainly concerned about the total risk because it is directly related to the probability that they will go out of business and lose much of their wealth.
10
The Fundamentals of Capital Budgeting
Learning Objectives
Discuss why capital budgeting decisions are the most important investment decisions made by a firm’s management.
Explain the benefits of using the net present value (NPV) method to analyze capital expenditure decisions and calculate the NPV for a capital project.
Describe the strengths and weaknesses of the payback period as a capital expenditure decision-making tool and compute the payback period for a capital project.
Explain why the accounting rate of return (ARR) is not recommended for use as a capital expenditure decision-making tool.
Compute the internal rate of return (IRR) for a capital project and discuss the conditions under which the IRR technique and the NPV technique produce different results.
Explain the benefits of postaudit and ongoing reviews of capital projects.
In October 2010, Intel Corporation’s newly built 300-mm wafer fabrication plant in Dalian China began regular production. The Dalian plant is Intel’s first chipset factory in Asia and is part of a network of eight such facilities worldwide. A $2.5 billion project that was expected to eventually employ 4,000 people, the Dalian plant is Intel’s single largest investment in China.
Even before construction of the fabrication plant was completed, Intel was developing plans for a $3.5 billion addition to the project. This second phase would consist of investments in supporting facilities, which include four assembly and testing plants, an R&D center, and a sales division. To place the anticipated $6.0 billion cost of the two Dalian project phases in perspective, Intel’s total worldwide capital expenditures during fiscal year 2008 were $5.2 billion.
Intel’s investment in the Dalian fabrication plant illustrates not only the large amount of corporate resources that can be committed to a major capital project, but also the strategic importance such an investment can have. When the project was originally announced in March 2007, Intel’s president and CEO, Paul Otellini, pointed out that China was Intel’s fastest-growing major market. It was imperative, said Otellini, that Intel make investments in markets that will provide for future growth.
The Dalian project involves significant risks for Intel. Construction of the fabrication plant was a three-year endeavor that unexpectedly coincided with a worldwide financial and economic crisis. A project of Dalian’s scale represented a potential source of excess capacity for Intel as the growth in demand for chipsets declined worldwide during 2008 and 2009. In addition, since international investments are subject to country risk, political or economic instability in China could have severe financial consequences for Intel.
While few companies make capital investments as large as Intel’s investment in Dalian China, all companies must routinely invest capital in projects that are critical to their success. These investment opportunities must be carefully scrutinized, and their costs and benefits carefully weighed. How do firms make these important capital budgeting decisions? In this chapter we examine this decision-making process and introduce some financial models used to make investment decisions.
CHAPTER PREVIEW
This chapter is about capital budgeting, a topic we first visited in
Chapter 1
. Capital budgeting is the process of deciding which capital investments the firm should make.
We begin the chapter with a discussion of the types of capital projects that firms undertake and how the capital budgeting process is managed within the firm. When making capital investment decisions, management’s goal is to select projects that will increase the value of the firm.
Next we examine some of the techniques used to evaluate capital budgeting decisions. We first discuss the net present value (NPV) method, which is the capital budgeting approach recommended in this book. The NPV method takes into account the time value of money and provides a direct measure of how much a capital project will increase the value of the firm.
We then examine the payback method and the accounting rate of return. As methods of selecting capital projects, both methods have some serious deficiencies. Finally, we discuss the internal rate of return (IRR), which is the expected rate of return for a capital project when the project’s NPV is equal to zero. The IRR is a very popular and important alternative to the NPV technique. However, in certain circumstances, the IRR can lead to incorrect decisions. We close by discussing evidence on techniques financial managers actually use when making capital budgeting decisions.
10.1 AN INTRODUCTION TO CAPITAL BUDGETING
·
We begin with an overview of capital budgeting, followed by a discussion of some important concepts you will need to understand in this and later chapters.
The Importance of Capital Budgeting
the process of choosing the productive assets in which the firm will invest
Capital budgeting decisions are the most important investment decisions made by management. The objective of these decisions is to select investments in productive assets that will increase the value of the firm. These investments create value when they are worth more than they cost. Capital investments are important because they can involve substantial cash outlays and, once made, are not easily reversed. They also define what the company is all about—the firm’s lines of business and its inherent business risk. For better or worse, capital investments produce most of a typical firm’s revenues for years to come.
Capital budgeting techniques help management systematically analyze potential business opportunities in order to decide which are worth undertaking. As you will see, not all capital budgeting techniques are equal. The best techniques are those that determine the value of a capital project by discounting all of the cash flows generated by the project and thus account for the time value of money. We focus on these techniques in this chapter.
In the final analysis, capital budgeting is really about management’s search for the best capital projects—those that add the greatest value to the firm. Over the long term, the most successful firms are those whose managements consistently search for and find capital investment opportunities that increase firm value.
The Capital Budgeting Process
The capital budgeting process starts with a firm’s strategic plan, which spells out its strategy for the next three to five years. Division managers then convert the firm’s strategic objectives into business plans. These plans have a one- to two-year time horizon, provide a detailed description of what each division should accomplish during the period covered by the plan, and have quantifiable targets that each division is expected to achieve. Behind each division’s business plan is a capital budget that details the resources management believes it needs to get the job done.
The capital budget is generally prepared jointly by the CFO’s staff and financial staff s at the divisional and lower levels and reflects, in large part, the activities outlined in the divisional business plans. Many of these proposed expenditures are routine in nature, such as the repair or purchase of new equipment at existing facilities. Less frequently, firms face broader strategic decisions, such as whether to launch a new product, build a new plant, enter a new market, or buy a business.
Exhibit 10.1
identifies some reasons that firms initiate capital projects.
EXHIBIT 10.1 Key Reasons for Making Capital Expenditures
Capital budgeting decisions are the most important investment decisions made by management. Many of these decisions are routine in nature, but from time to time, managers face broader strategic decisions that call for significant capital investments.
Where does a firm get all of the information it needs to make capital budgeting decisions? Most of the information is generated within the firm, and, for expansion decisions, it often starts with sales representatives and marketing managers who are in the marketplace talking to potential and current customers on a day-to-day basis. For example, a sales manager with a new product idea might present the idea to management and the marketing research group. If the product looks promising, the marketing research group will estimate the size of the market and a market price. If the product requires new technology, the firm’s research and development group must decide whether to develop the technology or to buy it. Next, cost accountants and production engineers determine the cost of producing the product and any capital expenditures necessary to manufacture it. Finally, the CFO’s staff takes the data and estimates the cost of the project and the cash flows it will generate over time. The project is a viable candidate for the capital budget if the present value of the expected cash benefits exceeds the project’s cost.
Classification of Investment Projects
Potential capital budgeting projects can be classified into three types: (1) independent projects, (2) mutually exclusive projects, and (3) contingent projects.
Independent Projects
Projects are independent when their cash flows are unrelated. With independent projects, accepting or rejecting one project does not eliminate other projects from consideration (assuming the firm has unlimited funds to invest). For example, suppose a firm has unlimited funding and management wants to: (1) build a new parking ramp at its headquarters; (2) acquire a small competitor; and (3) add manufacturing capacity to one of its plants. Since the cash flows for each project are unrelated, accepting or rejecting one of the projects will have no effect on the others.
independent projects
projects whose cash flows are unrelated
Mutually Exclusive Projects
projects for which acceptance of one precludes acceptance of the other
When projects are mutually exclusive, acceptance of one project precludes acceptance of others. Typically, mutually exclusive projects perform the same function, and thus, only one project needs to be accepted. For example, when BMW decided to manufacture automobiles in the United States, it considered three possible manufacturing sites (or capital projects). Once BMW management selected the Spartanburg, South Carolina, site, the other two possible locations were out of the running. Since some projects are mutually exclusive, it is very important that a capital budgeting method allow us to choose the best project when we are faced with two or more alternatives.
Contingent Projects
With contingent projects, the acceptance of one project is contingent on the acceptance of another. There are two types of contingency situations. In the first type of situation, the contingent product is mandatory. For example, when a public utility company (such as your local electric company) builds a power plant, it must also invest in suitable pollution control equipment to meet federal environmental standards. The pollution control investment is a mandatory contingent project. When faced with mandatory contingent projects, it is best to treat all of the projects as a single investment for the purpose of evaluation. This provides management with the best measure of the value created by these projects.
In the second type of situation, the contingent project is optional. For example, suppose Dell invests in a new computer for the home market. This computer has a feature that allows Dell to bundle a proprietary gaming system. The gaming system is a contingent project but is an optional add-on to the new computer. In these situations, the optional contingent project should be evaluated independently and should be accepted or rejected on its own merits.
contingent projects
projects whose acceptance depends on the acceptance of other projects
Basic Capital Budgeting Terms
In this section we briefly introduce two terms that you will need to be familiar with—cost of capital and capital rationing.
the required rate of return for a capital investment
The cost of capital is the rate of return that a capital project must earn to be accepted by management. The cost of capital can be thought of as an opportunity cost. Recall from
Chapter 8
that an opportunity cost is the value of the most valuable alternative given up if a particular investment is made.
Let’s consider the opportunity cost concept in the context of capital budgeting decisions. When investors buy shares of stock in a company or loan money to a company, they are giving management money to invest on their behalf. Thus, when a firm’s management makes capital investments in productive assets such as plant and equipment they are investing stockholders’ and creditors’ money in real assets. Since stockholders and creditors could have invested their money in financial assets, the minimum rate of return they are willing to accept on an investment in a real asset is the rate they could have earned investing in financial assets that have similar risk. The rate of return that investors can earn on financial assets with similar risk is an opportunity cost because investors lose the opportunity to earn that rate if the money is invested in a real asset instead. It is therefore the rate of return that investors will require for an investment in a capital project. In other words, this rate is the cost of capital. It is also known as the opportunity cost of capital.
Chapter 13
discusses how we estimate the opportunity cost of capital in practice.
BUILDING INTUITION INVESTMENT DECISIONS HAVE OPPORTUNITY COSTS
When any investment is made, the opportunity to earn a return from an alternative investment is lost. The lost return can be viewed as a cost that arises from a lost opportunity. For this reason, it is called an opportunity cost. The opportunity cost of capital is the return an investor gives up when his or her money is invested in one asset rather than the best alternative asset. For example, suppose that a firm invests in a piece of equipment rather than returning money to stockholders. If stockholders could have earned an annual return of 12 percent on a stock with cash flows that are as risky as the cash flows the equipment will produce, this is the opportunity cost of capital associated with the investment in the piece of equipment.
opportunity cost of capital
the return an investor gives up when his or her money is invested in one asset rather than the best alternative asset
Capital Rationing
When a firm has all the money it needs to invest in all the capital projects that meet its capital selection criteria, the firm is said to be operating without a funding constraint, or resource constraint. Firms are rarely in this position, especially growth firms. Typically, a firm has a fixed number of dollars available for capital expenditures, and the number of qualified projects that need funding exceeds the funds that are available. This funding constraint on investments means that some projects will be mutually exclusive, since investing in one project exhausts resources that might otherwise be invested in another. When faced with a resource constraint, the firm must allocate its funds to the subset of projects that provides the largest increase in stockholder value. The process of limiting, or rationing, capital expenditures in this way is called capital rationing. Capital rationing and its implications for capital budgeting are discussed in
Chapter 12
.
capital rationing
a situation where a firm does not have enough capital to invest in all attractive projects and must therefore ration capital
> BEFORE YOU GO ON
1. Why are capital investments the most important decisions made by a firm’s management?
2. What are the differences between capital projects that are independent, mutually exclusive, and contingent?
10.2 NET PRESENT VALUE
·
In this section we discuss a capital budgeting method that is consistent with this goal of financial management—to maximize the wealth of the firm’s owners. It is called the net present value (NPV) method, and it is one of the most basic analytical methods underlying corporate finance. The NPV method tells us the amount by which the benefits from a capital expenditure exceed its costs. It is the capital budgeting technique recommended in this book.
net present value (NPV) method
a method of evaluating a capital investment project which measures the difference between its cost and the present value of its expected cash flows
Valuation of Real Assets
Throughout the book, we have emphasized that the value of any asset is the present value of its future cash flows. In
Chapters 8
and
9
, we developed valuation models for financial assets, such as bonds, preferred stock, and common stock. We now extend our discussion of valuation models from financial to real assets. The steps used in valuing an asset are the same whether the asset is real or financial:
1. Estimate the future cash flows.
2. Determine the required rate of return, or discount rate, which depends on the riskiness of the future cash flows.
3. Compute the present value of the future cash flows to determine what the asset is worth.
The valuation of real assets, however, is less straightforward than the valuation of financial assets, for several reasons.
First, in many cases, cash flows for financial assets are well documented in a legal contract. If they are not, we are at least able to make some reasonable assumptions about what they are. For real assets, much less information exists. Specialists within the firm, usually from the finance, marketing, and production groups, often prepare estimates of future cash flows for capital projects with only limited information.
Second, many financial securities are traded in public markets, and these markets are reasonably efficient. Thus, market data on rates of return are accessible. For real assets, no such markets exist. As a result, we must estimate required rates of return on real assets (opportunity costs) from market data on financial assets; this can be difficult to do.
NPV—The Basic Concept
The NPV of a project is the difference between the present value of the project’s future cash flows and the present value of its cost. The NPV can be expressed as follows:
NPV = PV (Project’s future cash flows) − PV (Cost of the project)
To illustrate these important points, consider an example. Suppose a firm is considering building a new marina for pleasure boats. The firm has a genie that can tell the future with perfect certainty. The finance staff estimates that the marina will cost $3.50 million. The genie volunteers that the present value of the future cash flows from the marina is $4.25 million.
Assuming this information is correct, the NPV for the marina project is a positive $750,000 ($4.25 million − $3.50 million = $0.75 million). Management should accept the project because the excess of the value of the cash flows over cost increases the value of the firm by $750,000. Why is a positive NPV a direct measure of how much a capital project will increase the value of the firm? If management wanted to, the firm could sell the marina for $4.25 million, pay the $3.50 million in expenses, and deposit $750,000 in the bank. The value of the firm would increase by the $750,000 deposited in the bank. In sum, the NPV method tells us which capital projects to select and how much value they add to the firm.
NPV and Value Creation
We have just said that any project with a positive NPV should be accepted because it will increase the value of the firm. Let’s take a moment to think about this proposition. What makes a capital asset worth more than it costs? In other words, how does management create value with capital investments?
How Value Is Created
Suppose that when you were in college, you worked part time at a successful pizza parlor near campus. During this time, you learned a lot about the pizza business. After graduation, you purchased a pizza parlor for $100,000 that was in a good location but had been forced to close because of a lack of business. The owners had let the restaurant and the quality of the pizzas deteriorate, and the wait staff had been rude, especially to college students. Once you purchased the restaurant, you immediately invested $40,000 to fix it up: you painted the building, spruced up the interior, replaced some of the dining room furniture, and added an eye-catching, 1950s-style neon sign to attract attention. You also spent $15,000 for a one-time advertising blitz to quickly build a customer base. More important, you improved the quality of the pizzas you sold, and you built a profitable takeout business. Finally, you hired your wait staff carefully and trained them to be customer friendly.
Almost immediately the restaurant was earning a substantial profit and generating substantial cash flows. The really good news was that several owners of local pizzerias wanted to buy your restaurant. After intense negotiations with several of the potential buyers, you accepted a cash offer of $475,000 for the business shortly after you purchased it.
What is the NPV for the pizza parlor? For this investment, the NPV is easy to calculate. We do not need to estimate future cash flows and discount them because we already have an estimate of the present value of the cash flows the pizza parlor is expected to produce—$475,000. Someone is willing to pay you $475,000 because he or she believes the future cash flows are worth that amount. The cost of your investment includes the purchase price of the restaurant, the cost to fix it up, and the cost of the initial advertising campaign, which totals $155,000 ($100,000 $40,000 $15,000). Thus, the NPV for the pizza parlor is:
Where did the $320,000 in value you created go? The NPV of your investment is the amount that your personal net worth increased because of the investment. For an ongoing business, the result would have been a $320,000 increase in the value of the firm.
How about the original owners? Why would they sell a business worth $475,000 to you for $100,000? The answer is simple; if they could have transformed the business as you did, they would have done so. Instead, when they ran the business, it lost money! They sold it to you because you offered them a price reflecting its value to them.
Market Data versus Discounted Cash Flows
Our pizza parlor example is greatly simplified by the fact that we can observe the price that someone is willing to pay for the asset. In most capital project analyses, we have to estimate the market value of the asset by forecasting its future cash flows and discounting them by the cost of capital. The discounted value of a project’s future cash flows is an estimate of its value, or the market price for which it can be sold.
Framework for Calculating NPV
We now describe a framework for analyzing capital budgeting decisions using the NPV method. As you will see, the NPV technique uses the discounted cash flow technique developed in
Chapters 5
and
6
and applied in
Chapters 8
and
9
. The good news, then, is that the NPV method requires only the application of what you already know.
In addition to following the five-step framework for solving NPV analysis problems, we recommend that you use a worksheet with a time line like the one shown here to help you determine the proper cash flows for each period.
The five-step framework discussed in this section and the accompanying cash flow work-sheet (
Exhibit 10.2
) can help you systematically organize a project’s cash flow data and compute its NPV. Most mistakes people make when working capital budgeting problems result from problems with cash flows: not identifying a cash flow, getting a cash flow in the wrong time period, or assigning the wrong sign to a cash flow. What can make cash flow analysis difficult in capital budgeting is this: there are often multiple cash flows in a single time period, and some are cash inflows and others are cash outflows.
As always, we recommend that you prepare a time line when doing capital budgeting problems. A sample time line is shown in
Exhibit 10.2
, along with an identification of the cash flows for each period. Our goal is to compute the net cash flow (NCF) for each time period t, where NCF
t
= (Cash inflows − Cash outflows) for the period t. For a capital project, the time periods (t) are usually in years, and t varies from the current period (t = 0) to some finite time period that is the estimated life of the project (t = n). Recall that getting the correct sign on each cash flow is critical to getting the correct answer to a problem. As you have seen in earlier chapters, the convention in finance problem solving is that cash inflows carry a positive sign and cash outflows carry a negative sign. Finally, note that all cash flows in this chapter are on an after-tax basis. We will make adjustments for tax consequences on specific transactions such as the calculation of a project’s salvage value.
CCH Business Owner’s Toolkit is a valuable Web source for information about running a business, including capital budget analysis. Go to
http://www.toolkit.cch.com/text/p06_6500.asp
.
Our five-step framework for analysis is as follows:
1. Determine the initial cost of starting the project. We first need to identify and add up all the cash flows related to the initial cost of starting the project. In most cases, the initial cost of a project is incurred at the start; hence the cash flows are already in current dollars. These cash flows typically include any property, plant, and equipment outlays for production as well as employee hiring and training costs. In some cases, like Intel’s investment in the Dalian fabrication plant, these initial outlays can be made over several years before the project is up and running. Of course, any future cash flows must be discounted to obtain their present value. Turning to
Exhibit 10.2
, we have incurred a single negative cash flow (−CF0) as our initial cost of starting the project; thus NCF0 has a negative value.
2. Estimate the project’s future cash flows over its expected life. Once they are up-and-running, capital projects typically generate some cash inflows from revenues (CIF
t
) for each period, along with some cash outflows (COF
t
) associated with costs incurred to generate the revenues. In most cases revenues exceed costs, and thus, NCF
t
is positive. However, this may not always be the case. For example, if the project is the purchase of a piece of equipment, it is possible for NCF3 to have a negative value (CIF3 < COF3) if the equipment is projected to need a major overhaul or must be replaced during the third year. Finally, you also need to pay attention to a project's final cash flow, which is t = 5 in
Exhibit 10.2
. There may be a salvage value (SV) at the end of the project, which is a cash inflow. In that case NCF5 = (CIF5 − COF5 + SV). The important point is that for each time period, we must identify all the cash flows that take place, assign each cash flow its proper sign, and add up all the cash flows.
3. Determine the riskiness of the project and the appropriate cost of capital. The third step is to identify for each project its risk-adjusted cost of capital, which takes into account the riskiness of the project's cash flows. The riskier the project, the higher its cost of capital. The cost of capital is the discount rate used in determining the present value of the future expected cash flows. In this chapter, the cost of capital and any risk adjustments will be supplied, and no calculations will be required for this step.
4. Compute the project's NPV. The NPV, as you know, is the present value of the net cash flows the project is expected to generate minus the cost of the project.
5. Make a decision. If the NPV is positive, the project should be accepted because all projects with a positive NPV will increase the value of the firm. If the NPV is negative, the project should be rejected; projects with negative NPVs will decrease the value of the firm.
You might be wondering about how to handle a capital project with an NPV of 0. Technically, management should be indifferent to accepting or rejecting projects such as this because they neither increase nor decrease the value of the firm. When the NPV 0, the project is generating returns that are just equal to the opportunity cost of capital. At a practical level, projects rarely have an NPV equal to 0, and most firms have more good capital projects (with NPV > 0) than they can fund. Thus, this is not an issue that generates much interest among practitioners.
Net Present Value Techniques
The NPV of a capital project can be stated in equation form as the present value of all net cash flows (cash inflows − cash outflows) connected with the project, whether in the current period or in the future. The NPV equation can be written as follows:
When analyzing capital budgeting problems, we typically have a lot of data to sort through. The worksheet approach introduced in
Exhibit 10.2
is helpful in keeping track of the data in an organized format.
Exhibit 10.3
shows the time line and relevant cash flows for the pocket pizza project. The steps in analyzing the project’s cash flows and determining its NPV are as follows:
1. Determine the cost of the project. The cost of the project is the cost to modify the existing production line, which is $300,000. This is a cash outflow (negative sign).
2. Estimate the project’s future cash flows over its expected life. The project’s future cash inflows come from sales of the new product. Sales are estimated at $300,000 per year (positive sign). The cash outflows are the costs to manufacture and distribute the new product, which are $220,000 per year (negative sign). The life of the project is five years. The project has a salvage value of $30,000, which is a cash inflow (positive sign). The net cash flow (NCF) in a particular time period is just the sum of the cash inflows and cash outflows for that period. For example, the NCF for period t 0 is $300,000 the NCF for period t 1 is $80,000, and so on, as you can see in
Exhibit 10.3
.
The worksheet approach introduced in
Exhibit 10.2
is helpful in organizing the data given for the pocket pizza project.
3. Determine the riskiness of the project and appropriate cost of capital. The discount rate is the cost of capital, which is 15 percent.
4. Compute the project’s NPV. To compute the project’s NPV, we apply Equation 10.1 by plugging in the NCF values for each time period and using the cost of capital, 15 percent, as the discount rate. The equation looks like this (the figures are in thousands of dollars):
5. Make a decision. The pocket pizza project has a negative NPV, which indicates that the project is not a good investment and should be rejected. If management undertook this project, the value of the firm would decrease by $16,910; and, if the firm had one hundred thousand shares of stock outstanding, we can estimate that the project would decrease the value of each share by about 17 cents ($16,910/100,000 shares $0.1691 per share).
Calculating NPV with a Financial Calculator
Using a financial calculator is an easier way to calculate the present value of the future cash flows. In this example you should recognize that the cash flow pattern is a five-year ordinary annuity with an additional cash inflow in the fifth year. This is exactly the cash pattern for a bond with annual coupon payments and payment of principal at maturity we saw in
Chapter 8
. We can find the present value using a financial calculator, with $80 being the annuity stream for five years and $30 the salvage value at year 5:
The cost is $300,000 to modify the production line. Sales of the new product are estimated at $200,000 for the first year, $300,000 for the next two years, and $500,000 for the final two years. It is estimated that production, sales, and advertising costs will be $250,000 for the first year and will then decline to a constant $200,000 per year. There is no salvage value at the end of the product’s life, and the appropriate cost of capital is 15 percent. Is the project, as proposed, economically viable?
APPROACH: To solve the problem, work through the steps for NPV analysis given in the text.
SOLUTION:
Exhibit 10.4
shows the project’s cash flows.
1. The cost to modify the production line is $300,000, which is a cash outflow in Year 0 and the cost of the project.
2. The future cash flows over the expected life of the project are laid out on the time line in
Exhibit 10.4
. The project’s life is five years. The NCFs for the capital project are negative at the beginning of the project and in the first year ($300,000 and $50,000) and thereafter are positive.
3. The appropriate cost of capital is 15 percent.
4. The values are substituted into Equation 10.1 to calculate the NPV:
The worksheet shows the time line and cash flows for the self-rising pizza dough project in Learning by Doing Application 10.1. As always, it is important to assign each cash flow to the appropriate year and to give it the proper sign. Once you have computed the net cash flow for each time period, solving for NPV is just a matter of plugging the data into the NPV formula.
USING EXCEL NET PRESENT VALUE
Net present value problems are most commonly solved using a spreadsheet program. The program’s design is good for keeping track of all the cash flows and the periods in which they occur. The spreadsheet setup for Learning by Doing Application 10.1, presented on the right, shows how to calculate the NPV for the self-rising pizza dough machine:
Notice that the NPV formula does not take into account the cash flow in year zero. Therefore, you only enter into the NPV formula the cash flows in years 1 through 5, along with the discount rate. You then add the cash flow in year zero to the total from the NPV formula calculation to get the NPV for the investment.
Recall that investments are mutually exclusive if, by making one, another will not be undertaken. Projects may be mutually exclusive because they are substitutes for one another or because the firm has a funding constraint. A project’s NPV provides an objective measure of its incremental value to the firm’s investors, and thus makes it simple to choose between two or more mutually exclusive projects. When faced with such a choice, managers should allocate capital to the project that has the most positive dollar impact on the value of the firm, in other words, the project with the highest NPV.
EXAMPLE 10.1 DECISION MAKING
The IS Department’s Capital Projects
DECISION: If the projects are independent, you should accept projects 1 and 2, both of which have a positive NPV, and reject project 4. Project 3, with an NPV of zero, could be either accepted or rejected. If the projects are mutually exclusive and you can accept only one of them, it should be project 1, which has the largest NPV.
Concluding Comments on NPV
Some concluding comments about the NPV method are in order. First, as you may have noticed, the NPV computations are rather mechanical once we have estimated the cash flows and the cost of capital. The real difficulty is estimating or forecasting the future cash flows. Although this may seem to be a daunting task, managers with experience in producing and selling a particular type of product can usually generate fairly accurate estimates of sales volumes, prices, and production costs. Most business managers are routinely required to make decisions that involve expectations about future events. In fact, that is what business is really all about—dealing with uncertainty and making decisions that involve risk.
Second, estimating project cash flows over a long forecast period requires skill and judgment. There is nothing wrong with using estimates to make business decisions as long as they are based on informed judgments and not guesses. Problems can arise with the cash flow estimates when a project team becomes overly enamored with a project. In wanting a particular project to succeed, a project team can be too optimistic about the cash flow projections. It is therefore very important that capital budgeting decisions be subject to ongoing and post-audit review.
In conclusion, the NPV approach is the method we recommend for making capital investment decisions. It provides a direct (dollar) measure of how much a project will increase the value of the firm. NPV also makes it possible to correctly choose between mutually exclusive projects. The accompanying table summarizes NPV decision rules and the method’s key advantages and disadvantages.
1. What is the NPV of a project?
2. If a firm accepts a project with a $10,000 NPV, what is the effect on the value of the firm?
3. What are the five steps used in NPV analysis?
10.3 THE PAYBACK PERIOD
·
The payback period is one of the most widely used tools for evaluating capital projects. The payback period is defined as the number of years it takes for the cash flows from a project to recover the project’s initial investment. With the payback method for evaluating projects, a project is accepted if its payback period is below some specified threshold. Although it has serious weaknesses, this method does provide some insight into a project’s risk; the more quickly you recover the cash, the less risky is the project.
payback period
the length of time required to recover a project’s initial cost
Computing the Payback Period
To compute the payback period, we need to know the project’s cost and estimate its future net cash flows. The net cash flows and the project cost are the same values that we use to compute the NPV. The payback (PB) equation can be expressed as follows:
The exhibit shows the net and cumulative net cash flows for a proposed capital project with an initial cost of $70,000. The cash flow data are used to compute the payback period, which is 2.5 years.
shows the net cash flows (row 1) and cumulative net cash flows (row 2) for a proposed capital project with an initial cost of $70,000. The payback period calculation for our example is:
Exhibit 10.5
that the firm recovers cash flows of $30,000 in the first year and $30,000 in the second year, for a total of $60,000 over the two years. During the third year, the firm needs to recover only $10,000 ($70,000 $60,000 $10,000) to pay back the full cost of the project. The third-year cash flow is $20,000, so we will have to wait 0.5 year ($10,000/$20,000 = 0.5) to recover the final amount. Thus, the payback period for this project is 2.5 years (2 years + 0.5 year = 2.5 years).
The idea behind the payback period method is simple: the shorter the payback period, the faster the firm gets its money back and the more desirable the project. However, there is no economic rationale that links the payback method to stockholder value maximization. Firms that use the payback method accept all projects having a payback period under some threshold and reject those with a payback period over this threshold. If a firm has a number of projects that are mutually exclusive, the projects are selected in order of their payback rank: projects with the shortest payback period are selected first.
APPLICATION 10.2 LEARNING BY DOING
APPROACH: Use Equation 10.2 to calculate the number of years it takes for the cash flows from each project to recover the project’s initial investment. If the two projects are independent, you should accept the projects that have a payback period that is less than or equal to two years. If the projects are mutually exclusive, you should accept the project with the shortest payback period if that payback period is less than or equal to two years.
SOLUTION: The payback for project A requires only that we calculate the first term in Equation 10.2—Years before recovery: the first year recovers $100, the second year $200, and the third year $200, for a total of $500 ($100 + $200 + $200 = $500). Thus, in three years, the $500 investment is fully recovered, so PBA = 3.00.
For project B, the first year recovers $400 and the second year $300. Since we need only part of the second-year cash flow to recover the initial cost, we calculate both terms in Equation 10.2 to obtain the payback period.
How the Payback Period Performs
We have worked through some simple examples of how the payback period is computed. Now we will consider several more complex situations to see how well the payback period performs as a capital budgeting rule.
Exhibit 10.6
illustrates five different capital budgeting projects. The projects all have an initial investment of $500, but each one has a different cash flow pattern. The bottom part of the exhibit shows each project’s payback period, along with its net present value for comparison. We will assume that management has set a payback period of two years as the cutoff point for an acceptable project.
Project A: The cash flows for project A are $200 in the first year and $300 in the second, for a total of $500; thus, the project’s payback period is two years. Under our acceptance criterion, management should accept this project. Project A also has a positive NPV of $450, so the two capital budgeting decision rules agree.
Project B: Project B never generates enough cash flows to pay off the original investment of $500: $300 + $100 + $50 = $450. Thus, the project payback period is infinite. With an infinite payback period, the project should be rejected. Also, as you would expect, project B’s NPV is negative. So far, the payback period and NPV methods have agreed on which projects to accept.
Project C: Project C has a payback period of two years: $250 + $250 = $500. Thus, according to the payback criteria, it should be accepted. However, the project’s NPV is a negative $115, which indicates that the project should be rejected. Why the conflict? Look at the cash flows after the payback period of two years. In year 3 the project requires an additional investment of $250 (a cash outflow) and now is in a deficit position; that is, the cumulative net cash balance is now only $250 ($250 + $250 − $250 = $250). Then, in the final year, the project earns an additional $250, recovering the cost of the total investment. The project’s payback is really four years. Th e payback period analysis can lead to erroneous decisions because the rule does not consider cash flows after the payback period.
EXHIBIT 10.6 Payback Period with Various Cash Flow Patterns
Each of the five capital budgeting projects shown in the exhibit calls for an initial investment of $500, but all have different cash flow patterns. The bottom part of the exhibit shows each project’s payback period, along with its net present value for comparison. at the cash flows after the payback period of two years. In year 3 the project requires an additional investment of $250 (a cash outflow) and now is in a deficit position; that is, the cumulative net cash balance is now only $250 ($250 $250 $250 $250). Then, in the final year, the project earns an additional $250, recovering the cost of the total investment. The project’s payback is really four years. The payback period analysis can lead to erroneous decisions because the rule does not consider cash flows after the payback period.
Discounted Payback Period
Another weakness of the ordinary payback period criteria is that it does not take into account the time value of money. All dollars received before the cutoff period are given equal weight. To address this problem, some financial managers use a variant of the payback period called the discounted payback period. This payback calculation is similar to the ordinary payback calculation except that the future cash flows are discounted by the cost of capital.
discounted payback period
the length of time required to recover a project’s initial cost, accounting for the time value of money
The major advantage of the discounted payback approach is that it tells management how long it takes a project to reach an NPV of zero. Thus, any capital project that meets a firm’s decision rule must also have a positive NPV. This is an improvement over the standard payback calculation, which can lead to accepting projects with negative NPVs. Regardless of the improvement, the discounted payback method is not widely used by businesses, and it also ignores all cash flows after the arbitrary cutoff period, which is a major flaw.
To see how the discounted payback period is calculated, turn to
Exhibit 10.7
. The exhibit shows the net cash flows for a proposed capital project along with both the cumulative and discounted cumulative cash flows; thus, we can compute both the ordinary and the discounted payback periods for the project and then compare them. The cost of capital is 10 percent.
The first two rows show the nondiscounted cash flows, and we can see by inspection that the ordinary payback period is two years. We do not need to make any additional calculations because the cumulative cash flows equal zero at precisely two years. Now let’s turn our attention to the lower two rows, which show the project’s discounted and cumulative discounted cash flows. Note that the first year’s cash flow is $20,000 and its discounted value is $18,182 ($20,000 × 0.9091 = $18,182), and the second year’s cash flow is also $20,000 and its discounted value is $16,529 ($20,000 × 0.8264 = $16,529). Now, looking at the cumulative discounted cash flows row, notice that it turns positive between two and three years. This means that the discounted payback period is two years plus some fraction of the third year’s discounted cash flow. The exact discounted payback period computed value is 2 years + ($5,289/$15,026 per year) = 2 years + 0.35 years = 2.35 years.
The exhibit shows the net and cumulative net cash flows for a proposed capital project with an initial cost of $40,000. The cash flow data are used to compute the discounted payback period for a 10 percent cost of capital, which is 2.35 years.
As expected, the discounted payback period is longer than the ordinary payback period (2 years < 2.35 years), and in 2.35 years the project will reach a NPV = 0. The project NPV is positive (NPV = $9,737); therefore, we should accept the project. But notice that the payback decision criteria are ambiguous. If we use 2.0 years as the payback criterion, we reject the project and if we use 2.5 or 3.0 years as criterion, the project is accepted. The lack of a definitive decision rule remains a major problem with the payback period as a capital budgeting tool.
Evaluating the Payback Rule
The standard payback period is often calculated for projects because it provides an intuitive and simple measure of a project's liquidity risk. This makes sense because projects that pay for themselves quickly are less risky than projects whose paybacks occur farther in the future. There is a strong feeling in business that “getting your money back quickly” is an important standard when making capital investments. This intuition can be economically justified if the firm faces payments to creditors before the payback date, or if estimates of project cash flows beyond the payback date are very uncertain. Probably the greatest advantage of the payback period is its simplicity; it is easy to calculate and easy to understand.
When compared with the NPV method, however, the payback methods have some serious shortcomings. First, the standard payback calculation does not adjust or account for the timing or risk associated with future cash flows. Second, there is little economic justification for the choice of the payback cutoff criteria other than a liquidity motive. Who is to say that a particular cutoff, such as two years, is optimal with regard to maximizing stockholder value? Finally, perhaps the greatest shortcoming of the payback method is its failure to consider cash flows after the payback period. As a result, the payback method is biased toward shorter-term projects and may cause managers to reject important positive NPV projects where cash inflows tend to occur farther in the future, such as research and development investments, new product launches, and entry into new lines of business.
While the payback period is relatively simple to calculate, it is important to note that payback requires forecasts of future project cash flows up to the established cutoff period. Furthermore, discounted payback requires that managers identify a project's discount rate. Thus the inputs into the payback and NPV methods are virtually identical. Consequently, using a payback method may not even save much time and effort in evaluating a project. The table below summarizes key advantages and disadvantages of the payback method.
> BEFORE YOU GO ON
1. What is the payback period?
2. Why does the payback period provide a measure of a project’s liquidity risk?
3. What are the main shortcomings of the payback method?
10.4 THE ACCOUNTING RATE OF RETURN
·
a rate of return on a capital project based on average net income divided by average book value over the project’s life; also called the book value rate of return
We turn next to a capital budgeting technique based on the accounting rate of return (ARR), sometimes called the book value rate of return. This method computes the return on a capital project using accounting numbers—the project’s net income (NI) and book value (BV)—rather than cash flow data. The ARR can be calculated in a number of ways, but the most common definition is:
Although ARR is fairly easy to understand and calculate, as you probably guessed, it has a number of major flaws as a tool for evaluating capital expenditure decisions. Besides the fact that AAR is based on accounting numbers rather than cash flows, it is not really even an accounting-based rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income statement and balance sheet. Thus, the ARR ignores the time value of money. Also, as with the payback method, there is no economic rationale that links a particular acceptance criterion to the goal of maximizing stockholder value.
Because of these major shortcomings, the ARR technique should not be used to evaluate the viability of capital projects under any circumstances. You may wonder why we even included the ARR technique in the book if it is a poor criterion for evaluating projects. The reason is simply that we want to be sure that if you run across the ARR method at work, you will recognize it and be aware of its shortcomings.
> BEFORE YOU GO ON
1. What are the major shortcomings of using the ARR method as a capital budgeting method?
10.5 INTERNAL RATE OF RETURN
·
The internal rate of return, known in practice as the IRR, is an important alternative to the NPV method. The NPV and IRR techniques are closely related in that both involve discounting the cash flows from a project; thus, both account for the time value of money. When we use the NPV method to evaluate a capital project, the discount rate is the rate of return required by investors for investments with similar risk, which is the project’s opportunity cost of capital. When we use the IRR, we are looking for the rate of return associated with a project so that we can determine whether this rate is higher or lower than the project’s discount rate.
internal rate of return (IRR)
the discount rate at which the present value of a project’s expected cash inflows equals the present value of the project’s outflows; it is the discount rate at which the project’s NPV equals zero
We can define the IRR as the discount rate that equates the present value of a project’s cost to the present value of its expected cash inflows:
Calculating the IRR
The IRR is an expected rate of return much like the yield to maturity we calculated for bonds in
Chapter 8
. Thus, in calculating the IRR, we need to apply the same trial-and-error method we used in
Chapter 8
. We will begin by doing some IRR calculations by trial and error so that you understand the process, and then we will switch to the financial calculator and computer spreadsheets, which provide an answer more quickly.
Suppose that Ford Motor Company has an investment opportunity with cash flows as shown in
Exhibit 10.8
and that the cost of capital is 12 percent. We want to find the IRR for this project. Using Equation 10.4, we will substitute various values for IRR into the equation to compute the project’s IRR by trial and error. We continue this process until we find the IRR value that makes Equation 10.4 equal zero.
A good starting point is to use the cost of capital as the discount rate. Note that when we discount the NCFs by the cost of capital, we are calculating the project’s NPV:
The cash flow data in the exhibit are used to compute the project’s IRR. The project’s NPV is a positive $16,440, which indicates that the IRR is greater than the cost of capital of 12 percent.
Good guess! This means that the NPV of Ford’s capital project is zero at a discount rate of 13.7 percent. The required rate of return is the cost of capital, which is 12.0 percent. Since the project’s IRR of 13.7 percent exceeds the cost of capital, the IRR criterion indicates that the project should be accepted.
The project’s NPV is a positive $16,440, which also indicates that Ford should go ahead with the project. Thus, both the IRR and NPV suggest the same conclusion.
APPLICATION 10.3 LEARNING BY DOING
APPROACH: The IRR for an investment is the discount rate at which the NPV is zero. Thus, we can use Equation 10.4 to solve for the IRR and then compare this value with Larry’s cost of capital. If the IRR is greater than the cost of capital, the project has a positive NPV and should be accepted.
SOLUTION: The total cost of the machine is $6,750 ($5,000 $1,750 $6,750), and the final cash flow in year 10 is $2,400 ($2,000 $400 $2,400).
Because the project’s IRR exceeds Larry’s cost of capital of 15 percent, the project should be accepted. Larry is wrong.
USING EXCEL INTERNAL RATE OF RETURN
You know that calculating IRR by hand can be tedious. The trial-and-error method can take a long time and can be quite frustrating. Knowing all the cash flows and an approximate rate will allow you to use a spreadsheet formula to get an answer instantly.
The accompanying spreadsheet shows the setup for calculating the IRR for the low-cal yogurt machine at Larry’s Ice Cream Parlor that is described in Learning by Doing Application 10.3.
Here are a couple of important points to note about IRR calculations using spreadsheet programs:
1. Unlike the NPV formula, the IRR formula accounts for all cash flows, including the initial investment in year 0, so there is no need to add this cash flow later.
2. The syntax of the IRR function requires that you first provide the project’s cash flows in order beginning at time zero. To calculate the IRR, you will also need to provide a “guess” value, or a number you estimate is close to the IRR. A good value to start with is the cost of capital. To learn more about why this value is needed, you should go to your spreadsheet’s help manual and search for “IRR.”
In the Ford example, the IRR and NPV methods agree. The two methods will always agree when you are evaluating independent projects and the projects’ cash flows are conventional. As discussed earlier, an independent project is one that can be selected with no effect on the viability of any other project. A project with conventional cash flows is one with an initial cash outflow followed by one or more future cash inflows. Put another way, after the initial investment is made (cash outflow), the net cash flow in each future year is positive (inflows). For example, the purchase of a bond involves conventional cash flows. You purchase the bond for a price (cash outflow), and in the future you receive coupon payments and a principal payment at maturity (cash inflows).
conventional cash flow
a cash flow pattern consisting of an initial cash outflow that is followed by one or more cash inflows
Let’s look more closely at the kinds of situations in which the NPV and the IRR methods agree. A good way to visualize the relation between the IRR and NPV methods is to graph NPV as a function of the discount rate. The graph, called an NPV profile, shows the NPV of the project at various costs of capital.
NPV profile
a graph showing NPV as a function of the discount rate
shows the NPV profile for the Ford project. We have placed the NPVs on the vertical axis, or y-axis, and the discount rates on the horizontal axis, or x-axis. We used the calculations from our earlier example and made some additional NPV calculations at various discount rates as follows:
In the NPV profile for the Ford project, the NPV value is on the vertical (y) axis and the discount rate is on the horizontal (x) axis. You can see that as the discount rate increases, the NPV profile curve declines smoothly and intersects the x-axis at the point where the NPV is 0. The IRR of 13.7 percent is the point at which the NPV changes from a positive to a negative value. The NPV and IRR methods lead to identical accept-or-reject decisions for the Ford project.
As you can see, a discount rate of 0 percent corresponds with an NPV of $160,000; a discount rate of 5 percent with an NPV of $94,000; and so forth. As the discount rate increases, the NPV curve declines smoothly. Not surprisingly, the curve intersects the x-axis at precisely the point where the NPV is 0 and the IRR is 13.7 percent.
The NPV profile in
Exhibit 10.9
illustrates why the NPV and IRR methods lead to identical accept-reject decisions for the Ford project. The IRR of 13.7 percent precisely marks the point at which the NPV changes from a positive to a negative value. Whenever a project is independent and has conventional cash flows, the result will be as shown in the exhibit. The NPV will decline as the discount rate increases, and the IRR and the NPV methods will result in the same capital expenditure decision.
When the NPV and IRR Methods Disagree
We have seen that the IRR and NPV methods lead to identical investment decisions for capital projects that are independent and that have conventional cash flows. However, if either of these conditions is not met, the IRR and NPV methods can produce different accept-reject decisions.
Unconventional Cash Flows
Unconventional cash flows can cause a conflict between the NPV and IRR decision rules. In some instances the cash flows for an unconventional project are just the reverse of those of a conventional project: the initial cash flow is positive, and all subsequent cash flows are negative. In this case, we need only reverse the IRR decision rule and accept the project if the IRR is less than the cost of capital to make the IRR and NPV methods agree.
When a project’s future cash flows include both positive and negative cash flows, the situation is more complicated. An example of such a project is an assembly line that will require one or more major renovations over its lifetime. Another common business situation is a project that has conventional cash flows except for the final cash flow, which is negative. The final cash flow might be negative because extensive environmental cleanup is required at the end of the project, such as the cost for decommissioning a nuclear power plant, or because the equipment originally purchased has little or no salvage value and is expensive to remove.
Consider an example. Suppose a firm invests in a gold-mining operation that costs $55 million and has an expected life of two years. In the first year, the project generates a cash inflow of $150 million. In the second year, extensive environmental and site restoration is required, so the expected cash flow is a negative $100 million. The time line for these cash flows follows.
Exhibit 10.10
:
The NPV profile in
Exhibit 10.10
shows the results of this pattern: we have two IRRs, one at 16.05 percent and the other at 55.65 percent. Which is the correct IRR, or are both correct? Actually, there is no correct answer; the results are meaningless, and you should not try to interpret them. Thus, in this situation, the IRR technique provides information that should not be used for decision making.
How many IRR solutions can there be for a given cash flow? The maximum number of IRR solutions is equal to the number of sign reversals in the cash flow stream. For a project with a conventional cash flow, there is only one cash flow sign reversal; thus, there is only one IRR solution. In our mining example, there are two cash flow sign reversals; thus, there are two IRR solutions.
Finally, for some cash flow patterns, it is impossible to compute an IRR. These situations can occur when the initial cash flow (t = 0) is either a cash inflow or outflow and is followed by cash flows with two or more sign reversals. An example of such a cash flow pattern is NCF0 = $15, NCF1 −$25, and NCF2 = $20. This type of cash flow pattern might occur on a building project where the contractor is given a prepayment, usually the cost of materials and supplies ($15); then does the construction and pays the labor cost ($25); and finally, upon completion of the work, receives the final payment ($20). Note that when it is not possible to compute an IRR, the project either has a positive NPV or a negative NPV for all possible discount rates. In this example, the NPV is always positive.
The gold-mining operation has unconventional cash flows. Because there are two cash flow sign reversals, we end up with two IRRs––16.05 percent and 55.65 percent––neither of them correct. In situations like this, the IRR provides a solution that is meaningless, and therefore, the results should not be used for capital budgeting decisions.
Mutually Exclusive Projects
The other situation in which the IRR can lead to incorrect decisions is when capital projects are mutually exclusive—that is, when accepting one project means rejecting the other. For example, suppose you own a small store in the business district of Seattle that is currently vacant. You are looking at two business opportunities: opening an upscale coffee house or opening a copy center. Since you cannot pursue both projects at the same location they are mutually exclusive.
When you have mutually exclusive projects, how do you select the best alternative? If you are using the NPV method, the answer is easy. You select the project that has the highest NPV because it will increase the value of the firm by the largest amount. If you are using the IRR method, it would seem logical to select the project with the highest IRR. In this case, though, the logic is wrong! You cannot tell which mutually exclusive project to select just by looking at the projects’ IRRs.
Let’s consider another example to illustrate the problem. The cash flows for two projects, A and B, are as follows:
The following table shows the NPVs for the two projects at several discount rates:
To read an article that warns finance managers using the IRR about the method’s pitfalls, visit
www.cfo.com/printable/article.cfm/3304945?f=options
.
The relative IRR and NPV rankings change in this way because the cash inflows of project B arrive later than those of project A. Thus, higher discount rates have more of an impact on the value of project B. In other words, changes in relative IRR and NPV rankings result from differences in the timing of project cash flows.
The NPV profiles for two projects often cross over each other. When evaluating mutually exclusive projects, it is helpful to know where this crossover point is. For projects A and B in the exhibit, the crossover point is at 14.3 percent. For any cost of capital above 14.3 percent but below 20.7 percent the NPV for project A is higher than that for project B and is positive; thus, project A should be selected. For any cost of capital below the crossover point, the NPV of project B is higher, and project B should be selected.
Now take a look at
Exhibit 10.11
, which shows the NPV profiles for projects A and B. As you can see, there is a point, called the crossover point, at which the NPV profiles for projects A and B intersect. The crossover point here is at a discount rate of 14.3 percent. For any cost of capital above 14.3 percent, the NPV for project A is higher than that for project B; thus, project A should be selected if its NPV is positive. For any cost of capital below the crossover point, project B should be selected.
Another conflict involving mutually exclusive projects concerns comparisons of projects that have significantly different costs. The IRR does not adjust for these differences in the scale of projects. What the IRR gives us is a rate of return on each dollar invested. In contrast, the NPV method computes the total dollar value created by the project. The difference in results can be significant, as can be seen in Decision-Making Example 10.2 on the next page.
crossover point
the discount rate at which the NPV profiles of two projects cross and, thus, at which the NPVs of the projects are equal
Modified Internal Rate of Return (MIRR)
A major weakness of the IRR method compared with the NPV method concerns the rate at which the cash flows generated by a capital project are reinvested. The NPV method assumes that cash flows from a project are reinvested at the cost of capital, whereas the IRR technique assumes they are reinvested at the IRR. Determining which is the better assumption depends on which rate better represents the rate that firms can actually earn when they reinvest a project’s cash flows over time. It is generally believed that the cost of capital, which is often lower than the IRR, better reflects the rate that firms are likely to earn. Using the IRR may thus involve overly optimistic assumptions regarding reinvestment rates.
modified internal rate of return (MIRR)
an internal rate of return (IRR) measure which assumes that cash inflows are reinvested at the opportunity cost of capital until the end of the project
To eliminate the reinvestment rate assumption of the IRR, some practitioners prefer to calculate the modified internal rate of return (MIRR). In this approach, each operating cash flow is converted to a future value at the end of the project’s life, compounded at the cost of capital. These values are then summed up to get the project’s terminal value (TV). The MIRR is the interest rate that equates the project’s cost (PVCost), or cash outflows, with the future value of the project’s cash inflows at the end of the project (PVTV).
1
Because each future value is computed using the cost of capital as the interest rate, the reinvestment rate problem is eliminated.
EXAMPLE 10.2 DECISION MAKING
The Lemonade Stand versus the Convenience Store
DECISION: Your boss, who favors the IRR method, looks at the analysis and declares his son a genius. The IRR decision rule suggests that the lemonade stand, with its 76.2 percent rate of return, is the project to choose! You point out that the goal of capital budgeting is to select projects, or combinations of projects, that maximize the value of the firm, his business. The convenience store adds by far the greater value: $179,190 compared with only $1,694 for the lemonade stand. Although the lemonade stand has a high rate of return, its small size precludes it from being competitive against the larger project.
We can set up the equation for the MIRR in the same way we set up Equation 10.4 for the IRR:
Second, we need to compute the terminal value (TV). To do this, we find the future value of each operating cash flow at the end of the project’s life, compounded at the cost of capital. We then sum up these future values to get the project’s TV. Mathematically, the TV can be expressed as:
To illustrate, let’s return to the Ford Motor Company example shown in
Exhibit 10.8
. Recall that the cost of the project is $560, incurred at t 0, and that the discount rate is 12 percent. To determine the MIRR for the project, we start by calculating the terminal value of the cash flows, as shown on the following time line:
IRR versus NPV: A Final Comment
The IRR method, as noted, is an important alternative to the NPV method. As we have seen, it accounts for the time value of money, which is not true of methods such as the payback period and accounting rate of return. Furthermore, the IRR technique has great intuitive appeal. Many business practitioners are in the habit of thinking in terms of rates of return, whether the rates relate to their common-stock portfolios or their firms’ capital expenditures. To these practitioners, the IRR method just seems to make sense. Indeed, we suspect that the IRR’s popularity with business managers results more from its simple intuitive appeal than from its merit.
On the downside, we have seen that the IRR method has several flaws. For example, IRR can’t be used effectively for projects with unconventional cash flows, and IRR can lead to incorrect investment decisions when it is used to choose between mutually exclusive projects. MIRR addresses some of the shortcomings of IRR; namely, it does not assume that project proceeds are reinvested at the IRR, and it eliminates issues associated with unconventional project cash flows. Nonetheless, we believe that NPV should be the primary method used to make capital budgeting decisions. Investment decisions made using NPV are always consistent with the goal of maximizing the value of the firm, even when discriminating between mutually exclusive projects. Finally, it is important to note that the IRR, MIRR, and NPV methods all require a set of projected cash flows over the life of the project and a discount rate. Thus, using IRR or MIRR, rather than NPV, does not require less effort from financial managers.
1. What is the IRR method?
2. In capital budgeting, what is a conventional cash flow pattern?
3. Why should the NPV method be the primary decision tool used in making capital investment decisions?
10.6 CAPITAL BUDGETING IN PRACTICE
·
Capital expenditures are big-ticket items in the U.S. economy. According to the Department of Commerce, U.S. businesses invested $1.38 trillion in capital goods in 2008. Within the S&P 500, the sector with the largest total capital expenditures was the energy sector, in which expenditures totaled $170 billion in 2008. In contrast, expenditures in the healthcare sector were the smallest at $23 billion. Capital investments also represent large expenditures for individual firms, though the amount spent can vary widely from year to year. For example, AT&T expanded its wireless network capabilities rapidly in 2008 and its $19.6 billion in capital expenditures that year exceeded all other firms in the S&P 500 except Chevron, which spent $19.7 billion. More typical are the capital expenditure totals for Ford Motor Company, Cisco, and Kellogg Company, which are shown in the following table. Given the large dollar amounts and the strategic importance of capital expenditures, it is no surprise that corporate managers spend considerable time and energy analyzing capital projects.
The exhibit summarizes evidence from two studies that examined the use of capital budgeting techniques by businesses. As you can see, over time more firms have come to use the NPV and IRR techniques. Surprisingly, though, even in 1999, 20.3 percent still computed the accounting rate of return.
Practitioners’ Methods of Choice
Because of the importance of capital budgeting, over the years a number of surveys have asked financial managers what techniques they actually use in making capital investment decisions.
Exhibit 10.12
, which summarizes the results from two such studies, reveals significant changes over time. As shown, in 1981 only 16.5 percent of the financial managers surveyed frequently used the NPV approach, and the payback period and accounting rate of return approaches were used even less frequently. Most firms, 65.3 percent, used the IRR method. However, practices changed in the 1980s and 1990s. By 1999, 74.9 percent of the firms surveyed were frequently using the NPV technique, 75.7 percent were using the IRR, and 56.7 percent were using the payback period method. As you can see, the most recent findings reflect a much better alignment between what practitioners do and the theory discussed in this chapter. As you can also see, many financial managers use multiple capital budgeting tools.
An article that surveys the use of capital budgeting techniques by the CFOs of Fortune 500 companies can be found at
http://faculty.fuqua.duke.edu
/~jgraham/website/SurveyJACF .
Postaudit and Ongoing Reviews
Management should systematically review the status of all ongoing capital projects and perform postaudit reviews on all completed capital projects. In a postaudit review, management compares the actual performance of a project with what was projected in the capital budgeting proposal. For example, suppose a new microchip was expected to earn a 20 percent IRR, but the product’s actual IRR turned out to be 9 percent. A postaudit examination would determine why the project failed to achieve its expected financial goals. Project reviews keep all people involved in the capital budgeting process honest because they know that the project and their performance will be reviewed and that they will be held accountable for the results.
postaudit review
an audit to compare actual project results with the results projected in the capital budgeting proposal
Managers should also conduct ongoing reviews of capital projects in progress. Such a review should challenge the business plan, including the cash flow projections and the operating cost assumptions. For example, Intel has periodically reviewed the viability of its wafer fabrication plant in China and has made adjustments to reflect changing business conditions. Business plans are management’s best estimates of future events at the time they are prepared, but as new information becomes available, the decision to undertake a capital project and the nature of that project must be reassessed.
Management must also evaluate people responsible for implementing a capital project. They should monitor whether the project’s revenues and expenses are meeting projections. If the project is not proceeding according to plan, the difficult task for management is to determine whether the problem is a flawed plan or poor execution by the implementation team. Good plans can fail if they are poorly executed at the operating level.
> BEFORE YOU GO ON
1. What changes have taken place in the capital budgeting techniques used by U.S. companies?
SUMMARY OF Learning Objectives
Capital budgeting is the process by which management decides which productive assets the firm should invest in. Because capital expenditures involve large amounts of money, are critical to achieving the firm’s strategic plan, define the firm’s line of business over the long term, and determine the firm’s profitability for years to come, they are considered the most important investment decisions made by management.
Explain the benefits of using the net present value (NPV) method to analyze capital expenditure decisions and calculate the NPV for a capital project.
The net present value (NPV) method leads to better investment decisions than other techniques because it: (1) uses the discounted cash flow valuation approach, which accounts for the time value of money, and (2) provides a direct measure of how much a capital project is expected to increase the dollar value of the firm. Thus, NPV is consistent with the top management goal of maximizing stockholder value. NPV calculations are described in Section 10.2 and Learning by Doing Application 10.1.
Describe the strengths and weaknesses of the payback period as a capital expenditure decision-making tool and compute the payback period for a capital project.
The payback period is the length of time it will take for the cash flows from a project to recover the cost of the project. The payback period is widely used, mainly because it is simple to apply and easy to understand. It also provides a simple measure of liquidity risk because it tells management how quickly the firm will get its money back. The payback period has a number of shortcomings, however. For one thing, the payback period, as most commonly computed, ignores the time value of money. We can overcome this objection by using discounted cash flows to calculate the payback period. Regardless of how the payback period is calculated, however, it fails to take account of cash flows recovered after the payback period. Thus, the payback period is biased in favor of short-lived projects. Also, the hurdle rate used to identify what payback period is acceptable is arbitrarily determined. Payback period calculations are described in Section 10.3 and Learning by Doing Application 10.2.
Explain why the accounting rate of return (ARR) is not recommended as a capital expenditure decision-making tool.
The ARR is based on accounting numbers, such as book value and net income, rather than cash flow data. As such, it is not a true rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income statement and balance sheet. Furthermore, as with the payback method, there is no economic rationale for establishing the hurdle rate. Finally, the ARR does not account for the size of the projects when a choice between two projects of different sizes must be made.
Compute the internal rate of return (IRR) for a capital project and discuss the conditions under which the IRR technique and the NPV technique produce different results.
The IRR is the expected rate of return for an investment project; it is calculated as the discount rate that equates the present value of a project’s expected cash inflows to the present value of the project’s outflows—in other words, as the discount rate at which the NPV is equal to zero. Calculations are shown in Section 10.5 and Learning by Doing Application 10.3. If a project’s IRR is greater than the required rate of return, the cost of capital, the project is accepted. The IRR rule often gives the same investment decision for a project as the NPV rule. However, the IRR method does have operational pitfalls that can lead to incorrect decisions. Specifically, when a project’s cash flows are unconventional, the IRR calculation may yield no solution or more than one IRR. In addition, the IRR technique cannot be used to rank projects that are mutually exclusive because the project with the highest IRR may not be the project that would add the greatest value to the firm if accepted—that is, the project with the highest NPV.
Explain the benefits of postaudit and ongoing reviews of capital projects.
A postaudit review enables managers to determine whether a project’s goals were met and to quantify the actual benefits or costs of the project. By conducting these reviews, managers can avoid making similar mistakes in future projects, learn to better recognize opportunities, and keep people involved in the budgeting process honest. An ongoing review enables managers to assess the impact of changing information and market conditions on the value of a project that is already underway. Unexpected changes in conditions can affect the viability of continuing such a project as originally conceived.
SUMMARY OF Key Equations
· 10.1 Premium Manufacturing Company is evaluating two forklift systems to use in its plant that produces the towers for a windmill power farm. The costs and the cash flows from these systems are shown below. If the company uses a 12 percent discount rate for all projects, determine which forklift system should be purchased using the net present value (NPV) approach.
· 10.3 Perryman Crafts Corp. is evaluating two independent capital projects that will each cost the company $250,000. The two projects will provide the following cash flows:
· 10.4 Terrell Corp. is looking into purchasing a machine for its business that will cost $117,250 and will be depreciated on a straight-line basis over a five-year period. The sales and expenses (excluding depreciation) for the next five years are shown in the following table. The company’s tax rate is 34 percent.
· 10.5 Refer to Problem 10.1. Compute the IRR for each of the two systems. Is the investment decision different from the one indicated by NPV?
Solutions to Self-Study Problems
· 10.1 NPVs for two forklift systems:
NPV for Otis Forklifts:
· 10.2 Payback period for Rutledge project:
· 10.4 Evaluation of Terrell Corp. project:
· 10.5 IRRs for two forklift systems:
Otis Forklifts:
First compute the IRR by the trial-and-error approach.
NPV (Otis) = $337,075 > 0
We should use a higher discount rate to get NPV = 0.
At k = 15 percent:
Craigmore Forklifts:
First compute the IRR using the trial-and-error approach.
NPV (Craigmore) = $90,606 > 0
We should use a higher discount rate to get NPV = 0.
At k = 15 percent:
Critical Thinking Questions
· 10.1 Explain why the cost of capital is referred to as the “hurdle” rate in capital budgeting.
· 10.2 a. A company is building a new plant on the outskirts of Smallesville. The town has offered to donate the land, and as part of the agreement, the company will have to build an access road from the main highway to the plant. How will the project of building of the road be classified in capital budgeting analysis?
b. Sykes, Inc., is considering two projects: a plant expansion and a new computer system for the firm’s production department. Classify these projects as independent, mutually exclusive, or contingent projects and explain your reasoning.
c. Your firm is currently considering the upgrading of the operating systems of all the firm’s computers. One alternative is to choose the Linux operating system that a local computer services firm has offered to install and maintain. Microsoft has also put in a bid to install the new Windows Vista operating system for businesses. What types of projects are these?
· 10.3 In the context of capital budgeting, what is “capital rationing”?
· 10.4 Provide two conditions under which a set of projects might be characterized as mutually exclusive.
· 10.5 a. A firm invests in a project that is expected to earn a return of 12 percent. If the appropriate cost of capital is also 12 percent, did the firm make the right decision? Explain.
b. What is the impact on the firm if it accepts a project with a negative NPV?
· 10.6 Identify the weaknesses of the payback period method.
· 10.7 What are the strengths and weaknesses of the accounting rate of return approach?
· 10.8 Under what circumstances might the IRR and NPV approaches have conflicting results?
· 10.9 The modified IRR (MIRR) alleviates two concerns with using the IRR method for evaluating capital investments. What are they?
· 10.10 Elkridge Construction Company has an overall (composite) cost of capital of 12 percent. This cost of capital reflects the cost of capital for an Elkridge Construction project with average risk. However, the firm takes on projects of various risk levels. The company experience suggests that low-risk projects have a cost of capital of 10 percent and high-risk projects have a cost of capital of 15 percent. Which of the following projects should the company select to maximize shareholder wealth?
· 10.2 Net present value: Kingston, Inc. management is considering purchasing a new machine at a cost of $4,133,250. They expect this equipment to produce cash flows of $814,322, $863,275, $937,250, $1,017,112, $1,212,960, and $1,225,000 over the next six years. If the appropriate discount rate is 15 percent, what is the NPV of this investment?
· 10.3 Net present value: Crescent Industries management is planning to replace some existing machinery in its plant. The cost of the new equipment and the resulting cash flows are shown in the accompanying table. If the firm uses an 18 percent discount rate for projects like this, should management go ahead with the project?
· 10.5 Net present value: Blanda Incorporated management is considering investing in two alternative production systems. The systems are mutually exclusive, and the cost of the new equipment and the resulting cash flows are shown in the accompanying table. If the firm uses a 9 percent discount rate for their production systems, in which system should the firm invest?
· 10.7 Payback: Quebec, Inc., is purchasing machinery at a cost of $3,768,966. The company’s management expects the machinery to produce cash flows of $979,225, $1,158,886, and $1,881,497 over the next three years, respectively. What is the payback period?
· 10.8 Payback: Northern Specialties just purchased inventory-management computer software at a cost of $1,645,276. Cost savings from the investment over the next six years will produce the following cash flow stream: $212,455, $292,333, $387,479, $516,345, $645,766, and $618,325. What is the payback period on this investment?
· 10.9 Payback: Nakamichi Bancorp has made an investment in banking software at a cost of $1,875,000. Management expects productivity gains and cost savings over the next several years. If, as a result of this investment, the firm is expected to generate additional cash flows of $586,212, $713,277, $431,199, and $318,697 over the next four years, what is the investment’s payback period?
· 10.10 Average accounting rate of return (ARR): Capitol Corp. management is expecting a project to generate after-tax income of $63,435 in each of the next three years. The average book value of the project’s equipment over that period will be $212,500. If the firm’s acceptance decision on any project is based on an ARR of 37.5 percent, should this project be accepted?
· 10.11 Internal rate of return: Refer to Problem 10.4. What is the IRR that Franklin Mints management can expect on this project?
· 10.12 Internal rate of return: Hathaway, Inc., a resort company, is refurbishing one of its hotels at a cost of $7.8 million. Management expects that this will lead to additional cash flows of $1.8 million for the next six years. What is the IRR of this project? If the appropriate cost of capital is 12 percent, should Hathway go ahead with this project?
· 10.14 Net present value: Briarcrest Condiments is a spice-making firm. Recently, it developed a new process for producing spices. The process requires new machinery that would cost $1,968,450, have a life of five years, and would produce the cash flows shown in the following table. What is the NPV if the discount rate is 15.9 percent?
15. What are the NPVs of the two projects?
15. Should both projects be accepted? or either? or neither? Explain your reasoning.
1. 10.18 Discounted payback: Timeline Manufacturing Co. is evaluating two projects. The company uses payback criteria of three years or less. Project A has a cost of $912,855, and project B’s cost is $1,175,000. Cash flows from both projects are given in the following table. What are their discounted payback periods and which will be accepted with a discount rate of 8 percent?
1. 10.22 Modified internal rate of return (MIRR): Sycamore Home Furnishings is considering acquiring a new machine that can create customized window treatments. The equipment will cost $263,400 and will generate cash flows of $85,000 over each of the next six years. If the cost of capital is 12 percent, what is the MIRR on this project?
1. 10.23 Internal rate of return: Great Flights, Inc., an aviation firm, is considering purchasing three aircraft for a total cost of $161 million. The company would lease the aircraft to an airline. Cash flows from the proposed leases are shown in the following table. What is the IRR of this project?
1. 10.25 Internal rate of return: Ancala Corporation is considering investments in two new golf apparel lines for next season: golf hats and belts. Due to a funding constraint, these lines are mutually exclusive. A summary of each project’s estimated cash flows over its three-year life, as well as the IRR and NPV of each, are outlined below. The CFO of the firm has decided to manufacture the belts; however, the CEO is questioning this decision given that the IRR is higher for manufacturing hats. Explain to the CEO why the IRRs and NPVs of the belt and hat projects disagree? Is the CFO’s decision correct?
26. An initial investment of $25,000 followed by a single cash flow of $37,450 in year 6.
26. An initial investment of $1 million followed by a single cash flow of $1,650,000 in year 4.
26. An initial investment of $2 million followed by cash flows of $1,650,000 and $1,250,000 in years 2 and 4, respectively.
1. 10.27 Internal rate of return: Compute the IRR for the following project cash flows:
27. An initial outlay of $3,125,000 followed by annual cash flows of $565,325 for the next eight years.
27. An initial investment of $33,750 followed by annual cash flows of $9,430 for the next five years.
27. An initial outlay of $10,000 followed by annual cash flows of $2,500 for the next seven years.
29. What are the IRRs for the projects?
29. Does the IRR criterion indicate a different decision than the NPV criterion?
29. Explain how you would expect the management of Draconian Measures to decide.
1. 10.30 Dravid, Inc., is currently evaluating three projects that are independent. The cost of funds can be either 13.6 percent or 14.8 percent depending on their financing plan. All three projects cost the same at $500,000. Expected cash flow streams are shown in the following table. Which projects would be accepted at a discount rate of 14.8 percent? What if the discount rate was 13.6 percent?
35. What is the payback period?
35. What is the NPV for this project?
35. What is the IRR?
1. 10.36 Quasar Tech Co. is investing $6 million in new machinery that will produce the next-generation routers. Sales to its customers will amount to $1,750,000 for the next three years and then increase to $2.4 million for three more years. The project is expected to last six years and operating costs, excluding depreciation, will be $898,620 annually. The machinery will be depreciated to a salvage value of $0 over 6 years using the straight-line method. The company’s tax rate is 30 percent, and the cost of capital is 16 percent.
36. What is the payback period?
36. What is the average accounting return (ARR)?
36. Calculate the project NPV.
36. What is the IRR for the project?
1. 10.37 Skywards, Inc., an airline caterer, is purchasing refrigerated trucks at a total cost of $3.25 million. After-tax net income from this investment is expected to be $750,000 for the next five years. Annual depreciation expense will be $650,000. The cost of capital is 17 percent.
37. What is the discounted payback period?
37. Compute the ARR.
37. What is the NPV on this investment?
37. Calculate the IRR.
1. 10.38 Trident Corp. is evaluating two independent projects. The costs and expected cash flows are given in the following table. The cost of capital is 10 percent.
38. Calculate the projects’ IRR.
38. Which project should be chosen based on NPV? Based on IRR? Is there a conflict?
38. If you are the decision maker for the firm, which project or projects will be accepted? Explain your reasoning.
1. 10.39 Tyler, Inc., is considering switching to a new production technology. The cost of the required equipment will be $4 million. The discount rate is 12 percent. The cash flows that the firm expects the new technology to generate are as follows.
39. What is the NPV for the project? Should the firm go ahead with the project?
39. What is the IRR, and what would be the decision based on the IRR?
41. 0.16 year longer than the payback period.
41. 0.80 year longer than the payback period.
41. 1.01 years longer than the payback period.
41. 1.85 years longer than the payback period.
1. 10.42 An investment of $100 generates after-tax cash flows of $40 in Year 1, $80 in Year 2, and $120 in Year 3. The required rate of return is 20 percent. The net present value is closest to
a. $42.22 b. $58.33
c. $68.52 d. $98.95
1. 10.43 An investment of $150,000 is expected to generate an after-tax cash flow of $100,000 in one year and another $120,000 in two years. The cost of capital is 10 percent. What is the internal rate of return?
a. 28.19 percent b. 28.39 percent
c. 28.59 percent d. 28.79 percent
1. 10.44 An investment requires an outlay of $100 and produces after-tax cash flows of $40 annually for four years. A project enhancement increases the required outlay by $15 and the annual after-tax cash flows by $5. How will the enhancement affect the project’s NPV profile? The vertical intercept of the NPV profile of the project shifts:
44. Up and the horizontal intercept shifts left.
44. Up and the horizontal intercept shifts right.
44. Down and the horizontal intercept shifts left.
44. Down and the horizontal intercept shifts right.
Sample Test Problems
· 10.1 Net present value: Techno Corp. is considering developing new computer software. The cost of development will be $675,000 and management expects the net cash flow from sale of the software to be $195,000 for each of the next six years. If the discount rate is 14 percent, what is the net present value of this project?
· 10.2 Payback method: Parker Office Supplies management is considering replacing the company’s outdated inventory-management software. The cost of the new software will be $168,000. Cost savings are expected to be $43,500 for each of the first three years and then drop to $36,875 for the following two years. What is the payback period for this project?
· 10.3 Accounting rate of return: Fresno, Inc. is expecting a project to generate after-tax income of $156,435 in each of the next three years. The average book value of its equipment over that period will be $322,500. If the firm’s acceptance decision on any project is based on an ARR of 40 percent, should this project be accepted?
· 10.4 Internal rate of return: Refer to Sample Test Problem 10.1. What is the IRR on this project?
· 10.5 Net present value: Raycom, Inc. needs a new overhead crane and two alternatives are available. Crane T costs $1.35 million and will produce cost savings of $765,000 in each of the next three years. Crane R will cost $1.675 million and will yield annual cost savings of $815,000 for the next three years. The required rate of return is 15 percent. Which of the two options should Raycom choose based on NPV criteria, and why?
As we pointed out in
Chapter 5
, financial decision-making problems can be solved either by discounting cash flows to the beginning of the project or by using compounding to find the future value of cash flows at the end of a project’s life.
13
The Cost of Capital
Learning Objectives
Explain what the weighted average cost of capital for a firm is and why it is often used as a discount rate to evaluate projects.
Calculate the cost of debt for a firm.
Calculate the cost of common stock and the cost of preferred stock for a firm.
Calculate the weighted average cost of capital for a firm, explain the limitations of using a firm’s weighted average cost of capital as the discount rate when evaluating a project, and discuss the alternatives to the firm’s weighted average cost of capital that are available.
The Walt Disney Company announced in May 2010 that it would build a new hotel at Walt Disney World, its first new hotel at that theme park in seven years. The hotel, which is to be opened in several phases beginning in 2012, has been named “Disney’s Art of Animation Resort.” It will be built on a 65-acre parcel of land across the lake from Disney’s Pop Century Resort and will have 1,120 suites and 864 traditional hotel rooms. Disney executives anticipate that the rooms in the Art of Animation Resort will be priced comparably to those at the Pop Century Resort, which begin at less than $100 per night.
As you can imagine, the cost of financing a project like this is substantial. Disney is a highly sophisticated and successful hotel and theme park developer and operator. Before the company announced the construction of the Art of Animation Resort, you can be sure that the managers at Disney carefully considered the financial aspects of the project. They evaluated the required investment, what revenues the new hotel was likely to generate, and how much it would cost to operate and maintain. They also estimated what it would cost to finance the project—how much they would pay for the debt and the returns equity investors would require for an investment with this level of risk. This “cost of capital” would be incorporated into their NPV analysis through the discounting process.
Doing a good job of estimating the cost of capital is especially important for a capitalintensive project such as a hotel. The cost of financing a hotel like the one that Disney is building can easily total $50 or more per room rental. In other words, if an average room rents for $100, the cost of financing the project can consume 50 percent or more of the revenue the hotel receives from renting a room!
From this example, you can see how important it is to get the cost of capital right. If Disney managers had estimated the cost of capital to be 7 percent when it was really 9 percent, they might have ended up investing in a project with a large negative NPV. How did they approach this important task? In this chapter we discuss how managers estimate the cost of capital they use to evaluate projects.
CHAPTER PREVIEW
discussed the general concept of risk and described what financial analysts mean when they talk about the risk associated with a project’s cash flows. It also explained how this risk is related to expected returns. With this background, we are ready to discuss the methods that financial managers use to estimate discount rates, the reasons they use these methods, and the shortcomings of each method.
We start this chapter by introducing the weighted average cost of capital and explaining how this concept is related to the discount rates that many financial managers use to evaluate projects. Then we describe various methods that are used to estimate the three broad types of financing that firms use to acquire assets—debt, common stock, and preferred stock—as well as the overall weighted average cost of capital for the firm.
We next discuss the circumstances under which it is appropriate to use the weighted average cost of capital for a firm as the discount rate for a project and outline the types of problems that can arise when the weighted average cost of capital is used inappropriately. Finally, we examine alternatives to using the weighted average cost of capital as a discount rate.
13.1 THE FIRM’s OVERALL COST OF CAPITAL
·
Our discussions of investment analysis up to this point have focused on evaluating individual projects. We have assumed that the rate used to discount the cash flows for a project reflects the risks associated with the incremental after-tax free cash flows from that project. In
Chapter 7
, we saw that unsystematic risk can be eliminated by holding a diversified portfolio. Therefore, systematic risk is the only risk that investors require compensation for bearing. With this insight, we concluded that we could use Equation 7.10, to estimate the expected rate of return for a particular investment:
i
) is the expected return on project i, Rrf is the risk-free rate of return, b
i
is the beta for project i, and E (Rm) is the expected return on the market. Recall that the difference between the expected return on the market and the risk-free rate [E (Rm) − Rrf] is known as the market risk premium.
Although these ideas help us better understand the discount rate on a conceptual level, they can be difficult to implement in practice. Firms do not issue publicly traded shares for individual projects. This means that analysts do not have the stock returns necessary to use a regression analysis like that illustrated in
Exhibit 7.10
to estimate the beta (β) for an individual project. As a result, they have no way to directly estimate the discount rate that reflects the systematic risk of the incremental cash flows from a particular project.
In many firms, senior financial managers deal with this problem by estimating the cost of capital for the firm as a whole and then requiring analysts within the firm to use this cost of capital to discount the cash flows for all projects.
1
A problem with this approach is that it ignores the fact that a firm is really a collection of projects with different levels of risk. A firm’s overall cost of capital is actually a weighted average of the costs of capital for these projects, where the weights reflect the relative values of the projects.
To see why a firm is a collection of projects, consider The Boeing Company. Boeing manufactures a number of different models of civilian and military aircraft. If you have ever flown on a commercial airline, chances are that you have been on a Boeing 737, 747, 757, 767, or 777 aircraft. Boeing manufactures several versions of each of these aircraft models to meet the needs of its customers. These versions have different ranges, seat configurations, numbers of seats, and so on. Some are designed exclusively to haul freight for companies such as UPS and FedEx. Every version of every model of aircraft at Boeing was, at some point in time, a new project. The assets owned by Boeing today and its expected cash flows are just the sum of the assets and cash flows from all of these individual projects plus the other projects at the firm, such as those involving military aircraft.
2
This means that the overall systematic risk associated with Boeing’s cash flows and the company’s cost of capital are weighted averages of the systematic risks and the costs of capital for its individual projects.
If the risk of an individual project differs from the average risk of the firm, the firm’s overall cost of capital is not the ideal discount rate to use when evaluating that project. Nevertheless, since this is the discount rate that is commonly used, we begin by discussing how a firm’s overall cost of capital is estimated. We then discuss alternatives to using the firm’s cost of capital as the discount rate in evaluating a project.
The Finance Balance Sheet
To understand how financial analysts estimate their firms’ costs of capital, you must be familiar with a concept that we call the finance balance sheet. The finance balance sheet is like the accounting balance sheet from
Chapter 3
. The main difference is that it is based on market values rather than book values. Recall that the total book value of the assets reported on an accounting balance sheet does not necessarily reflect the total market value of those assets. This is because the book value is largely based on historical costs, while the total market value of the assets equals the present value of the total cash flows that those assets are expected to generate in the future. The market value can be greater than or less than the book value but is rarely the same.
finance balance sheet
a balance sheet that is based on market values of expected cash flows
While the left-hand side of the accounting balance sheet reports the book values of a firm’s assets, the right-hand side reports how those assets were financed. Firms finance the purchase of their assets using debt and equity.
3
Since the cost of the assets must equal the total value of the debt and equity that was used to purchase them, the book value of the assets must equal the book value of the liabilities plus the book value of the equity on the accounting balance sheet. In
Chapter 3
we called this equality the balance sheet identity.
Just as the total book value of the assets at a firm does not generally equal the total market value of those assets, the book value of total liabilities plus stockholders’ equity does not usually equal the market value of these claims. In fact, the total market value of the debt and equity claims differ from their book values by exactly the same amount that the market values of a firm’s assets differ from their book values. This is because the total market value of the debt and the equity at a firm equals the present value of the cash flows that the debt holders and the stockholders have the right to receive. These cash flows are the cash flows that the assets in the firm are expected to generate. In other words, the people who have lent money to a firm and the people who have purchased the firm’s stock have the right to receive all of the cash flows that the firm is expected to generate in the future. The value of the claims they hold must equal the value of the cash flows that they have a right to receive.
The fact that the market value of the assets must equal the value of the cash flows that these assets are expected to generate, combined with the fact that the value of the expected cash flows also equals the total market value of the firm’s total liabilities and equity, means that we can write the market value (MV) of assets as follows:
The market value of a firm’s assets, which equals the present value of the cash flows those assets are expected to generate in the future, must equal the market value of the claims on those cash flows—the firm’s liabilities and equity.
Equation 13.1 is just like the accounting balance sheet identity. The only difference is that Equation 13.1 is based on market values. This relation is illustrated in
Exhibit 13.1
.
To see why the market value of the assets must equal the total market value of the liabilities and equity, consider a firm whose only business is to own and manage an apartment building that was purchased 20 years ago for $1,000,000. Suppose that there is currently a mortgage on the building that is worth $300,000, the firm has no other liabilities, and the current market value of the building, based on the expected cash flows from future rents, is $4,000,000. What is the market value of all of the equity (stock) in this firm?
The fact that you paid $1,000,000 20 years ago is not relevant to this question. What matters in finance is the value of the expected cash flows from future rents, the $4,000,000. This is the market value of the firm’s assets—the left-hand side of the balance sheet in
Exhibit 13.1
. Since we know that the firm owes $300,000, we can substitute into Equation 13.1 and solve for the market value of the equity:
If the concept of a balance sheet based on market values seems familiar to you, it is because the idea of preparing an actual balance sheet based on market values was discussed in
Chapter 3
. In that chapter we pointed out that such a balance sheet would be more useful to financial decision makers than the ordinary accounting balance sheet. Financial managers are much more concerned about the future than the past when they make decisions. You might revisit the discussion of sunk costs in
Chapter 11
to remind yourself of why this is true.
BUILDING INTUITION THE MARKET VALUE OF A FIRM’s ASSETS EQUALS THE MARKET VALUE OF THE CLAIMS ON THOSE ASSETS
The market value of the debt and equity claims against the cash flows of a firm must equal the present value of the cash flows that the firm’s assets are expected to generate. This is because, between them, the debt holders and the stockholders have the legal right to receive all of those cash flows.
How Firms Estimate Their Cost of Capital
Now that we have discussed the basic idea of the finance balance sheet, consider the challenge that financial analysts face when they want to estimate the cost of capital for a firm. If analysts at a firm could estimate the betas for each of the firm’s individual projects, they could estimate the beta for the entire firm as a weighted average of the betas for the individual projects. They could do this because, as we discussed earlier, the firm is simply a collection (portfolio) of projects. This calculation would just be an application of Equation 7.11:
i
is the beta for project i and xi
is the fraction of the total firm value represented by project i.
The analysts could then use the beta for the firm in Equation 7.10:
Instead, analysts must use their knowledge of the finance balance sheet, along with the concept of market efficiency, which we discussed in
Chapter 2
, to estimate the cost of capital for the firm. Rather than using Equations 7.11 and 7.10 to perform the calculations for the individual projects represented on the left-hand side of the finance balance sheet, analysts perform a similar set of calculations for the different types of financing (debt and equity) on the right-hand side of the finance balance sheet. They can do this because, as we said earlier, the people who finance the firm have the right to receive all of the cash flows on the left-hand side. This means that the systematic risk associated with the total assets on the left-hand side is the same as the systematic risk associated with the total financing on the right-hand side. In other words, the weighted average of the betas for the different claims on the assets must equal a weighted average of the betas for the individual assets (projects).
Analysts do not need to estimate betas for each type of financing that the firm has. As long as they can estimate the cost of each type of financing—either directly, by observing that cost in the capital markets, or by using Equation 7.10—they can compute the cost of capital for the firm using the following equation:
Firm is the cost of capital for the firm, ki
is the cost of financing type i, and xi
is the fraction of the total market value of the financing (or of the assets) of the firm represented by financing type i. This formula simply says that the overall cost of capital for the firm is a weighted average of the cost of each different type of financing used by the firm.
4
Note that since we are specifically talking about the cost of capital, we use the symbol ki
to represent this cost, rather than the more general notation E (R
i
) that we used in
Chapter 7
.
The similarity between Equation 13.2 and Equation 7.11 is not an accident. Both are applications of the basic idea that the systematic risk of a portfolio of assets is a weighted average of the systematic risks of the individual assets. Because Rrf and E (Rm) in Equation 7.10 are the same for all assets, when we substitute Equation 7.10 into Equation 13.2 (remember that E (R
i
) in Equation 7.10 is the same as ki
in Equation 13.2) and cancel out Rrf and E(Rm), we get Equation 7.11. We will not prove this here, but you might do so to convince yourself that what we are saying is true.
To see how Equation 13.2 is applied, let’s return to the example of the firm whose only business is to manage an apartment building. Recall that the total value of this firm is $4,000,000 and that it has $300,000 in debt. If the firm has only one loan and one type of stock, then the fractions of the total value represented by those two types of financing are as follows:
5
Firm. From this point on, we will use the abbreviation WACC to represent the firm’s overall cost of capital.
weighted average cost of capital (WACC)
the weighted average of the costs of the different types of capital (debt and equity) that have been used to finance a firm; the cost of each type of capital is weighted by the proportion of the total capital that it represents
BUILDING INTUITION A FIRM’s COST OF CAPITAL IS A WEIGHTED AVERAGE OF ALL OF ITS FINANCING COSTS
The cost of capital for a firm is a weighted average of the costs of the different types of financing used by a firm. The weights are the proportions of the total firm value represented by the different types of financing. By weighting the costs of the individual financing types in this way, we obtain the overall average opportunity cost of each dollar invested in the firm.
APPLICATION 13.1 LEARNING BY DOING
APPROACH: You can use Equation 13.2 to calculate the WACC for this firm. Since you are planning to finance the purchase using capital from three different sources—two loans and your own equity investment—the right-hand side of Equation 13.2 will have three terms.
SOLUTION:
We begin by calculating the weights for the different types of financing:
> BEFORE YOU GO ON
1. Why does the market value of the claims on the assets of a firm equal the market value of the assets?
2. How is the WACC for a firm calculated?
3. What does the WACC for a firm tell us?
13.2 THE COST OF DEBT
·
In our discussion of how the WACC for a firm is calculated, we assumed that the costs of the different types of financing were known. This assumption allowed us to simply plug those costs into Equation 13.2 once we had calculated the weight for each type of financing. Unfortunately, life is not that simple. In the real world, analysts have to estimate each of the individual costs. In other words, the discussion in the preceding section glossed over a number of concepts and issues that you should be familiar with. This section and Section 13.3 discuss those concepts and issues and show how the costs of the different types of financing can be estimated.
Before we move on to the specifics of how to estimate the costs of different types of financing, we must stress an important point: All of these calculations depend in some part on financial markets being efficient. We suggested this in the last section when we mentioned that analysts have to rely on the concept of market efficiency to estimate the WACC. The reason is that analysts often cannot directly observe the rate of return that investors require for a particular type of financing. Instead, analysts must rely on the security prices they can observe in the financial markets to estimate the required rate.
It makes sense to rely on security prices only if you believe that the financial markets are reasonably efficient at incorporating new information into these prices. If the markets were not efficient, estimates of expected returns that were based on market security prices would be unreliable. Of course, if the returns that are plugged into Equation 13.2 are bad, the resulting estimate for WACC will also be bad. With this caveat, we can now discuss how to estimate the costs of the various types of financing.
Key Concepts for Estimating the Cost of Debt
Virtually all firms use some form of debt financing. The financial managers at firms typically arrange for revolving lines of credit to finance working capital items such as inventories or accounts receivable. These lines of credit are very much like the lines of credit that come with your credit cards. Firms also obtain private fixed-term loans, such as bank loans, or sell bonds to the public to finance ongoing operations or the purchase of long-term assets—just as you would finance your living expenses while you are in school with a student loan or a car with a car loan. For example, an electric utility firm, such as FPL Group in Florida, will sell bonds to finance a new power plant, and a rapidly growing retailer, such as Target, will use debt to finance new stores and distribution centers. As mentioned earlier, we will discuss how firms finance themselves in more detail in
Chapters 15
and
16
, but for now it is sufficient to recognize that firms use these three general types of debt financing: lines of credit, private fixed-term loans, and bonds that are sold in the public markets.
There is a cost associated with each type of debt that a firm uses. However, when we estimate the cost of capital for a firm, we are particularly interested in the cost of the firm’s long-term debt. Firms generally use long-term debt to finance their long-term assets, and it is the long-term assets that concern us when we think about the value of a firm’s assets. By long-term debt, we usually mean the debt that, when it was borrowed, was set to mature in more than one year. This typically includes fixed-term bank loans used to finance ongoing operations or long-term assets, as well as the bonds that a firm sells in the public debt markets.
Although one year is not an especially long time, debt with a maturity of more than one year is typically viewed as permanent debt. This is because firms often borrow the money to pay off this debt when it matures.
We do not normally worry about revolving lines of credit when calculating the cost of debt because these lines tend to be temporary. Banks typically require that the outstanding balances be periodically paid down to $0 (just as we are sure you pay your entire credit card balance from time to time).
When analysts estimate the cost of a firm’s long-term debt, they are estimating the cost on a particular date—the date on which they are doing the analysis. This is a very important point to keep in mind because the interest rate that the firm is paying on its outstanding debt does not necessarily reflect its current cost of debt. Interest rates change over time, and so does the cost of debt for a firm. The rate a firm was charged three years ago for a five-year loan is unlikely to be the same rate that it would be charged today for a new five-year loan. For example, suppose that FPL Group issued bonds five years ago for 7 percent. Since then, interest rates have fallen, so the same bonds could be sold at par value today for 6 percent. The cost of debt today is 6 percent, not 7 percent, and 6 percent is the cost of debt that management will use in WACC calculations. If you looked in the firm’s financial statements, you would see that the firm is paying an interest rate of 7 percent. This is what the financial managers of the firm agreed to pay five years ago, not what it would cost to sell the same bonds today. The accounting statements reflect the cost of debt that was sold at some time in the past.
BUILDING INTUITION THE CURRENT COST OF LONG-TERM DEBT IS WHAT MATTERS WHEN CALCULATING WACC
The current cost of long-term debt is the appropriate cost of debt for WACC calculations. This is because the WACC we use in capital budgeting is the opportunity cost of capital for the firm’s investors as of today. This means we must use today’s costs of debt and equity when we calculate the WACC. Historical costs do not belong in WACC calculations.
Estimating the Current Cost of a Bond or an Outstanding Loan
We have now seen that we should not use historical costs of debt in WACC calculations. Let’s discuss how we can estimate the current costs of bonds and other fixed-term loans by using market information.
The Current Cost of a Bond
You may not realize it, but we have already discussed how to estimate the current cost of debt for a publicly traded bond. This cost is estimated using the yield to maturity calculation. Recall that in
Chapter 8
we defined the yield to maturity as the discount rate that makes the present value of the coupon and principal payments equal to the price of the bond.
For example, consider a 10-year bond with a $1,000 face value that was issued five years ago. This bond has five years remaining before it matures. If the bond has an annual coupon rate of 7 percent, pays coupon interest semiannually, and is currently selling for $1,042.65, we can calculate its yield to maturity by using Equation 8.1 and solving for i or by using a financial calculator. Let’s use Equation 8.1 for this example.
To do this, as was discussed in the section on semiannual compounding in
Chapter 8
, we first convert the bond data to reflect semiannual compounding: (1) the total number of coupon payments is 10 (2 payments per year × 5 years = 10 payments), and (2) the semiannual coupon payment is $35 [($1,000 × 7 percent per year)/2 periods per year = $70/2 = $35]. We can now use Equation 8.1 and solve for i to find the yield to maturity:
i = k
Bond = 0.030, or 3.0%
This semiannual rate would be quoted as an annual rate of 6 percent (2 periods per year × 0.03 = 0.06, or 6 percent) in financial markets. However, as explained in
Chapter 8
, this annual rate fails to account for the effects of compounding. We must therefore use Equation 6.7 to calculate the effective annual interest rate (EAR) in order to obtain the actual current annual cost of this debt:
Notice that the above calculation takes into account the interest payments, the face value of the debt (the amount that will be repaid in five years), and the current price at which the bond is selling. It is necessary to account for all of these characteristics of the bond. The return received by someone who buys the bond today will be determined by both the interest income and the capital appreciation (or capital depreciation in this case, since the price is higher than the face value).
We must account for one other factor when we calculate the current cost of bond financing to a company—the cost of issuing the bond. In the above example, we calculated the return that someone who buys the bond can expect to receive. Since a company must pay fees to investment bankers, lawyers, and accountants, along with various other costs, to actually issue a bond, the cost to the company is higher than 6.1 percent.
6
Therefore, in order to obtain an accurate estimate of the cost of a bond to the issuing firm, analysts must incorporate issuance costs into their calculations. Issuance costs are an example of direct out-of-pocket costs, the actual out-of-pocket costs that a firm incurs when it raises capital.
The way in which issuance costs are incorporated into the calculation of the cost of a bond is straightforward. Analysts use the net proceeds that the company receives from the bond, rather than the price that is paid by the investor, on the left-hand side of Equation 8.1. Suppose the company in our example sold 5-year bonds with a 7 percent coupon today and paid issuance costs equal to 2 percent of the total value of the bonds. After paying the issuance costs, the company would receive only 98 percent of the price paid by the investors. Therefore, the company would actually receive only $1,042.65 × (1 − 0.02) = $1,021.80 for each bond it sold and the semiannual cost to the company would be:
EAR = 11.032422 − 1 = 0.066, or 6.6%
In this example the issuance costs increase the effective cost of the bonds from 6.1 percent to 6.6 percent per year.
The Current Cost of an Outstanding Loan
Conceptually, calculating the current cost of long-term bank or other private debt is not as straightforward as estimating the current cost of a public bond because financial analysts cannot observe the market price of private debt. Fortunately, analysts do not typically have to do this. Instead, they can simply call their banker and ask what rate the bank would charge if they decided to refinance the debt today. A rate quote from a banker provides a good estimate of the current cost of a private loan.
Taxes and the Cost of Debt
It is very important that you understand one additional concept concerning the cost of debt: In the United States, firms can deduct interest payments for tax purposes. In other words, every dollar a firm pays in interest reduces the firm’s taxable income by one dollar. Thus, if the firm’s marginal tax rate is 35 percent, the firm’s total tax bill will be reduced by 35 cents for every dollar of interest it pays. A dollar of interest would actually cost this firm only 65 cents because the firm would save 35 cents on its taxes.
More generally, the after-tax cost of interest payments equals the pretax cost times 1 minus the tax rate. This means that the after-tax cost of debt is:
Taxes affect the cost of debt in much the same way that the interest tax deduction on a home mortgage affects the cost of financing a house. For example, assume that you borrow $200,000 at 6 percent to buy a house on January 1 and your interest payments total $12,000 in the first year. Under the tax law, you can deduct this $12,000 from your taxable income when you calculate your taxes for the year.
7
Debt pretax at 6 percent and t at 20 percent, Equation 13.3 gives us:
Most firms have several different debt issues outstanding at any particular point in time. Just as you might have both a car loan and a school loan, a firm might have several bank loans and bond issues outstanding. To estimate the firm’s overall cost of debt when it has several debt issues outstanding we must first estimate the costs of the individual debt issues and then calculate a weighted average of these costs.
To see how this is done, let’s consider an example. Suppose that your pizza parlor business has grown dramatically in the past three years from a single restaurant to 30 restaurants. To finance this growth, two years ago you sold $25 million of five-year bonds. These bonds pay interest annually and have a coupon rate of 8 percent. They are currently selling for $1,026.24 per $1,000 bond. Just today, you also borrowed $5 million from your local bank at an interest rate of 6 percent. Assume that this is all the long-term debt that you have and that there are no issuance costs. What is the overall average after-tax cost of your debt if your business’s marginal tax rate is 35 percent?
The pretax cost of the bonds as of today is the effective annual yield on those bonds. Since the bonds were sold two years ago, they will mature three years from now. Using Equation 8.1, we find that the effective annual yield (which equals the yield to maturity in this example) for these bonds is:
Now that we know the pretax costs of the two types of debt that your business has outstanding, we can calculate the overall average cost of your debt by calculating the weighted average of their two costs. Since the weights for the two types of debt are based on their current market values we must first determine these values. Because the bonds are currently selling above their par value we know that their current market value is greater than their $25 million face value. In fact, it equals:
1$1,026.24/$1,0002 × $25,000,000 = $25,656,000
Since the bank loan was just made today, its value simply equals the amount borrowed or $5 million. The weights for the two types of debt are therefore:
APPROACH: The overall after-tax cost of debt can be calculated using the following three-step process: (1) Calculate the fraction of the total debt (weight) for each individual debt issue. (2) Using these weights, calculate the weighted average pretax cost of debt. (3) Use Equation 13.3 to calculate the after-tax average cost of debt.
SOLUTION:
(1) The weights for the three types of debt are as follows:
Using the Cost of Debt in Decision Making
DECISION: This is an easy decision. You should choose the least expensive alternative—the loan from your local bank. In this example, you can directly compare the pretax costs of the two alternatives. You do not need to calculate the after-tax costs because multiplying each pretax cost by the same number, 1 − t, will not change your decision.
> BEFORE YOU GO ON
1. Why do analysts care about the current cost of long-term debt when estimating a firm’s cost of capital?
2. How do you estimate the cost of debt for a firm with more than one type of debt?
3. How do taxes affect the cost of debt?
13.3 THE COST OF EQUITY
·
The cost of equity (stock) for a firm is a weighted average of the costs of the different types of stock that the firm has outstanding at a particular point in time. We saw in
Chapter 9
that some firms have both preferred stock and common stock outstanding. In order to calculate the cost of equity for these firms, we have to know how to calculate the cost of both common stock and preferred stock. In this section, we discuss how financial analysts can estimate the costs associated with these two different types of stock.
Common Stock
Just as information about market rates of return is used to estimate the cost of debt, market information is also used to estimate the cost of equity. There are several ways to do this. The particular approach a financial analyst chooses will depend on what information is available and how reliable the analyst believes it is. Next we discuss three alternative methods for estimating the cost of common stock. It is important to remember throughout this discussion that the “cost” we are referring to is the rate of return that investors require for investing in the stock at a particular point in time, given its systematic risk.
Method 1: Using the Capital Asset Pricing Model (CAPM)
The first method for estimating the cost of common equity is one that we discussed in
Chapter 7
. This method uses Equation 7.10:
If we recognize that E(R
i
) in Equation 7.10 is the cost of the common stock capital used by the firm (k
cs) when we are calculating the cost of equity and that [E (Rm) − Rrf] is the market risk premium, we can rewrite Equation 7.10 as follows:
Chapter 7
. In those examples you were given the current risk-free rate, the beta for the stock, and the market risk premium and were asked to calculate k
cs using the equation. Now we turn our attention to some practical considerations that you must be concerned with when choosing the appropriate risk-free rate, beta, and market risk premium for this calculation.
The Risk-Free Rate. First, let’s consider the risk-free rate. The current effective annual yield on a risk-free asset should always be used in Equation 13.4.
8
This is because the risk-free rate at a particular point in time reflects the rate of inflation that the market expects in the future. Since the expected rate of inflation changes over time, an old risk-free rate might not reflect current inflation expectations.
When analysts select a risk-free rate, they must choose between using a short-term rate, such as that for Treasury bills, or a longer-term rate, such as those for Treasury notes or bonds. Which of these choices is most appropriate? This question has been hotly debated by finance professionals for many years. We recommend that you use the risk-free rate on a long-term Treasury security when you estimate the cost of equity capital because the equity claim is a long-term claim on the firm’s cash flows. As you saw in
Chapter 9
, the stockholders have a claim on the cash flows of the firm in perpetuity. By using a long-term Treasury security, you are matching a long-term risk-free rate with a long-term claim. A long-term risk-free rate better reflects long-term inflation expectations and the cost of getting investors to part with their money for a long period of time than a short-term rate.
You can find current yields on Treasury bills, notes, and bonds at the Web site of the U.S. Federal Reserve Bank at
http://www.federalreserve.gov/releases/H15/update
.
The Beta. If the common stock of a company is publicly traded, then you can estimate the beta for that stock using a regression analysis similar to that illustrated in
Exhibit 7.10
. However, identifying the appropriate beta is much more complicated if the common stock is not publicly traded. Since most companies in the United States are privately owned and do not have publicly traded stock, this is a problem that arises quite often when someone wants to estimate the cost of common equity for a firm.
Financial analysts often overcome this problem by identifying a “comparable” company with publicly traded stock that is in the same business and that has a similar amount of debt. For example, suppose you are trying to estimate the beta for your pizza business. The company has now grown to include more than 2,000 restaurants throughout the world. The frozen-foods business, however, was never successful and had to be shut down. You know that Domino’s Pizza, Inc., one of your major competitors, has publicly traded equity and that the proportion of debt to equity for Domino’s is similar to the proportion for your firm. Since Domino’s overall business is similar to yours, in that it is only in the pizza business and competes in similar geographic areas, it would be reasonable to consider Domino’s a comparable company.
Companies with publicly traded equity usually provide a lot of information about their businesses and financial performance on their Web sites. The Domino’s Pizza Web site is a good example. Go to
http://phx.corporate-ir.net/phoenix.zhtml?c=135383&p=irol-irhome
.
The systematic risk associated with the stock of a comparable company is likely to be similar to the systematic risk for the private firm because systematic risk is determined by the nature of the firm’s business and the amount of debt that it uses. If you are able to identify a good comparable company, such as Domino’s Pizza, you can use its beta in Equation 13.4 to estimate the cost of equity capital for your firm. Even when a good comparable company cannot be identified, it is sometimes possible to use an average of the betas for the public firms in the same industry.
The Market Risk Premium. It is not possible to directly observe the market risk premium. We just do not know what rate of return investors expect for the market portfolio, E(Rm), at a particular point in time. Therefore, we cannot simply calculate the market risk premium as the difference between the expected return on the market and the risk-free rate, [E (Rm) − Rrf]. For this reason, financial analysts generally use a measure of the average risk premium investors have actually earned in the past as an indication of the risk premium they might require today.
For example, from 1926 through the end of 2009, actual returns on the U.S. stock market exceeded actual returns on long-term U.S. government bonds by an average of 6.01percent per year. If, on average, investors earned the risk premium that they expected, this figure reflects the average market risk premium over the period from 1926 to 2009. If a financial analyst believes that the market risk premium in the past is a reasonable estimate of the risk premium today, then he or she might use 6.01 percent as the market risk premium in Equation 13.4.
With this background, let’s work an example to illustrate how Equation 13.4 is used in practice to estimate the cost of common stock for a firm. Suppose that it is November 19, 2010, and we want to estimate the cost of the common stock for the oil company ConocoPhillips. Using yields reported in the Wall Street Journal on that day, we determine that 30-day Treasury bills have an effective annual yield of 0.13 percent and that 20-year Treasury bonds have an effective annual yield of 3.95 percent. From the MSN Money web site (
http://moneycentral.msn.com
), we find that the beta for ConocoPhillips stock is 1.15. We know that the market risk premium averaged 6.01 percent from 1926 to 2009. What is the expected rate of return on ConocoPhillips common stock?
Since we are estimating the expected rate of return on common stock, and common stock is a long-term asset, we use the long-term Treasury bond yield of 3.95 percent in the calculation. Notice that the Treasury bill and Treasury bond rates differed by 3.82 percent (3.95 − 0.13 = 3.82) on November 19, 2010. They often differ by this amount or more, so the choice of which rate to use can make quite a difference in the estimated cost of equity.
Once we have selected the appropriate risk-free rate, we can plug it, along with the beta and market risk premium values, into Equation 13.4 to calculate the cost of common equity for ConocoPhillips:
APPLICATION 13.3 LEARNING BY DOING
Calculating the Cost of Equity Using a Stock’s Beta
http://finance.yahoo.com
). This beta is 1.36. What do you estimate the cost of common equity in your pizza business to be?
APPROACH: Method 1 for calculating the cost of equity is to use the Capital Asset Pricing Model (CAPM). Therefore, in this example we will use Equation 13.4.
SOLUTION:
cs = Rrf + (βcs × Market risk premium2 = 0.0395 + 11.36 × 0.06012 = 0.121, or 12.1%
Method 2: Using the Constant-Growth Dividend Model
In
Chapter 9
we noted that if the dividends received by the owner of a share of common stock are expected to grow at a constant rate in perpetuity, then the value of that share today can be calculated using Equation 9.4:
We can replace the R in Equation 9.4 with k
cs since we are specifically estimating the expected rate of return for investing in common stock (also the cost of equity if the firm has no other types of stock outstanding). We can then rearrange this equation to solve for k
cs:
Consider an example. Suppose that the current price for the common stock at Sprigg Lane Company is $20, that the firm is expected to pay a dividend of $2 per share to its common stockholders next year, and that the dividend is expected to grow at a rate of 3 percent in perpetuity after next year. Equation 13.5 tells us that the required rate of return for Sprigg Lane’s stock is:
You might be asking yourself at this point where you would get P0, D1, and g in order to use Equation 13.5 for a particular stock. You can get the current price of a share of stock as well as the dividend that a firm is expected to pay next year quite easily from many different Web sites on the Internet—for example, MSN Money and Yahoo! Finance, which were both mentioned earlier. The financial information includes the dollar value of dividends paid in the past year and the dividend that the firm is expected to pay in the next year.
Estimating the long-term rate of growth in dividends is more difficult, but there are some guidelines that can help. As we discussed in
Chapter 9
, the first rule is that dividends cannot grow faster than the long-term growth rate of the economy in a perpetuity model such as Equation 9.4 or 13.5. Assuming dividends will grow faster than the economy is the same as assuming that dividends will eventually become larger than the economy itself! We know this is impossible.
What is the long-term growth rate of the economy? Well, historically it has been the rate of inflation plus about 3 percent. This means that if inflation is expected to be 3 percent in the long run, then a reasonable estimate for the long-term growth rate in the economy is 6 percent (3 percent inflation plus 3 percent real growth). This tells us that g in Equation 13.5 will not be greater than 6 percent. What exactly it will be depends on the nature of the business and the industry it is in. If it is a declining industry, then g might be negative. If the industry is expected to grow with the economy and the particular firm you are evaluating is expected to retain its market share, then a reasonable estimate for g might be 5 or 6 percent.
You can obtain recent stock prices and financial information for a large number of firms from MSN Money at
http://moneycentral.msn.com
or from Yahoo! Finance at
http://finance.yahoo.com/
.
Method 3: Using a Multistage-Growth Dividend Model
Using a multistage-growth dividend model to estimate the cost of equity for a firm is very similar to using a constant-growth dividend model. The difference is that a multistage-growth dividend model allows for faster dividend growth rates in the near term, followed by a constant long-term growth rate. If this concept sounds familiar, that is because it is the idea behind the mixed (supernormal) growth dividend model discussed in
Chapter 9
. In Equation 9.6 this model was written as:
multistage-growth dividend model
a model that allows for varying dividend growth rates in the near term, followed by a constant long-term growth rate; another term used to describe the mixed (supernormal) dividend growth model discussed in
Chapter 9
. In Equation 9.6 this model was written as:
i
is the dividend in period i, P
t
is the value of constant-growth dividend payments in period t, and R is the required rate of return.
To refresh your memory of how this model works, let’s consider a three-stage example. Suppose that a firm will pay a dividend one year from today (D1) and that this dividend will increase at a rate of g
1 the following year, g
2 the year after that, and g
3 per year thereafter. The value of a share of this stock today thus equals:
cs since we are specifically estimating the expected rate of return for common stock. We have also written all of the dividends in terms of D1 to illustrate how the different growth rates will affect the dividends in each year. Finally, we have written P
t
in terms of the constant-growth model. If we substitute D1, D2, D3, and D4 where appropriate, you can see that this is really just Equation 9.6, where we have replaced R with k
cs and written P
t
in terms of the constant-growth model:
Exhibit 13.2
illustrates how cash flows relate to the four terms in the equation.
In the three-stage dividend growth model shown here, the price of a share of stock is equal to the present value of dividends expected to be received at the end of years 1, 2, and 3, plus the present value of a growing perpetuity that begins in year 4 and whose dividends are assumed to grow at a constant rate g
3 forever.
Note that the fourth term in
Exhibit 13.2
is discounted only three years because, as we saw in
Chapters 6
and
9
, the constant-growth model gives you the present value of a growing perpetuity as of the year before the first cash flow. In this case since the first cash flow is D4, the model gives you the value of the growing perpetuity as of year 3.
A multistage-growth dividend model is much more flexible than the constant-growth dividend model because we do not have to assume that dividends grow at the same rate forever. We can use a model such as this to estimate the cost of common stock, k
cs, by plugging P0, D1, and the appropriate growth rates into the model and solving for k
cs using trial and error—just as we solved for the yield to maturity of bonds in
Chapter 8
and earlier in this chapter. The major issues we have to be concerned about when we use a growth dividend model are (1) that we have chosen the right model, meaning that we have included enough stages or growth rates, and (2) that our estimates of the growth rates are reasonable.
Let’s work an example to illustrate how this model is used to calculate the cost of common stock. Suppose that we want to estimate the cost of common stock for a firm that is expected to pay a dividend of $1.50 per share next year. This dividend is expected to increase 15 percent the following year, 10 percent the year after that, 7 percent the year after that, and 5 percent annually thereafter. If the firm’s common stock is currently selling for $24 per share, what is the rate of return that investors require for investing in this stock?
Because there are four different growth rates in this example, we have to solve a formula with five terms:
cs:
cs using trial and error. When we do this, we find that k
cs is 12.2 percent. This is the rate of return at which the present value of the cash flows equals $24. Therefore, it is the rate that investors currently require for investing in this stock.
USING EXCEL SOLVING FOR kCS USING A MULTISTAGE-GROWTH DIVIDEND MODEL
Because trial and error calculations can be somewhat tedious when you perform them by hand, you may find it helpful to use a spreadsheet program. If you would like to use a spreadsheet program to solve the preceding problem yourself, the output from the spreadsheet below shows you how to do it using trial and error.
Once you input the indicated numbers and formulas into cells B3 through B14, you can then vary the number in cell B2 until the number in cell B8 equals $24. Once you have built the model, you can also use the “goal seek” or “solver” functions in Excel to avoid having to manually solve the problem by trial and error. See the “Help” feature in Excel for information on how to use these functions.
We now have discussed three methods of estimating the cost of common equity for a firm. You might be asking yourself how you are supposed to know which method to use. The short answer is that, in practice, most people use the CAPM (Method 1) to estimate the cost of common equity if the result is going to be used in the discount rate for evaluating a project. One reason is that, assuming the theory is valid, CAPM tells managers what rate of return investors should require for equity having the same level of systematic risk that the firm’s equity has. This is the appropriate opportunity cost of equity capital for an NPV analysis if the project has the same risk as the firm and will have similar leverage. Furthermore, CAPM does not require financial analysts to make assumptions about future growth rates in dividends, as Methods 2 and 3 do.
Used properly, Methods 2 and 3 provide an estimate of the rate of return that is implied by the current price of a firm’s stock at a particular point in time. If the stock markets are efficient, then this should be the same as the number that we would estimate using CAPM. However, to the extent that the firm’s stock is mispriced—for example, because investors are not informed or have misinterpreted the future prospects for the firm—deriving the cost of equity from the price at one point in time can yield a bad estimate of the true cost of equity.
Preferred Stock
As we discussed in
Chapter 9
, preferred stock is a form of equity that has a stated value and specified dividend rate. For example, a share of preferred stock might have a stated value of $100 and a 5 percent dividend rate. The owner of such a share would be entitled to receive a dividend of $5 ($100 × 0.05 = $5) each year. Another key feature of preferred stock is that it does not have an expiration date. In other words, preferred stock continues to pay the specified dividend in perpetuity, unless the firm repurchases it or goes out of business.
These characteristics of preferred stock allow us to use the perpetuity model, Equation 6.3, to estimate the cost of preferred equity. For example, suppose that investors would pay $85 for a share of the preferred stock mentioned above. We can rewrite Equation 6.3:
ps is the cost of the preferred stock. Rearranging the formula to solve for k
ps yields:
ps for the preferred stock in our example is:
It is easy to incorporate issuance costs into the above calculation to obtain the cost of the preferred stock to the firm that issues it. As in the earlier bond calculations, we use the net proceeds from the sale rather than the price that is paid by the investor in the calculation. For example, suppose that in order for a firm to sell the above preferred stock, it must pay an investment banker 5 percent of the amount of money raised. If there are no other issuance costs, the company would receive $85 × (1 − 0.05) = $80.75 for each share sold, and the total cost of this financing to the firm would be:
Estimating the Cost of Preferred Stock
ps because you cannot get a good estimate of the beta using regression analysis. How else can you estimate the cost of this preferred stock, and what is this cost?
APPROACH: You can also use Equation 13.6 to estimate the cost of preferred stock.
SOLUTION: First, you must find the annual dividend that someone who owns a share of this stock will receive. This preferred stock issue pays an annual dividend (for simplicity we are assuming one dividend payment per year) that equals 8 percent of $1,000 or $1,000 × 0.8 = $80. Substituting the annual dividend and the market price into Equation 13.6 yields:
Chapter 9
that certain characteristics of preferred stock look a lot like those of debt. The equation Pps = Dps/k
ps shows that the value of preferred stock also varies with market rates of return in the same way as debt. Because k
ps is in the denominator of the fraction on the right-hand side of the equation, whenever k
ps increases, Pps decreases, and whenever k
ps decreases, Pps increases. That is, the value of preferred stock is negatively related to market rates.
It is also important to recognize that the CAPM can be used to estimate the cost of preferred equity, just as it can be used to estimate the cost of common equity. A financial analyst can simply substitute k
ps for k
cs and βps for βcs in Equation 13.4 and use it to estimate the cost of preferred stock. Remember from
Chapter 7
that the CAPM does not apply only to common stock; rather, it applies to any asset. Therefore, we can use it to calculate the rate of return on any asset if we can estimate the beta for that asset.
> BEFORE YOU GO ON
1. What information is needed to use the CAPM to estimate k
cs or k
ps?
2. Under what circumstances can you use the constant-growth dividend formula to estimate k
cs?
3. What is the advantage of using a multistage-growth dividend model, rather than the constant-growth dividend model, to estimate k
cs?
13.4 USING THE WACC IN PRACTICE
·
We have now covered the basic concepts and computational tools that are used to estimate the WACC. At this point, we are ready to talk about some of the practical issues that arise when financial analysts calculate the WACC for their firms.
When financial analysts think about calculating the WACC, they usually think of it as a weighted average of the firm’s after-tax cost of debt, cost of preferred stock, and cost of common equity. Equation 13.2 is usually written as:
Debt + x
ps + x
cs = 1. If the firm has more than one type of debt outstanding or more than one type of preferred or common stock, analysts will calculate a weighted average for each of those types of securities and then plug those averages into Equation 13.7. Financial analysts will also use the market values, rather than the accounting book values, of the debt, preferred stock, and common stock to calculate the weights (the x’s) in Equation 13.7. This is because, as we have already seen, the theory underlying the discounting process requires that the costs of the different types of financing be weighted by their relative market values. Accounting book values have no place in these calculations unless they just happen to equal the market values.
Calculating WACC: An Example
An example provides a useful way of illustrating how the theories and tools that we have discussed are used in practice. Assume that you are a financial analyst at a manufacturing company that has used three types of debt, preferred stock, and common stock to finance its investments.
Debt: The debt includes a $4 million bank loan that is secured by machinery and equipment. This loan has an interest rate of 6 percent, and your firm could expect to pay the same rate if the loan were refinanced today. Your firm also has a second bank loan (a $3 million mortgage on your manufacturing plant) with an interest rate of 5.5 percent. The rate would also be 5.5 percent today if you refinanced this loan. The third type of debt is a bond issue that the firm sold two years ago for $11 million. The market value of these bonds today is $10 million. Using the approach we discussed earlier, you have estimated that the effective annual yield on the bonds is 7 percent.
Preferred Stock: The preferred stock pays an annual dividend of 4.5 percent on a stated value of $100. A share of this stock is currently selling for $60, and there are 100,000 shares outstanding.
Common Stock: There are 1 million shares of common stock outstanding, and they are currently selling for $21 each. Using a regression analysis, you have estimated that the beta of these shares is 0.95.
The 20-year Treasury bond rate is currently 3.95 percent, and you have estimated the market risk premium to be 6.01 percent using the returns on stocks and Treasury bonds from the 1926 to 2009 period. Your firm’s marginal tax rate is 35 percent. What is the WACC for your firm?
The first step in computing the WACC is to calculate the pretax cost of debt. Since the market value of the firm’s debt is $17 million ($4 million + $3 million + $10 million = $17 million), we can calculate the pretax cost of debt as follows:
We next calculate the cost of the preferred stock using Equation 13.6, as follows:
http://www.pwcglobal.com/Extweb/pwcpublications.nsf/docid/748F5814D61CC2618525693A007EC870
.
We are now ready to use Equation 13.7 to calculate the firm’s WACC. Since the firm has $17 million of debt, $6 million of preferred stock ($60 × 100,000 shares = $6 million), and $21 million of common equity ($21 × 1,000,000 shares = 21 million), the total market value of its capital is $44 million ($17 million + $6 million + $21 million = $44 million). The firm’s WACC is therefore:
In order to finance the 2,000 restaurants that are now part of your company, you have sold three different bond issues. Based on the current prices of the bonds from these issues and the issue characteristics (face values and coupon rates), you have estimated the market values and effective annual yields to be:
APPROACH: You can use Equation 13.7 to solve for the WACC for your pizza business. To do so, you must first calculate the weighted average cost of debt. You can then plug the weights and costs for the debt and common equity into Equation 13.7. Since your business has no preferred stock, the value for this term in Equation 13.7 will equal $0.
SOLUTION: The weighted average cost of the debt is:
Interpreting the WACC
cs will decline from 9.7 percent to 8.2 percent:
DECISION: You should politely point out that she is making the wrong comparison. Since the refinancing will result in the firm being financed entirely with equity, k
cs will equal the firm’s WACC. Therefore, the 8.2 percent should really be compared with the 7.3 percent WACC. If your manager goes through with the refinancing, she will be making a bad decision. The average after-tax cost of the capital that your firm uses will increase from 7.3 percent to 8.2 percent.
Limitations of WACC as a Discount Rate for Evaluating Projects
At the beginning of this chapter, we told you that financial managers often require analysts within the firm to use the firm’s current cost of capital to discount the cash flows for individual projects. They do so because it is very difficult to directly estimate the discount rate for individual projects. You should recognize by now that the WACC is the discount rate that analysts are often required to use. Using the WACC to discount the cash flows for a project can make sense under certain circumstances. However, in other circumstances, it can be very dangerous. The rest of this section discusses when it makes sense to use the WACC as a discount rate and the problems that can occur when the WACC is used incorrectly.
discussed how an analyst forecasting the cash flows for a project is forecasting the incremental after-tax free cash flows at the firm level. These cash flows represent the difference between the cash flows that the firm will generate if the project is adopted and the cash flows that the firm will generate if the project is not adopted.
Financial theory tells us that the rate that should be used to discount these incremental cash flows is the rate that reflects their systematic risk. This means that the WACC is going to be the appropriate discount rate for evaluating a project only when the project has cash flows with systematic risks that are exactly the same as those for the firm as a whole. Unfortunately, this is not true for most projects. The firm itself is a portfolio of projects with varying degrees of risk.
When a single rate, such as the WACC, is used to discount cash flows for projects with varying levels of risk, the discount rate will be too low in some cases and too high in others. When the discount rate is too low, the firm runs the risk of accepting a negative NPV project. To see how this might happen, assume that you work at a company that manufactures soft drinks and that the managers at your company are concerned about all the competition in the core soft drink business. They are thinking about expanding into the manufacture and sale of exotic tropical beverages. The managers believe that entering this market would allow the firm to better differentiate its products and earn higher profits. Suppose also that the appropriate beta for soft drink projects is 1.2, while the appropriate beta for tropical beverage projects is 1.5. Since your firm is only in the soft drink business right now, the beta for its overall cash flows is 1.2.
Exhibit 13.3
illustrates the problem that could arise if your firm’s WACC is used to evaluate a tropical beverage project.
In the exhibit, you can see that since the beta of the tropical beverage project is larger than the beta of the firm as a whole, the expected return (or discount rate) for the tropical beverage project should be higher than the firm’s WACC. The Security Market Line indicates what this expected return should be. Now, if the firm’s WACC is used to discount the expected cash flows for this project, and the expected return on the project is above the firm’s WACC, then the estimated NPV will be positive. So far, so good. However, as illustrated in the exhibit, some projects may have an expected return that is above the WACC but below the SML. For projects such as those, using the WACC as the discount rate may actually cause the firm to accept a negative NPV project! The estimated NPV will be positive even though the true NPV is negative. The negative NPV projects that would be accepted in those situations have returns that fall in the red shaded area below the SML, above the WACC line, and to the right of the firm’s beta.
Two types of problems can arise when the WACC for a firm is used to evaluate individual projects: a positive NPV project may be rejected or a negative NPV project may be accepted. For the tropical beverage example, if the expected return on that project was below the level indicated by the SML, but above the firm’s WACC, the project might be accepted even though it would have a negative NPV.
In
Exhibit 13.3
you can also see that using the WACC to discount expected cash flows for low-risk projects can result in managers at the firm rejecting projects that have positive NPVs. This problem is, in some sense, the mirror image of the case where the WACC is lower than the correct discount rate. Financial managers run the risk of turning down positive NPV projects whenever the WACC is higher than the correct discount rate. The positive NPV projects that would be rejected are those that fall into the green shaded area that is below the WACC but above the SML and to the left of the firm’s beta.
To see how these types of problems arise, consider a project that requires an initial investment of $100 and that is expected to produce cash inflows of $40 per year for three years. If the correct discount rate for this project is 8 percent, its NPV will be:
Suppose, however, that the financial managers of the firm considering this project require that all projects be evaluated using the firm’s WACC of 11 percent. When the cash flows are discounted using a rate of 11 percent, the NPV is:
It is also important to recognize that when a firm uses a single rate to evaluate all of its projects, there will be a bias toward accepting more risky projects. The average risk of the firm’s assets will tend to increase over time. Furthermore, because some positive NPV projects are likely to be rejected and some negative NPV projects are likely to be accepted, new projects on the whole will probably create less value for stockholders than if the appropriate discount rate had been used to evaluate all projects. This, in turn, can put the firm at a disadvantage when compared with its competitors and adversely affect the value of its existing projects.
The key point to take away from this discussion is that it is only really correct to use a firm’s WACC to discount the cash flows for a project if the expected cash flows from that project have the same systematic risk as the expected cash flows from the firm as a whole. You might be wondering how you can tell when this condition exists. The answer is that we never know for sure. Nevertheless, there are some guidelines that you can use when assessing whether the systematic risk for a particular project is similar to that for the firm as a whole.
The systematic risk of the cash flows from a project depend on the nature of the business. Revenues and expenses in some businesses are affected more by changes in general economic conditions than revenues and expenses in other businesses. For example, consider the differences between a company that makes bread and a company that makes recreational vehicles. The demand for bread will be relatively constant in good economic conditions and in bad. The demand for recreational vehicles will be more volatile. People buy fewer recreational vehicles during recessions than when the economy is doing well. Furthermore, as we discussed in
Chapter 12
, operating leverage magnifies volatility in revenue. Therefore, if the recreational vehicle manufacturing process has more fixed costs than the bread manufacturing business, the difference in the volatilities of the pretax operating cash flows will be even greater than the difference in the volatilities of the revenues.
While total volatility is not the same as systematic volatility, we find that businesses with more total volatility (uncertainty or risk) typically have more systematic volatility. Since beta is a measure of systematic risk, and systematic risk is a key factor in determining a firm’s WACC, this suggests that the firm’s WACC should be used only for projects with business risks similar to those for the firm as a whole. Since financial managers usually think of systematic risk when they think of underlying business risks, we can restate this condition as follows:
Condition 1: A firm’s WACC should be used to evaluate the cash flows for a new project only if the level of systematic risk for the project is the same as that for the portfolio of projects that currently comprise the firm.
You have to consider one other factor when you decide whether it is appropriate to use a firm’s WACC to discount the cash flows for a project. That is the way in which the project will be financed and how this financing compares with the way the firm’s assets are financed. To better understand why this is important, consider Equation 13.7:
Condition 2: A firm’s WACC should be used to evaluate a project only if that project uses the same financing mix—the same proportions of debt, preferred shares, and common shares—used to finance the firm as a whole.
In summary, WACC is a measure of the current cost of the capital that the firm has used to finance its projects. It is an appropriate discount rate for evaluating projects only if (1) the project’s systematic risk is the same as that of the firm’s current portfolio of projects and (2) the project will be financed with the same mix of debt and equity as the firm’s current portfolio of projects. If either of these two conditions does not hold, then managers should be careful in using the firm’s current WACC to evaluate a project.
Alternatives to Using WACC for Evaluating Projects
Financial managers understand the limitations of using a firm’s WACC to evaluate projects, but they also know that there are no perfect alternatives. As we noted earlier in this chapter, there is no publicly traded common stock for most individual projects within a firm. It is, therefore, not possible to directly estimate the beta for the common stock used to finance an individual project.
9
Although it might be possible to obtain an estimate of the cost of debt from the firm’s bankers, without an estimate of the common stock beta—and, therefore, the cost of common stock—it is not possible to obtain a direct estimate of the appropriate discount rate for a project using Equation 13.7.
If the discount rate for a project cannot be estimated directly, a financial analyst might try to find a public firm that is in a business that is similar to that of the project. For example, in our exotic tropical beverage example, an analyst at the soft drink company might look for a company that produces only exotic tropical beverages and that also has publicly traded stock. This public company would be what financial analysts call a pure-play comparable because it is exactly like the project. The returns on the pure-play company’s stock could be used to estimate the expected return on the equity that is used to finance the project. Unfortunately, this approach is generally not feasible due to the difficulty of finding a public firm that is only in the business represented by the project. If the public firm is in other businesses as well, then we run into the same sorts of problems that we face when we use the firm’s WACC.
pure-play comparable
a comparable company that is in exactly the same business as the project or business being analyzed
From a practical standpoint, financial managers, such as company treasurers and chief financial officers, do not like letting analysts estimate the discount rates for their projects. Different analysts tend to make different assumptions or use different approaches, which can lead to inconsistencies that make it difficult to compare projects. In addition, analysts may be tempted to manipulate discount rates in order to make pet projects look more attractive.
In an effort to use discount rates that reflect project risks better than the firm’s WACC, while retaining control of the process through which discount rates are set, financial managers sometimes classify projects into categories based on their systematic risks. They then specify a discount rate that is to be used to discount the cash flows for all projects within each category. The idea is that each category of projects has a different level of systematic risk and therefore a different discount rate should be used for each.
Exhibit 13.4
illustrates such a classification scheme.
The scheme illustrated in
Exhibit 13.4
includes four project categories:
1. Efficiency projects, such as the implementation of a new production technology that reduces manufacturing costs for an existing product.
2. Product extension projects, such as those in which Boeing created variations of its aircraft, like the Boeing 737, to help meet customer needs.
3. Market extension projects, in which existing products are sold in new markets, such as when Texas Instruments considers selling a new version of a computer chip that has been used in digital phones to digital camera manufacturers.
4. New product projects, in which entirely new products are being considered.
When using the scheme illustrated in
Exhibit 13.4
, the financial manager would assign a discount rate for each category that reflects the typical beta in the indicated range of betas. Such an approach is attractive because it is not generally difficult for analysts to figure out in which of the four categories particular projects belong, and it limits their discretion in choosing discount rates. Most important, it can reduce the possibility of accepting negative NPV projects or rejecting positive NPV projects. We can see the latter benefit by comparing the shaded areas in the figures in
Exhibits 13.3
and
13.4
. The total size of the shaded areas, which represents the possibility of making an error, is much smaller in
Exhibit 13.4
.
The potential for errors—either rejecting a positive NPV project or accepting a negative NPV project—is smaller when discount rates better reflect the risk of the projects that they are used to evaluate. You can see this by noting that the total size of the shaded areas in this figure is smaller than the size of the shaded areas in
Exhibit 13.3
. In the ideal situation, where the correct discount rate is used for each project, there would be no shaded area at all in a figure like this.
> BEFORE YOU GO ON
1. Do analysts use book values or market values to calculate the weights when they use Equation 13.7? Why?
2. What kinds of errors can be made when the WACC for a firm is used as the discount rate for evaluating all projects in the firm?
3. Under what conditions is the WACC the appropriate discount rate for a project?
SUMMARY OF Learning Objectives
The weighted average cost of capital (WACC) for a firm is a weighted average of the current costs of the different types of financing that a firm has used to finance the purchase of its assets. When the WACC is calculated, the cost of each type of financing is weighted according to the fraction of the total firm value represented by that type of financing. The WACC is often used as a discount rate in evaluating projects because it is not possible to directly estimate the appropriate discount rate for many projects. As we also discuss in Section 13.4, having a single discount rate reduces inconsistencies that can arise when different analysts in the firm use different methods to estimate the discount rate and can also limit the ability of analysts to manipulate discount rates to favor pet projects.
Calculate the cost of debt for a firm.
The cost of debt can be calculated by solving for the yield to maturity of the debt using the bond pricing model (Equation 8.1), computing the effective annual yield, and adjusting for taxes using Equation 13.3.
Calculate the cost of common stock and the cost of preferred stock for a firm.
The cost of common stock can be estimated using the CAPM, the constant-growth dividend formula, and a multistage-growth dividend formula. The cost of preferred stock can be calculated using the perpetuity model for the present value of cash flows.
Calculate the weighted average cost of capital for a firm, explain the limitations of using a firm’s weighted average cost of capital as the discount rate when evaluating a project, and discuss the alternatives to the firm’s weighted average cost of capital that are available.
The weighted average cost of capital is estimated using either Equation 13.2 or Equation 13.7, with the cost of each individual type of financing estimated using the appropriate method.
When a firm uses a single rate to discount the cash flows for all of its projects, some project cash flows will be discounted using a rate that is too high and other project cash flows will be discounted using a rate that is too low. This can result in the firm rejecting some positive NPV projects and accepting some negative NPV projects. It will bias the firm toward accepting more risky projects and can cause the firm to create less value for stockholders than it would have if the appropriate discount rates had been used.
One alternative to using the WACC as a discount rate is to identify a firm that engages in business activities that are similar to those associated with the project under consideration and that has publicly traded stock. The returns from this pure-play firm’s stock can then be used to estimate the common stock beta for the project. In instances where pure-play firms are not available, another alternative is for the financial manager to classify projects according to their systematic risks and use a different discount rate for each class of project. This is the type of classification scheme illustrated in
Exhibit 13.4
.
SUMMARY OF Key Equations
· 13.1 The market value of a firm’s assets is $3 billion. If the market value of the firm’s liabilities is $2 billion, what is the market value of the stockholders’ investment and why?
· 13.2 Berron Comics, Inc., has borrowed $100 million and is required to pay its lenders $8 million in interest this year. If Berron is in the 35 percent marginal tax bracket, then what is the after-tax cost of debt (in dollars as well as in annual interest) to Berron.
· 13.3 Explain why the after-tax cost of equity (common or preferred) does not have to be adjusted by the marginal income tax rate for the firm.
· 13.4 Mike’s T-Shirts, Inc., has debt claims of $400 (market value) and equity claims of $600 (market value). If the after-tax cost of debt financing is 11 percent and the cost of equity is 17 percent, what is Mike’s weighted average cost of capital?
· 13.5 You are analyzing a firm that is financed with 60 percent debt and 40 percent equity. The current cost of debt financing is 10 percent, but due to a recent downgrade by the rating agencies, the firm’s cost of debt is expected to increase to 12 percent immediately. How will this change the firm’s weighted average cost of capital if you ignore taxes?
Solutions to Self-Study Problems
· 13.1 Since the identity that Assets = Liabilities + Equity holds for market values as well as book values, we know that the market value of the firm’s equity is $3 billion − $2 billion, or $1 billion.
· 13.2 Because Berron enjoys a tax deduction for its interest charges, the after-tax interest expense for Berron is $8 million × (1 − 0.35) = $5.2 million, which translates into an annual after-tax interest expense of $5.2/$100 = 0.052, or 5.2 percent.
· 13.3 The U.S. tax code allows a deduction for interest expense incurred on borrowing. Preferred and common shares are not considered debt and, thus, do not benefit from an interest deduction. As a result, there is no distinction between the before-tax and after-tax cost of equity capital.
· 13.4 Mike’s T-Shirts’s total firm value = $400 + $600 = $1,000. Therefore,
Debt × k
Debt, and since the firm’s pretax cost of debt is expected to increase by 2 percent, we know that the effect on WACC (pretax) will be 0.6 × 0.02 = 0.012, or 1.2 percent. Incidentally, if we assume that the firm is subject to the 40 percent marginal tax rate, then the after-tax increase in the cost of capital for the firm would be 0.012 × (1 − 0.4) = 0.0072, or 0.72 percent.
Critical Thinking Questions
· 13.1 Explain why the required rate of return on a firm’s assets must be equal to the weighted average cost of capital associated with its liabilities and equity.
· 13.2 Which is easier to calculate directly, the expected rate of return on the assets of a firm or the expected rate of return on the firm’s debt and equity? Assume that you are an outsider to the firm.
· 13.3 With respect to the level of risk and the required return for a firm’s portfolio of projects, discuss how the market and a firm’s management can have inconsistent information and expectations.
· 13.4 Your friend has recently told you that the federal government effectively subsidizes the use of debt financing (vs. equity financing) for corporations. Do you agree with that statement? Explain.
· 13.5 Your firm will have a fixed interest expense for the next 10 years. You recently found out that the marginal income tax rate for the firm will change from 30 percent to 40 percent next year. Describe how the change will affect the cash flow available to investors.
· 13.6 Describe why it is not usually appropriate to use the coupon rate on a firm’s bonds to estimate the pretax cost of debt for the firm.
· 13.7 Maltese Falcone, Inc., has not checked its weighted average cost of capital for four years. Firm management claims that since Maltese has not had to raise capital for new projects in four years, they should not have to worry about their current weighted average cost of capital. They argue that they have essentially locked in their cost of capital. Critique management’s statements.
· 13.8 Ten years ago, the Edson Water Company issued preferred stock at a price equal to the par value of $100. If the dividend yield on that issue was 12 percent, explain why the firm’s current cost of preferred capital is not likely to equal 12 percent.
· 13.9 Discuss under what circumstances you might be able to use a model that assumes constant growth in dividends to calculate the current cost of equity capital for a firm.
· 13.10 Your boss just finished computing your firm’s weighted average cost of capital. He is relieved because he says that he can now use that cost of capital to evaluate all projects that the firm is considering for the next four years. Evaluate that statement.
Questions and Problems
· 13.2 WACC: What is the weighted average cost of capital?
· 13.3 Taxes and the cost of debt: How are taxes accounted for when we calculate the cost of debt?
· 13.4 Cost of common stock: List and describe each of the three methods used to calculate the cost of common stock.
· 13.5 Cost of common stock: Whitewall Tire Co. just paid an annual dividend of $1.60 on its common shares. If Whitewall is expected to increase its annual dividend by 2 percent per year into the foreseeable future and the current price of Whitewall’s common shares is $11.66, what is the cost of common stock for Whitewall?
· 13.6 Cost of common stock: Seerex Wok Co. is expected to pay a dividend of $1.10 one year from today on its common shares. That dividend is expected to increase by 5 percent every year thereafter. If the price of Seerex is $13.75, what is Seerex’s cost of common stock?
· 13.7 Cost of common stock: Two-Stage Rocket paid an annual dividend of $1.25 yesterday, and it is commonly known that the firm’s management expects to increase its dividend by 8 percent for the next two years and by 2 percent thereafter. If the current price of Two-Stage’s common stock is $17.80, what is the cost of common equity capital for the firm?
· 13.8 Cost of preferred stock: Fjord Luxury Liners has preferred shares outstanding that pay an annual dividend equal to $15 per year. If the current price of Fjord preferred shares is $107.14, what is the after-tax cost of preferred stock for Fjord?
· 13.9 Cost of preferred stock: Kresler Autos has preferred shares outstanding that pay annual dividends of $12, and the current price of the shares is $80. What is the after-tax cost of new preferred shares for Kresler if the flotation (issuance) costs for preferred are 5 percent?
· 13.10 WACC: Describe the alternatives to using a firm’s WACC as a discount rate when evaluating a project.
· 13.11 WACC for a firm: Capital Co. has a capital structure, based on current market values, that consists of 50 percent debt, 10 percent preferred stock, and 40 percent common stock. If the returns required by investors are 8 percent, 10 percent, and 15 percent for the debt, preferred stock, and common stock, respectively, what is Capital’s after-tax WACC? Assume that the firm’s marginal tax rate is 40 percent.
13.12 WACC: What are direct out-of-pocket costs?
· 13.14 Finance balance sheet: Explain why the cost of capital for a firm is equal to the expected rate of return to the investors in the firm.
· 13.15 Current cost of a bond: You know that the after-tax cost of debt capital for Bubbles Champagne is 7 percent. If the firm has only one issue of five-year bonds outstanding, what is the current price of the bonds if the coupon rate on those bonds is 10 percent? Assume the bonds make semiannual coupon payments and the marginal tax rate is 30 percent.
· 13.16 Current cost of a bond: Perpetual Ltd. has issued bonds that never require the principal amount to be repaid to investors. Correspondingly, Perpetual must make interest payments into the infinite future. If the bondholders receive annual payments of $75 and the current price of the bonds is $882.35, what is the after-tax cost of this debt for Perpetual if the firm is in the 40 percent marginal tax rate?
· 13.17 Current cost of a bond: You are analyzing the cost of debt for a firm. You know that the firm’s 14-year maturity, 8.5 percent coupon bonds are selling at a price of $823.48. The bonds pay interest semiannually. If these bonds are the only debt outstanding, what is the after-tax cost of debt for this firm if it has a 30 percent marginal and average tax rate?
· 13.18 Taxes and the cost of debt: Holding all other things constant, does a decrease in the marginal tax rate for a firm provide incentive for the firm to increase or decrease its use of debt?
· 13.19 Cost of debt for a firm: You are analyzing the after-tax cost of debt for a firm. You know that the firm’s 12-year maturity, 9.5 percent semi-annual coupon bonds are selling at a price of $1,200. If these bonds are the only debt outstanding for the firm, what is the after-tax cost of debt for this firm if it has a marginal tax rate of 34 percent? What if the bonds are selling at par?
· 13.20 Cost of common stock: Underestimated Inc.’s common shares currently sell for $36 each. The firm’s management believes that its shares should really sell for $54 each. If the firm just paid an annual dividend of $2 per share and management expects those dividends to increase by 8 percent per year forever (and this is common knowledge to the market), what is the current cost of common equity for the firm and what does management believe is a more appropriate cost of common equity for the firm?
· 13.21 Cost of common stock: Write out the general equation for the price of the stock for a firm that will grow dividends very rapidly at a constant rate for the four years after the next dividend is paid and will grow dividends thereafter at a constant, but lower rate. Discuss the problems in estimating the cost of equity capital for such a stock.
· 13.22 Cost of common stock: You have calculated the cost of common stock using all three methods described in the chapter. Unfortunately, all three methods have yielded different answers. Describe which answer (if any) is most appropriate.
· 13.23 WACC for a firm: The managers of a firm financed entirely with common stock are evaluating two distinct projects. The first project has a large amount of unsystematic risk and a small amount of systematic risk. The second project has a small amount of unsystematic risk and a large amount of systematic risk. Which project, if taken, is more likely to increase the firm’s cost of capital?
· 13.24 WACC for a firm: The Imaginary Products Co. currently has debt with a market value of $300 million outstanding. The debt consists of 9 percent coupon bonds (semiannual coupon payments) which have a maturity of 15 years and are currently priced at $1,440.03 per bond. The firm also has an issue of 2 million preferred shares outstanding with a market price of $12.00. The preferred shares pay an annual dividend of $1.20. Imaginary also has 14 million shares of common stock outstanding with a price of $20.00 per share. The firm is expected to pay a $2.20 common dividend one year from today, and that dividend is expected to increase by 5 percent per year forever. If Imaginary is subject to a 40 percent marginal tax rate, then what is the firm’s weighted average cost of capital?
· 13.25 Choosing a discount rate: For the Imaginary Products firm in Problem 13.24, calculate the appropriate cost of capital for a new project that is financed with the same proportion of debt, preferred shares, and common shares as the firm’s current capital structure. Also assume that the project has the same degree of systematic risk as the average project that the firm is currently undertaking (the project is also in the same general industry as the firm’s current line of business).
· 13.26 Choosing a discount rate: If a firm anticipates financing a project with a capital mix different than its current capital structure, describe in realistic terms how the firm is subjecting itself to a calculation error if its historical WACC is used to evaluate the project.
· 13.28 You are an external financial analyst evaluating the merits of a stock. Since you are using a dividend discount model approach to evaluate a cost of equity capital, you need to estimate the dividend growth rate for the firm in the future. Describe how you might go about doing this.
· 13.29 You know that the return of Momentum Cyclicals common shares is 1.6 times as sensitive to macroeconomic information as the return of the market. If the risk-free rate of return is 4 percent and market risk premium is 6 percent, what is Momentum Cyclicals’s cost of common equity capital?
· 13.30 In your analysis of the cost of capital for a common stock, you calculate a cost of capital using a dividend discount model that is much lower than the calculation for the cost of capital using the CAPM model. Explain a possible source for the discrepancy.
· 13.31 RetRyder Hand Trucks has a preferred share issue outstanding that pays a dividend of $1.30 per year. The current cost of preferred equity for RetRyder is 9 percent. If RetRyder issues additional preferred shares that pay exactly the same dividend and the investment banker retains 8 percent of the sale price, what is the cost of the new preferred shares for RetRyder?
· 13.32 Enigma Corporation’s management believes that the firm’s cost of capital (WACC) is too high because the firm has been too secretive with the market concerning its operations. Evaluate that statement.
· 13.33 Discuss what valuable information would be lost if you decided to use book values in order to calculate the cost of each of the capital components within a firm’s capital structure.
· 13.34 Hurricane Corporation is financed with debt, preferred equity, and common equity with market values of $20 million, $10 million, and $30 million, respectively. The betas for the debt, preferred stock, and common stock are 0.2, 0.5, and 1.1, respectively. If the risk-free rate is 3.95 percent, the market risk premium is 6.01 percent, and Hurricane’s average and marginal tax rates are both 30 percent, what is the company’s weighted average cost of capital?
· 13.35 You are working as an intern at Coral Gables Products, a privately owned manufacturing company. Shortly after you read
Chapter 13
in this book, you got into a discussion with the Chief Financial Officer (CFO) at Coral Gables about weighted average cost of capital calculations. She pointed out that, just as the beta of the assets of a firm equals a weighted average of the betas for the individual assets, as shown in Equation 7.11:
· 13.36 The CFO described in Problem 13.35 asks you to estimate the beta for Coral Gables’s common stock. Since the common stock is not publicly traded, you do not have the data necessary to estimate the beta using regression analysis. However, you have found a company with publicly traded stock that has operations which are exactly like those at Coral Gables. Using stock returns for this pure-play comparable firm, you estimate the beta for the comparable company’s stock to be 1.06. The market value of that company’s common equity is $45 million, and it has one debt issue outstanding with a market value of $15 million and an annual pretax cost of 4.85 percent. The comparable company has no preferred stock.
35. If the risk-free rate is 3.95 percent and the market risk premium is 6.01 percent, what is the beta of the assets of the comparable company?
35. If the total market value of Coral Gables’ financing consists of 35 percent debt and 65 percent equity (this is what the CFO estimates the market values to be) and the pretax cost of its debt is 5.45 percent, what is the beta for Coral Gables’s common stock?
1. 13.37 Estimate the weighted average cost of capital for Coral Gables using your estimated beta and the information in the problem statement in Problem 13.36. Assume that the average and marginal tax rates for Coral Gables are both 25 percent.
37. Expected market return.
37. Rate of return required by stockholders.
37. Cost of retained earnings plus dividends.
37. Risk the company incurs when financing.
1. 13.39
Dot.Com
has determined that it could issue $1,000 face value bonds with an 8 percent coupon paid semiannually and a five-year maturity at $900 per bond. If
Dot.Com
‘s marginal tax rate is 38 percent, its after-tax cost of debt is closest to:
38. 6.2 percent.
38. 6.4 percent.
38. 6.6 percent.
38. 6.8 percent.
1. 13.40 Morgan Insurance Ltd. issued a fixed-rate perpetual preferred stock three years ago and placed it privately with institutional investors. The stock was issued at $25.00 per share with a $1.75 dividend. If the company were to issue preferred stock today, the yield would be 6.5 percent. The stock’s current value is:
39. $25.00.
39. $26.92.
39. $37.31.
39. $40.18.
1. 13.41 The Gearing Company has an after-tax cost of debt capital of 4 percent, a cost of preferred stock of 8 percent, a cost of equity capital of 10 percent, and a weighted average cost of capital of 7 percent. Gearing intends to maintain its current capital structure as it raises additional capital. In making its capital-budgeting decisions for the average-risk project, the relevant cost of capital is:
40. 4 percent.
40. 7 percent.
40. 8 percent.
40. 10 percent.
1. 13.42 Suppose the cost of capital of the Gadget Company is 10 percent. If Gadget has a capital structure that is 50 percent debt and 50 percent equity, its before-tax cost of debt is 5 percent, and its marginal tax rate is 20 percent, then its cost of equity capital is closest to:
41. 10 percent.
41. 12 percent.
41. 14 percent.
41. 16 percent.
Sample Test Problems
· 13.1 The Balanced, Inc., has three different product lines. Its least risky product line has a beta of 1.7, while its middle-risk product line has a beta of 1.8, and its most risky product line has a beta of 2.1. The market value of the assets invested in these lines is $1 billion for the least risky line, $3 billion for the middle-risk line, and $7 billion for the riskiest product line. What is the beta of the assets of The Balanced, Inc.? (Hint: see problem 13.35 on page 439.)
· 13.2 Ellwood Corp. has a five-year bond issue outstanding with a coupon rate of 10 percent and a price of $1,039.56. If the bonds pay coupons semiannually, what is the pretax cost of the debt and what is the after-tax cost of the debt? Assume the marginal tax rate for the firm is 40 percent.
· 13.3 Miron’s Copper Corp. management expects its common stock dividends to grow 1.5 percent per year for the indefinite future. The firm’s shares are currently selling for $18.45, and the firm just paid a dividend of $3.00 yesterday. What is the cost of common stock for this firm?
· 13.4 Micah’s Time Portals has a preferred stock issue outstanding that pays an annual dividend of $2.50 per year and is currently selling for $27.78 a share. What is the cost of preferred stock for this firm?
· 13.5 The Old Time New Age Co. has a portfolio of projects with a beta of 1.25. The firm is currently evaluating a new project that involves a new product in a new competitive market. Briefly discuss what adjustment Old Time New Age might make to its 1.25 beta in order to evaluate this new project.
Surveys of capital budgeting practices at major public firms in the United States indicate that a large percentage (possibly as high as 80 percent) of firms use the cost of capital for a firm or a division in capital budgeting calculations. For a discussion of this evidence, see the article titled “Best Practices in Estimating the Cost of Capital: Survey and Synthesis,” by R. F. Bruner, K. M. Eades, R. S. Harris, and R. C. Higgins, which was published in the Spring/Summer 1998 issue of Financial Practice and Education.
The total expected cash flows at Boeing also include cash flows from projects that the firm is expected to undertake in the future, or what are often referred to as growth opportunities. This idea is discussed in detail in later chapters. For our immediate purposes, we will assume that these cash flows are expected to equal $0.
We will discuss how firms finance their assets in more detail in
Chapters 15
and
16
. For the time being, we will simply assume that a firm uses some combination of debt and equity. Here we use the term debt in the broadest sense to refer to all liabilities, including liabilities on which the firm does not pay interest, such as accounts payable. As is common practice, we focus only on long-term interest-bearing debt, such as bank loans and bonds, in the cost of capital calculations. The reason for this is discussed in the next section.
As we will discuss in Section 13.2, if markets are efficient, the prices we observe in the markets will reflect the true costs of the different securities that the firm has outstanding.
We are ignoring the effect of taxes on the cost of debt financing for the time being. This effect is discussed in detail in Section 13.2 and explicitly incorporated into subsequent calculations.
These types of costs are incurred by firms whenever they raise capital. We only show how to include them in the cost of bond financing and, later, in estimating the cost of preferred stock, but they should also be included in calculations of the costs of capital from other sources, such as bank loans and common equity.
There is a limit on the total amount of home loan interest payments that you can deduct when you calculate your taxable income. For instance, in 2011 you could deduct interest payments on loans with a total face value of $1,100,000 ($1,000,000 mortgage plus $100,000 home equity loan).
We use the term “risk-free” here to refer to assets that have no default risk. Investors in the assets can still face interest rate risk as described in
Chapter 8
.
Some firms issue a type of stock that has an equity claim on only part of their business. If a project is similar to the part of the business for which “tracking stock” like this has been sold, the returns on the tracking stock can be used to estimate the beta for the common stock used to finance a project.
Risk and Return
Learning Objectives
Explain the relation between risk and return.
Describe the two components of a total holding period return, and calculate this return for an asset.
Explain what an expected return is and calculate the expected return for an asset.
Explain what the standard deviation of returns is and why it is very useful in finance, and calculate it for an asset.
Explain the concept of diversification.
Discuss which type of risk matters to investors and why.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate whether the expected return of an asset is sufficient to compensate an investor for the risks associated with that asset.
When Blockbuster Inc. filed for bankruptcy protection on Thursday, September 23, 2010, its days as the dominant video rental firm were long gone. Netflix had become the most successful competitor in the video rental market through its strategy of renting videos exclusively online and avoiding the high costs associated with operating video rental stores.
The bankruptcy filing passed control of Blockbuster to a group of bondholders, including the famous billionaire investor Carl Icahn, and the shares owned by the old stockholders became virtually worthless. The bondholders planned to reorganize the company and restructure its financing so that it had a chance of competing more effectively with Netflix in the future.
Over the previous five years, Blockbuster stockholders had watched the value of their shares steadily decline as, year after year, the company failed to respond effectively to the threat posed by Netflix. From September 23, 2005 to September 23, 2010, the price of Blockbuster shares fell from $4.50 to $0.04. In contrast, the price of Netflix shares rose from $24.17 to $160.47 over the same period. While the Blockbuster stockholders were losing almost 100 percent of their investments, Netflix stockholders were earning an average return of 46 percent per year!
This chapter discusses risk, return, and the relation between them. The difference in the returns earned by Blockbuster and Netflix stockholders from 2005 to 2010 illustrates a challenge faced by all investors. The shares of both of these companies were viewed as risky investments in 2005, and yet an investor who put all of his or her money in Blockbuster lost virtually everything, while an investor who put all of his or her money in Netflix earned a very high return. How should have investors viewed the risks of investing in these companys’ shares in 2005? How is risk related to the returns that investors might expect to earn? How does diversification reduce the overall risk of an investor’s portfolio? These are among the topics that we discuss in this chapter.
CHAPTER PREVIEW
Up to this point, we have often mentioned the rate of return that we use to discount cash flows, but we have not explained how that rate is determined. We have now reached the point where it is time to examine key concepts underlying the discount rate. This chapter introduces a quantitative framework for measuring risk and return. This framework will help you develop an intuitive understanding of how risk and return are related and what risks matter to investors. The relation between risk and return has implications for the rate we use to discount cash flows because the time value of money that we discussed in
Chapters 5
and
6
is directly related to the returns that investors require. We must understand these concepts in order to determine the correct present value for a series of cash flows and to be able to make investment decisions that create value for stockholders.
We begin this chapter with a discussion of the general relation between risk and return to introduce the idea that investors require a higher rate of return from riskier assets. This is one of the most fundamental relations in finance. We next develop the statistical concepts required to quantify holding period returns, expected returns, and risk. We then apply these concepts to portfolios with a single asset and with more than one asset to illustrate the benefit of diversification. From this discussion, you will see how investing in more than one asset enables an investor to reduce the total risk associated with his or her investment portfolio, and you will learn how to quantify this benefit.
Once we have discussed the concept of diversification, we examine what it means for the relation between risk and return. We find that the total risk associated with an investment consists of two components: (1) unsystematic risk and (2) systematic risk. Diversification enables investors to eliminate the unsystematic risk associated with an individual asset. Investors do not require higher returns for the unsystematic risk that they can eliminate through diversification. Only systematic risk—risk that cannot be diversified away—affects expected returns on an investment. The distinction between unsystematic and systematic risk and the recognition that unsystematic risk can be diversified away are extremely important in finance. After reading this chapter, you will understand precisely what the term risk means in finance and how it is related to the rates of return that investors require.
7.1 RISK AND RETURN
·
The rate of return that investors require for an investment depends on the risk associated with that investment. The greater the risk, the larger the return investors require as compensation for bearing that risk. This is one of the most fundamental relations in finance. The rate of return is what you earn on an investment, stated in percentage terms. We will be more specific later, but for now you might think of risk as a measure of how certain you are that you will receive a particular return. Higher risk means you are less certain.
To get a better understanding of how risk and return are related, consider an example. You are trying to select the best investment from among the following three stocks:
The greater the risk associated with an investment, the greater the return investors expect from it. A corollary to this idea is that investors want the highest return for a given level of risk or the lowest risk for a given level of return. When choosing between two investments that have the same level of risk, investors prefer the investment with the higher return. Alternatively, if two investments have the same expected return, investors prefer the less risky alternative.
Which would you choose? If you were comparing only Stocks A and B, you should choose Stock A. Both stocks have the same expected return, but Stock A has less risk. It does not make sense to invest in the riskier stock if the expected return is the same. Similarly, you can see that Stock C is clearly superior to Stock B. Stocks B and C have the same level of risk, but Stock C has a higher expected return. It would not make sense to accept a lower return for taking on the same level of risk.
But what about the choice between Stocks A and C? This choice is less obvious. Making it requires understanding the concepts that we discuss in the rest of this chapter.
7.2 QUANTITATIVE MEASURES OF RETURN
·
Before we begin a detailed discussion of the relation between risk and return, we should define more precisely what these terms mean. We begin with measures of return.
Holding Period Returns
the total return on an asset over a specific period of time or holding period
When people refer to the return from an investment, they are generally referring to the total return over some investment period, or holding period. The total holding period return consists of two components: (1) capital appreciation and (2) income. The capital appreciation component of a return, RCA, arises from a change in the price of the asset over the investment or holding period and is calculated as follows:
The income component of a return arises from income that an investor receives from the asset while he or she owns it. For example, when a firm pays a cash dividend on its stock, the income component of the return on that stock, RI, is calculated as follows:
The total holding period return, RT, is simply the sum of the capital appreciation and income components of return:
http://www.callan.com/research/periodic
/.
Let’s consider an example of calculating the total holding period return on an investment. One year ago today, you purchased a share of Dell Inc. stock for $12.50. Today it is worth $13.90. Dell paid no dividend on its stock. What total return did you earn on this stock over the past year?
If Dell paid no dividend and you received no other income from holding the stock, the total return for the year equals the return from the capital appreciation. The total return is calculated as follows:
APPLICATION 7.1 LEARNING BY DOING
APPROACH: Use Equation 7.1 to calculate the total holding period return. To calculate RT using Equation 7.1, you must know P0, P1, and CF1. In this problem, you can assume that the $7,000 was spent at the time you bought the car to purchase parts and materials. Therefore, your initial investment, P0, was $1,500 $7,000 $8,500. Since there were no other cash inflows or outflows between the time that you bought the car and the time that you sold it, CF1 equals $0.
SOLUTION: The total holding period return is:
·
Suppose that you are a senior who plays college baseball and that your team is in the College World Series. Furthermore, suppose that you have been drafted by the Washington Nationals and are coming up for what you expect to be your last at-bat as a college player. The fact that you expect this to be your last at-bat is important because you just signed a very unusual contract with the Nationals. Your signing bonus will be determined solely by whether you get a hit in your final collegiate at-bat. If you get a hit, then your signing bonus will be $800,000. Otherwise, it will be $400,000. This past season, you got a hit 32.5 percent of the times you were at bat (you did not get a hit 67.5 percent of the time), and you believe this percentage reflects the likelihood that you will get a hit in your last collegiate at-bat.
1
H is the probability of a hit, p
NH is the probability of no hit, BH is the bonus you receive if you get a hit, and BNH is the bonus you receive if you get no hit. Since p
H equals 0.325, p
NH equals 0.675, BH equals $800,000, and BNH equals $400,000, the expected value of your bonus is:
What would your expected payoff be if you got a hit 99 percent of the time? We intuitively know that the expected bonus should be much closer to $800,000 in this case. In fact, it is:
We calculate an expected return in finance in the same way that we calculate any expected value. The expected return is a weighted average of the possible returns from an investment, where each of these returns is weighted by the probability that it will occur. In general terms, the expected return on an asset, E (RAsset), is calculated as follows:
an average of the possible returns from an investment, where each return is weighted by the probability that it will occur
where R
i
is possible return i and p
i
is the probability that you will actually earn R
i
. The summation symbol in this equation
It is important to make sure that the sum of the n individual probabilities, the pi
‘s, always equals 1, or 100 percent, when you calculate an expected value. The sum of the probabilities cannot be less than 100 percent because you must account for all possible outcomes in the calculation. On the other hand, as you may recall from statistics, the sum of the probabilities of all possible outcomes cannot exceed 100 percent. For example, notice that the sum of the pi
‘s equals 1 in each of the expected bonus calculations that we discussed earlier (0.325 0.625 in the first calculation, 0.5 0.5 in the second, and 0.99 0.01 in the third).
The expected return on an asset reflects the return that you can expect to receive from investing in that asset over the period that you plan to own it. It is your best estimate of this return, given the possible outcomes and their associated probabilities.
Note that if each of the possible outcomes is equally likely (that is, p
1 = p
2 = p
3 = … = pn
= p = 1/n), this formula reduces to the formula for a simple (equally weighted) average of the possible returns:
Since Dell pays no dividends, the total return on its stock equals the return from capital appreciation:
‘s equals 1.
APPLICATION 7.2 LEARNING BY DOING
Given your oil price prediction, you estimate that there is a 30 percent chance that the value of your railroad cars will increase by 15 percent, a 40 percent chance that their value will increase by 25 percent, and a 30 percent chance that their value will increase by 30 percent in the next year. In addition to appreciation in the value of your cars, you expect to earn 10 percent on your investment over the next year (after expenses) from leasing the railroad cars. What total return do you expect to earn on your railroad car investment over the next year?
APPROACH: Use Equation 7.1 first to calculate the total return that you would earn under each of the three possible outcomes. Next use these total return values, along with the associated probabilities, in Equation 7.2 to calculate the expected total return.
SOLUTION: To calculate the total returns using Equation 7.1,
EXAMPLE 7.1 DECISION MAKING
Using Expected Values in Decision Making
There is some uncertainty regarding how many new customers will visit your restaurant after hearing one of your radio ads. You estimate that there is a 30 percent chance that 35 people will visit, a 45 percent chance that 50 people will visit, and a 25 percent chance that 60 people will visit. Therefore, you expect the following number of new customers to visit your restaurant in response to each radio ad:
Similarly, you estimate that there is a 20 percent chance you will get 20 new customers in response to an ad placed on a taxi, a 30 percent chance you will get 30 new customers, a 30 percent chance that you will get 40 new customers, and a 20 percent chance that you will get 50 new customers. Therefore, you expect the following number of new customers in response to each ad that you place on a taxi:
Which of these two advertising options is more attractive? Is it cost effective?
DECISION: You should advertise on taxicabs. For a monthly cost of $1,000, you expect to attract 1,400 new customers with taxicab advertisements but only 960 new customers if you advertise on the radio.
The answer to the question of whether advertising on taxicabs is cost effective depends on how much the gross profits (profits after variable costs) of your business are increased by those 1,400 customers. Monthly gross profits will have to increase by $1,000, or average 72 cents per new customer ($1,000/1,400 $0.72) to cover the cost of the advertising campaign.
> BEFORE YOU GO ON
1. What are the two components of a total holding period return?
2. How is the expected return on an investment calculated?
7.3 THE VARIANCE AND STANDARD DEVIATION AS MEASURES OF RISK
·
We turn next to a discussion of the two most basic measures of risk used in finance—the variance and the standard deviation. These are the same variance and standard deviation measures that you studied if you took a course in statistics.
Calculating the Variance and Standard Deviation
Let’s begin by returning to our College World Series example. Recall that you will receive a bonus of $800,000 if you get a hit in your final collegiate at-bat and a bonus of $400,000 if you do not. The expected value of your bonus is $530,000. Suppose you want to measure the risk, or uncertainty, associated with the bonus. How can you do this? One approach would be to compute a measure of how much, on average, the bonus payoffs deviate from the expected value. The underlying intuition here is that the greater the difference between the actual bonus and the expected value, the greater the risk. For example, you might calculate the difference between each possible bonus payment and the expected value, and sum these differences. If you do this, you will get the following result:
variance (σ2)
a measure of the uncertainty associated with an outcome
A better approach would be to square the differences (squaring the differences makes all the numbers positive) and multiply each squared difference by its associated probability before summing them up. This calculation yields the variance (σ2) of the possible outcomes. The variance does not suffer from the two problems mentioned earlier and provides a measure of risk that has a consistent interpretation across different situations or assets. For the original bonus arrangement, the variance is:
Because it is somewhat awkward to work with units of squared dollars, in a calculation such as this we would typically take the square root of the variance. The square root gives us the standard deviation (σ) of the possible outcomes. For our example, the standard deviation is:
the square root of the variance
As you will see when we discuss the normal distribution, the standard deviation has a natural interpretation that is very useful for assessing investment risks.
The general formula for calculating the variance of returns can be written as follows:
Interpreting the Variance and Standard Deviation
The variance and standard deviation are especially useful measures of risk for variables that are normally distributed—those that can be represented by a normal distribution. The normal distribution is a symmetric frequency distribution that is completely described by its mean (average) and standard deviation.
Exhibit 7.1
illustrates what this distribution looks like. Even if you have never taken a statistics course, you have already encountered the normal distribution. It is the “bell curve” on which instructors often base their grade distributions. SAT scores and IQ scores are also based on normal distributions.
normal distribution
a symmetric frequency distribution that is completely described by its mean and standard deviation; also known as a bell curve due to its shape
This distribution is very useful in finance because the returns for many assets are approximately normally distributed. This makes the variance and standard deviation practical measures of the uncertainty associated with investment returns. Since the standard deviation is more easily interpreted than the variance, we will focus on the standard deviation as we discuss the normal distribution and its application in finance.
In
Exhibit 7.1
, you can see that the normal distribution is symmetric: the left and right sides are mirror images of each other. The mean falls directly in the center of the distribution, and the probability that an outcome is less than or greater than a particular distance from the mean is the same whether the outcome is on the left or the right side of the distribution. For example, if the mean is 0, the probability that a particular outcome is 3 or less is the same as the probability that it is + 3 or more (both are 3 or more units from the mean). This enables us to use a single measure of risk for the normal distribution. That measure is the standard deviation.
The normal distribution is a symmetric distribution that is completely described by its mean and standard deviation. The mean is the value that defines the center of the distribution, and the standard deviation, s, describes the dispersion of the values centered around the mean.
The standard deviation tells us everything we need to know about the width of the normal distribution or, in other words, the variation in the individual values. This variation is what we mean when we talk about risk in finance. In general terms, risk is a measure of the range of potential outcomes. The standard deviation is an especially useful measure of risk because it tells us the probability that an outcome will fall a particular distance from the mean, or within a particular range. You can see this in the following table, which shows the fraction of all observations in a normal distribution that are within the indicated number of standard deviations from the mean.
For example, if the expected return for a real estate investment in Miami, Florida, is 10 percent with a standard deviation of 2 percent, there is a 90 percent chance that the actual return will be within 3.29 percent of 10 percent. How do we know this? As shown in the table, 90 percent of all outcomes in a normal distribution have a value that is within 1.645 standard deviations of the mean value, and 1.645 × 2 percent = 3.29 percent. This tells us that there is a 90 percent chance that the realized return on the investment in Miami will be between 6.71 percent (10 percent − 3.29 percent = 6.71 percent) and 13.29 percent (10 percent + 3.29 percent = 13.29 percent), a range of 6.58 percent (13.29 percent − 6.71 percent = 6.58 percent).
You may be wondering what is standard about the standard deviation. The answer is that this statistic is standard in the sense that it can be used to directly compare the uncertainties (risks) associated with the returns on different investments. For instance, suppose you are comparing the real estate investment in Miami with a real estate investment in Fresno, California. Assume that the expected return on the Fresno investment is also 10 percent. If the standard deviation for the returns on the Fresno investment is 3 percent, there is a 90 percent chance that the actual return is within 4.935 percent (1.645 × 3 percent = 4.935 percent) of 10 percent. In other words, 90 percent of the time, the return will be between 5.065 percent (10 percent − 4.935 percent = 5.065 percent) and 14.935 percent (10 percent + 4.935 percent = 14.935 percent), a range of 9.87 percent (14.935 percent − 5.065 percent = 9.87 percent).
This range is exactly 9.87 percent/6.58 percent = 1.5 times as large as the range for the Miami investment opportunity. Notice that the ratio of the two standard deviations also equals 1.5 (3 percent/2 percent = 1.5). This is not a coincidence. We could have used the standard deviations to directly compute the relative uncertainty associated with the Fresno and Miami investment returns. The relation between the standard deviation of returns and the width of a normal distribution (the uncertainty) is illustrated in
Exhibit 7.2
.
Let’s consider another example of how the standard deviation is interpreted. Suppose customers at your pizza restaurant have complained that there is no consistency in the number of slices of pepperoni that your cooks are putting on large pepperoni pizzas. One night you decide to work in the area where the pizzas are made so that you can count the number of pepperoni slices on the large pizzas to get a better idea of just how much variation there is. After counting the slices of pepperoni on 50 pizzas, you estimate that, on average, your pies have 18 slices of pepperoni and that the standard deviation is 3 slices.
With this information, you estimate that 95 percent of the large pepperoni pizzas sold in your restaurant have between 12.12 and 23.88 slices. You are able to estimate this range because you know that 95 percent of the observations in a normal distribution fall within 1.96 standard deviations of the mean. With a standard deviation of three slices, this implies that the number of pepperoni slices on 95 percent of your pizzas is within 5.88 slices of the mean (3 slices × 1.96 = 5.88 slices). This, in turn, indicates a range of 12.12 (18 − 5.88 = 12.12) to 23.88 (18 + 5.88 = 23.88) slices.
Since you put only whole slices of pepperoni on your pizzas, 95 percent of the time the number of slices is somewhere between 12 and 24. No wonder your customers are up in arms! In response to this information, you decide to implement a standard policy regarding the number of pepperoni slices that go on each type of pizza.
The larger standard deviation for the return on the Fresno investment means that the Fresno investment is riskier than the Miami investment. The actual return for the Fresno investment is more likely to be further from its expected return.
APPLICATION 7.3 LEARNING BY DOING
APPROACH: Use the values in the previous table or
Exhibit 7.1
to compute the range within which Google’s stock return will fall 90 percent of the time. First, find the number of standard deviations associated with a 90 percent level of confidence in the table or
Exhibit 7.1
and then multiply this number by the standard deviation of the annual return for Google’s stock. Then subtract the resulting value from the expected return (mean) to obtain the lower end of the range and add it to the expected return to obtain the upper end.
SOLUTION: From the table, you can see that we would expect the return over the next year to be within 1.645 standard deviations of the mean 90 percent of the time. Multiplying this value by the standard deviation of Google’s stock (23 percent) yields 23 percent × 1.645 = 37.835 percent. This means that there is a 90 percent chance that the return will be between −33.165 percent (4.67 percent − 37.835 percent = −33.165 percent) and 42.505 percent (4.67 percent + 37.835 percent = 42.505 percent).
While the expected return of 4.67 percent is relatively low, the returns on Google stock vary considerably, and there is a reasonable chance that the stock return in the next year could be quite high or quite low (even negative). As you will see shortly, this wide range of possible returns is similar to the range we observe for typical shares in the U.S. stock market.
Historical Market Performance
Now that we have discussed how returns and risks can be measured, we are ready to examine the characteristics of the historical returns earned by securities such as stocks and bonds.
Exhibit 7.3
illustrates the distributions of historical returns for some securities in the United States and shows the average and standard deviations of these annual returns for the period from 1926 to 2009.
Note that the statistics reported in
Exhibit 7.3
are for indexes that represent total average returns for the indicated types of securities, not total returns on individual securities. We generally use indexes to represent the performance of the stock or bond markets. For instance, when news services report on the performance of the stock market, they often report that the Dow Jones Industrial Average (an index based on 30 large stocks), the S&P 500 Index (an index based on 500 large stocks), or the NASDAQ Composite Index (an index based on all stocks that are traded on NASDAQ) went up or down on a particular day. These and other indexes are discussed in
Chapter 9
.
The plots in
Exhibit 7.3
are arranged in order of decreasing risk, which is indicated by the decreasing standard deviation of the annual returns. The top plot shows returns for a small-stock index that represents the 10 percent of U.S. firms that have the lowest total equity value (number of shares multiplied by price per share). The second plot shows returns for the S&P 500 Index, representing large U.S. stocks. The remaining plots show three different types of government debt: Long-term government bonds that mature in 20 years, intermediate-term government bonds that mature in five years, and U.S. Treasury bills, which are short-term debts of the U.S. government, that mature in 30 days.
Higher standard deviations of returns have historically been associated with higher returns. For example, between 1926 and 2009, the standard deviation of the annual returns for small stocks was higher than the standard deviations of the returns earned by other types of securities, and the average return that investors earned from small stocks was also higher. At the other end of the spectrum, the returns on Treasury bills had the smallest standard deviation, and Treasury bills earned the smallest average return.
Source: Data from Morningstar, 2010 SBBI Yearbook
The key point to note in
Exhibit 7.3
is that, on average, annual returns have been higher for riskier securities. Small stocks, which have the largest standard deviation of total returns, at 32.79 percent, also have the largest average annual return, 16.57 percent. On the other end of the spectrum, Treasury bills have the smallest standard deviation, 3.08 percent, and the smallest average annual return, 3.71 percent. Returns for small stocks in any particular year may have been higher or lower than returns for the other types of securities, but on average, they were higher. This is evidence that investors require higher returns for investments with greater risks.
The statistics in
Exhibit 7.3
describe actual investment returns, as opposed to expected returns. In other words, they represent what has happened in the past. Financial analysts often use historical numbers such as these to estimate the returns that might be expected in the future. That is exactly what we did in the baseball example earlier in this chapter. We used the percentage of at-bats in which you got a hit this past season to estimate the likelihood that you would get a hit in your last collegiate at-bat. We assumed that your past performance was a reasonable indicator of your future performance.
To see how historical numbers are used in finance, let’s suppose that you are considering investing in a fund that mimics the S&P 500 Index (this is what we call an index fund) and that you want to estimate what the returns on the S&P 500 Index are likely to be in the future. If you believe that the 1926 to 2009 period provides a reasonable indication of what we can expect in the future, then the average historical return on the S&P 500 Index of 11.84 percent provides a perfectly reasonable estimate of the return you can expect from your investment in the S&P 500 Index fund. In
Chapter 13
we will explore in detail how historical data can be used in this way to estimate the discount rate used to evaluate projects in the capital budgeting process.
Comparing the historical returns for an individual stock with the historical returns for an index can also be instructive.
Exhibit 7.4
shows such a comparison for Apple Inc. and the S&P 500 Index using monthly returns for the period from September 2005 to September 2010. Notice in the exhibit that the returns on Apple stock are much more volatile than the average returns on the firms represented in the S&P 500 Index. In other words, the standard deviation of returns for Apple stock is higher than that for the S&P 500 Index. This is not a coincidence; we will discuss shortly why returns on individual stocks tend to be riskier than returns on indexes.
One last point is worth noting while we are examining historical returns: the value of a $1.00 investment in 1926 would have varied greatly by 2009, depending on where that dollar was invested.
Exhibit 7.5
shows that $1.00 invested in U.S. Treasury bills in 1926 would have been worth $20.53 by 2009. In contrast, that same $1.00 invested in small stocks would have been worth $12,231.13 by 2009!
2
Over a long period of time, earning higher rates of return can have a dramatic impact on the value of an investment. This huge difference reflects the impact of compounding of returns (returns earned on returns), much like the compounding of interest we discussed in
Chapter 5
.
The returns on shares of individual stocks tend to be much more volatile than the returns on portfolios of stocks, such as the S&P 500.
The value of a $1 investment in stocks, small or large, grew much more rapidly than the value of a $1 investment in bonds or Treasury bills over the 1926 to 2009 period. This graph illustrates how earning a higher rate of return over a long period of time can affect the value of an investment portfolio. Although annual stock returns were less certain between 1926 and 2009, the returns on stock investments were much greater.
Source: Data from Morningstar, 2010 SBBI Yearbook
> BEFORE YOU GO ON
1. What is the relation between the variance and the standard deviation?
2. What relation do we generally observe between risk and return when we examine historical returns?
3. How would we expect the standard deviation of the return on an individual stock to compare with the standard deviation of the return on a stock index?
7.4 RISK AND DIVERSIFICATION
·
It does not generally make sense to invest all of your money in a single asset. The reason is directly related to the fact that returns on individual stocks tend to be riskier than returns on indexes. By investing in two or more assets whose values do not always move in the same direction at the same time, an investor can reduce the risk of his or her collection of investments, or portfolio. This is the idea behind the concept of diversification.
portfolio
the collection of assets an investor owns
diversification
Reducing risk by investing in two or more assets whose values do not always move in the same direction at the same time
This section develops the tools necessary to evaluate the benefits of diversification. We begin with a discussion of how to quantify risk and return for a single-asset portfolio, and then we discuss more realistic and complicated portfolios that have two or more assets. Although our discussion focuses on stock portfolios, it is important to recognize that the concepts discussed apply equally well to portfolios that include a range of assets, such as stocks, bonds, gold, art, and real estate, among others.
Single-Asset Portfolios
Returns for individual stocks from one day to the next have been found to be largely independent of each other and approximately normally distributed. In other words, the return for a stock on one day is largely independent of the return on that same stock the next day, two days later, three days later, and so on. Each daily return can be viewed as having been randomly drawn from a normal distribution where the probability associated with the return depends on how far it is from the expected value. If we know what the expected value and standard deviation are for the distribution of returns for a stock, it is possible to quantify the risks and expected returns that an investment in the stock might yield in the future.
To see how we might do this, assume that you are considering investing in one of two stocks for the next year: Advanced Micro Devices (AMD) or Intel. Also, to keep things simple, assume that there are only three possible economic conditions (outcomes) a year from now and that the returns on AMD and Intel under each of these outcomes are as follows:
The coefficient of variation (CV) is a measure that can help us in making comparisons such as that between Stocks A and C. The coefficient of variation for stock i is calculated as:
coefficient of variation (CV)
a measure of the risk associated with an investment for each one percent of expected return
Recall that Stock A has an expected return of 12 percent and a risk level of 12 percent, while Stock C has an expected return of 16 percent and a risk level of 16 percent. If we assume that the risk level given for each stock is equal to the standard deviation of its return, we can find the coefficients of variation for the stocks as follows:
While this analysis appears to make sense, there is a conceptual problem with using the coefficient of variation to compute the amount of risk an investor can expect to realize for each 1 percent of expected return. This problem arises because investors expect to earn a positive return even when assets are completely risk free. For example, as shown in
Exhibit 7.3
, from 1926 to 2009 investors earned an average return of 3.71 percent each year on 30-day Treasury bills, which are considered to be risk free.
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If investors can earn a positive risk-free rate without bearing any risk, then it really only makes sense to compare the risk of the investment, s
Ri
, with the return that investors expect to earn over and above the risk-free rate. As we will discuss in detail in Section 7.6, the expected return over and above the risk-free rate is a measure of the return that investors expect to earn for bearing risk.
This suggests that we should use the difference between the expected return, E (R
i
), and the risk-free rate, Rrf, instead of E (R
i
) alone in the coefficient of variation calculation. With this change, Equation 7.4 would be written as:
Let’s compute this modified coefficient of variation for the AMD and Intel example. If the risk-free rate equals 0.03, or 3 percent, the modified coefficients of variation for the two stocks are:
A popular version of this modified coefficient of variation calculation is known as the Sharpe Ratio. This ratio is named after 1990 Nobel Prize Laureate William Sharpe who developed the concept and was one of the originators of the capital asset pricing model which is discussed in Section 7.7. The Sharpe Ratio is simply the inverse of the modified coefficient of variation:
A measure of the return per unit of risk for an investment
For the stocks of AMD and Intel, the Sharpe Ratios are:
http://en.wikipedia.org/wiki/sharpe-ratio
.
This tells us that investors in AMD stock can expect to earn 0.524 percent for each one standard deviation of return while investors in Intel stock can expect to earn 0.458 percent for each one standard deviation of return. Many people find the Sharpe Ratio to be a more intuitive measure than the coefficient of variation because they find it easier to think about the return per unit of risk than risk per unit of return.
APPLICATION 7.4 LEARNING BY DOING
Calculating and Interpreting the Sharpe Ratio
APPROACH: Use Equation 7.5 to compute the Sharpe Ratios for the two investments.
SOLUTION: The Sharpe Ratios are:
Portfolios with More Than One Asset
It may seem like a good idea to evaluate investments by calculating a measure of risk for each 1 percent of expected return or the expected return per unit of risk. However, the coefficient of variation and the Sharpe Ratio have a critical shortcoming that is not quite evident when we are considering only a single asset. In order to explain this shortcoming, we must discuss the more realistic setting in which an investor has constructed a portfolio with more than one asset.
Expected Return on a Portfolio with more than one Asset
Suppose that you own a portfolio that consists of $500 of AMD stock and $500 of Intel stock and that over the next year you expect to earn returns on the AMD and Intel shares of 9.9 percent and 8.1 percent, respectively. How would you calculate the expected return for the overall portfolio?
Let’s try to answer this question using our intuition. If half of your funds are invested in each stock, it would seem reasonable that the expected return for this portfolio should be a 50-50 mixture of the expected returns from the two stocks, or:
i
represents the fraction of the portfolio invested in asset i. The corresponding equation for a portfolio with n assets is:
To see how Equation 7.6 is used to calculate the expected return on a portfolio with more than two assets, consider an example. Suppose that you were recently awarded a $500,000 grant from a national foundation to pursue your interest in advancing the art of noodling—a popular pastime in some parts of the country in which people catch 40- to 50-pound catfish by putting their hands into catfish holes and wiggling their fingers like noodles to attract the fish.
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Since your grant is intended to support your activities for five years, you kept $100,000 to cover your expenses for the next year and invested the remaining $400,000 in U.S. Treasury bonds and stocks. Specifically, you invested $100,000 in Treasury bonds (TB) that yield 4.5 percent; $150,000 in Procter & Gamble stock (P&G), which has an expected return of 7.5 percent; and $150,000 in Exxon Mobil Corporation stock (EMC), which has an expected return of 9.0 percent. What is the expected return on this $400,000 portfolio?
In order to use Equation 7.6, we must first calculate xi, the fraction of the portfolio invested in asset i, for each investment. These fractions are as follows:
Now that we have calculated the expected return on a portfolio with more than one asset, the next question is how to quantify the risk of such a portfolio. Before we discuss the mechanics of how to do this, it is important to have some intuitive understanding of how volatility in the returns for different assets interact to determine the volatility of the overall portfolio.
APPLICATION 7.5 LEARNING BY DOING
Calculating the Expected Return on a Portfolio
APPROACH: First, calculate the fraction of your portfolio that will be invested in each type of asset after you have diversified. Then use Equation 7.6 to calculate the expected return on the portfolio.
SOLUTION: After you have diversified, 55 percent (100 percent − 45 percent = 55 percent) of your portfolio will be invested in your restaurant, 22.5 percent (45 percent × 0.50 = 22.5 percent) will be invested in the stock market index fund, and 22.5 percent (45 percent × 0.50 = 22.5 percent) will be invested in the bond market index fund. Therefore, from Equation 7.6, we know that the expected return for your portfolio is:
The prices of two stocks in a portfolio will rarely, if ever, change by the same amount and in the same direction at the same time. Normally, the price of one stock will change by more than the price of the other. In fact, the prices of two stocks will frequently move in different directions. These differences in price movements affect the total volatility of the returns for a portfolio.
shows monthly returns for the stock of Southwest Airlines and Netflix over the period from September 2005 through September 2010. Notice that the returns on these shares are generally different and that the prices of the shares can move in different directions in a given month (one stock has a positive return when the other has a negative return). When the stock prices move in opposite directions, the change in the price of one stock off sets at least some of the change in the price of the other stock. As a result, the level of risk for a portfolio of the two stocks is less than the average of the risks associated with the individual shares.
This means that we cannot calculate the variance of a portfolio containing two assets simply by calculating the weighted average of the variances of the individual stocks. We have to account for the fact that the returns on different shares in a portfolio tend to partially offset each other. For a two-asset portfolio, we calculate the variance of the returns using the following formula:
i
represents the fraction of the portfolio invested in stock i, s2R i
is the variance of the return of stock i, and sR1,2 is the covariance of the returns between stocks 1 and 2. The covariance of returns is a measure of how the returns on two assets covary, or move together. The third term in Equation 7.7 accounts for the fact that returns from the two assets will offset each other to some extent. The covariance of returns is calculated using the following formula:
a measure of how the returns on two assets covary, or move together
The returns on two stocks are generally different. In some periods, the return on one stock is positive, while the return on the other is negative. Even when the returns on both are positive or negative, they are rarely exactly the same.
where i represents outcomes rather than assets. Compare this equation with Equation 7.3, reproduced here:
Just as it is difficult to directly interpret the variance of the returns for an asset—recall that the variance is in units of squared dollars—it is difficult to directly interpret the covariance of returns between two assets. We get around this problem by dividing the covariance of returns by the product of the standard deviations of the returns for the two assets. This gives us the correlation, r, between the returns on those assets:
Let’s work an example to see how Equation 7.7 is used to calculate the variance of the returns on a portfolio that consists of 50 percent Southwest Airlines stock and 50 percent Netflix stock. Using the data plotted in
Exhibit 7.6
, we can calculate the variance of the annual returns for the Southwest Airlines and Netflix stocks, , to be 0.1065 and 0.2021, respectively. The covariance between the annual returns on these two stocks is 0.0070. We do not show the calculations for the variances and the covariance because each of these numbers was calculated using 60 different monthly returns. These calculations are too cumbersome to illustrate. Rest assured, however, that they were calculated using Equations 7.3 and 7.8.
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With these values, we can calculate the variance of a portfolio that consists of 50 percent Southwest Airlines (SW) stock and 50 percent Netflix stock as:
If we calculate the standard deviations by taking the square roots of the variances, we find that the standard deviations for Southwest Airlines stock, Netflix stock, and the portfolio consisting of those two stocks are 0.326 (32.6 percent), 0.450 (45.0 percent), and 0.284 (28.4 percent), respectively.
illustrates the monthly returns for the portfolio of Southwest Airlines and Netflix stock, along with the monthly returns for the individual stocks. You can see in this exhibit that, while the returns on the portfolio vary quite a bit, this variation is less than that for the individual company shares.
The variation in the returns from a portfolio that consists of Southwest Airlines and Netflix stock in equal proportions is less than the variation in the returns from either of those stocks alone.
Using Equation 7.9, we can calculate the correlation of the returns between Southwest Airlines and Netflix stock as:
As we add more and more assets to a portfolio, calculating the variance using the approach illustrated in Equation 7.7 becomes increasingly complex. The reason for this is that we have to account for the covariance between each pair of assets. These more extensive calculations are beyond the scope of this book, but they are conceptually the same as those for a portfolio with two assets.
APPLICATION 7.6 LEARNING BY DOING
APPROACH: Use Equation 7.7 to calculate the variance of the portfolio returns and then take the square root of this value to obtain the standard deviation.
SOLUTION: The variance of the portfolio returns is:
Comparing the variance of the portfolio returns of 0.0394 with the variances of the restaurant returns, 0.0625, and the stock market index fund returns, 0.0400, shows once again that a portfolio with two or more assets can have a smaller variance of returns (and thus a smaller standard deviation of returns) than any of the individual assets in the portfolio.
The Limits of Diversification
BUILDING INTUITION DIVERSIFIED PORTFOLIOS ARE LESS RISKY
Diversified portfolios generally have less risk for a given level of return than the individual risky assets in the portfolio. This is because the values of individual assets rarely change by the same amount and in the same direction at the same time. As a result, some of the variation in an asset’s value can be diversified away by owning another asset at the same time. This is important because it tells us that investors can eliminate some of the risk associated with individual investments by holding them in a diversified portfolio.
In the sample calculations for the portfolio containing Southwest Airlines and Netflix stock, we saw that the standard deviation of the returns for a portfolio consisting of equal investments in those two stocks was 28.4 percent from September 2005 through September 2010 and that this figure was lower than the standard deviation of returns for either of the individual stocks (32.6 percent and 45.0 percent). You might wonder how the standard deviation for the portfolio is likely to change if we increase the number of assets in the portfolio. The answer is simple. If the returns on the individual stocks added to our portfolio do not all change in the same way, then increasing the number of stocks in the portfolio will reduce the standard deviation of the portfolio returns even further.
Let’s consider a simple example to illustrate this point. Suppose that all assets have a standard deviation of returns that is equal to 40 percent and that the covariance between the returns for each pair of assets is 0.048. If we form a portfolio in which we have an equal investment in two assets, the standard deviation of returns for the portfolio will be 32.25 percent. If we add a third asset, the portfolio standard deviation of returns will decrease to 29.21 percent. It will be even lower, at 27.57 percent, for a four-asset portfolio.
Exhibit 7.8
illustrates how the standard deviation for the portfolio declines as more stocks are added.
In addition to showing how increasing the number of assets decreases the overall risk of a portfolio,
Exhibit 7.8
illustrates three other very important points. First, the decrease in the standard deviation for the portfolio gets smaller and smaller as more assets are added. You can see this effect by looking at the distance between the straight horizontal line and the plot of the standard deviation of the portfolio returns.
The second important point is that, as the number of assets becomes very large, the portfolio standard deviation does not approach zero. It decreases only so far. In the example in
Exhibit 7.8
, it approaches 21.9 percent. The standard deviation does not approach zero because we are assuming that the variations in the asset returns do not completely cancel each other out. This is a realistic assumption because in practice investors can rarely diversify away all risk. They can diversify away risk that is unique to the individual assets, but they cannot diversify away risk that is common to all assets. The risk that can be diversified away is called unsystematic or diversifiable risk, and the risk that cannot be diversified away is called systematic or nondiversifiable risk. In the next section, we will discuss systematic risk in detail.
unsystematic or diversifiable risk
risk that can be eliminated through diversification
systematic or nondiversifiable risk
risk that cannot be eliminated through diversification
The total risk of a portfolio decreases as the number of assets increases. This is because the amount of unsystematic risk in the portfolio decreases. The diversification benefit from adding another asset declines as the total number of assets in the portfolio increases and the unsystematic risk approaches zero. Most of the diversification benefit can often be achieved with as few as 15 or 20 assets.
The third key point illustrated in
Exhibit 7.8
is that most of the risk-reduction benefits from diversification can be achieved in a portfolio with 15 to 20 assets. Of course, the number of assets required to achieve a high level of diversification depends on the covariances between the assets in the portfolio. However, in general, it is not necessary to invest in a very large number of different assets.
> BEFORE YOU GO ON
1. What does the coefficient of variation tell us, and how is it related to the Sharpe Ratio?
2. What are the two components of total risk?
3. Why does the total risk of a portfolio not approach zero as the number of assets in a portfolio becomes very large?
7.5 SYSTEMATIC RISK
·
The objective of diversification is to eliminate variation in returns that is unique to individual assets. We diversify our investments across a number of different assets in the hope that these unique variations will cancel each other out. With complete diversification, all of the unsystematic risk is eliminated from the portfolio. An investor with a diversified portfolio still faces systematic risk, however, and we now turn our attention to that form of risk.
Why Systematic Risk Is All That Matters
The idea that unsystematic risk can be diversified away has direct implications for the relation between risk and return. If the transaction costs associated with constructing a diversified portfolio are relatively low, then rational, informed investors, such as the students who are taking this class, will prefer to hold diversified portfolios.
Diversified investors face only systematic risk, whereas investors whose portfolios are not well diversified face systematic risk plus unsystematic risk. Because they face less risk, the diversified investors will be willing to pay higher prices for individual assets than the other investors. Therefore, expected returns on individual assets will be lower than the total risk (systematic plus unsystematic risk) of those assets suggests they should be.
To illustrate, consider two individual investors, Emily and Jane. Each of them is trying to decide if she should purchase stock in your pizza restaurant. Emily holds a diversified portfolio and Jane does not. Assume your restaurant’s stock has five units of systematic risk and nine units of total risk. You can see that Emily faces less risk than Jane and will require a lower expected rate of return. Consequently, Emily will be willing to pay a higher price than Jane.
BUILDING INTUITION SYSTEMATIC RISK IS THE RISK THAT MATTERS
The required rate of return on an asset depends only on the systematic risk associated with that asset. Because unsystematic risk can be diversified away, investors can and will eliminate their exposure to this risk. Competition among diversified investors will drive the prices of assets to the point where the expected returns will compensate investors for only the systematic risk that they bear.
If the market includes a large number of diversified investors such as Emily, competition among these investors will drive the price of your restaurant’s shares up further. This competition will ultimately push the price up to the point where the expected return just compensates all investors for the systematic risk associated with your stock. The bottom line is that, because of competition among diversified investors, all investors are only rewarded for bearing systematic risk in asset markets. For this reason, we are concerned only about systematic risk when we think about the relation between risk and return in finance.
Measuring Systematic Risk
If systematic risk is all that matters when we think about expected returns, then we cannot use the standard deviation as a measure of risk.
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The standard deviation is a measure of total risk. We need a way of quantifying the systematic risk of individual assets.
A natural starting point for doing this is to recognize that the most diversified portfolio possible will come closest to eliminating all unsystematic risk. Such a portfolio provides a natural benchmark against which we can measure the systematic risk of an individual asset. What is the most diversified portfolio possible? The answer is simple. It is the portfolio that consists of all assets, including stocks, bonds, real estate, precious metals, commodities, art, baseball cards, and so forth from all over the world. In finance, we call this the market portfolio.
market portfolio
the portfolio of all assets
Unfortunately, we do not have very good data for most of these assets for most of the world, so we use the next best thing: the U.S. public stock market. A large number of stocks from a broad range of industries trade in this market. The companies that issue these stocks own a wide range of assets all over the world. These characteristics, combined with the facts that the U.S. market has been operating for a very long time and that we have very reliable and detailed information on prices for U.S. stocks, make the U.S. stock market a natural benchmark for estimating systematic risk.
Since systematic risk is, by definition, risk that cannot be diversified away, the systematic risk of an individual asset is really just a measure of the relation between the returns on the individual asset and the returns on the market. In fact, systematic risk is often referred to as market risk. To see how we might use data from the U.S. public stock market to estimate the systematic risk of an individual asset, look at
Exhibit 7.9
, which plots 60 historical monthly returns for General Electric Company (GE) against the corresponding monthly returns for the S&P 500 index (a proxy for the U.S. stock market). In this plot, you can see that returns on GE stock tend to be higher when returns on the S&P 500 tend to be higher. The measure of systematic risk that we use in finance is a statistical measure of this relation.
market risk
a term commonly used to refer to nondiversifiable, or systematic, risk
We quantify the relation between the returns on GE stock and the market by finding the slope of the line that best represents the relation illustrated in
Exhibit 7.9
. Specifically, we estimate the slope of the line of best fit. We do this using the statistical technique called regression analysis. If you are not familiar with regression analysis, don’t worry; the details are beyond the scope of this course. All you have to know is that this technique gives us the line that fits the data best.
The monthly returns on General Electric stock are positively related to the returns on the S&P 500 index. In other words, the return on General Electric’s stock tends to be higher when the return on the S&P 500 Index is higher and lower when the return on the S&P 500 index is lower.
The line shown In the exhibit best represents the relation between the monthly returns on General Electric stock and the returns on the S&P 500 index. The slope of this line, which equals 1.61, indicates that the return on General Electric stock tends to equal about 1.61 times the return on the S&P 500 index.
illustrates the line that was estimated for the data in
EXHIBIT 7.9
using regression analysis. Note that the slope of this line is 1.61. Recall from your math classes that the slope of a line equals the ratio of the rise (vertical distance) divided by the corresponding run (horizontal distance). In this case, the slope is the change in the return on GE stock divided by the change in the return on the U.S. stock market. A slope of 1.61 therefore means that, on average, the change in the return on GE stock was 1.61 times as large as the change in the return on the S&P 500 index. Thus, if the S&P 500 index goes up 1 percent, the average increase in GE’s stock is 1.61 percent. This is a measure of systematic risk because it tells us that the volatility of the returns on GE stock is 1.61 times as large as that for the S&P 500 as a whole.
To explore this idea more completely, let’s consider another, simpler example. Suppose that you have data for Nike stock and for the U.S. stock market (S&P 500 index) for each of the past two years. In the first year, the return on the market was 10 percent, and the return on Nike stock was 15 percent. In the second year, the return on the market was 12 percent, and the return on Nike stock was 19 percent. From this information, we know that the return on Nike stock increased by 4 percent while the return on the market increased 2 percent. If we plotted the returns for Nike stock and for the market for each of the last two periods, as we did for GE stock and the market in
Exhibits 7.9
and
7.10
, and estimated the line that best fit the data, it would be a line that connected the dots for the two periods. The slope of this line would equal 2, calculated as follows:
In finance, we call the slope of the line of best fit beta. Often we simply use the corresponding Greek letter, β, to refer to this measure of systematic risk. As shown below, a beta of 1 tells us that an asset has just as much systematic risk as the market. A beta higher than or lower than 1 tells us that the asset has more or less systematic risk than the market, respectively. A beta of 0 indicates a risk-free security, such as a U.S. Treasury bill.
beta (β)
a measure of nondiversifiable, systematic, or market, risk
Exhibit 7.10
, the line of best fit does not go through each data point. That is because some of the change in GE’s stock price each month reflected information that did not affect the S&P 500 as a whole. That information is the unsystematic component of the risk of GE’s stock. The vertical distance between each data point and the line of best fit represents variation in GE’s stock return that can be attributed to this unsystematic risk.
A convenient place to find betas for individual companies is MSN Money Central at
http://moneycentral.msn.com
. Just enter the stock symbol in the “Get Quote” box near the top of the page and hit “Enter” on your computer (try the railroad company CSX, for example). You will get prices, an estimate of the beta, and other financial information.
The positive slope (β) of the regression line in
Exhibit 7.10
tells us that returns for the S&P 500 and for GE stock will tend to move in the same direction. The return on the S&P 500 and the return on GE’s stock will not always change in the same direction, however, because the unsystematic risk associated with GE stock can more than offset the effect of the market in any particular period. In the next section, we will discuss the implications of beta for the level (as opposed to the change) in the expected return for a stock such as GE.
> BEFORE YOU GO ON
1. Why are returns on the stock market used as a benchmark in measuring systematic risk?
2. How is beta estimated?
3. How would you interpret a beta of 1.5 for an asset? A beta of 0.75?
7.6 COMPENSATION FOR BEARING SYSTEMATIC RISK
Now that we have identified the measure of the risk that diversified investors care about—systematic risk—we are in a position to examine how this measure relates to expected returns. Earlier, in our discussion of the coefficient of variation and the Sharpe Ratio, we asserted that the expected return over and above the risk-free rate is the return that investors expect to earn for bearing risk. To see why this must be true, think about the rate of return that you would require for an investment. First, you would want to make sure that you were compensated for inflation. It would not make sense to invest if you expected the investment to return an amount that did not at least allow you to have the same purchasing power that the money you invested had when you made the investment. Second, you would want some compensation for the fact that you are giving up the use of your money for a period of time. This compensation may be very small if you are forgoing the use of your money for only a short time, such as when you invest in a 30-day Treasury bill, but it might be relatively large if you are investing for several years. Finally, you would also require compensation for the systematic risk associated with the investment.
When you invest in a U.S. government security such as a Treasury bill, note, or bond, you are investing in a security that has no risk of default. After all, the U.S. government can always increase taxes or print more money to pay you back. Changes in economic conditions and other factors that affect the returns on other assets do not affect the default risk of U.S. government securities. As a result, these securities do not have systematic risk, and their returns can be viewed as risk free. In other words, returns on government bonds reflect the compensation required by investors to account for the impact of inflation on purchasing power and for their inability to use the money during the life of the investment.
It follows that the difference between required returns on government securities and required returns for risky investments represents the compensation investors require for taking risk. Recognizing this allows us to write the expected return for an asset i as:
i
is the number of units of systematic risk associated with asset i. Finally, if beta, β, is the appropriate measure for the number of units of systematic risk, we can also define compensation for taking risk as follows:
i
is the beta for asset i.
Remember that beta is a measure of systematic risk that is directly related to the risk of the market as a whole. If the beta for an asset is 2, that asset has twice as much systematic risk as the market. If the beta for an asset is 0.5, then the asset has half as much systematic risk as the market. Recognizing this natural interpretation of beta suggests that the appropriate “unit of systematic risk” is the level of risk in the market as a whole and that the appropriate “compensation per unit of systematic risk” is the expected return required for the level of systematic risk in the market as a whole. The required rate of return on the market, over and above that of the risk-free return, represents compensation required by investors for bearing a market (systematic) risk. This suggests that:
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In deriving Equation 7.10, we intuitively arrived at the Capital Asset Pricing Model (CAPM). Equation 7.10 is the CAPM, a model that describes the relation between risk and expected return. We will discuss the predictions of the CAPM in more detail shortly, but first let’s look more closely at how it works.
Capital Asset Pricing Model (CAPM)
a model that describes the relation between risk and expected return
Suppose that you want to estimate the expected return for a stock that has a beta of 1.5 and that the expected return on the market and risk-free rate are 10 percent and 4 percent, respectively. We can use Equation 7.10 (the CAPM) to find the expected return for this stock:
While the expected return on the market is known in the above example, we actually cannot observe it in practice. For this reason, financial analysts estimate the market risk premium using historical data. We discuss how they do this in
Chapter 13
.
The Security Market Line
a plot of the relation between expected return and systematic risk
displays a plot of Equation 7.10 to illustrate how the expected return on an asset varies with systematic risk. This plot shows that the relation between the expected return on an asset and beta is positive and linear. In other words, it is a straight line with a positive slope. The line in
Exhibit 7.11
is known as the Security Market Line (SML).
The Security Market Line (SML) is the line that shows the relation between expected return and systematic risk, as measured by beta. When beta equals zero and there is no systematic risk, the expected return equals the risk-free rate. As systematic risk (beta) increases, the expected return increases. This is an illustration of the positive relation between risk and return. The SML shows that it is systematic risk that matters to investors.
In
Exhibit 7.11
you can see that the expected rate of return equals the risk-free rate when beta equals 0. This makes sense because when investors do not face systematic risk, they will only require a return that reflects the expected rate of inflation and the fact that they are giving up the use of their money for a period of time.
Exhibit 7.11
also shows that the expected return on an asset equals the expected return on the market when beta equals 1. This is not surprising given that both the asset and the market would have the same level of systematic risk if this were the case.
It is important to recognize that the SML illustrates what the CAPM predicts the expected total return should be for various values of beta. The actual expected total return depends on the price of the asset. You can see this from Equation 7.1:
Exhibit 7.11
. This means that the asset’s price is lower than the CAPM suggests it should be. Conversely, if the expected return on an asset plots below the SML, this implies that the asset’s price is higher than the CAPM suggests it should be. The point at which a particular asset plots relative to the SML, then, tells us something about whether the price of that asset might be low or high. Recognizing this fact can be helpful in evaluating the attractiveness of an investment such as the General Electric stock in Learning by Doing Application 7.7.
APPLICATION 7.7 LEARNING BY DOING
Expected Returns and Systematic Risk
APPROACH: Use Equation 7.10 to calculate the expected return on General Electric stock.
SOLUTION: The expected return is:
The expected return for a portfolio can also be predicted using the CAPM. The expected return on a portfolio with n assets is calculated using the relation:
The fact that the SML is a straight line turns out to be rather convenient if we want to estimate the beta for a portfolio. Recall that the equation for the expected return for a portfolio with n assets was given by Equation 7.6:
i
is the proportion of the portfolio value that is invested in asset i, β i is the beta of asset i, and n is the number of assets in the portfolio. This formula makes it simple to calculate the beta of any portfolio of assets once you know the betas of the individual assets. As an exercise, you might prove this to yourself by using Equations 7.6 and 7.10 to derive Equation 7.11.
Let’s consider an example to see how Equation 7.11 is used. Suppose that you invested 25 percent of your wealth in a fully diversified market fund, 25 percent in risk-free Treasury bills, and 50 percent in a house with twice as much systematic risk as the market. What is the beta of your overall portfolio? What rate of return would you expect to earn from this portfolio if the risk-free rate is 4 percent and the market risk premium is 6 percent?
We know that the beta for the market must equal 1 by definition and that the beta for a risk-free asset equals 0. The beta for your home must be 2 since it has twice the systematic risk of the market. Therefore, the beta of your portfolio is:
Chapter 13
we will explore the relation between the expected return and the rate used to discount project cash flows in much more detail. By the time we finish that discussion, you will understand thoroughly how businesses determine the rate that they use to discount the cash flows from their investments.
APPLICATION 7.8 LEARNING BY DOING
While the cottages cost the same, they are different distances from campus and in different neighborhoods. You believe that this causes them to have different levels of systematic risk, and you estimate that the betas for the individual cottages are 1.2, 1.3, and 1.5. If the risk-free rate is 4 percent and the market risk premium is 6 percent, what will be the expected return on your real estate portfolio after you make all three investments?
APPROACH: There are two approaches that you can use to solve this problem. First, you can estimate the expected return for each cottage using Equation 7.10 and then calculate the expected return on the portfolio using Equation 7.6. Alternatively, you can calculate the beta for the portfolio using Equation 7.11 and then use Equation 7.10 to calculate the expected return.
SOLUTION: Using the first approach, we find that Equation 7.10 gives us the following expected returns:
Choosing between Two Investments
DECISION: You should not invest in either stock. The expected returns for both of them are below the values predicted by the CAPM for investments with the same level of risk. In other words, both would plot below the line in
Exhibit 7.11
. This implies that they are both overpriced.
> BEFORE YOU GO ON
1. How is the expected return on an asset related to its systematic risk?
2. What name is given to the relation between risk and expected return implied by the CAPM?
3. If an asset’s expected return does not plot on the line in question 2 above, what does that imply about its price?
SUMMARY OF Learning Objectives
Investors require greater returns for taking greater risk. They prefer the investment with the highest possible return for a given level of risk or the investment with the lowest risk for a given level of return.
Describe the two components of a total holding period return, and calculate this return for an asset.
The total holding period return on an investment consists of a capital appreciation component and an income component. This return is calculated using Equation 7.1. It is important to recognize that investors do not care whether they receive a dollar of return through capital appreciation or as a cash dividend. Investors value both sources of return equally.
Explain what an expected return is and calculate the expected return for an asset.
The expected return is a weighted average of the possible returns from an investment, where each of these returns is weighted by the probability that it will occur. It is calculated using Equation 7.2.
Explain what the standard deviation of returns is and why it is very useful in finance, and calculate it for an asset.
The standard deviation of returns is a measure of the total risk associated with the returns from an asset. It is useful in evaluating returns in finance because the returns on many assets tend to be normally distributed. The standard deviation of returns provides a convenient measure of the dispersion of returns. In other words, it tells us about the probability that a return will fall within a particular distance from the expected value or within a particular range. To calculate the standard deviation, the variance is first calculated using Equation 7.3. The standard deviation of returns is then calculated by taking the square root of the variance.
Explain the concept of diversification.
Diversification is reducing risk by investing in two or more assets whose values do not always move in the same direction at the same time. Investing in a portfolio containing assets whose prices do not always move together reduces risk because some of the changes in the prices of individual assets offset each other. This can cause the overall volatility in the value of an investor’s portfolio to be lower than if it consisted of only a single asset.
Discuss which type of risk matters to investors and why.
Investors care about only systematic risk. This is because they can eliminate unsystematic risk by holding a diversified portfolio. Diversified investors will bid up prices for assets to the point at which they are just being compensated for the systematic risks they must bear.
Describe what the Capital Asset Pricing Model (CAPM) tells us and how to use it to evaluate whether the expected return of an asset is sufficient to compensate an investor for the risks associated with that asset.
The CAPM tells us that the relation between systematic risk and return is linear and that the risk-free rate of return is the appropriate return for an asset with no systematic risk. From the CAPM we know what rate of return investors will require for an investment with a particular amount of systematic risk (beta). This means that we can use the expected return predicted by the CAPM as a benchmark for evaluating whether expected returns for individual assets are sufficient. If the expected return for an asset is less than that predicted by the CAPM, then the asset is an unattractive investment because its return is lower than the CAPM indicates it should be. By the same token, if the expected return for an asset is greater than that predicted by the CAPM, then the asset is an attractive investment because its return is higher than it should be.
SUMMARY OF Key Equations
· 7.1 Kaaran made a friendly wager with a colleague that involves the result from flipping a coin. If heads comes up, Kaaran must pay her colleague $15; otherwise, her colleague will pay Kaaran $15. What is Kaaran’s expected cash flow, and what is the variance of that cash flow if the coin has an equal probability of coming up heads or tails? Suppose Kaaran’s colleague is willing to handicap the bet by paying her $20 if the coin toss results in tails. If everything else remains the same, what are Kaaran’s expected cash flow and the variance of that cash flow?
· 7.2 You know that the price of CFI, Inc., stock will be $12 exactly one year from today. Today the price of the stock is $11. Describe what must happen to the price of CFI, Inc., today in order for an investor to generate a 20 percent return over the next year. Assume that CFI does not pay dividends.
· 7.3 The expected value of a normal distribution of prices for a stock is $50. If you are 90 percent sure that the price of the stock will be between $40 and $60, then what is the variance of the stock price?
· 7.4 You must choose between investing in stock A or stock B. You have already used CAPM to calculate the rate of return you should expect to receive for each stock given their systematic risk and decided that the expected return for both exceeds that predicted by CAPM by the same amount. In other words, both are equally attractive investments for a diversified investor. However, since you are still in school and do not have a lot of money, your investment portfolio is not diversified. You have decided to invest in the stock that has the highest expected return per unit of total risk. If the expected return and standard deviation of returns for stock A are 10 percent and 25 percent, respectively, and the expected return and standard deviation of returns for stock B are 15 percent and 40 percent, respectively, which should you choose? Assume that the risk-free rate is 5 percent.
· 7.5 CSB, Inc., has a beta of 1.35. If the expected market return is 14.5 percent and the risk-free rate is 5.5 percent, what is the appropriate required return of CSB (using the CAPM)?
Solutions to Self-Study Problems
· 7.1
· 7.2 The expected return for CFI based on today’s stock price is ($12 − $11)/$11 = 9.09 percent, which is lower than 20 percent. Since the stock price one year from today is fixed, the only way that you will generate a 20 percent return is if the price of the stock drops today. Consequently, the price of the stock today must drop to $10. It is found by solving the following: 0.2 = ($12 − x)/x, or x = $10.
· 7.3 Since you know that 1.645 standard deviations around the expected return captures 90 percent of the distribution, you can set up either of the following equations:
· 7.5 E(RCSB = Rrf + β
CSB[E(RM) − Rrf] = 0.055 + [1.35 × (0.145 − 0.055)] = 0.1765 or, 17.65%
Critical Thinking Questions
· 7.1 Given that you know the risk as well as the expected return for two stocks, discuss what process you might utilize to determine which of the two stocks is a better buy. You may assume that the two stocks will be the only assets held in your portfolio.
· 7.2 What is the difference between the expected rate of return and the required rate of return? What does it mean if they are different for a particular asset at a particular point in time?
· 7.3 Suppose that the standard deviation of the returns on the shares of stock at two different companies is exactly the same. Does this mean that the required rate of return will be the same for these two stocks? How might the required rate of return on the stock of a third company be greater than the required rates of return on the stocks of the first two companies even if the standard deviation of the returns of the third company’s stock is lower?
· 7.4 The correlation between stocks A and B is 0.50, while the correlation between stocks A and C is −0.5. You already own stock A and are thinking of buying either stock B or stock C. If you want your portfolio to have the lowest possible risk, would you buy stock B or C? Would you expect the stock you choose to affect the return that you earn on your portfolio?
· 7.5 The idea that we can know the return on a security for each possible outcome is overly simplistic. However, even though we cannot possibly predict all possible outcomes, this fact has little bearing on the risk-free return. Explain why.
· 7.6 Which investment category has shown the greatest degree of risk in the United States since 1926? Explain why that makes sense in a world where the value of an asset in this investment category is likely to be more adversely affected by a particular negative event than the price of a corporate bond. Use the same type of explanation to help explain other investment choices since 1926.
· 7.7 You are concerned about one of the investments in your fully diversified portfolio. You just have an uneasy feeling about the CFO, I am Shifty, of that particular firm. You do believe, however, that the firm makes a good product and that it is appropriately priced by the market. Should you be concerned about the effect on your portfolio if Shifty embezzles a portion of the firm’s cash?
· 7.8 The CAPM is used to price the risk in any asset. Our examples have focused on stocks, but we could also price the expected rate of return for bonds. Explain how debt securities are also subject to systematic risk.
· 7.9 In recent years, investors have correctly agreed that the market portfolio consists of more than just a group of U.S. stocks and bonds. If you are an investor who invests in only U.S. stocks, describe the effects on the risk in your portfolio.
· 7.10 You may have heard the statement that you should not include your home as an asset in your investment portfolio. Assume that your house will comprise up to 75 percent of your assets in the early part of your investment life. Evaluate the implications of omitting it from your portfolio when calculating the risk of your overall investment portfolio.
Questions and Problems
· 7.2 Expected returns: John is watching an old game show rerun on television called Let’s Make a Deal in which the contestant chooses a prize behind one of two curtains. Behind one of the curtains is a gag prize worth $150, and behind the other is a round-the-world trip worth $7,200. The game show has placed a subliminal message on the curtain containing the gag prize, which makes the probability of choosing the gag prize equal to 75 percent. What is the expected value of the selection, and what is the standard deviation of that selection?
· 7.3 Expected returns: You have chosen biology as your college major because you would like to be a medical doctor. However, you find that the probability of being accepted to medical school is about 10 percent. If you are accepted to medical school, then your starting salary when you graduate will be $300,000 per year. However, if you are not accepted, then you would choose to work in a zoo, where you will earn $40,000 per year. Without considering the additional educational years or the time value of money, what is your expected starting salary as well as the standard deviation of that starting salary?
· 7.4 Historical market: Describe the general relation between risk and return that we observe in the historical bond and stock market data.
· 7.5 Single-asset portfolios: Stocks A, B, and C have expected returns of 15 percent, 15 percent, and 12 percent, respectively, while their standard deviations are 45 percent, 30 percent, and 30 percent, respectively. If you were considering the purchase of each of these stocks as the only holding in your portfolio and the risk-free rate is 0 percent, which stock should you choose?
· 7.6 Diversification: Describe how investing in more than one asset can reduce risk through diversification.
· 7.7 Systematic risk: Define systematic risk.
· 7.8 Measuring systematic risk: Susan is expecting the returns on the market portfolio to be negative in the near term. Since she is managing a stock mutual fund, she must remain invested in a portfolio of stocks. However, she is allowed to adjust the beta of her portfolio. What kind of beta would you recommend for Susan’s portfolio?
· 7.9 Measuring systematic risk: Describe and justify what the value of the beta of a U.S. Treasury bill should be.
· 7.10 Measuring systematic risk: If the expected rate of return for the market is not much greater than the risk-free rate of return, what is the general level of compensation for bearing systematic risk?
· 7.11 CAPM: Describe the Capital Asset Pricing Model (CAPM) and what it tells us.
· 7.12 The Security market line: If the expected return on the market is 10 percent and the risk-free rate is 4 percent, what is the expected return for a stock with a beta equal to 1.5? What is the market risk premium for the set of circumstances described?
· 7.14 Interpreting the variance and standard deviation: The distribution of grades in an introductory finance class is normally distributed, with an expected grade of 75. If the standard deviation of grades is 7, in what range would you expect 95 percent of the grades to fall?
· 7.15 Calculating the variance and standard deviation: Kate recently invested in real estate with the intention of selling the property one year from today. She has modeled the returns on that investment based on three economic scenarios. She believes that if the economy stays healthy, then her investment will generate a 30 percent return. However, if the economy softens, as predicted, the return will be 10 percent, while the return will be 25 percent if the economy slips into a recession. If the probabilities of the healthy, soft, and recessionary states are 0.4, 0.5, and 0.1, respectively, then what are the expected return and the standard deviation of the return on Kate’s investment?
· 7.16 Calculating the variance and standard deviation: Barbara is considering investing in a stock and is aware that the return on that investment is particularly sensitive to how the economy is performing. Her analysis suggests that four states of the economy can affect the return on the investment. Using the table of returns and probabilities below, find the expected return and the standard deviation of the return on Barbara’s investment.
· 7.19 Portfolios with more than one asset: Emmy is analyzing a two-stock portfolio that consists of a Utility stock and a Commodity stock. She knows that the return on the Utility stock has a standard deviation of 40 percent and the return on the Commodity stock has a standard deviation of 30 percent. However, she does not know the exact covariance in the returns of the two stocks. Emmy would like to plot the variance of the portfolio for each of three cases—covariance of 0.12, 0, and 0.12—in order to understand how the variance of such a portfolio would react. Do the calculation for all three cases (0.12, 0, and 0.12), assuming an equal proportion of each stock in the portfolio.
· 7.20 Portfolios with more than one asset: Given the returns and probabilities for the three possible states listed below, calculate the covariance between the returns of Stock A and Stock B. For convenience, assume that the expected returns of Stock A and Stock B are 11.75 percent and 18 percent, respectively.
· 7.22 Compensation for bearing systematic risk: Write out the equation for the covariance in the returns of two assets, Asset 1 and Asset 2. Using that equation, explain the easiest way for the two asset returns to have a covariance of zero.
· 7.23 Compensation for bearing systematic risk: Evaluate the following statement: By fully diversifying a portfolio, such as by buying every asset in the market, we can completely eliminate all types of risk, thereby creating a synthetic Treasury bill.
· 7.24 CAPM: Damien knows that the beta of his portfolio is equal to 1, but he does not know the risk-free rate of return or the market risk premium. He also knows that the expected return on the market is 8 percent. What is the expected return on Damien’s portfolio?
· 7.25 CAPM: In February 2011 the risk-free rate was 4.75 percent, the market risk premium was 6 percent, and the beta for Dell stock was 1.31. What is the expected return that was consistent with the systematic risk associated with the returns on Dell stock?
· 7.26 CAPM: The market risk premium is 6 percent, and the risk-free rate is 5 percent. If the expected return on a bond is 6.5 percent, what is its beta?
· 7.28 Sumeet knows that the covariance in the return on two assets is 0.0025. Without knowing the expected return of the two assets, explain what that covariance means.
· 7.29 In order to fund her retirement, Glenda requires a portfolio with an expected return of 12 percent per year over the next 30 years. She has decided to invest in Stocks 1, 2, and 3, with 25 percent in Stock 1, 50 percent in Stock 2, and 25 percent in Stock 3. If Stocks 1 and 2 have expected returns of 9 percent and 10 percent per year, respectively, then what is the minimum expected annual return for Stock 3 that will enable Glenda to achieve her investment requirement?
· 7.30 Tonalli is putting together a portfolio of 10 stocks in equal proportions. What is the relative importance of the variance for each stock versus the covariance for the pairs of stocks in her portfolio? For this exercise, ignore the actual values of the variance and covariance terms and explain their importance conceptually.
· 7.31 Explain why investors who have diversified their portfolios will determine the price and, consequently, the expected return on an asset.
· 7.32 Brad is about to purchase an additional asset for his well-diversified portfolio. He notices that when he plots the historical returns of the asset against those of the market portfolio, the line of best fit tends to have a large amount of prediction error for each data point (the scatter plot is not very tight around the line of best fit). Do you think that this will have a large or a small impact on the beta of the asset? Explain your opinion.
· 7.33 The beta of an asset is equal to 0. Discuss what the asset must be.
· 7.34 The expected return on the market portfolio is 15 percent, and the return on the risk-free security is 5 percent. What is the expected return on a portfolio with a beta equal to 0.5?
· 7.35 Draw the Security Market Line (SML) for the case where the market risk premium is 5 percent and the risk-free rate is 7 percent. Now suppose an asset has a beta of 1.0 and an expected return of 4 percent. Plot it on your graph. Is the security properly priced? If not, explain what we might expect to happen to the price of this security in the market. Next, suppose another asset has a beta of 3.0 and an expected return of 20 percent. Plot it on the graph. Is this security properly priced? If not, explain what we might expect to happen to the price of this security in the market.
· 7.36 If the CAPM describes the relation between systematic risk and expected returns, can both an individual asset and the market portfolio of all risky assets have negative expected real rates of return? Why or why not?
· 7.37 You have been provided the following data on the securities of three firms and the market:
Sample Test Problems
· 7.1 Friendly Airlines stock is selling at a current price of $37.50 per share. If the stock does not pay a dividend and has a 12 percent expected return, what is the expected price of the stock one year from today?
· 7.2 Stefan’s parents are about to invest their nest egg in a stock that he has estimated to have an expected return of 9 percent over the next year. If the return on the stock is normally distributed with a 3 percent standard deviation, in what range will the stock return fall 95 percent of the time?
· 7.3 Elaine has narrowed her investment alternatives to two stocks (at this time she is not worried about diversifying): Stock M, which has a 23 percent expected return, and Stock Y, which has an 8 percent expected return. If Elaine requires a 16 percent return on her total investment, then what proportion of her portfolio will she invest in each stock?
· 7.4 You have just prepared a graph similar to
Exhibit 7.9
, comparing historical data for Pear Computer Corp. and the general market. When you plot the line of best fit for these data, you find that the slope of that line is 2.5. If you know that the market generated a return of 12 percent and that the risk-free rate is 5 percent, then what would your best estimate be for the return of Pear Computer during that same time period?
· 7.5 The CAPM predicts that the return of MoonBucks Tea Corp. is 23.6 percent. If the risk-free rate of return is 8 percent and the expected return on the market is 20 percent, then what is MoonBucks’ beta?
For simplicity, let’s ignore the possibility of your hitting a sacrifice fly and other such outcomes.
From a practical standpoint, it would not really have been possible to grow $1.00 to $12,231.13 by investing in small U.S. stocks because this increase assumes that an investor is able to rebalance the stock portfolio by buying and selling shares as necessary at no cost. Since buying and selling shares is costly, the final wealth would have been lower. Nevertheless, even after transaction costs, it would have been much more profitable to invest in small stocks than in U.S. Treasury bills.
On August 5, 2011, Standard and Poor’s, the credit rating agency, lowered its rating on U.S. Treasury securities from AAA to AA+, indicating that it considered these securities to have a very small amount of default risk. The other two large credit rating agencies, Moody’s and Fitch, decided not to lower their ratings of U.S. Treasury securities at that time, suggesting that if there was any default risk, it remained extremely small.
For more information on noodling, see the April 21, 2006, New York Times article titled “In the Jaws of a Catfish,” by Ethan Todras-Whitehill and the May 16, 2011, Wall Street Journal article titled “Long Arm of the Law Penalizes Texans Who Nab Catfish by Hand,” by Ana Campoy.
The only adjustment that we had to make was to account for the fact that our calculations used monthly returns rather than annual returns. This adjustment simply required us to multiply each number we calculated by 12 because there are 12 months in a year.
This statement is true in the context of how expected returns are determined. However, the standard deviation is still a very useful measure of the risk faced by an individual investor who does not hold a diversified portfolio. For example, the owners of most small businesses have much of their personal wealth tied up in their businesses. They are certainly concerned about the total risk because it is directly related to the probability that they will go out of business and lose much of their wealth.
10
The Fundamentals of Capital Budgeting
Learning Objectives
Discuss why capital budgeting decisions are the most important investment decisions made by a firm’s management.
Explain the benefits of using the net present value (NPV) method to analyze capital expenditure decisions and calculate the NPV for a capital project.
Describe the strengths and weaknesses of the payback period as a capital expenditure decision-making tool and compute the payback period for a capital project.
Explain why the accounting rate of return (ARR) is not recommended for use as a capital expenditure decision-making tool.
Compute the internal rate of return (IRR) for a capital project and discuss the conditions under which the IRR technique and the NPV technique produce different results.
Explain the benefits of postaudit and ongoing reviews of capital projects.
In October 2010, Intel Corporation’s newly built 300-mm wafer fabrication plant in Dalian China began regular production. The Dalian plant is Intel’s first chipset factory in Asia and is part of a network of eight such facilities worldwide. A $2.5 billion project that was expected to eventually employ 4,000 people, the Dalian plant is Intel’s single largest investment in China.
Even before construction of the fabrication plant was completed, Intel was developing plans for a $3.5 billion addition to the project. This second phase would consist of investments in supporting facilities, which include four assembly and testing plants, an R&D center, and a sales division. To place the anticipated $6.0 billion cost of the two Dalian project phases in perspective, Intel’s total worldwide capital expenditures during fiscal year 2008 were $5.2 billion.
Intel’s investment in the Dalian fabrication plant illustrates not only the large amount of corporate resources that can be committed to a major capital project, but also the strategic importance such an investment can have. When the project was originally announced in March 2007, Intel’s president and CEO, Paul Otellini, pointed out that China was Intel’s fastest-growing major market. It was imperative, said Otellini, that Intel make investments in markets that will provide for future growth.
The Dalian project involves significant risks for Intel. Construction of the fabrication plant was a three-year endeavor that unexpectedly coincided with a worldwide financial and economic crisis. A project of Dalian’s scale represented a potential source of excess capacity for Intel as the growth in demand for chipsets declined worldwide during 2008 and 2009. In addition, since international investments are subject to country risk, political or economic instability in China could have severe financial consequences for Intel.
While few companies make capital investments as large as Intel’s investment in Dalian China, all companies must routinely invest capital in projects that are critical to their success. These investment opportunities must be carefully scrutinized, and their costs and benefits carefully weighed. How do firms make these important capital budgeting decisions? In this chapter we examine this decision-making process and introduce some financial models used to make investment decisions.
CHAPTER PREVIEW
This chapter is about capital budgeting, a topic we first visited in
Chapter 1
. Capital budgeting is the process of deciding which capital investments the firm should make.
We begin the chapter with a discussion of the types of capital projects that firms undertake and how the capital budgeting process is managed within the firm. When making capital investment decisions, management’s goal is to select projects that will increase the value of the firm.
Next we examine some of the techniques used to evaluate capital budgeting decisions. We first discuss the net present value (NPV) method, which is the capital budgeting approach recommended in this book. The NPV method takes into account the time value of money and provides a direct measure of how much a capital project will increase the value of the firm.
We then examine the payback method and the accounting rate of return. As methods of selecting capital projects, both methods have some serious deficiencies. Finally, we discuss the internal rate of return (IRR), which is the expected rate of return for a capital project when the project’s NPV is equal to zero. The IRR is a very popular and important alternative to the NPV technique. However, in certain circumstances, the IRR can lead to incorrect decisions. We close by discussing evidence on techniques financial managers actually use when making capital budgeting decisions.
10.1 AN INTRODUCTION TO CAPITAL BUDGETING
·
We begin with an overview of capital budgeting, followed by a discussion of some important concepts you will need to understand in this and later chapters.
The Importance of Capital Budgeting
the process of choosing the productive assets in which the firm will invest
Capital budgeting decisions are the most important investment decisions made by management. The objective of these decisions is to select investments in productive assets that will increase the value of the firm. These investments create value when they are worth more than they cost. Capital investments are important because they can involve substantial cash outlays and, once made, are not easily reversed. They also define what the company is all about—the firm’s lines of business and its inherent business risk. For better or worse, capital investments produce most of a typical firm’s revenues for years to come.
Capital budgeting techniques help management systematically analyze potential business opportunities in order to decide which are worth undertaking. As you will see, not all capital budgeting techniques are equal. The best techniques are those that determine the value of a capital project by discounting all of the cash flows generated by the project and thus account for the time value of money. We focus on these techniques in this chapter.
In the final analysis, capital budgeting is really about management’s search for the best capital projects—those that add the greatest value to the firm. Over the long term, the most successful firms are those whose managements consistently search for and find capital investment opportunities that increase firm value.
The Capital Budgeting Process
The capital budgeting process starts with a firm’s strategic plan, which spells out its strategy for the next three to five years. Division managers then convert the firm’s strategic objectives into business plans. These plans have a one- to two-year time horizon, provide a detailed description of what each division should accomplish during the period covered by the plan, and have quantifiable targets that each division is expected to achieve. Behind each division’s business plan is a capital budget that details the resources management believes it needs to get the job done.
The capital budget is generally prepared jointly by the CFO’s staff and financial staff s at the divisional and lower levels and reflects, in large part, the activities outlined in the divisional business plans. Many of these proposed expenditures are routine in nature, such as the repair or purchase of new equipment at existing facilities. Less frequently, firms face broader strategic decisions, such as whether to launch a new product, build a new plant, enter a new market, or buy a business.
Exhibit 10.1
identifies some reasons that firms initiate capital projects.
EXHIBIT 10.1 Key Reasons for Making Capital Expenditures
Capital budgeting decisions are the most important investment decisions made by management. Many of these decisions are routine in nature, but from time to time, managers face broader strategic decisions that call for significant capital investments.
Where does a firm get all of the information it needs to make capital budgeting decisions? Most of the information is generated within the firm, and, for expansion decisions, it often starts with sales representatives and marketing managers who are in the marketplace talking to potential and current customers on a day-to-day basis. For example, a sales manager with a new product idea might present the idea to management and the marketing research group. If the product looks promising, the marketing research group will estimate the size of the market and a market price. If the product requires new technology, the firm’s research and development group must decide whether to develop the technology or to buy it. Next, cost accountants and production engineers determine the cost of producing the product and any capital expenditures necessary to manufacture it. Finally, the CFO’s staff takes the data and estimates the cost of the project and the cash flows it will generate over time. The project is a viable candidate for the capital budget if the present value of the expected cash benefits exceeds the project’s cost.
Classification of Investment Projects
Potential capital budgeting projects can be classified into three types: (1) independent projects, (2) mutually exclusive projects, and (3) contingent projects.
Independent Projects
Projects are independent when their cash flows are unrelated. With independent projects, accepting or rejecting one project does not eliminate other projects from consideration (assuming the firm has unlimited funds to invest). For example, suppose a firm has unlimited funding and management wants to: (1) build a new parking ramp at its headquarters; (2) acquire a small competitor; and (3) add manufacturing capacity to one of its plants. Since the cash flows for each project are unrelated, accepting or rejecting one of the projects will have no effect on the others.
independent projects
projects whose cash flows are unrelated
Mutually Exclusive Projects
projects for which acceptance of one precludes acceptance of the other
When projects are mutually exclusive, acceptance of one project precludes acceptance of others. Typically, mutually exclusive projects perform the same function, and thus, only one project needs to be accepted. For example, when BMW decided to manufacture automobiles in the United States, it considered three possible manufacturing sites (or capital projects). Once BMW management selected the Spartanburg, South Carolina, site, the other two possible locations were out of the running. Since some projects are mutually exclusive, it is very important that a capital budgeting method allow us to choose the best project when we are faced with two or more alternatives.
Contingent Projects
With contingent projects, the acceptance of one project is contingent on the acceptance of another. There are two types of contingency situations. In the first type of situation, the contingent product is mandatory. For example, when a public utility company (such as your local electric company) builds a power plant, it must also invest in suitable pollution control equipment to meet federal environmental standards. The pollution control investment is a mandatory contingent project. When faced with mandatory contingent projects, it is best to treat all of the projects as a single investment for the purpose of evaluation. This provides management with the best measure of the value created by these projects.
In the second type of situation, the contingent project is optional. For example, suppose Dell invests in a new computer for the home market. This computer has a feature that allows Dell to bundle a proprietary gaming system. The gaming system is a contingent project but is an optional add-on to the new computer. In these situations, the optional contingent project should be evaluated independently and should be accepted or rejected on its own merits.
contingent projects
projects whose acceptance depends on the acceptance of other projects
Basic Capital Budgeting Terms
In this section we briefly introduce two terms that you will need to be familiar with—cost of capital and capital rationing.
the required rate of return for a capital investment
The cost of capital is the rate of return that a capital project must earn to be accepted by management. The cost of capital can be thought of as an opportunity cost. Recall from
Chapter 8
that an opportunity cost is the value of the most valuable alternative given up if a particular investment is made.
Let’s consider the opportunity cost concept in the context of capital budgeting decisions. When investors buy shares of stock in a company or loan money to a company, they are giving management money to invest on their behalf. Thus, when a firm’s management makes capital investments in productive assets such as plant and equipment they are investing stockholders’ and creditors’ money in real assets. Since stockholders and creditors could have invested their money in financial assets, the minimum rate of return they are willing to accept on an investment in a real asset is the rate they could have earned investing in financial assets that have similar risk. The rate of return that investors can earn on financial assets with similar risk is an opportunity cost because investors lose the opportunity to earn that rate if the money is invested in a real asset instead. It is therefore the rate of return that investors will require for an investment in a capital project. In other words, this rate is the cost of capital. It is also known as the opportunity cost of capital.
Chapter 13
discusses how we estimate the opportunity cost of capital in practice.
BUILDING INTUITION INVESTMENT DECISIONS HAVE OPPORTUNITY COSTS
When any investment is made, the opportunity to earn a return from an alternative investment is lost. The lost return can be viewed as a cost that arises from a lost opportunity. For this reason, it is called an opportunity cost. The opportunity cost of capital is the return an investor gives up when his or her money is invested in one asset rather than the best alternative asset. For example, suppose that a firm invests in a piece of equipment rather than returning money to stockholders. If stockholders could have earned an annual return of 12 percent on a stock with cash flows that are as risky as the cash flows the equipment will produce, this is the opportunity cost of capital associated with the investment in the piece of equipment.
opportunity cost of capital
the return an investor gives up when his or her money is invested in one asset rather than the best alternative asset
Capital Rationing
When a firm has all the money it needs to invest in all the capital projects that meet its capital selection criteria, the firm is said to be operating without a funding constraint, or resource constraint. Firms are rarely in this position, especially growth firms. Typically, a firm has a fixed number of dollars available for capital expenditures, and the number of qualified projects that need funding exceeds the funds that are available. This funding constraint on investments means that some projects will be mutually exclusive, since investing in one project exhausts resources that might otherwise be invested in another. When faced with a resource constraint, the firm must allocate its funds to the subset of projects that provides the largest increase in stockholder value. The process of limiting, or rationing, capital expenditures in this way is called capital rationing. Capital rationing and its implications for capital budgeting are discussed in
Chapter 12
.
capital rationing
a situation where a firm does not have enough capital to invest in all attractive projects and must therefore ration capital
> BEFORE YOU GO ON
1. Why are capital investments the most important decisions made by a firm’s management?
2. What are the differences between capital projects that are independent, mutually exclusive, and contingent?
10.2 NET PRESENT VALUE
·
In this section we discuss a capital budgeting method that is consistent with this goal of financial management—to maximize the wealth of the firm’s owners. It is called the net present value (NPV) method, and it is one of the most basic analytical methods underlying corporate finance. The NPV method tells us the amount by which the benefits from a capital expenditure exceed its costs. It is the capital budgeting technique recommended in this book.
net present value (NPV) method
a method of evaluating a capital investment project which measures the difference between its cost and the present value of its expected cash flows
Valuation of Real Assets
Throughout the book, we have emphasized that the value of any asset is the present value of its future cash flows. In
Chapters 8
and
9
, we developed valuation models for financial assets, such as bonds, preferred stock, and common stock. We now extend our discussion of valuation models from financial to real assets. The steps used in valuing an asset are the same whether the asset is real or financial:
1. Estimate the future cash flows.
2. Determine the required rate of return, or discount rate, which depends on the riskiness of the future cash flows.
3. Compute the present value of the future cash flows to determine what the asset is worth.
The valuation of real assets, however, is less straightforward than the valuation of financial assets, for several reasons.
First, in many cases, cash flows for financial assets are well documented in a legal contract. If they are not, we are at least able to make some reasonable assumptions about what they are. For real assets, much less information exists. Specialists within the firm, usually from the finance, marketing, and production groups, often prepare estimates of future cash flows for capital projects with only limited information.
Second, many financial securities are traded in public markets, and these markets are reasonably efficient. Thus, market data on rates of return are accessible. For real assets, no such markets exist. As a result, we must estimate required rates of return on real assets (opportunity costs) from market data on financial assets; this can be difficult to do.
NPV—The Basic Concept
The NPV of a project is the difference between the present value of the project’s future cash flows and the present value of its cost. The NPV can be expressed as follows:
NPV = PV (Project’s future cash flows) − PV (Cost of the project)
To illustrate these important points, consider an example. Suppose a firm is considering building a new marina for pleasure boats. The firm has a genie that can tell the future with perfect certainty. The finance staff estimates that the marina will cost $3.50 million. The genie volunteers that the present value of the future cash flows from the marina is $4.25 million.
Assuming this information is correct, the NPV for the marina project is a positive $750,000 ($4.25 million − $3.50 million = $0.75 million). Management should accept the project because the excess of the value of the cash flows over cost increases the value of the firm by $750,000. Why is a positive NPV a direct measure of how much a capital project will increase the value of the firm? If management wanted to, the firm could sell the marina for $4.25 million, pay the $3.50 million in expenses, and deposit $750,000 in the bank. The value of the firm would increase by the $750,000 deposited in the bank. In sum, the NPV method tells us which capital projects to select and how much value they add to the firm.
NPV and Value Creation
We have just said that any project with a positive NPV should be accepted because it will increase the value of the firm. Let’s take a moment to think about this proposition. What makes a capital asset worth more than it costs? In other words, how does management create value with capital investments?
How Value Is Created
Suppose that when you were in college, you worked part time at a successful pizza parlor near campus. During this time, you learned a lot about the pizza business. After graduation, you purchased a pizza parlor for $100,000 that was in a good location but had been forced to close because of a lack of business. The owners had let the restaurant and the quality of the pizzas deteriorate, and the wait staff had been rude, especially to college students. Once you purchased the restaurant, you immediately invested $40,000 to fix it up: you painted the building, spruced up the interior, replaced some of the dining room furniture, and added an eye-catching, 1950s-style neon sign to attract attention. You also spent $15,000 for a one-time advertising blitz to quickly build a customer base. More important, you improved the quality of the pizzas you sold, and you built a profitable takeout business. Finally, you hired your wait staff carefully and trained them to be customer friendly.
Almost immediately the restaurant was earning a substantial profit and generating substantial cash flows. The really good news was that several owners of local pizzerias wanted to buy your restaurant. After intense negotiations with several of the potential buyers, you accepted a cash offer of $475,000 for the business shortly after you purchased it.
What is the NPV for the pizza parlor? For this investment, the NPV is easy to calculate. We do not need to estimate future cash flows and discount them because we already have an estimate of the present value of the cash flows the pizza parlor is expected to produce—$475,000. Someone is willing to pay you $475,000 because he or she believes the future cash flows are worth that amount. The cost of your investment includes the purchase price of the restaurant, the cost to fix it up, and the cost of the initial advertising campaign, which totals $155,000 ($100,000 $40,000 $15,000). Thus, the NPV for the pizza parlor is:
Where did the $320,000 in value you created go? The NPV of your investment is the amount that your personal net worth increased because of the investment. For an ongoing business, the result would have been a $320,000 increase in the value of the firm.
How about the original owners? Why would they sell a business worth $475,000 to you for $100,000? The answer is simple; if they could have transformed the business as you did, they would have done so. Instead, when they ran the business, it lost money! They sold it to you because you offered them a price reflecting its value to them.
Market Data versus Discounted Cash Flows
Our pizza parlor example is greatly simplified by the fact that we can observe the price that someone is willing to pay for the asset. In most capital project analyses, we have to estimate the market value of the asset by forecasting its future cash flows and discounting them by the cost of capital. The discounted value of a project’s future cash flows is an estimate of its value, or the market price for which it can be sold.
Framework for Calculating NPV
We now describe a framework for analyzing capital budgeting decisions using the NPV method. As you will see, the NPV technique uses the discounted cash flow technique developed in
Chapters 5
and
6
and applied in
Chapters 8
and
9
. The good news, then, is that the NPV method requires only the application of what you already know.
In addition to following the five-step framework for solving NPV analysis problems, we recommend that you use a worksheet with a time line like the one shown here to help you determine the proper cash flows for each period.
The five-step framework discussed in this section and the accompanying cash flow work-sheet (
Exhibit 10.2
) can help you systematically organize a project’s cash flow data and compute its NPV. Most mistakes people make when working capital budgeting problems result from problems with cash flows: not identifying a cash flow, getting a cash flow in the wrong time period, or assigning the wrong sign to a cash flow. What can make cash flow analysis difficult in capital budgeting is this: there are often multiple cash flows in a single time period, and some are cash inflows and others are cash outflows.
As always, we recommend that you prepare a time line when doing capital budgeting problems. A sample time line is shown in
Exhibit 10.2
, along with an identification of the cash flows for each period. Our goal is to compute the net cash flow (NCF) for each time period t, where NCF
t
= (Cash inflows − Cash outflows) for the period t. For a capital project, the time periods (t) are usually in years, and t varies from the current period (t = 0) to some finite time period that is the estimated life of the project (t = n). Recall that getting the correct sign on each cash flow is critical to getting the correct answer to a problem. As you have seen in earlier chapters, the convention in finance problem solving is that cash inflows carry a positive sign and cash outflows carry a negative sign. Finally, note that all cash flows in this chapter are on an after-tax basis. We will make adjustments for tax consequences on specific transactions such as the calculation of a project’s salvage value.
CCH Business Owner’s Toolkit is a valuable Web source for information about running a business, including capital budget analysis. Go to
http://www.toolkit.cch.com/text/p06_6500.asp
.
Our five-step framework for analysis is as follows:
1. Determine the initial cost of starting the project. We first need to identify and add up all the cash flows related to the initial cost of starting the project. In most cases, the initial cost of a project is incurred at the start; hence the cash flows are already in current dollars. These cash flows typically include any property, plant, and equipment outlays for production as well as employee hiring and training costs. In some cases, like Intel’s investment in the Dalian fabrication plant, these initial outlays can be made over several years before the project is up and running. Of course, any future cash flows must be discounted to obtain their present value. Turning to
Exhibit 10.2
, we have incurred a single negative cash flow (−CF0) as our initial cost of starting the project; thus NCF0 has a negative value.
2. Estimate the project’s future cash flows over its expected life. Once they are up-and-running, capital projects typically generate some cash inflows from revenues (CIF
t
) for each period, along with some cash outflows (COF
t
) associated with costs incurred to generate the revenues. In most cases revenues exceed costs, and thus, NCF
t
is positive. However, this may not always be the case. For example, if the project is the purchase of a piece of equipment, it is possible for NCF3 to have a negative value (CIF3 < COF3) if the equipment is projected to need a major overhaul or must be replaced during the third year. Finally, you also need to pay attention to a project's final cash flow, which is t = 5 in
Exhibit 10.2
. There may be a salvage value (SV) at the end of the project, which is a cash inflow. In that case NCF5 = (CIF5 − COF5 + SV). The important point is that for each time period, we must identify all the cash flows that take place, assign each cash flow its proper sign, and add up all the cash flows.
3. Determine the riskiness of the project and the appropriate cost of capital. The third step is to identify for each project its risk-adjusted cost of capital, which takes into account the riskiness of the project's cash flows. The riskier the project, the higher its cost of capital. The cost of capital is the discount rate used in determining the present value of the future expected cash flows. In this chapter, the cost of capital and any risk adjustments will be supplied, and no calculations will be required for this step.
4. Compute the project's NPV. The NPV, as you know, is the present value of the net cash flows the project is expected to generate minus the cost of the project.
5. Make a decision. If the NPV is positive, the project should be accepted because all projects with a positive NPV will increase the value of the firm. If the NPV is negative, the project should be rejected; projects with negative NPVs will decrease the value of the firm.
You might be wondering about how to handle a capital project with an NPV of 0. Technically, management should be indifferent to accepting or rejecting projects such as this because they neither increase nor decrease the value of the firm. When the NPV 0, the project is generating returns that are just equal to the opportunity cost of capital. At a practical level, projects rarely have an NPV equal to 0, and most firms have more good capital projects (with NPV > 0) than they can fund. Thus, this is not an issue that generates much interest among practitioners.
Net Present Value Techniques
The NPV of a capital project can be stated in equation form as the present value of all net cash flows (cash inflows − cash outflows) connected with the project, whether in the current period or in the future. The NPV equation can be written as follows:
When analyzing capital budgeting problems, we typically have a lot of data to sort through. The worksheet approach introduced in
Exhibit 10.2
is helpful in keeping track of the data in an organized format.
Exhibit 10.3
shows the time line and relevant cash flows for the pocket pizza project. The steps in analyzing the project’s cash flows and determining its NPV are as follows:
1. Determine the cost of the project. The cost of the project is the cost to modify the existing production line, which is $300,000. This is a cash outflow (negative sign).
2. Estimate the project’s future cash flows over its expected life. The project’s future cash inflows come from sales of the new product. Sales are estimated at $300,000 per year (positive sign). The cash outflows are the costs to manufacture and distribute the new product, which are $220,000 per year (negative sign). The life of the project is five years. The project has a salvage value of $30,000, which is a cash inflow (positive sign). The net cash flow (NCF) in a particular time period is just the sum of the cash inflows and cash outflows for that period. For example, the NCF for period t 0 is $300,000 the NCF for period t 1 is $80,000, and so on, as you can see in
Exhibit 10.3
.
The worksheet approach introduced in
Exhibit 10.2
is helpful in organizing the data given for the pocket pizza project.
3. Determine the riskiness of the project and appropriate cost of capital. The discount rate is the cost of capital, which is 15 percent.
4. Compute the project’s NPV. To compute the project’s NPV, we apply Equation 10.1 by plugging in the NCF values for each time period and using the cost of capital, 15 percent, as the discount rate. The equation looks like this (the figures are in thousands of dollars):
5. Make a decision. The pocket pizza project has a negative NPV, which indicates that the project is not a good investment and should be rejected. If management undertook this project, the value of the firm would decrease by $16,910; and, if the firm had one hundred thousand shares of stock outstanding, we can estimate that the project would decrease the value of each share by about 17 cents ($16,910/100,000 shares $0.1691 per share).
Calculating NPV with a Financial Calculator
Using a financial calculator is an easier way to calculate the present value of the future cash flows. In this example you should recognize that the cash flow pattern is a five-year ordinary annuity with an additional cash inflow in the fifth year. This is exactly the cash pattern for a bond with annual coupon payments and payment of principal at maturity we saw in
Chapter 8
. We can find the present value using a financial calculator, with $80 being the annuity stream for five years and $30 the salvage value at year 5:
The cost is $300,000 to modify the production line. Sales of the new product are estimated at $200,000 for the first year, $300,000 for the next two years, and $500,000 for the final two years. It is estimated that production, sales, and advertising costs will be $250,000 for the first year and will then decline to a constant $200,000 per year. There is no salvage value at the end of the product’s life, and the appropriate cost of capital is 15 percent. Is the project, as proposed, economically viable?
APPROACH: To solve the problem, work through the steps for NPV analysis given in the text.
SOLUTION:
Exhibit 10.4
shows the project’s cash flows.
1. The cost to modify the production line is $300,000, which is a cash outflow in Year 0 and the cost of the project.
2. The future cash flows over the expected life of the project are laid out on the time line in
Exhibit 10.4
. The project’s life is five years. The NCFs for the capital project are negative at the beginning of the project and in the first year ($300,000 and $50,000) and thereafter are positive.
3. The appropriate cost of capital is 15 percent.
4. The values are substituted into Equation 10.1 to calculate the NPV:
The worksheet shows the time line and cash flows for the self-rising pizza dough project in Learning by Doing Application 10.1. As always, it is important to assign each cash flow to the appropriate year and to give it the proper sign. Once you have computed the net cash flow for each time period, solving for NPV is just a matter of plugging the data into the NPV formula.
USING EXCEL NET PRESENT VALUE
Net present value problems are most commonly solved using a spreadsheet program. The program’s design is good for keeping track of all the cash flows and the periods in which they occur. The spreadsheet setup for Learning by Doing Application 10.1, presented on the right, shows how to calculate the NPV for the self-rising pizza dough machine:
Notice that the NPV formula does not take into account the cash flow in year zero. Therefore, you only enter into the NPV formula the cash flows in years 1 through 5, along with the discount rate. You then add the cash flow in year zero to the total from the NPV formula calculation to get the NPV for the investment.
Recall that investments are mutually exclusive if, by making one, another will not be undertaken. Projects may be mutually exclusive because they are substitutes for one another or because the firm has a funding constraint. A project’s NPV provides an objective measure of its incremental value to the firm’s investors, and thus makes it simple to choose between two or more mutually exclusive projects. When faced with such a choice, managers should allocate capital to the project that has the most positive dollar impact on the value of the firm, in other words, the project with the highest NPV.
EXAMPLE 10.1 DECISION MAKING
The IS Department’s Capital Projects
DECISION: If the projects are independent, you should accept projects 1 and 2, both of which have a positive NPV, and reject project 4. Project 3, with an NPV of zero, could be either accepted or rejected. If the projects are mutually exclusive and you can accept only one of them, it should be project 1, which has the largest NPV.
Concluding Comments on NPV
Some concluding comments about the NPV method are in order. First, as you may have noticed, the NPV computations are rather mechanical once we have estimated the cash flows and the cost of capital. The real difficulty is estimating or forecasting the future cash flows. Although this may seem to be a daunting task, managers with experience in producing and selling a particular type of product can usually generate fairly accurate estimates of sales volumes, prices, and production costs. Most business managers are routinely required to make decisions that involve expectations about future events. In fact, that is what business is really all about—dealing with uncertainty and making decisions that involve risk.
Second, estimating project cash flows over a long forecast period requires skill and judgment. There is nothing wrong with using estimates to make business decisions as long as they are based on informed judgments and not guesses. Problems can arise with the cash flow estimates when a project team becomes overly enamored with a project. In wanting a particular project to succeed, a project team can be too optimistic about the cash flow projections. It is therefore very important that capital budgeting decisions be subject to ongoing and post-audit review.
In conclusion, the NPV approach is the method we recommend for making capital investment decisions. It provides a direct (dollar) measure of how much a project will increase the value of the firm. NPV also makes it possible to correctly choose between mutually exclusive projects. The accompanying table summarizes NPV decision rules and the method’s key advantages and disadvantages.
1. What is the NPV of a project?
2. If a firm accepts a project with a $10,000 NPV, what is the effect on the value of the firm?
3. What are the five steps used in NPV analysis?
10.3 THE PAYBACK PERIOD
·
The payback period is one of the most widely used tools for evaluating capital projects. The payback period is defined as the number of years it takes for the cash flows from a project to recover the project’s initial investment. With the payback method for evaluating projects, a project is accepted if its payback period is below some specified threshold. Although it has serious weaknesses, this method does provide some insight into a project’s risk; the more quickly you recover the cash, the less risky is the project.
payback period
the length of time required to recover a project’s initial cost
Computing the Payback Period
To compute the payback period, we need to know the project’s cost and estimate its future net cash flows. The net cash flows and the project cost are the same values that we use to compute the NPV. The payback (PB) equation can be expressed as follows:
The exhibit shows the net and cumulative net cash flows for a proposed capital project with an initial cost of $70,000. The cash flow data are used to compute the payback period, which is 2.5 years.
shows the net cash flows (row 1) and cumulative net cash flows (row 2) for a proposed capital project with an initial cost of $70,000. The payback period calculation for our example is:
Exhibit 10.5
that the firm recovers cash flows of $30,000 in the first year and $30,000 in the second year, for a total of $60,000 over the two years. During the third year, the firm needs to recover only $10,000 ($70,000 $60,000 $10,000) to pay back the full cost of the project. The third-year cash flow is $20,000, so we will have to wait 0.5 year ($10,000/$20,000 = 0.5) to recover the final amount. Thus, the payback period for this project is 2.5 years (2 years + 0.5 year = 2.5 years).
The idea behind the payback period method is simple: the shorter the payback period, the faster the firm gets its money back and the more desirable the project. However, there is no economic rationale that links the payback method to stockholder value maximization. Firms that use the payback method accept all projects having a payback period under some threshold and reject those with a payback period over this threshold. If a firm has a number of projects that are mutually exclusive, the projects are selected in order of their payback rank: projects with the shortest payback period are selected first.
APPLICATION 10.2 LEARNING BY DOING
APPROACH: Use Equation 10.2 to calculate the number of years it takes for the cash flows from each project to recover the project’s initial investment. If the two projects are independent, you should accept the projects that have a payback period that is less than or equal to two years. If the projects are mutually exclusive, you should accept the project with the shortest payback period if that payback period is less than or equal to two years.
SOLUTION: The payback for project A requires only that we calculate the first term in Equation 10.2—Years before recovery: the first year recovers $100, the second year $200, and the third year $200, for a total of $500 ($100 + $200 + $200 = $500). Thus, in three years, the $500 investment is fully recovered, so PBA = 3.00.
For project B, the first year recovers $400 and the second year $300. Since we need only part of the second-year cash flow to recover the initial cost, we calculate both terms in Equation 10.2 to obtain the payback period.
How the Payback Period Performs
We have worked through some simple examples of how the payback period is computed. Now we will consider several more complex situations to see how well the payback period performs as a capital budgeting rule.
Exhibit 10.6
illustrates five different capital budgeting projects. The projects all have an initial investment of $500, but each one has a different cash flow pattern. The bottom part of the exhibit shows each project’s payback period, along with its net present value for comparison. We will assume that management has set a payback period of two years as the cutoff point for an acceptable project.
Project A: The cash flows for project A are $200 in the first year and $300 in the second, for a total of $500; thus, the project’s payback period is two years. Under our acceptance criterion, management should accept this project. Project A also has a positive NPV of $450, so the two capital budgeting decision rules agree.
Project B: Project B never generates enough cash flows to pay off the original investment of $500: $300 + $100 + $50 = $450. Thus, the project payback period is infinite. With an infinite payback period, the project should be rejected. Also, as you would expect, project B’s NPV is negative. So far, the payback period and NPV methods have agreed on which projects to accept.
Project C: Project C has a payback period of two years: $250 + $250 = $500. Thus, according to the payback criteria, it should be accepted. However, the project’s NPV is a negative $115, which indicates that the project should be rejected. Why the conflict? Look at the cash flows after the payback period of two years. In year 3 the project requires an additional investment of $250 (a cash outflow) and now is in a deficit position; that is, the cumulative net cash balance is now only $250 ($250 + $250 − $250 = $250). Then, in the final year, the project earns an additional $250, recovering the cost of the total investment. The project’s payback is really four years. Th e payback period analysis can lead to erroneous decisions because the rule does not consider cash flows after the payback period.
EXHIBIT 10.6 Payback Period with Various Cash Flow Patterns
Each of the five capital budgeting projects shown in the exhibit calls for an initial investment of $500, but all have different cash flow patterns. The bottom part of the exhibit shows each project’s payback period, along with its net present value for comparison. at the cash flows after the payback period of two years. In year 3 the project requires an additional investment of $250 (a cash outflow) and now is in a deficit position; that is, the cumulative net cash balance is now only $250 ($250 $250 $250 $250). Then, in the final year, the project earns an additional $250, recovering the cost of the total investment. The project’s payback is really four years. The payback period analysis can lead to erroneous decisions because the rule does not consider cash flows after the payback period.
Discounted Payback Period
Another weakness of the ordinary payback period criteria is that it does not take into account the time value of money. All dollars received before the cutoff period are given equal weight. To address this problem, some financial managers use a variant of the payback period called the discounted payback period. This payback calculation is similar to the ordinary payback calculation except that the future cash flows are discounted by the cost of capital.
discounted payback period
the length of time required to recover a project’s initial cost, accounting for the time value of money
The major advantage of the discounted payback approach is that it tells management how long it takes a project to reach an NPV of zero. Thus, any capital project that meets a firm’s decision rule must also have a positive NPV. This is an improvement over the standard payback calculation, which can lead to accepting projects with negative NPVs. Regardless of the improvement, the discounted payback method is not widely used by businesses, and it also ignores all cash flows after the arbitrary cutoff period, which is a major flaw.
To see how the discounted payback period is calculated, turn to
Exhibit 10.7
. The exhibit shows the net cash flows for a proposed capital project along with both the cumulative and discounted cumulative cash flows; thus, we can compute both the ordinary and the discounted payback periods for the project and then compare them. The cost of capital is 10 percent.
The first two rows show the nondiscounted cash flows, and we can see by inspection that the ordinary payback period is two years. We do not need to make any additional calculations because the cumulative cash flows equal zero at precisely two years. Now let’s turn our attention to the lower two rows, which show the project’s discounted and cumulative discounted cash flows. Note that the first year’s cash flow is $20,000 and its discounted value is $18,182 ($20,000 × 0.9091 = $18,182), and the second year’s cash flow is also $20,000 and its discounted value is $16,529 ($20,000 × 0.8264 = $16,529). Now, looking at the cumulative discounted cash flows row, notice that it turns positive between two and three years. This means that the discounted payback period is two years plus some fraction of the third year’s discounted cash flow. The exact discounted payback period computed value is 2 years + ($5,289/$15,026 per year) = 2 years + 0.35 years = 2.35 years.
The exhibit shows the net and cumulative net cash flows for a proposed capital project with an initial cost of $40,000. The cash flow data are used to compute the discounted payback period for a 10 percent cost of capital, which is 2.35 years.
As expected, the discounted payback period is longer than the ordinary payback period (2 years < 2.35 years), and in 2.35 years the project will reach a NPV = 0. The project NPV is positive (NPV = $9,737); therefore, we should accept the project. But notice that the payback decision criteria are ambiguous. If we use 2.0 years as the payback criterion, we reject the project and if we use 2.5 or 3.0 years as criterion, the project is accepted. The lack of a definitive decision rule remains a major problem with the payback period as a capital budgeting tool.
Evaluating the Payback Rule
The standard payback period is often calculated for projects because it provides an intuitive and simple measure of a project's liquidity risk. This makes sense because projects that pay for themselves quickly are less risky than projects whose paybacks occur farther in the future. There is a strong feeling in business that “getting your money back quickly” is an important standard when making capital investments. This intuition can be economically justified if the firm faces payments to creditors before the payback date, or if estimates of project cash flows beyond the payback date are very uncertain. Probably the greatest advantage of the payback period is its simplicity; it is easy to calculate and easy to understand.
When compared with the NPV method, however, the payback methods have some serious shortcomings. First, the standard payback calculation does not adjust or account for the timing or risk associated with future cash flows. Second, there is little economic justification for the choice of the payback cutoff criteria other than a liquidity motive. Who is to say that a particular cutoff, such as two years, is optimal with regard to maximizing stockholder value? Finally, perhaps the greatest shortcoming of the payback method is its failure to consider cash flows after the payback period. As a result, the payback method is biased toward shorter-term projects and may cause managers to reject important positive NPV projects where cash inflows tend to occur farther in the future, such as research and development investments, new product launches, and entry into new lines of business.
While the payback period is relatively simple to calculate, it is important to note that payback requires forecasts of future project cash flows up to the established cutoff period. Furthermore, discounted payback requires that managers identify a project's discount rate. Thus the inputs into the payback and NPV methods are virtually identical. Consequently, using a payback method may not even save much time and effort in evaluating a project. The table below summarizes key advantages and disadvantages of the payback method.
> BEFORE YOU GO ON
1. What is the payback period?
2. Why does the payback period provide a measure of a project’s liquidity risk?
3. What are the main shortcomings of the payback method?
10.4 THE ACCOUNTING RATE OF RETURN
·
a rate of return on a capital project based on average net income divided by average book value over the project’s life; also called the book value rate of return
We turn next to a capital budgeting technique based on the accounting rate of return (ARR), sometimes called the book value rate of return. This method computes the return on a capital project using accounting numbers—the project’s net income (NI) and book value (BV)—rather than cash flow data. The ARR can be calculated in a number of ways, but the most common definition is:
Although ARR is fairly easy to understand and calculate, as you probably guessed, it has a number of major flaws as a tool for evaluating capital expenditure decisions. Besides the fact that AAR is based on accounting numbers rather than cash flows, it is not really even an accounting-based rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income statement and balance sheet. Thus, the ARR ignores the time value of money. Also, as with the payback method, there is no economic rationale that links a particular acceptance criterion to the goal of maximizing stockholder value.
Because of these major shortcomings, the ARR technique should not be used to evaluate the viability of capital projects under any circumstances. You may wonder why we even included the ARR technique in the book if it is a poor criterion for evaluating projects. The reason is simply that we want to be sure that if you run across the ARR method at work, you will recognize it and be aware of its shortcomings.
> BEFORE YOU GO ON
1. What are the major shortcomings of using the ARR method as a capital budgeting method?
10.5 INTERNAL RATE OF RETURN
·
The internal rate of return, known in practice as the IRR, is an important alternative to the NPV method. The NPV and IRR techniques are closely related in that both involve discounting the cash flows from a project; thus, both account for the time value of money. When we use the NPV method to evaluate a capital project, the discount rate is the rate of return required by investors for investments with similar risk, which is the project’s opportunity cost of capital. When we use the IRR, we are looking for the rate of return associated with a project so that we can determine whether this rate is higher or lower than the project’s discount rate.
internal rate of return (IRR)
the discount rate at which the present value of a project’s expected cash inflows equals the present value of the project’s outflows; it is the discount rate at which the project’s NPV equals zero
We can define the IRR as the discount rate that equates the present value of a project’s cost to the present value of its expected cash inflows:
Calculating the IRR
The IRR is an expected rate of return much like the yield to maturity we calculated for bonds in
Chapter 8
. Thus, in calculating the IRR, we need to apply the same trial-and-error method we used in
Chapter 8
. We will begin by doing some IRR calculations by trial and error so that you understand the process, and then we will switch to the financial calculator and computer spreadsheets, which provide an answer more quickly.
Suppose that Ford Motor Company has an investment opportunity with cash flows as shown in
Exhibit 10.8
and that the cost of capital is 12 percent. We want to find the IRR for this project. Using Equation 10.4, we will substitute various values for IRR into the equation to compute the project’s IRR by trial and error. We continue this process until we find the IRR value that makes Equation 10.4 equal zero.
A good starting point is to use the cost of capital as the discount rate. Note that when we discount the NCFs by the cost of capital, we are calculating the project’s NPV:
The cash flow data in the exhibit are used to compute the project’s IRR. The project’s NPV is a positive $16,440, which indicates that the IRR is greater than the cost of capital of 12 percent.
Good guess! This means that the NPV of Ford’s capital project is zero at a discount rate of 13.7 percent. The required rate of return is the cost of capital, which is 12.0 percent. Since the project’s IRR of 13.7 percent exceeds the cost of capital, the IRR criterion indicates that the project should be accepted.
The project’s NPV is a positive $16,440, which also indicates that Ford should go ahead with the project. Thus, both the IRR and NPV suggest the same conclusion.
APPLICATION 10.3 LEARNING BY DOING
APPROACH: The IRR for an investment is the discount rate at which the NPV is zero. Thus, we can use Equation 10.4 to solve for the IRR and then compare this value with Larry’s cost of capital. If the IRR is greater than the cost of capital, the project has a positive NPV and should be accepted.
SOLUTION: The total cost of the machine is $6,750 ($5,000 $1,750 $6,750), and the final cash flow in year 10 is $2,400 ($2,000 $400 $2,400).
Because the project’s IRR exceeds Larry’s cost of capital of 15 percent, the project should be accepted. Larry is wrong.
USING EXCEL INTERNAL RATE OF RETURN
You know that calculating IRR by hand can be tedious. The trial-and-error method can take a long time and can be quite frustrating. Knowing all the cash flows and an approximate rate will allow you to use a spreadsheet formula to get an answer instantly.
The accompanying spreadsheet shows the setup for calculating the IRR for the low-cal yogurt machine at Larry’s Ice Cream Parlor that is described in Learning by Doing Application 10.3.
Here are a couple of important points to note about IRR calculations using spreadsheet programs:
1. Unlike the NPV formula, the IRR formula accounts for all cash flows, including the initial investment in year 0, so there is no need to add this cash flow later.
2. The syntax of the IRR function requires that you first provide the project’s cash flows in order beginning at time zero. To calculate the IRR, you will also need to provide a “guess” value, or a number you estimate is close to the IRR. A good value to start with is the cost of capital. To learn more about why this value is needed, you should go to your spreadsheet’s help manual and search for “IRR.”
In the Ford example, the IRR and NPV methods agree. The two methods will always agree when you are evaluating independent projects and the projects’ cash flows are conventional. As discussed earlier, an independent project is one that can be selected with no effect on the viability of any other project. A project with conventional cash flows is one with an initial cash outflow followed by one or more future cash inflows. Put another way, after the initial investment is made (cash outflow), the net cash flow in each future year is positive (inflows). For example, the purchase of a bond involves conventional cash flows. You purchase the bond for a price (cash outflow), and in the future you receive coupon payments and a principal payment at maturity (cash inflows).
conventional cash flow
a cash flow pattern consisting of an initial cash outflow that is followed by one or more cash inflows
Let’s look more closely at the kinds of situations in which the NPV and the IRR methods agree. A good way to visualize the relation between the IRR and NPV methods is to graph NPV as a function of the discount rate. The graph, called an NPV profile, shows the NPV of the project at various costs of capital.
NPV profile
a graph showing NPV as a function of the discount rate
shows the NPV profile for the Ford project. We have placed the NPVs on the vertical axis, or y-axis, and the discount rates on the horizontal axis, or x-axis. We used the calculations from our earlier example and made some additional NPV calculations at various discount rates as follows:
In the NPV profile for the Ford project, the NPV value is on the vertical (y) axis and the discount rate is on the horizontal (x) axis. You can see that as the discount rate increases, the NPV profile curve declines smoothly and intersects the x-axis at the point where the NPV is 0. The IRR of 13.7 percent is the point at which the NPV changes from a positive to a negative value. The NPV and IRR methods lead to identical accept-or-reject decisions for the Ford project.
As you can see, a discount rate of 0 percent corresponds with an NPV of $160,000; a discount rate of 5 percent with an NPV of $94,000; and so forth. As the discount rate increases, the NPV curve declines smoothly. Not surprisingly, the curve intersects the x-axis at precisely the point where the NPV is 0 and the IRR is 13.7 percent.
The NPV profile in
Exhibit 10.9
illustrates why the NPV and IRR methods lead to identical accept-reject decisions for the Ford project. The IRR of 13.7 percent precisely marks the point at which the NPV changes from a positive to a negative value. Whenever a project is independent and has conventional cash flows, the result will be as shown in the exhibit. The NPV will decline as the discount rate increases, and the IRR and the NPV methods will result in the same capital expenditure decision.
When the NPV and IRR Methods Disagree
We have seen that the IRR and NPV methods lead to identical investment decisions for capital projects that are independent and that have conventional cash flows. However, if either of these conditions is not met, the IRR and NPV methods can produce different accept-reject decisions.
Unconventional Cash Flows
Unconventional cash flows can cause a conflict between the NPV and IRR decision rules. In some instances the cash flows for an unconventional project are just the reverse of those of a conventional project: the initial cash flow is positive, and all subsequent cash flows are negative. In this case, we need only reverse the IRR decision rule and accept the project if the IRR is less than the cost of capital to make the IRR and NPV methods agree.
When a project’s future cash flows include both positive and negative cash flows, the situation is more complicated. An example of such a project is an assembly line that will require one or more major renovations over its lifetime. Another common business situation is a project that has conventional cash flows except for the final cash flow, which is negative. The final cash flow might be negative because extensive environmental cleanup is required at the end of the project, such as the cost for decommissioning a nuclear power plant, or because the equipment originally purchased has little or no salvage value and is expensive to remove.
Consider an example. Suppose a firm invests in a gold-mining operation that costs $55 million and has an expected life of two years. In the first year, the project generates a cash inflow of $150 million. In the second year, extensive environmental and site restoration is required, so the expected cash flow is a negative $100 million. The time line for these cash flows follows.
Exhibit 10.10
:
The NPV profile in
Exhibit 10.10
shows the results of this pattern: we have two IRRs, one at 16.05 percent and the other at 55.65 percent. Which is the correct IRR, or are both correct? Actually, there is no correct answer; the results are meaningless, and you should not try to interpret them. Thus, in this situation, the IRR technique provides information that should not be used for decision making.
How many IRR solutions can there be for a given cash flow? The maximum number of IRR solutions is equal to the number of sign reversals in the cash flow stream. For a project with a conventional cash flow, there is only one cash flow sign reversal; thus, there is only one IRR solution. In our mining example, there are two cash flow sign reversals; thus, there are two IRR solutions.
Finally, for some cash flow patterns, it is impossible to compute an IRR. These situations can occur when the initial cash flow (t = 0) is either a cash inflow or outflow and is followed by cash flows with two or more sign reversals. An example of such a cash flow pattern is NCF0 = $15, NCF1 −$25, and NCF2 = $20. This type of cash flow pattern might occur on a building project where the contractor is given a prepayment, usually the cost of materials and supplies ($15); then does the construction and pays the labor cost ($25); and finally, upon completion of the work, receives the final payment ($20). Note that when it is not possible to compute an IRR, the project either has a positive NPV or a negative NPV for all possible discount rates. In this example, the NPV is always positive.
The gold-mining operation has unconventional cash flows. Because there are two cash flow sign reversals, we end up with two IRRs––16.05 percent and 55.65 percent––neither of them correct. In situations like this, the IRR provides a solution that is meaningless, and therefore, the results should not be used for capital budgeting decisions.
Mutually Exclusive Projects
The other situation in which the IRR can lead to incorrect decisions is when capital projects are mutually exclusive—that is, when accepting one project means rejecting the other. For example, suppose you own a small store in the business district of Seattle that is currently vacant. You are looking at two business opportunities: opening an upscale coffee house or opening a copy center. Since you cannot pursue both projects at the same location they are mutually exclusive.
When you have mutually exclusive projects, how do you select the best alternative? If you are using the NPV method, the answer is easy. You select the project that has the highest NPV because it will increase the value of the firm by the largest amount. If you are using the IRR method, it would seem logical to select the project with the highest IRR. In this case, though, the logic is wrong! You cannot tell which mutually exclusive project to select just by looking at the projects’ IRRs.
Let’s consider another example to illustrate the problem. The cash flows for two projects, A and B, are as follows:
The following table shows the NPVs for the two projects at several discount rates:
To read an article that warns finance managers using the IRR about the method’s pitfalls, visit
www.cfo.com/printable/article.cfm/3304945?f=options
.
The relative IRR and NPV rankings change in this way because the cash inflows of project B arrive later than those of project A. Thus, higher discount rates have more of an impact on the value of project B. In other words, changes in relative IRR and NPV rankings result from differences in the timing of project cash flows.
The NPV profiles for two projects often cross over each other. When evaluating mutually exclusive projects, it is helpful to know where this crossover point is. For projects A and B in the exhibit, the crossover point is at 14.3 percent. For any cost of capital above 14.3 percent but below 20.7 percent the NPV for project A is higher than that for project B and is positive; thus, project A should be selected. For any cost of capital below the crossover point, the NPV of project B is higher, and project B should be selected.
Now take a look at
Exhibit 10.11
, which shows the NPV profiles for projects A and B. As you can see, there is a point, called the crossover point, at which the NPV profiles for projects A and B intersect. The crossover point here is at a discount rate of 14.3 percent. For any cost of capital above 14.3 percent, the NPV for project A is higher than that for project B; thus, project A should be selected if its NPV is positive. For any cost of capital below the crossover point, project B should be selected.
Another conflict involving mutually exclusive projects concerns comparisons of projects that have significantly different costs. The IRR does not adjust for these differences in the scale of projects. What the IRR gives us is a rate of return on each dollar invested. In contrast, the NPV method computes the total dollar value created by the project. The difference in results can be significant, as can be seen in Decision-Making Example 10.2 on the next page.
crossover point
the discount rate at which the NPV profiles of two projects cross and, thus, at which the NPVs of the projects are equal
Modified Internal Rate of Return (MIRR)
A major weakness of the IRR method compared with the NPV method concerns the rate at which the cash flows generated by a capital project are reinvested. The NPV method assumes that cash flows from a project are reinvested at the cost of capital, whereas the IRR technique assumes they are reinvested at the IRR. Determining which is the better assumption depends on which rate better represents the rate that firms can actually earn when they reinvest a project’s cash flows over time. It is generally believed that the cost of capital, which is often lower than the IRR, better reflects the rate that firms are likely to earn. Using the IRR may thus involve overly optimistic assumptions regarding reinvestment rates.
modified internal rate of return (MIRR)
an internal rate of return (IRR) measure which assumes that cash inflows are reinvested at the opportunity cost of capital until the end of the project
To eliminate the reinvestment rate assumption of the IRR, some practitioners prefer to calculate the modified internal rate of return (MIRR). In this approach, each operating cash flow is converted to a future value at the end of the project’s life, compounded at the cost of capital. These values are then summed up to get the project’s terminal value (TV). The MIRR is the interest rate that equates the project’s cost (PVCost), or cash outflows, with the future value of the project’s cash inflows at the end of the project (PVTV).
1
Because each future value is computed using the cost of capital as the interest rate, the reinvestment rate problem is eliminated.
EXAMPLE 10.2 DECISION MAKING
The Lemonade Stand versus the Convenience Store
DECISION: Your boss, who favors the IRR method, looks at the analysis and declares his son a genius. The IRR decision rule suggests that the lemonade stand, with its 76.2 percent rate of return, is the project to choose! You point out that the goal of capital budgeting is to select projects, or combinations of projects, that maximize the value of the firm, his business. The convenience store adds by far the greater value: $179,190 compared with only $1,694 for the lemonade stand. Although the lemonade stand has a high rate of return, its small size precludes it from being competitive against the larger project.
We can set up the equation for the MIRR in the same way we set up Equation 10.4 for the IRR:
Second, we need to compute the terminal value (TV). To do this, we find the future value of each operating cash flow at the end of the project’s life, compounded at the cost of capital. We then sum up these future values to get the project’s TV. Mathematically, the TV can be expressed as:
To illustrate, let’s return to the Ford Motor Company example shown in
Exhibit 10.8
. Recall that the cost of the project is $560, incurred at t 0, and that the discount rate is 12 percent. To determine the MIRR for the project, we start by calculating the terminal value of the cash flows, as shown on the following time line:
IRR versus NPV: A Final Comment
The IRR method, as noted, is an important alternative to the NPV method. As we have seen, it accounts for the time value of money, which is not true of methods such as the payback period and accounting rate of return. Furthermore, the IRR technique has great intuitive appeal. Many business practitioners are in the habit of thinking in terms of rates of return, whether the rates relate to their common-stock portfolios or their firms’ capital expenditures. To these practitioners, the IRR method just seems to make sense. Indeed, we suspect that the IRR’s popularity with business managers results more from its simple intuitive appeal than from its merit.
On the downside, we have seen that the IRR method has several flaws. For example, IRR can’t be used effectively for projects with unconventional cash flows, and IRR can lead to incorrect investment decisions when it is used to choose between mutually exclusive projects. MIRR addresses some of the shortcomings of IRR; namely, it does not assume that project proceeds are reinvested at the IRR, and it eliminates issues associated with unconventional project cash flows. Nonetheless, we believe that NPV should be the primary method used to make capital budgeting decisions. Investment decisions made using NPV are always consistent with the goal of maximizing the value of the firm, even when discriminating between mutually exclusive projects. Finally, it is important to note that the IRR, MIRR, and NPV methods all require a set of projected cash flows over the life of the project and a discount rate. Thus, using IRR or MIRR, rather than NPV, does not require less effort from financial managers.
1. What is the IRR method?
2. In capital budgeting, what is a conventional cash flow pattern?
3. Why should the NPV method be the primary decision tool used in making capital investment decisions?
10.6 CAPITAL BUDGETING IN PRACTICE
·
Capital expenditures are big-ticket items in the U.S. economy. According to the Department of Commerce, U.S. businesses invested $1.38 trillion in capital goods in 2008. Within the S&P 500, the sector with the largest total capital expenditures was the energy sector, in which expenditures totaled $170 billion in 2008. In contrast, expenditures in the healthcare sector were the smallest at $23 billion. Capital investments also represent large expenditures for individual firms, though the amount spent can vary widely from year to year. For example, AT&T expanded its wireless network capabilities rapidly in 2008 and its $19.6 billion in capital expenditures that year exceeded all other firms in the S&P 500 except Chevron, which spent $19.7 billion. More typical are the capital expenditure totals for Ford Motor Company, Cisco, and Kellogg Company, which are shown in the following table. Given the large dollar amounts and the strategic importance of capital expenditures, it is no surprise that corporate managers spend considerable time and energy analyzing capital projects.
The exhibit summarizes evidence from two studies that examined the use of capital budgeting techniques by businesses. As you can see, over time more firms have come to use the NPV and IRR techniques. Surprisingly, though, even in 1999, 20.3 percent still computed the accounting rate of return.
Practitioners’ Methods of Choice
Because of the importance of capital budgeting, over the years a number of surveys have asked financial managers what techniques they actually use in making capital investment decisions.
Exhibit 10.12
, which summarizes the results from two such studies, reveals significant changes over time. As shown, in 1981 only 16.5 percent of the financial managers surveyed frequently used the NPV approach, and the payback period and accounting rate of return approaches were used even less frequently. Most firms, 65.3 percent, used the IRR method. However, practices changed in the 1980s and 1990s. By 1999, 74.9 percent of the firms surveyed were frequently using the NPV technique, 75.7 percent were using the IRR, and 56.7 percent were using the payback period method. As you can see, the most recent findings reflect a much better alignment between what practitioners do and the theory discussed in this chapter. As you can also see, many financial managers use multiple capital budgeting tools.
An article that surveys the use of capital budgeting techniques by the CFOs of Fortune 500 companies can be found at
http://faculty.fuqua.duke.edu
/~jgraham/website/SurveyJACF .
Postaudit and Ongoing Reviews
Management should systematically review the status of all ongoing capital projects and perform postaudit reviews on all completed capital projects. In a postaudit review, management compares the actual performance of a project with what was projected in the capital budgeting proposal. For example, suppose a new microchip was expected to earn a 20 percent IRR, but the product’s actual IRR turned out to be 9 percent. A postaudit examination would determine why the project failed to achieve its expected financial goals. Project reviews keep all people involved in the capital budgeting process honest because they know that the project and their performance will be reviewed and that they will be held accountable for the results.
postaudit review
an audit to compare actual project results with the results projected in the capital budgeting proposal
Managers should also conduct ongoing reviews of capital projects in progress. Such a review should challenge the business plan, including the cash flow projections and the operating cost assumptions. For example, Intel has periodically reviewed the viability of its wafer fabrication plant in China and has made adjustments to reflect changing business conditions. Business plans are management’s best estimates of future events at the time they are prepared, but as new information becomes available, the decision to undertake a capital project and the nature of that project must be reassessed.
Management must also evaluate people responsible for implementing a capital project. They should monitor whether the project’s revenues and expenses are meeting projections. If the project is not proceeding according to plan, the difficult task for management is to determine whether the problem is a flawed plan or poor execution by the implementation team. Good plans can fail if they are poorly executed at the operating level.
> BEFORE YOU GO ON
1. What changes have taken place in the capital budgeting techniques used by U.S. companies?
SUMMARY OF Learning Objectives
Capital budgeting is the process by which management decides which productive assets the firm should invest in. Because capital expenditures involve large amounts of money, are critical to achieving the firm’s strategic plan, define the firm’s line of business over the long term, and determine the firm’s profitability for years to come, they are considered the most important investment decisions made by management.
Explain the benefits of using the net present value (NPV) method to analyze capital expenditure decisions and calculate the NPV for a capital project.
The net present value (NPV) method leads to better investment decisions than other techniques because it: (1) uses the discounted cash flow valuation approach, which accounts for the time value of money, and (2) provides a direct measure of how much a capital project is expected to increase the dollar value of the firm. Thus, NPV is consistent with the top management goal of maximizing stockholder value. NPV calculations are described in Section 10.2 and Learning by Doing Application 10.1.
Describe the strengths and weaknesses of the payback period as a capital expenditure decision-making tool and compute the payback period for a capital project.
The payback period is the length of time it will take for the cash flows from a project to recover the cost of the project. The payback period is widely used, mainly because it is simple to apply and easy to understand. It also provides a simple measure of liquidity risk because it tells management how quickly the firm will get its money back. The payback period has a number of shortcomings, however. For one thing, the payback period, as most commonly computed, ignores the time value of money. We can overcome this objection by using discounted cash flows to calculate the payback period. Regardless of how the payback period is calculated, however, it fails to take account of cash flows recovered after the payback period. Thus, the payback period is biased in favor of short-lived projects. Also, the hurdle rate used to identify what payback period is acceptable is arbitrarily determined. Payback period calculations are described in Section 10.3 and Learning by Doing Application 10.2.
Explain why the accounting rate of return (ARR) is not recommended as a capital expenditure decision-making tool.
The ARR is based on accounting numbers, such as book value and net income, rather than cash flow data. As such, it is not a true rate of return. Instead of discounting a project’s cash flows over time, it simply gives us a number based on average figures from the income statement and balance sheet. Furthermore, as with the payback method, there is no economic rationale for establishing the hurdle rate. Finally, the ARR does not account for the size of the projects when a choice between two projects of different sizes must be made.
Compute the internal rate of return (IRR) for a capital project and discuss the conditions under which the IRR technique and the NPV technique produce different results.
The IRR is the expected rate of return for an investment project; it is calculated as the discount rate that equates the present value of a project’s expected cash inflows to the present value of the project’s outflows—in other words, as the discount rate at which the NPV is equal to zero. Calculations are shown in Section 10.5 and Learning by Doing Application 10.3. If a project’s IRR is greater than the required rate of return, the cost of capital, the project is accepted. The IRR rule often gives the same investment decision for a project as the NPV rule. However, the IRR method does have operational pitfalls that can lead to incorrect decisions. Specifically, when a project’s cash flows are unconventional, the IRR calculation may yield no solution or more than one IRR. In addition, the IRR technique cannot be used to rank projects that are mutually exclusive because the project with the highest IRR may not be the project that would add the greatest value to the firm if accepted—that is, the project with the highest NPV.
Explain the benefits of postaudit and ongoing reviews of capital projects.
A postaudit review enables managers to determine whether a project’s goals were met and to quantify the actual benefits or costs of the project. By conducting these reviews, managers can avoid making similar mistakes in future projects, learn to better recognize opportunities, and keep people involved in the budgeting process honest. An ongoing review enables managers to assess the impact of changing information and market conditions on the value of a project that is already underway. Unexpected changes in conditions can affect the viability of continuing such a project as originally conceived.
SUMMARY OF Key Equations
· 10.1 Premium Manufacturing Company is evaluating two forklift systems to use in its plant that produces the towers for a windmill power farm. The costs and the cash flows from these systems are shown below. If the company uses a 12 percent discount rate for all projects, determine which forklift system should be purchased using the net present value (NPV) approach.
· 10.3 Perryman Crafts Corp. is evaluating two independent capital projects that will each cost the company $250,000. The two projects will provide the following cash flows:
· 10.4 Terrell Corp. is looking into purchasing a machine for its business that will cost $117,250 and will be depreciated on a straight-line basis over a five-year period. The sales and expenses (excluding depreciation) for the next five years are shown in the following table. The company’s tax rate is 34 percent.
· 10.5 Refer to Problem 10.1. Compute the IRR for each of the two systems. Is the investment decision different from the one indicated by NPV?
Solutions to Self-Study Problems
· 10.1 NPVs for two forklift systems:
NPV for Otis Forklifts:
· 10.2 Payback period for Rutledge project:
· 10.4 Evaluation of Terrell Corp. project:
· 10.5 IRRs for two forklift systems:
Otis Forklifts:
First compute the IRR by the trial-and-error approach.
NPV (Otis) = $337,075 > 0
We should use a higher discount rate to get NPV = 0.
At k = 15 percent:
Craigmore Forklifts:
First compute the IRR using the trial-and-error approach.
NPV (Craigmore) = $90,606 > 0
We should use a higher discount rate to get NPV = 0.
At k = 15 percent:
Critical Thinking Questions
· 10.1 Explain why the cost of capital is referred to as the “hurdle” rate in capital budgeting.
· 10.2 a. A company is building a new plant on the outskirts of Smallesville. The town has offered to donate the land, and as part of the agreement, the company will have to build an access road from the main highway to the plant. How will the project of building of the road be classified in capital budgeting analysis?
b. Sykes, Inc., is considering two projects: a plant expansion and a new computer system for the firm’s production department. Classify these projects as independent, mutually exclusive, or contingent projects and explain your reasoning.
c. Your firm is currently considering the upgrading of the operating systems of all the firm’s computers. One alternative is to choose the Linux operating system that a local computer services firm has offered to install and maintain. Microsoft has also put in a bid to install the new Windows Vista operating system for businesses. What types of projects are these?
· 10.3 In the context of capital budgeting, what is “capital rationing”?
· 10.4 Provide two conditions under which a set of projects might be characterized as mutually exclusive.
· 10.5 a. A firm invests in a project that is expected to earn a return of 12 percent. If the appropriate cost of capital is also 12 percent, did the firm make the right decision? Explain.
b. What is the impact on the firm if it accepts a project with a negative NPV?
· 10.6 Identify the weaknesses of the payback period method.
· 10.7 What are the strengths and weaknesses of the accounting rate of return approach?
· 10.8 Under what circumstances might the IRR and NPV approaches have conflicting results?
· 10.9 The modified IRR (MIRR) alleviates two concerns with using the IRR method for evaluating capital investments. What are they?
· 10.10 Elkridge Construction Company has an overall (composite) cost of capital of 12 percent. This cost of capital reflects the cost of capital for an Elkridge Construction project with average risk. However, the firm takes on projects of various risk levels. The company experience suggests that low-risk projects have a cost of capital of 10 percent and high-risk projects have a cost of capital of 15 percent. Which of the following projects should the company select to maximize shareholder wealth?
· 10.2 Net present value: Kingston, Inc. management is considering purchasing a new machine at a cost of $4,133,250. They expect this equipment to produce cash flows of $814,322, $863,275, $937,250, $1,017,112, $1,212,960, and $1,225,000 over the next six years. If the appropriate discount rate is 15 percent, what is the NPV of this investment?
· 10.3 Net present value: Crescent Industries management is planning to replace some existing machinery in its plant. The cost of the new equipment and the resulting cash flows are shown in the accompanying table. If the firm uses an 18 percent discount rate for projects like this, should management go ahead with the project?
· 10.5 Net present value: Blanda Incorporated management is considering investing in two alternative production systems. The systems are mutually exclusive, and the cost of the new equipment and the resulting cash flows are shown in the accompanying table. If the firm uses a 9 percent discount rate for their production systems, in which system should the firm invest?
· 10.7 Payback: Quebec, Inc., is purchasing machinery at a cost of $3,768,966. The company’s management expects the machinery to produce cash flows of $979,225, $1,158,886, and $1,881,497 over the next three years, respectively. What is the payback period?
· 10.8 Payback: Northern Specialties just purchased inventory-management computer software at a cost of $1,645,276. Cost savings from the investment over the next six years will produce the following cash flow stream: $212,455, $292,333, $387,479, $516,345, $645,766, and $618,325. What is the payback period on this investment?
· 10.9 Payback: Nakamichi Bancorp has made an investment in banking software at a cost of $1,875,000. Management expects productivity gains and cost savings over the next several years. If, as a result of this investment, the firm is expected to generate additional cash flows of $586,212, $713,277, $431,199, and $318,697 over the next four years, what is the investment’s payback period?
· 10.10 Average accounting rate of return (ARR): Capitol Corp. management is expecting a project to generate after-tax income of $63,435 in each of the next three years. The average book value of the project’s equipment over that period will be $212,500. If the firm’s acceptance decision on any project is based on an ARR of 37.5 percent, should this project be accepted?
· 10.11 Internal rate of return: Refer to Problem 10.4. What is the IRR that Franklin Mints management can expect on this project?
· 10.12 Internal rate of return: Hathaway, Inc., a resort company, is refurbishing one of its hotels at a cost of $7.8 million. Management expects that this will lead to additional cash flows of $1.8 million for the next six years. What is the IRR of this project? If the appropriate cost of capital is 12 percent, should Hathway go ahead with this project?
· 10.14 Net present value: Briarcrest Condiments is a spice-making firm. Recently, it developed a new process for producing spices. The process requires new machinery that would cost $1,968,450, have a life of five years, and would produce the cash flows shown in the following table. What is the NPV if the discount rate is 15.9 percent?
15. What are the NPVs of the two projects?
15. Should both projects be accepted? or either? or neither? Explain your reasoning.
1. 10.18 Discounted payback: Timeline Manufacturing Co. is evaluating two projects. The company uses payback criteria of three years or less. Project A has a cost of $912,855, and project B’s cost is $1,175,000. Cash flows from both projects are given in the following table. What are their discounted payback periods and which will be accepted with a discount rate of 8 percent?
1. 10.22 Modified internal rate of return (MIRR): Sycamore Home Furnishings is considering acquiring a new machine that can create customized window treatments. The equipment will cost $263,400 and will generate cash flows of $85,000 over each of the next six years. If the cost of capital is 12 percent, what is the MIRR on this project?
1. 10.23 Internal rate of return: Great Flights, Inc., an aviation firm, is considering purchasing three aircraft for a total cost of $161 million. The company would lease the aircraft to an airline. Cash flows from the proposed leases are shown in the following table. What is the IRR of this project?
1. 10.25 Internal rate of return: Ancala Corporation is considering investments in two new golf apparel lines for next season: golf hats and belts. Due to a funding constraint, these lines are mutually exclusive. A summary of each project’s estimated cash flows over its three-year life, as well as the IRR and NPV of each, are outlined below. The CFO of the firm has decided to manufacture the belts; however, the CEO is questioning this decision given that the IRR is higher for manufacturing hats. Explain to the CEO why the IRRs and NPVs of the belt and hat projects disagree? Is the CFO’s decision correct?
26. An initial investment of $25,000 followed by a single cash flow of $37,450 in year 6.
26. An initial investment of $1 million followed by a single cash flow of $1,650,000 in year 4.
26. An initial investment of $2 million followed by cash flows of $1,650,000 and $1,250,000 in years 2 and 4, respectively.
1. 10.27 Internal rate of return: Compute the IRR for the following project cash flows:
27. An initial outlay of $3,125,000 followed by annual cash flows of $565,325 for the next eight years.
27. An initial investment of $33,750 followed by annual cash flows of $9,430 for the next five years.
27. An initial outlay of $10,000 followed by annual cash flows of $2,500 for the next seven years.
29. What are the IRRs for the projects?
29. Does the IRR criterion indicate a different decision than the NPV criterion?
29. Explain how you would expect the management of Draconian Measures to decide.
1. 10.30 Dravid, Inc., is currently evaluating three projects that are independent. The cost of funds can be either 13.6 percent or 14.8 percent depending on their financing plan. All three projects cost the same at $500,000. Expected cash flow streams are shown in the following table. Which projects would be accepted at a discount rate of 14.8 percent? What if the discount rate was 13.6 percent?
35. What is the payback period?
35. What is the NPV for this project?
35. What is the IRR?
1. 10.36 Quasar Tech Co. is investing $6 million in new machinery that will produce the next-generation routers. Sales to its customers will amount to $1,750,000 for the next three years and then increase to $2.4 million for three more years. The project is expected to last six years and operating costs, excluding depreciation, will be $898,620 annually. The machinery will be depreciated to a salvage value of $0 over 6 years using the straight-line method. The company’s tax rate is 30 percent, and the cost of capital is 16 percent.
36. What is the payback period?
36. What is the average accounting return (ARR)?
36. Calculate the project NPV.
36. What is the IRR for the project?
1. 10.37 Skywards, Inc., an airline caterer, is purchasing refrigerated trucks at a total cost of $3.25 million. After-tax net income from this investment is expected to be $750,000 for the next five years. Annual depreciation expense will be $650,000. The cost of capital is 17 percent.
37. What is the discounted payback period?
37. Compute the ARR.
37. What is the NPV on this investment?
37. Calculate the IRR.
1. 10.38 Trident Corp. is evaluating two independent projects. The costs and expected cash flows are given in the following table. The cost of capital is 10 percent.
38. Calculate the projects’ IRR.
38. Which project should be chosen based on NPV? Based on IRR? Is there a conflict?
38. If you are the decision maker for the firm, which project or projects will be accepted? Explain your reasoning.
1. 10.39 Tyler, Inc., is considering switching to a new production technology. The cost of the required equipment will be $4 million. The discount rate is 12 percent. The cash flows that the firm expects the new technology to generate are as follows.
39. What is the NPV for the project? Should the firm go ahead with the project?
39. What is the IRR, and what would be the decision based on the IRR?
41. 0.16 year longer than the payback period.
41. 0.80 year longer than the payback period.
41. 1.01 years longer than the payback period.
41. 1.85 years longer than the payback period.
1. 10.42 An investment of $100 generates after-tax cash flows of $40 in Year 1, $80 in Year 2, and $120 in Year 3. The required rate of return is 20 percent. The net present value is closest to
a. $42.22 b. $58.33
c. $68.52 d. $98.95
1. 10.43 An investment of $150,000 is expected to generate an after-tax cash flow of $100,000 in one year and another $120,000 in two years. The cost of capital is 10 percent. What is the internal rate of return?
a. 28.19 percent b. 28.39 percent
c. 28.59 percent d. 28.79 percent
1. 10.44 An investment requires an outlay of $100 and produces after-tax cash flows of $40 annually for four years. A project enhancement increases the required outlay by $15 and the annual after-tax cash flows by $5. How will the enhancement affect the project’s NPV profile? The vertical intercept of the NPV profile of the project shifts:
44. Up and the horizontal intercept shifts left.
44. Up and the horizontal intercept shifts right.
44. Down and the horizontal intercept shifts left.
44. Down and the horizontal intercept shifts right.
Sample Test Problems
· 10.1 Net present value: Techno Corp. is considering developing new computer software. The cost of development will be $675,000 and management expects the net cash flow from sale of the software to be $195,000 for each of the next six years. If the discount rate is 14 percent, what is the net present value of this project?
· 10.2 Payback method: Parker Office Supplies management is considering replacing the company’s outdated inventory-management software. The cost of the new software will be $168,000. Cost savings are expected to be $43,500 for each of the first three years and then drop to $36,875 for the following two years. What is the payback period for this project?
· 10.3 Accounting rate of return: Fresno, Inc. is expecting a project to generate after-tax income of $156,435 in each of the next three years. The average book value of its equipment over that period will be $322,500. If the firm’s acceptance decision on any project is based on an ARR of 40 percent, should this project be accepted?
· 10.4 Internal rate of return: Refer to Sample Test Problem 10.1. What is the IRR on this project?
· 10.5 Net present value: Raycom, Inc. needs a new overhead crane and two alternatives are available. Crane T costs $1.35 million and will produce cost savings of $765,000 in each of the next three years. Crane R will cost $1.675 million and will yield annual cost savings of $815,000 for the next three years. The required rate of return is 15 percent. Which of the two options should Raycom choose based on NPV criteria, and why?
As we pointed out in
Chapter 5
, financial decision-making problems can be solved either by discounting cash flows to the beginning of the project or by using compounding to find the future value of cash flows at the end of a project’s life.
13
The Cost of Capital
Learning Objectives
Explain what the weighted average cost of capital for a firm is and why it is often used as a discount rate to evaluate projects.
Calculate the cost of debt for a firm.
Calculate the cost of common stock and the cost of preferred stock for a firm.
Calculate the weighted average cost of capital for a firm, explain the limitations of using a firm’s weighted average cost of capital as the discount rate when evaluating a project, and discuss the alternatives to the firm’s weighted average cost of capital that are available.
The Walt Disney Company announced in May 2010 that it would build a new hotel at Walt Disney World, its first new hotel at that theme park in seven years. The hotel, which is to be opened in several phases beginning in 2012, has been named “Disney’s Art of Animation Resort.” It will be built on a 65-acre parcel of land across the lake from Disney’s Pop Century Resort and will have 1,120 suites and 864 traditional hotel rooms. Disney executives anticipate that the rooms in the Art of Animation Resort will be priced comparably to those at the Pop Century Resort, which begin at less than $100 per night.
As you can imagine, the cost of financing a project like this is substantial. Disney is a highly sophisticated and successful hotel and theme park developer and operator. Before the company announced the construction of the Art of Animation Resort, you can be sure that the managers at Disney carefully considered the financial aspects of the project. They evaluated the required investment, what revenues the new hotel was likely to generate, and how much it would cost to operate and maintain. They also estimated what it would cost to finance the project—how much they would pay for the debt and the returns equity investors would require for an investment with this level of risk. This “cost of capital” would be incorporated into their NPV analysis through the discounting process.
Doing a good job of estimating the cost of capital is especially important for a capitalintensive project such as a hotel. The cost of financing a hotel like the one that Disney is building can easily total $50 or more per room rental. In other words, if an average room rents for $100, the cost of financing the project can consume 50 percent or more of the revenue the hotel receives from renting a room!
From this example, you can see how important it is to get the cost of capital right. If Disney managers had estimated the cost of capital to be 7 percent when it was really 9 percent, they might have ended up investing in a project with a large negative NPV. How did they approach this important task? In this chapter we discuss how managers estimate the cost of capital they use to evaluate projects.
CHAPTER PREVIEW
discussed the general concept of risk and described what financial analysts mean when they talk about the risk associated with a project’s cash flows. It also explained how this risk is related to expected returns. With this background, we are ready to discuss the methods that financial managers use to estimate discount rates, the reasons they use these methods, and the shortcomings of each method.
We start this chapter by introducing the weighted average cost of capital and explaining how this concept is related to the discount rates that many financial managers use to evaluate projects. Then we describe various methods that are used to estimate the three broad types of financing that firms use to acquire assets—debt, common stock, and preferred stock—as well as the overall weighted average cost of capital for the firm.
We next discuss the circumstances under which it is appropriate to use the weighted average cost of capital for a firm as the discount rate for a project and outline the types of problems that can arise when the weighted average cost of capital is used inappropriately. Finally, we examine alternatives to using the weighted average cost of capital as a discount rate.
13.1 THE FIRM’s OVERALL COST OF CAPITAL
·
Our discussions of investment analysis up to this point have focused on evaluating individual projects. We have assumed that the rate used to discount the cash flows for a project reflects the risks associated with the incremental after-tax free cash flows from that project. In
Chapter 7
, we saw that unsystematic risk can be eliminated by holding a diversified portfolio. Therefore, systematic risk is the only risk that investors require compensation for bearing. With this insight, we concluded that we could use Equation 7.10, to estimate the expected rate of return for a particular investment:
i
) is the expected return on project i, Rrf is the risk-free rate of return, b
i
is the beta for project i, and E (Rm) is the expected return on the market. Recall that the difference between the expected return on the market and the risk-free rate [E (Rm) − Rrf] is known as the market risk premium.
Although these ideas help us better understand the discount rate on a conceptual level, they can be difficult to implement in practice. Firms do not issue publicly traded shares for individual projects. This means that analysts do not have the stock returns necessary to use a regression analysis like that illustrated in
Exhibit 7.10
to estimate the beta (β) for an individual project. As a result, they have no way to directly estimate the discount rate that reflects the systematic risk of the incremental cash flows from a particular project.
In many firms, senior financial managers deal with this problem by estimating the cost of capital for the firm as a whole and then requiring analysts within the firm to use this cost of capital to discount the cash flows for all projects.
1
A problem with this approach is that it ignores the fact that a firm is really a collection of projects with different levels of risk. A firm’s overall cost of capital is actually a weighted average of the costs of capital for these projects, where the weights reflect the relative values of the projects.
To see why a firm is a collection of projects, consider The Boeing Company. Boeing manufactures a number of different models of civilian and military aircraft. If you have ever flown on a commercial airline, chances are that you have been on a Boeing 737, 747, 757, 767, or 777 aircraft. Boeing manufactures several versions of each of these aircraft models to meet the needs of its customers. These versions have different ranges, seat configurations, numbers of seats, and so on. Some are designed exclusively to haul freight for companies such as UPS and FedEx. Every version of every model of aircraft at Boeing was, at some point in time, a new project. The assets owned by Boeing today and its expected cash flows are just the sum of the assets and cash flows from all of these individual projects plus the other projects at the firm, such as those involving military aircraft.
2
This means that the overall systematic risk associated with Boeing’s cash flows and the company’s cost of capital are weighted averages of the systematic risks and the costs of capital for its individual projects.
If the risk of an individual project differs from the average risk of the firm, the firm’s overall cost of capital is not the ideal discount rate to use when evaluating that project. Nevertheless, since this is the discount rate that is commonly used, we begin by discussing how a firm’s overall cost of capital is estimated. We then discuss alternatives to using the firm’s cost of capital as the discount rate in evaluating a project.
The Finance Balance Sheet
To understand how financial analysts estimate their firms’ costs of capital, you must be familiar with a concept that we call the finance balance sheet. The finance balance sheet is like the accounting balance sheet from
Chapter 3
. The main difference is that it is based on market values rather than book values. Recall that the total book value of the assets reported on an accounting balance sheet does not necessarily reflect the total market value of those assets. This is because the book value is largely based on historical costs, while the total market value of the assets equals the present value of the total cash flows that those assets are expected to generate in the future. The market value can be greater than or less than the book value but is rarely the same.
finance balance sheet
a balance sheet that is based on market values of expected cash flows
While the left-hand side of the accounting balance sheet reports the book values of a firm’s assets, the right-hand side reports how those assets were financed. Firms finance the purchase of their assets using debt and equity.
3
Since the cost of the assets must equal the total value of the debt and equity that was used to purchase them, the book value of the assets must equal the book value of the liabilities plus the book value of the equity on the accounting balance sheet. In
Chapter 3
we called this equality the balance sheet identity.
Just as the total book value of the assets at a firm does not generally equal the total market value of those assets, the book value of total liabilities plus stockholders’ equity does not usually equal the market value of these claims. In fact, the total market value of the debt and equity claims differ from their book values by exactly the same amount that the market values of a firm’s assets differ from their book values. This is because the total market value of the debt and the equity at a firm equals the present value of the cash flows that the debt holders and the stockholders have the right to receive. These cash flows are the cash flows that the assets in the firm are expected to generate. In other words, the people who have lent money to a firm and the people who have purchased the firm’s stock have the right to receive all of the cash flows that the firm is expected to generate in the future. The value of the claims they hold must equal the value of the cash flows that they have a right to receive.
The fact that the market value of the assets must equal the value of the cash flows that these assets are expected to generate, combined with the fact that the value of the expected cash flows also equals the total market value of the firm’s total liabilities and equity, means that we can write the market value (MV) of assets as follows:
The market value of a firm’s assets, which equals the present value of the cash flows those assets are expected to generate in the future, must equal the market value of the claims on those cash flows—the firm’s liabilities and equity.
Equation 13.1 is just like the accounting balance sheet identity. The only difference is that Equation 13.1 is based on market values. This relation is illustrated in
Exhibit 13.1
.
To see why the market value of the assets must equal the total market value of the liabilities and equity, consider a firm whose only business is to own and manage an apartment building that was purchased 20 years ago for $1,000,000. Suppose that there is currently a mortgage on the building that is worth $300,000, the firm has no other liabilities, and the current market value of the building, based on the expected cash flows from future rents, is $4,000,000. What is the market value of all of the equity (stock) in this firm?
The fact that you paid $1,000,000 20 years ago is not relevant to this question. What matters in finance is the value of the expected cash flows from future rents, the $4,000,000. This is the market value of the firm’s assets—the left-hand side of the balance sheet in
Exhibit 13.1
. Since we know that the firm owes $300,000, we can substitute into Equation 13.1 and solve for the market value of the equity:
If the concept of a balance sheet based on market values seems familiar to you, it is because the idea of preparing an actual balance sheet based on market values was discussed in
Chapter 3
. In that chapter we pointed out that such a balance sheet would be more useful to financial decision makers than the ordinary accounting balance sheet. Financial managers are much more concerned about the future than the past when they make decisions. You might revisit the discussion of sunk costs in
Chapter 11
to remind yourself of why this is true.
BUILDING INTUITION THE MARKET VALUE OF A FIRM’s ASSETS EQUALS THE MARKET VALUE OF THE CLAIMS ON THOSE ASSETS
The market value of the debt and equity claims against the cash flows of a firm must equal the present value of the cash flows that the firm’s assets are expected to generate. This is because, between them, the debt holders and the stockholders have the legal right to receive all of those cash flows.
How Firms Estimate Their Cost of Capital
Now that we have discussed the basic idea of the finance balance sheet, consider the challenge that financial analysts face when they want to estimate the cost of capital for a firm. If analysts at a firm could estimate the betas for each of the firm’s individual projects, they could estimate the beta for the entire firm as a weighted average of the betas for the individual projects. They could do this because, as we discussed earlier, the firm is simply a collection (portfolio) of projects. This calculation would just be an application of Equation 7.11:
i
is the beta for project i and xi
is the fraction of the total firm value represented by project i.
The analysts could then use the beta for the firm in Equation 7.10:
Instead, analysts must use their knowledge of the finance balance sheet, along with the concept of market efficiency, which we discussed in
Chapter 2
, to estimate the cost of capital for the firm. Rather than using Equations 7.11 and 7.10 to perform the calculations for the individual projects represented on the left-hand side of the finance balance sheet, analysts perform a similar set of calculations for the different types of financing (debt and equity) on the right-hand side of the finance balance sheet. They can do this because, as we said earlier, the people who finance the firm have the right to receive all of the cash flows on the left-hand side. This means that the systematic risk associated with the total assets on the left-hand side is the same as the systematic risk associated with the total financing on the right-hand side. In other words, the weighted average of the betas for the different claims on the assets must equal a weighted average of the betas for the individual assets (projects).
Analysts do not need to estimate betas for each type of financing that the firm has. As long as they can estimate the cost of each type of financing—either directly, by observing that cost in the capital markets, or by using Equation 7.10—they can compute the cost of capital for the firm using the following equation:
Firm is the cost of capital for the firm, ki
is the cost of financing type i, and xi
is the fraction of the total market value of the financing (or of the assets) of the firm represented by financing type i. This formula simply says that the overall cost of capital for the firm is a weighted average of the cost of each different type of financing used by the firm.
4
Note that since we are specifically talking about the cost of capital, we use the symbol ki
to represent this cost, rather than the more general notation E (R
i
) that we used in
Chapter 7
.
The similarity between Equation 13.2 and Equation 7.11 is not an accident. Both are applications of the basic idea that the systematic risk of a portfolio of assets is a weighted average of the systematic risks of the individual assets. Because Rrf and E (Rm) in Equation 7.10 are the same for all assets, when we substitute Equation 7.10 into Equation 13.2 (remember that E (R
i
) in Equation 7.10 is the same as ki
in Equation 13.2) and cancel out Rrf and E(Rm), we get Equation 7.11. We will not prove this here, but you might do so to convince yourself that what we are saying is true.
To see how Equation 13.2 is applied, let’s return to the example of the firm whose only business is to manage an apartment building. Recall that the total value of this firm is $4,000,000 and that it has $300,000 in debt. If the firm has only one loan and one type of stock, then the fractions of the total value represented by those two types of financing are as follows:
5
Firm. From this point on, we will use the abbreviation WACC to represent the firm’s overall cost of capital.
weighted average cost of capital (WACC)
the weighted average of the costs of the different types of capital (debt and equity) that have been used to finance a firm; the cost of each type of capital is weighted by the proportion of the total capital that it represents
BUILDING INTUITION A FIRM’s COST OF CAPITAL IS A WEIGHTED AVERAGE OF ALL OF ITS FINANCING COSTS
The cost of capital for a firm is a weighted average of the costs of the different types of financing used by a firm. The weights are the proportions of the total firm value represented by the different types of financing. By weighting the costs of the individual financing types in this way, we obtain the overall average opportunity cost of each dollar invested in the firm.
APPLICATION 13.1 LEARNING BY DOING
APPROACH: You can use Equation 13.2 to calculate the WACC for this firm. Since you are planning to finance the purchase using capital from three different sources—two loans and your own equity investment—the right-hand side of Equation 13.2 will have three terms.
SOLUTION:
We begin by calculating the weights for the different types of financing:
> BEFORE YOU GO ON
1. Why does the market value of the claims on the assets of a firm equal the market value of the assets?
2. How is the WACC for a firm calculated?
3. What does the WACC for a firm tell us?
13.2 THE COST OF DEBT
·
In our discussion of how the WACC for a firm is calculated, we assumed that the costs of the different types of financing were known. This assumption allowed us to simply plug those costs into Equation 13.2 once we had calculated the weight for each type of financing. Unfortunately, life is not that simple. In the real world, analysts have to estimate each of the individual costs. In other words, the discussion in the preceding section glossed over a number of concepts and issues that you should be familiar with. This section and Section 13.3 discuss those concepts and issues and show how the costs of the different types of financing can be estimated.
Before we move on to the specifics of how to estimate the costs of different types of financing, we must stress an important point: All of these calculations depend in some part on financial markets being efficient. We suggested this in the last section when we mentioned that analysts have to rely on the concept of market efficiency to estimate the WACC. The reason is that analysts often cannot directly observe the rate of return that investors require for a particular type of financing. Instead, analysts must rely on the security prices they can observe in the financial markets to estimate the required rate.
It makes sense to rely on security prices only if you believe that the financial markets are reasonably efficient at incorporating new information into these prices. If the markets were not efficient, estimates of expected returns that were based on market security prices would be unreliable. Of course, if the returns that are plugged into Equation 13.2 are bad, the resulting estimate for WACC will also be bad. With this caveat, we can now discuss how to estimate the costs of the various types of financing.
Key Concepts for Estimating the Cost of Debt
Virtually all firms use some form of debt financing. The financial managers at firms typically arrange for revolving lines of credit to finance working capital items such as inventories or accounts receivable. These lines of credit are very much like the lines of credit that come with your credit cards. Firms also obtain private fixed-term loans, such as bank loans, or sell bonds to the public to finance ongoing operations or the purchase of long-term assets—just as you would finance your living expenses while you are in school with a student loan or a car with a car loan. For example, an electric utility firm, such as FPL Group in Florida, will sell bonds to finance a new power plant, and a rapidly growing retailer, such as Target, will use debt to finance new stores and distribution centers. As mentioned earlier, we will discuss how firms finance themselves in more detail in
Chapters 15
and
16
, but for now it is sufficient to recognize that firms use these three general types of debt financing: lines of credit, private fixed-term loans, and bonds that are sold in the public markets.
There is a cost associated with each type of debt that a firm uses. However, when we estimate the cost of capital for a firm, we are particularly interested in the cost of the firm’s long-term debt. Firms generally use long-term debt to finance their long-term assets, and it is the long-term assets that concern us when we think about the value of a firm’s assets. By long-term debt, we usually mean the debt that, when it was borrowed, was set to mature in more than one year. This typically includes fixed-term bank loans used to finance ongoing operations or long-term assets, as well as the bonds that a firm sells in the public debt markets.
Although one year is not an especially long time, debt with a maturity of more than one year is typically viewed as permanent debt. This is because firms often borrow the money to pay off this debt when it matures.
We do not normally worry about revolving lines of credit when calculating the cost of debt because these lines tend to be temporary. Banks typically require that the outstanding balances be periodically paid down to $0 (just as we are sure you pay your entire credit card balance from time to time).
When analysts estimate the cost of a firm’s long-term debt, they are estimating the cost on a particular date—the date on which they are doing the analysis. This is a very important point to keep in mind because the interest rate that the firm is paying on its outstanding debt does not necessarily reflect its current cost of debt. Interest rates change over time, and so does the cost of debt for a firm. The rate a firm was charged three years ago for a five-year loan is unlikely to be the same rate that it would be charged today for a new five-year loan. For example, suppose that FPL Group issued bonds five years ago for 7 percent. Since then, interest rates have fallen, so the same bonds could be sold at par value today for 6 percent. The cost of debt today is 6 percent, not 7 percent, and 6 percent is the cost of debt that management will use in WACC calculations. If you looked in the firm’s financial statements, you would see that the firm is paying an interest rate of 7 percent. This is what the financial managers of the firm agreed to pay five years ago, not what it would cost to sell the same bonds today. The accounting statements reflect the cost of debt that was sold at some time in the past.
BUILDING INTUITION THE CURRENT COST OF LONG-TERM DEBT IS WHAT MATTERS WHEN CALCULATING WACC
The current cost of long-term debt is the appropriate cost of debt for WACC calculations. This is because the WACC we use in capital budgeting is the opportunity cost of capital for the firm’s investors as of today. This means we must use today’s costs of debt and equity when we calculate the WACC. Historical costs do not belong in WACC calculations.
Estimating the Current Cost of a Bond or an Outstanding Loan
We have now seen that we should not use historical costs of debt in WACC calculations. Let’s discuss how we can estimate the current costs of bonds and other fixed-term loans by using market information.
The Current Cost of a Bond
You may not realize it, but we have already discussed how to estimate the current cost of debt for a publicly traded bond. This cost is estimated using the yield to maturity calculation. Recall that in
Chapter 8
we defined the yield to maturity as the discount rate that makes the present value of the coupon and principal payments equal to the price of the bond.
For example, consider a 10-year bond with a $1,000 face value that was issued five years ago. This bond has five years remaining before it matures. If the bond has an annual coupon rate of 7 percent, pays coupon interest semiannually, and is currently selling for $1,042.65, we can calculate its yield to maturity by using Equation 8.1 and solving for i or by using a financial calculator. Let’s use Equation 8.1 for this example.
To do this, as was discussed in the section on semiannual compounding in
Chapter 8
, we first convert the bond data to reflect semiannual compounding: (1) the total number of coupon payments is 10 (2 payments per year × 5 years = 10 payments), and (2) the semiannual coupon payment is $35 [($1,000 × 7 percent per year)/2 periods per year = $70/2 = $35]. We can now use Equation 8.1 and solve for i to find the yield to maturity:
i = k
Bond = 0.030, or 3.0%
This semiannual rate would be quoted as an annual rate of 6 percent (2 periods per year × 0.03 = 0.06, or 6 percent) in financial markets. However, as explained in
Chapter 8
, this annual rate fails to account for the effects of compounding. We must therefore use Equation 6.7 to calculate the effective annual interest rate (EAR) in order to obtain the actual current annual cost of this debt:
Notice that the above calculation takes into account the interest payments, the face value of the debt (the amount that will be repaid in five years), and the current price at which the bond is selling. It is necessary to account for all of these characteristics of the bond. The return received by someone who buys the bond today will be determined by both the interest income and the capital appreciation (or capital depreciation in this case, since the price is higher than the face value).
We must account for one other factor when we calculate the current cost of bond financing to a company—the cost of issuing the bond. In the above example, we calculated the return that someone who buys the bond can expect to receive. Since a company must pay fees to investment bankers, lawyers, and accountants, along with various other costs, to actually issue a bond, the cost to the company is higher than 6.1 percent.
6
Therefore, in order to obtain an accurate estimate of the cost of a bond to the issuing firm, analysts must incorporate issuance costs into their calculations. Issuance costs are an example of direct out-of-pocket costs, the actual out-of-pocket costs that a firm incurs when it raises capital.
The way in which issuance costs are incorporated into the calculation of the cost of a bond is straightforward. Analysts use the net proceeds that the company receives from the bond, rather than the price that is paid by the investor, on the left-hand side of Equation 8.1. Suppose the company in our example sold 5-year bonds with a 7 percent coupon today and paid issuance costs equal to 2 percent of the total value of the bonds. After paying the issuance costs, the company would receive only 98 percent of the price paid by the investors. Therefore, the company would actually receive only $1,042.65 × (1 − 0.02) = $1,021.80 for each bond it sold and the semiannual cost to the company would be:
EAR = 11.032422 − 1 = 0.066, or 6.6%
In this example the issuance costs increase the effective cost of the bonds from 6.1 percent to 6.6 percent per year.
The Current Cost of an Outstanding Loan
Conceptually, calculating the current cost of long-term bank or other private debt is not as straightforward as estimating the current cost of a public bond because financial analysts cannot observe the market price of private debt. Fortunately, analysts do not typically have to do this. Instead, they can simply call their banker and ask what rate the bank would charge if they decided to refinance the debt today. A rate quote from a banker provides a good estimate of the current cost of a private loan.
Taxes and the Cost of Debt
It is very important that you understand one additional concept concerning the cost of debt: In the United States, firms can deduct interest payments for tax purposes. In other words, every dollar a firm pays in interest reduces the firm’s taxable income by one dollar. Thus, if the firm’s marginal tax rate is 35 percent, the firm’s total tax bill will be reduced by 35 cents for every dollar of interest it pays. A dollar of interest would actually cost this firm only 65 cents because the firm would save 35 cents on its taxes.
More generally, the after-tax cost of interest payments equals the pretax cost times 1 minus the tax rate. This means that the after-tax cost of debt is:
Taxes affect the cost of debt in much the same way that the interest tax deduction on a home mortgage affects the cost of financing a house. For example, assume that you borrow $200,000 at 6 percent to buy a house on January 1 and your interest payments total $12,000 in the first year. Under the tax law, you can deduct this $12,000 from your taxable income when you calculate your taxes for the year.
7
Debt pretax at 6 percent and t at 20 percent, Equation 13.3 gives us:
Most firms have several different debt issues outstanding at any particular point in time. Just as you might have both a car loan and a school loan, a firm might have several bank loans and bond issues outstanding. To estimate the firm’s overall cost of debt when it has several debt issues outstanding we must first estimate the costs of the individual debt issues and then calculate a weighted average of these costs.
To see how this is done, let’s consider an example. Suppose that your pizza parlor business has grown dramatically in the past three years from a single restaurant to 30 restaurants. To finance this growth, two years ago you sold $25 million of five-year bonds. These bonds pay interest annually and have a coupon rate of 8 percent. They are currently selling for $1,026.24 per $1,000 bond. Just today, you also borrowed $5 million from your local bank at an interest rate of 6 percent. Assume that this is all the long-term debt that you have and that there are no issuance costs. What is the overall average after-tax cost of your debt if your business’s marginal tax rate is 35 percent?
The pretax cost of the bonds as of today is the effective annual yield on those bonds. Since the bonds were sold two years ago, they will mature three years from now. Using Equation 8.1, we find that the effective annual yield (which equals the yield to maturity in this example) for these bonds is:
Now that we know the pretax costs of the two types of debt that your business has outstanding, we can calculate the overall average cost of your debt by calculating the weighted average of their two costs. Since the weights for the two types of debt are based on their current market values we must first determine these values. Because the bonds are currently selling above their par value we know that their current market value is greater than their $25 million face value. In fact, it equals:
1$1,026.24/$1,0002 × $25,000,000 = $25,656,000
Since the bank loan was just made today, its value simply equals the amount borrowed or $5 million. The weights for the two types of debt are therefore:
APPROACH: The overall after-tax cost of debt can be calculated using the following three-step process: (1) Calculate the fraction of the total debt (weight) for each individual debt issue. (2) Using these weights, calculate the weighted average pretax cost of debt. (3) Use Equation 13.3 to calculate the after-tax average cost of debt.
SOLUTION:
(1) The weights for the three types of debt are as follows:
Using the Cost of Debt in Decision Making
DECISION: This is an easy decision. You should choose the least expensive alternative—the loan from your local bank. In this example, you can directly compare the pretax costs of the two alternatives. You do not need to calculate the after-tax costs because multiplying each pretax cost by the same number, 1 − t, will not change your decision.
> BEFORE YOU GO ON
1. Why do analysts care about the current cost of long-term debt when estimating a firm’s cost of capital?
2. How do you estimate the cost of debt for a firm with more than one type of debt?
3. How do taxes affect the cost of debt?
13.3 THE COST OF EQUITY
·
The cost of equity (stock) for a firm is a weighted average of the costs of the different types of stock that the firm has outstanding at a particular point in time. We saw in
Chapter 9
that some firms have both preferred stock and common stock outstanding. In order to calculate the cost of equity for these firms, we have to know how to calculate the cost of both common stock and preferred stock. In this section, we discuss how financial analysts can estimate the costs associated with these two different types of stock.
Common Stock
Just as information about market rates of return is used to estimate the cost of debt, market information is also used to estimate the cost of equity. There are several ways to do this. The particular approach a financial analyst chooses will depend on what information is available and how reliable the analyst believes it is. Next we discuss three alternative methods for estimating the cost of common stock. It is important to remember throughout this discussion that the “cost” we are referring to is the rate of return that investors require for investing in the stock at a particular point in time, given its systematic risk.
Method 1: Using the Capital Asset Pricing Model (CAPM)
The first method for estimating the cost of common equity is one that we discussed in
Chapter 7
. This method uses Equation 7.10:
If we recognize that E(R
i
) in Equation 7.10 is the cost of the common stock capital used by the firm (k
cs) when we are calculating the cost of equity and that [E (Rm) − Rrf] is the market risk premium, we can rewrite Equation 7.10 as follows:
Chapter 7
. In those examples you were given the current risk-free rate, the beta for the stock, and the market risk premium and were asked to calculate k
cs using the equation. Now we turn our attention to some practical considerations that you must be concerned with when choosing the appropriate risk-free rate, beta, and market risk premium for this calculation.
The Risk-Free Rate. First, let’s consider the risk-free rate. The current effective annual yield on a risk-free asset should always be used in Equation 13.4.
8
This is because the risk-free rate at a particular point in time reflects the rate of inflation that the market expects in the future. Since the expected rate of inflation changes over time, an old risk-free rate might not reflect current inflation expectations.
When analysts select a risk-free rate, they must choose between using a short-term rate, such as that for Treasury bills, or a longer-term rate, such as those for Treasury notes or bonds. Which of these choices is most appropriate? This question has been hotly debated by finance professionals for many years. We recommend that you use the risk-free rate on a long-term Treasury security when you estimate the cost of equity capital because the equity claim is a long-term claim on the firm’s cash flows. As you saw in
Chapter 9
, the stockholders have a claim on the cash flows of the firm in perpetuity. By using a long-term Treasury security, you are matching a long-term risk-free rate with a long-term claim. A long-term risk-free rate better reflects long-term inflation expectations and the cost of getting investors to part with their money for a long period of time than a short-term rate.
You can find current yields on Treasury bills, notes, and bonds at the Web site of the U.S. Federal Reserve Bank at
http://www.federalreserve.gov/releases/H15/update
.
The Beta. If the common stock of a company is publicly traded, then you can estimate the beta for that stock using a regression analysis similar to that illustrated in
Exhibit 7.10
. However, identifying the appropriate beta is much more complicated if the common stock is not publicly traded. Since most companies in the United States are privately owned and do not have publicly traded stock, this is a problem that arises quite often when someone wants to estimate the cost of common equity for a firm.
Financial analysts often overcome this problem by identifying a “comparable” company with publicly traded stock that is in the same business and that has a similar amount of debt. For example, suppose you are trying to estimate the beta for your pizza business. The company has now grown to include more than 2,000 restaurants throughout the world. The frozen-foods business, however, was never successful and had to be shut down. You know that Domino’s Pizza, Inc., one of your major competitors, has publicly traded equity and that the proportion of debt to equity for Domino’s is similar to the proportion for your firm. Since Domino’s overall business is similar to yours, in that it is only in the pizza business and competes in similar geographic areas, it would be reasonable to consider Domino’s a comparable company.
Companies with publicly traded equity usually provide a lot of information about their businesses and financial performance on their Web sites. The Domino’s Pizza Web site is a good example. Go to
http://phx.corporate-ir.net/phoenix.zhtml?c=135383&p=irol-irhome
.
The systematic risk associated with the stock of a comparable company is likely to be similar to the systematic risk for the private firm because systematic risk is determined by the nature of the firm’s business and the amount of debt that it uses. If you are able to identify a good comparable company, such as Domino’s Pizza, you can use its beta in Equation 13.4 to estimate the cost of equity capital for your firm. Even when a good comparable company cannot be identified, it is sometimes possible to use an average of the betas for the public firms in the same industry.
The Market Risk Premium. It is not possible to directly observe the market risk premium. We just do not know what rate of return investors expect for the market portfolio, E(Rm), at a particular point in time. Therefore, we cannot simply calculate the market risk premium as the difference between the expected return on the market and the risk-free rate, [E (Rm) − Rrf]. For this reason, financial analysts generally use a measure of the average risk premium investors have actually earned in the past as an indication of the risk premium they might require today.
For example, from 1926 through the end of 2009, actual returns on the U.S. stock market exceeded actual returns on long-term U.S. government bonds by an average of 6.01percent per year. If, on average, investors earned the risk premium that they expected, this figure reflects the average market risk premium over the period from 1926 to 2009. If a financial analyst believes that the market risk premium in the past is a reasonable estimate of the risk premium today, then he or she might use 6.01 percent as the market risk premium in Equation 13.4.
With this background, let’s work an example to illustrate how Equation 13.4 is used in practice to estimate the cost of common stock for a firm. Suppose that it is November 19, 2010, and we want to estimate the cost of the common stock for the oil company ConocoPhillips. Using yields reported in the Wall Street Journal on that day, we determine that 30-day Treasury bills have an effective annual yield of 0.13 percent and that 20-year Treasury bonds have an effective annual yield of 3.95 percent. From the MSN Money web site (
http://moneycentral.msn.com
), we find that the beta for ConocoPhillips stock is 1.15. We know that the market risk premium averaged 6.01 percent from 1926 to 2009. What is the expected rate of return on ConocoPhillips common stock?
Since we are estimating the expected rate of return on common stock, and common stock is a long-term asset, we use the long-term Treasury bond yield of 3.95 percent in the calculation. Notice that the Treasury bill and Treasury bond rates differed by 3.82 percent (3.95 − 0.13 = 3.82) on November 19, 2010. They often differ by this amount or more, so the choice of which rate to use can make quite a difference in the estimated cost of equity.
Once we have selected the appropriate risk-free rate, we can plug it, along with the beta and market risk premium values, into Equation 13.4 to calculate the cost of common equity for ConocoPhillips:
APPLICATION 13.3 LEARNING BY DOING
Calculating the Cost of Equity Using a Stock’s Beta
http://finance.yahoo.com
). This beta is 1.36. What do you estimate the cost of common equity in your pizza business to be?
APPROACH: Method 1 for calculating the cost of equity is to use the Capital Asset Pricing Model (CAPM). Therefore, in this example we will use Equation 13.4.
SOLUTION:
cs = Rrf + (βcs × Market risk premium2 = 0.0395 + 11.36 × 0.06012 = 0.121, or 12.1%
Method 2: Using the Constant-Growth Dividend Model
In
Chapter 9
we noted that if the dividends received by the owner of a share of common stock are expected to grow at a constant rate in perpetuity, then the value of that share today can be calculated using Equation 9.4:
We can replace the R in Equation 9.4 with k
cs since we are specifically estimating the expected rate of return for investing in common stock (also the cost of equity if the firm has no other types of stock outstanding). We can then rearrange this equation to solve for k
cs:
Consider an example. Suppose that the current price for the common stock at Sprigg Lane Company is $20, that the firm is expected to pay a dividend of $2 per share to its common stockholders next year, and that the dividend is expected to grow at a rate of 3 percent in perpetuity after next year. Equation 13.5 tells us that the required rate of return for Sprigg Lane’s stock is:
You might be asking yourself at this point where you would get P0, D1, and g in order to use Equation 13.5 for a particular stock. You can get the current price of a share of stock as well as the dividend that a firm is expected to pay next year quite easily from many different Web sites on the Internet—for example, MSN Money and Yahoo! Finance, which were both mentioned earlier. The financial information includes the dollar value of dividends paid in the past year and the dividend that the firm is expected to pay in the next year.
Estimating the long-term rate of growth in dividends is more difficult, but there are some guidelines that can help. As we discussed in
Chapter 9
, the first rule is that dividends cannot grow faster than the long-term growth rate of the economy in a perpetuity model such as Equation 9.4 or 13.5. Assuming dividends will grow faster than the economy is the same as assuming that dividends will eventually become larger than the economy itself! We know this is impossible.
What is the long-term growth rate of the economy? Well, historically it has been the rate of inflation plus about 3 percent. This means that if inflation is expected to be 3 percent in the long run, then a reasonable estimate for the long-term growth rate in the economy is 6 percent (3 percent inflation plus 3 percent real growth). This tells us that g in Equation 13.5 will not be greater than 6 percent. What exactly it will be depends on the nature of the business and the industry it is in. If it is a declining industry, then g might be negative. If the industry is expected to grow with the economy and the particular firm you are evaluating is expected to retain its market share, then a reasonable estimate for g might be 5 or 6 percent.
You can obtain recent stock prices and financial information for a large number of firms from MSN Money at
http://moneycentral.msn.com
or from Yahoo! Finance at
http://finance.yahoo.com/
.
Method 3: Using a Multistage-Growth Dividend Model
Using a multistage-growth dividend model to estimate the cost of equity for a firm is very similar to using a constant-growth dividend model. The difference is that a multistage-growth dividend model allows for faster dividend growth rates in the near term, followed by a constant long-term growth rate. If this concept sounds familiar, that is because it is the idea behind the mixed (supernormal) growth dividend model discussed in
Chapter 9
. In Equation 9.6 this model was written as:
multistage-growth dividend model
a model that allows for varying dividend growth rates in the near term, followed by a constant long-term growth rate; another term used to describe the mixed (supernormal) dividend growth model discussed in
Chapter 9
. In Equation 9.6 this model was written as:
i
is the dividend in period i, P
t
is the value of constant-growth dividend payments in period t, and R is the required rate of return.
To refresh your memory of how this model works, let’s consider a three-stage example. Suppose that a firm will pay a dividend one year from today (D1) and that this dividend will increase at a rate of g
1 the following year, g
2 the year after that, and g
3 per year thereafter. The value of a share of this stock today thus equals:
cs since we are specifically estimating the expected rate of return for common stock. We have also written all of the dividends in terms of D1 to illustrate how the different growth rates will affect the dividends in each year. Finally, we have written P
t
in terms of the constant-growth model. If we substitute D1, D2, D3, and D4 where appropriate, you can see that this is really just Equation 9.6, where we have replaced R with k
cs and written P
t
in terms of the constant-growth model:
Exhibit 13.2
illustrates how cash flows relate to the four terms in the equation.
In the three-stage dividend growth model shown here, the price of a share of stock is equal to the present value of dividends expected to be received at the end of years 1, 2, and 3, plus the present value of a growing perpetuity that begins in year 4 and whose dividends are assumed to grow at a constant rate g
3 forever.
Note that the fourth term in
Exhibit 13.2
is discounted only three years because, as we saw in
Chapters 6
and
9
, the constant-growth model gives you the present value of a growing perpetuity as of the year before the first cash flow. In this case since the first cash flow is D4, the model gives you the value of the growing perpetuity as of year 3.
A multistage-growth dividend model is much more flexible than the constant-growth dividend model because we do not have to assume that dividends grow at the same rate forever. We can use a model such as this to estimate the cost of common stock, k
cs, by plugging P0, D1, and the appropriate growth rates into the model and solving for k
cs using trial and error—just as we solved for the yield to maturity of bonds in
Chapter 8
and earlier in this chapter. The major issues we have to be concerned about when we use a growth dividend model are (1) that we have chosen the right model, meaning that we have included enough stages or growth rates, and (2) that our estimates of the growth rates are reasonable.
Let’s work an example to illustrate how this model is used to calculate the cost of common stock. Suppose that we want to estimate the cost of common stock for a firm that is expected to pay a dividend of $1.50 per share next year. This dividend is expected to increase 15 percent the following year, 10 percent the year after that, 7 percent the year after that, and 5 percent annually thereafter. If the firm’s common stock is currently selling for $24 per share, what is the rate of return that investors require for investing in this stock?
Because there are four different growth rates in this example, we have to solve a formula with five terms:
cs:
cs using trial and error. When we do this, we find that k
cs is 12.2 percent. This is the rate of return at which the present value of the cash flows equals $24. Therefore, it is the rate that investors currently require for investing in this stock.
USING EXCEL SOLVING FOR kCS USING A MULTISTAGE-GROWTH DIVIDEND MODEL
Because trial and error calculations can be somewhat tedious when you perform them by hand, you may find it helpful to use a spreadsheet program. If you would like to use a spreadsheet program to solve the preceding problem yourself, the output from the spreadsheet below shows you how to do it using trial and error.
Once you input the indicated numbers and formulas into cells B3 through B14, you can then vary the number in cell B2 until the number in cell B8 equals $24. Once you have built the model, you can also use the “goal seek” or “solver” functions in Excel to avoid having to manually solve the problem by trial and error. See the “Help” feature in Excel for information on how to use these functions.
We now have discussed three methods of estimating the cost of common equity for a firm. You might be asking yourself how you are supposed to know which method to use. The short answer is that, in practice, most people use the CAPM (Method 1) to estimate the cost of common equity if the result is going to be used in the discount rate for evaluating a project. One reason is that, assuming the theory is valid, CAPM tells managers what rate of return investors should require for equity having the same level of systematic risk that the firm’s equity has. This is the appropriate opportunity cost of equity capital for an NPV analysis if the project has the same risk as the firm and will have similar leverage. Furthermore, CAPM does not require financial analysts to make assumptions about future growth rates in dividends, as Methods 2 and 3 do.
Used properly, Methods 2 and 3 provide an estimate of the rate of return that is implied by the current price of a firm’s stock at a particular point in time. If the stock markets are efficient, then this should be the same as the number that we would estimate using CAPM. However, to the extent that the firm’s stock is mispriced—for example, because investors are not informed or have misinterpreted the future prospects for the firm—deriving the cost of equity from the price at one point in time can yield a bad estimate of the true cost of equity.
Preferred Stock
As we discussed in
Chapter 9
, preferred stock is a form of equity that has a stated value and specified dividend rate. For example, a share of preferred stock might have a stated value of $100 and a 5 percent dividend rate. The owner of such a share would be entitled to receive a dividend of $5 ($100 × 0.05 = $5) each year. Another key feature of preferred stock is that it does not have an expiration date. In other words, preferred stock continues to pay the specified dividend in perpetuity, unless the firm repurchases it or goes out of business.
These characteristics of preferred stock allow us to use the perpetuity model, Equation 6.3, to estimate the cost of preferred equity. For example, suppose that investors would pay $85 for a share of the preferred stock mentioned above. We can rewrite Equation 6.3:
ps is the cost of the preferred stock. Rearranging the formula to solve for k
ps yields:
ps for the preferred stock in our example is:
It is easy to incorporate issuance costs into the above calculation to obtain the cost of the preferred stock to the firm that issues it. As in the earlier bond calculations, we use the net proceeds from the sale rather than the price that is paid by the investor in the calculation. For example, suppose that in order for a firm to sell the above preferred stock, it must pay an investment banker 5 percent of the amount of money raised. If there are no other issuance costs, the company would receive $85 × (1 − 0.05) = $80.75 for each share sold, and the total cost of this financing to the firm would be:
Estimating the Cost of Preferred Stock
ps because you cannot get a good estimate of the beta using regression analysis. How else can you estimate the cost of this preferred stock, and what is this cost?
APPROACH: You can also use Equation 13.6 to estimate the cost of preferred stock.
SOLUTION: First, you must find the annual dividend that someone who owns a share of this stock will receive. This preferred stock issue pays an annual dividend (for simplicity we are assuming one dividend payment per year) that equals 8 percent of $1,000 or $1,000 × 0.8 = $80. Substituting the annual dividend and the market price into Equation 13.6 yields:
Chapter 9
that certain characteristics of preferred stock look a lot like those of debt. The equation Pps = Dps/k
ps shows that the value of preferred stock also varies with market rates of return in the same way as debt. Because k
ps is in the denominator of the fraction on the right-hand side of the equation, whenever k
ps increases, Pps decreases, and whenever k
ps decreases, Pps increases. That is, the value of preferred stock is negatively related to market rates.
It is also important to recognize that the CAPM can be used to estimate the cost of preferred equity, just as it can be used to estimate the cost of common equity. A financial analyst can simply substitute k
ps for k
cs and βps for βcs in Equation 13.4 and use it to estimate the cost of preferred stock. Remember from
Chapter 7
that the CAPM does not apply only to common stock; rather, it applies to any asset. Therefore, we can use it to calculate the rate of return on any asset if we can estimate the beta for that asset.
> BEFORE YOU GO ON
1. What information is needed to use the CAPM to estimate k
cs or k
ps?
2. Under what circumstances can you use the constant-growth dividend formula to estimate k
cs?
3. What is the advantage of using a multistage-growth dividend model, rather than the constant-growth dividend model, to estimate k
cs?
13.4 USING THE WACC IN PRACTICE
·
We have now covered the basic concepts and computational tools that are used to estimate the WACC. At this point, we are ready to talk about some of the practical issues that arise when financial analysts calculate the WACC for their firms.
When financial analysts think about calculating the WACC, they usually think of it as a weighted average of the firm’s after-tax cost of debt, cost of preferred stock, and cost of common equity. Equation 13.2 is usually written as:
Debt + x
ps + x
cs = 1. If the firm has more than one type of debt outstanding or more than one type of preferred or common stock, analysts will calculate a weighted average for each of those types of securities and then plug those averages into Equation 13.7. Financial analysts will also use the market values, rather than the accounting book values, of the debt, preferred stock, and common stock to calculate the weights (the x’s) in Equation 13.7. This is because, as we have already seen, the theory underlying the discounting process requires that the costs of the different types of financing be weighted by their relative market values. Accounting book values have no place in these calculations unless they just happen to equal the market values.
Calculating WACC: An Example
An example provides a useful way of illustrating how the theories and tools that we have discussed are used in practice. Assume that you are a financial analyst at a manufacturing company that has used three types of debt, preferred stock, and common stock to finance its investments.
Debt: The debt includes a $4 million bank loan that is secured by machinery and equipment. This loan has an interest rate of 6 percent, and your firm could expect to pay the same rate if the loan were refinanced today. Your firm also has a second bank loan (a $3 million mortgage on your manufacturing plant) with an interest rate of 5.5 percent. The rate would also be 5.5 percent today if you refinanced this loan. The third type of debt is a bond issue that the firm sold two years ago for $11 million. The market value of these bonds today is $10 million. Using the approach we discussed earlier, you have estimated that the effective annual yield on the bonds is 7 percent.
Preferred Stock: The preferred stock pays an annual dividend of 4.5 percent on a stated value of $100. A share of this stock is currently selling for $60, and there are 100,000 shares outstanding.
Common Stock: There are 1 million shares of common stock outstanding, and they are currently selling for $21 each. Using a regression analysis, you have estimated that the beta of these shares is 0.95.
The 20-year Treasury bond rate is currently 3.95 percent, and you have estimated the market risk premium to be 6.01 percent using the returns on stocks and Treasury bonds from the 1926 to 2009 period. Your firm’s marginal tax rate is 35 percent. What is the WACC for your firm?
The first step in computing the WACC is to calculate the pretax cost of debt. Since the market value of the firm’s debt is $17 million ($4 million + $3 million + $10 million = $17 million), we can calculate the pretax cost of debt as follows:
We next calculate the cost of the preferred stock using Equation 13.6, as follows:
http://www.pwcglobal.com/Extweb/pwcpublications.nsf/docid/748F5814D61CC2618525693A007EC870
.
We are now ready to use Equation 13.7 to calculate the firm’s WACC. Since the firm has $17 million of debt, $6 million of preferred stock ($60 × 100,000 shares = $6 million), and $21 million of common equity ($21 × 1,000,000 shares = 21 million), the total market value of its capital is $44 million ($17 million + $6 million + $21 million = $44 million). The firm’s WACC is therefore:
In order to finance the 2,000 restaurants that are now part of your company, you have sold three different bond issues. Based on the current prices of the bonds from these issues and the issue characteristics (face values and coupon rates), you have estimated the market values and effective annual yields to be:
APPROACH: You can use Equation 13.7 to solve for the WACC for your pizza business. To do so, you must first calculate the weighted average cost of debt. You can then plug the weights and costs for the debt and common equity into Equation 13.7. Since your business has no preferred stock, the value for this term in Equation 13.7 will equal $0.
SOLUTION: The weighted average cost of the debt is:
Interpreting the WACC
cs will decline from 9.7 percent to 8.2 percent:
DECISION: You should politely point out that she is making the wrong comparison. Since the refinancing will result in the firm being financed entirely with equity, k
cs will equal the firm’s WACC. Therefore, the 8.2 percent should really be compared with the 7.3 percent WACC. If your manager goes through with the refinancing, she will be making a bad decision. The average after-tax cost of the capital that your firm uses will increase from 7.3 percent to 8.2 percent.
Limitations of WACC as a Discount Rate for Evaluating Projects
At the beginning of this chapter, we told you that financial managers often require analysts within the firm to use the firm’s current cost of capital to discount the cash flows for individual projects. They do so because it is very difficult to directly estimate the discount rate for individual projects. You should recognize by now that the WACC is the discount rate that analysts are often required to use. Using the WACC to discount the cash flows for a project can make sense under certain circumstances. However, in other circumstances, it can be very dangerous. The rest of this section discusses when it makes sense to use the WACC as a discount rate and the problems that can occur when the WACC is used incorrectly.
discussed how an analyst forecasting the cash flows for a project is forecasting the incremental after-tax free cash flows at the firm level. These cash flows represent the difference between the cash flows that the firm will generate if the project is adopted and the cash flows that the firm will generate if the project is not adopted.
Financial theory tells us that the rate that should be used to discount these incremental cash flows is the rate that reflects their systematic risk. This means that the WACC is going to be the appropriate discount rate for evaluating a project only when the project has cash flows with systematic risks that are exactly the same as those for the firm as a whole. Unfortunately, this is not true for most projects. The firm itself is a portfolio of projects with varying degrees of risk.
When a single rate, such as the WACC, is used to discount cash flows for projects with varying levels of risk, the discount rate will be too low in some cases and too high in others. When the discount rate is too low, the firm runs the risk of accepting a negative NPV project. To see how this might happen, assume that you work at a company that manufactures soft drinks and that the managers at your company are concerned about all the competition in the core soft drink business. They are thinking about expanding into the manufacture and sale of exotic tropical beverages. The managers believe that entering this market would allow the firm to better differentiate its products and earn higher profits. Suppose also that the appropriate beta for soft drink projects is 1.2, while the appropriate beta for tropical beverage projects is 1.5. Since your firm is only in the soft drink business right now, the beta for its overall cash flows is 1.2.
Exhibit 13.3
illustrates the problem that could arise if your firm’s WACC is used to evaluate a tropical beverage project.
In the exhibit, you can see that since the beta of the tropical beverage project is larger than the beta of the firm as a whole, the expected return (or discount rate) for the tropical beverage project should be higher than the firm’s WACC. The Security Market Line indicates what this expected return should be. Now, if the firm’s WACC is used to discount the expected cash flows for this project, and the expected return on the project is above the firm’s WACC, then the estimated NPV will be positive. So far, so good. However, as illustrated in the exhibit, some projects may have an expected return that is above the WACC but below the SML. For projects such as those, using the WACC as the discount rate may actually cause the firm to accept a negative NPV project! The estimated NPV will be positive even though the true NPV is negative. The negative NPV projects that would be accepted in those situations have returns that fall in the red shaded area below the SML, above the WACC line, and to the right of the firm’s beta.
Two types of problems can arise when the WACC for a firm is used to evaluate individual projects: a positive NPV project may be rejected or a negative NPV project may be accepted. For the tropical beverage example, if the expected return on that project was below the level indicated by the SML, but above the firm’s WACC, the project might be accepted even though it would have a negative NPV.
In
Exhibit 13.3
you can also see that using the WACC to discount expected cash flows for low-risk projects can result in managers at the firm rejecting projects that have positive NPVs. This problem is, in some sense, the mirror image of the case where the WACC is lower than the correct discount rate. Financial managers run the risk of turning down positive NPV projects whenever the WACC is higher than the correct discount rate. The positive NPV projects that would be rejected are those that fall into the green shaded area that is below the WACC but above the SML and to the left of the firm’s beta.
To see how these types of problems arise, consider a project that requires an initial investment of $100 and that is expected to produce cash inflows of $40 per year for three years. If the correct discount rate for this project is 8 percent, its NPV will be:
Suppose, however, that the financial managers of the firm considering this project require that all projects be evaluated using the firm’s WACC of 11 percent. When the cash flows are discounted using a rate of 11 percent, the NPV is:
It is also important to recognize that when a firm uses a single rate to evaluate all of its projects, there will be a bias toward accepting more risky projects. The average risk of the firm’s assets will tend to increase over time. Furthermore, because some positive NPV projects are likely to be rejected and some negative NPV projects are likely to be accepted, new projects on the whole will probably create less value for stockholders than if the appropriate discount rate had been used to evaluate all projects. This, in turn, can put the firm at a disadvantage when compared with its competitors and adversely affect the value of its existing projects.
The key point to take away from this discussion is that it is only really correct to use a firm’s WACC to discount the cash flows for a project if the expected cash flows from that project have the same systematic risk as the expected cash flows from the firm as a whole. You might be wondering how you can tell when this condition exists. The answer is that we never know for sure. Nevertheless, there are some guidelines that you can use when assessing whether the systematic risk for a particular project is similar to that for the firm as a whole.
The systematic risk of the cash flows from a project depend on the nature of the business. Revenues and expenses in some businesses are affected more by changes in general economic conditions than revenues and expenses in other businesses. For example, consider the differences between a company that makes bread and a company that makes recreational vehicles. The demand for bread will be relatively constant in good economic conditions and in bad. The demand for recreational vehicles will be more volatile. People buy fewer recreational vehicles during recessions than when the economy is doing well. Furthermore, as we discussed in
Chapter 12
, operating leverage magnifies volatility in revenue. Therefore, if the recreational vehicle manufacturing process has more fixed costs than the bread manufacturing business, the difference in the volatilities of the pretax operating cash flows will be even greater than the difference in the volatilities of the revenues.
While total volatility is not the same as systematic volatility, we find that businesses with more total volatility (uncertainty or risk) typically have more systematic volatility. Since beta is a measure of systematic risk, and systematic risk is a key factor in determining a firm’s WACC, this suggests that the firm’s WACC should be used only for projects with business risks similar to those for the firm as a whole. Since financial managers usually think of systematic risk when they think of underlying business risks, we can restate this condition as follows:
Condition 1: A firm’s WACC should be used to evaluate the cash flows for a new project only if the level of systematic risk for the project is the same as that for the portfolio of projects that currently comprise the firm.
You have to consider one other factor when you decide whether it is appropriate to use a firm’s WACC to discount the cash flows for a project. That is the way in which the project will be financed and how this financing compares with the way the firm’s assets are financed. To better understand why this is important, consider Equation 13.7:
Condition 2: A firm’s WACC should be used to evaluate a project only if that project uses the same financing mix—the same proportions of debt, preferred shares, and common shares—used to finance the firm as a whole.
In summary, WACC is a measure of the current cost of the capital that the firm has used to finance its projects. It is an appropriate discount rate for evaluating projects only if (1) the project’s systematic risk is the same as that of the firm’s current portfolio of projects and (2) the project will be financed with the same mix of debt and equity as the firm’s current portfolio of projects. If either of these two conditions does not hold, then managers should be careful in using the firm’s current WACC to evaluate a project.
Alternatives to Using WACC for Evaluating Projects
Financial managers understand the limitations of using a firm’s WACC to evaluate projects, but they also know that there are no perfect alternatives. As we noted earlier in this chapter, there is no publicly traded common stock for most individual projects within a firm. It is, therefore, not possible to directly estimate the beta for the common stock used to finance an individual project.
9
Although it might be possible to obtain an estimate of the cost of debt from the firm’s bankers, without an estimate of the common stock beta—and, therefore, the cost of common stock—it is not possible to obtain a direct estimate of the appropriate discount rate for a project using Equation 13.7.
If the discount rate for a project cannot be estimated directly, a financial analyst might try to find a public firm that is in a business that is similar to that of the project. For example, in our exotic tropical beverage example, an analyst at the soft drink company might look for a company that produces only exotic tropical beverages and that also has publicly traded stock. This public company would be what financial analysts call a pure-play comparable because it is exactly like the project. The returns on the pure-play company’s stock could be used to estimate the expected return on the equity that is used to finance the project. Unfortunately, this approach is generally not feasible due to the difficulty of finding a public firm that is only in the business represented by the project. If the public firm is in other businesses as well, then we run into the same sorts of problems that we face when we use the firm’s WACC.
pure-play comparable
a comparable company that is in exactly the same business as the project or business being analyzed
From a practical standpoint, financial managers, such as company treasurers and chief financial officers, do not like letting analysts estimate the discount rates for their projects. Different analysts tend to make different assumptions or use different approaches, which can lead to inconsistencies that make it difficult to compare projects. In addition, analysts may be tempted to manipulate discount rates in order to make pet projects look more attractive.
In an effort to use discount rates that reflect project risks better than the firm’s WACC, while retaining control of the process through which discount rates are set, financial managers sometimes classify projects into categories based on their systematic risks. They then specify a discount rate that is to be used to discount the cash flows for all projects within each category. The idea is that each category of projects has a different level of systematic risk and therefore a different discount rate should be used for each.
Exhibit 13.4
illustrates such a classification scheme.
The scheme illustrated in
Exhibit 13.4
includes four project categories:
1. Efficiency projects, such as the implementation of a new production technology that reduces manufacturing costs for an existing product.
2. Product extension projects, such as those in which Boeing created variations of its aircraft, like the Boeing 737, to help meet customer needs.
3. Market extension projects, in which existing products are sold in new markets, such as when Texas Instruments considers selling a new version of a computer chip that has been used in digital phones to digital camera manufacturers.
4. New product projects, in which entirely new products are being considered.
When using the scheme illustrated in
Exhibit 13.4
, the financial manager would assign a discount rate for each category that reflects the typical beta in the indicated range of betas. Such an approach is attractive because it is not generally difficult for analysts to figure out in which of the four categories particular projects belong, and it limits their discretion in choosing discount rates. Most important, it can reduce the possibility of accepting negative NPV projects or rejecting positive NPV projects. We can see the latter benefit by comparing the shaded areas in the figures in
Exhibits 13.3
and
13.4
. The total size of the shaded areas, which represents the possibility of making an error, is much smaller in
Exhibit 13.4
.
The potential for errors—either rejecting a positive NPV project or accepting a negative NPV project—is smaller when discount rates better reflect the risk of the projects that they are used to evaluate. You can see this by noting that the total size of the shaded areas in this figure is smaller than the size of the shaded areas in
Exhibit 13.3
. In the ideal situation, where the correct discount rate is used for each project, there would be no shaded area at all in a figure like this.
> BEFORE YOU GO ON
1. Do analysts use book values or market values to calculate the weights when they use Equation 13.7? Why?
2. What kinds of errors can be made when the WACC for a firm is used as the discount rate for evaluating all projects in the firm?
3. Under what conditions is the WACC the appropriate discount rate for a project?
SUMMARY OF Learning Objectives
The weighted average cost of capital (WACC) for a firm is a weighted average of the current costs of the different types of financing that a firm has used to finance the purchase of its assets. When the WACC is calculated, the cost of each type of financing is weighted according to the fraction of the total firm value represented by that type of financing. The WACC is often used as a discount rate in evaluating projects because it is not possible to directly estimate the appropriate discount rate for many projects. As we also discuss in Section 13.4, having a single discount rate reduces inconsistencies that can arise when different analysts in the firm use different methods to estimate the discount rate and can also limit the ability of analysts to manipulate discount rates to favor pet projects.
Calculate the cost of debt for a firm.
The cost of debt can be calculated by solving for the yield to maturity of the debt using the bond pricing model (Equation 8.1), computing the effective annual yield, and adjusting for taxes using Equation 13.3.
Calculate the cost of common stock and the cost of preferred stock for a firm.
The cost of common stock can be estimated using the CAPM, the constant-growth dividend formula, and a multistage-growth dividend formula. The cost of preferred stock can be calculated using the perpetuity model for the present value of cash flows.
Calculate the weighted average cost of capital for a firm, explain the limitations of using a firm’s weighted average cost of capital as the discount rate when evaluating a project, and discuss the alternatives to the firm’s weighted average cost of capital that are available.
The weighted average cost of capital is estimated using either Equation 13.2 or Equation 13.7, with the cost of each individual type of financing estimated using the appropriate method.
When a firm uses a single rate to discount the cash flows for all of its projects, some project cash flows will be discounted using a rate that is too high and other project cash flows will be discounted using a rate that is too low. This can result in the firm rejecting some positive NPV projects and accepting some negative NPV projects. It will bias the firm toward accepting more risky projects and can cause the firm to create less value for stockholders than it would have if the appropriate discount rates had been used.
One alternative to using the WACC as a discount rate is to identify a firm that engages in business activities that are similar to those associated with the project under consideration and that has publicly traded stock. The returns from this pure-play firm’s stock can then be used to estimate the common stock beta for the project. In instances where pure-play firms are not available, another alternative is for the financial manager to classify projects according to their systematic risks and use a different discount rate for each class of project. This is the type of classification scheme illustrated in
Exhibit 13.4
.
SUMMARY OF Key Equations
· 13.1 The market value of a firm’s assets is $3 billion. If the market value of the firm’s liabilities is $2 billion, what is the market value of the stockholders’ investment and why?
· 13.2 Berron Comics, Inc., has borrowed $100 million and is required to pay its lenders $8 million in interest this year. If Berron is in the 35 percent marginal tax bracket, then what is the after-tax cost of debt (in dollars as well as in annual interest) to Berron.
· 13.3 Explain why the after-tax cost of equity (common or preferred) does not have to be adjusted by the marginal income tax rate for the firm.
· 13.4 Mike’s T-Shirts, Inc., has debt claims of $400 (market value) and equity claims of $600 (market value). If the after-tax cost of debt financing is 11 percent and the cost of equity is 17 percent, what is Mike’s weighted average cost of capital?
· 13.5 You are analyzing a firm that is financed with 60 percent debt and 40 percent equity. The current cost of debt financing is 10 percent, but due to a recent downgrade by the rating agencies, the firm’s cost of debt is expected to increase to 12 percent immediately. How will this change the firm’s weighted average cost of capital if you ignore taxes?
Solutions to Self-Study Problems
· 13.1 Since the identity that Assets = Liabilities + Equity holds for market values as well as book values, we know that the market value of the firm’s equity is $3 billion − $2 billion, or $1 billion.
· 13.2 Because Berron enjoys a tax deduction for its interest charges, the after-tax interest expense for Berron is $8 million × (1 − 0.35) = $5.2 million, which translates into an annual after-tax interest expense of $5.2/$100 = 0.052, or 5.2 percent.
· 13.3 The U.S. tax code allows a deduction for interest expense incurred on borrowing. Preferred and common shares are not considered debt and, thus, do not benefit from an interest deduction. As a result, there is no distinction between the before-tax and after-tax cost of equity capital.
· 13.4 Mike’s T-Shirts’s total firm value = $400 + $600 = $1,000. Therefore,
Debt × k
Debt, and since the firm’s pretax cost of debt is expected to increase by 2 percent, we know that the effect on WACC (pretax) will be 0.6 × 0.02 = 0.012, or 1.2 percent. Incidentally, if we assume that the firm is subject to the 40 percent marginal tax rate, then the after-tax increase in the cost of capital for the firm would be 0.012 × (1 − 0.4) = 0.0072, or 0.72 percent.
Critical Thinking Questions
· 13.1 Explain why the required rate of return on a firm’s assets must be equal to the weighted average cost of capital associated with its liabilities and equity.
· 13.2 Which is easier to calculate directly, the expected rate of return on the assets of a firm or the expected rate of return on the firm’s debt and equity? Assume that you are an outsider to the firm.
· 13.3 With respect to the level of risk and the required return for a firm’s portfolio of projects, discuss how the market and a firm’s management can have inconsistent information and expectations.
· 13.4 Your friend has recently told you that the federal government effectively subsidizes the use of debt financing (vs. equity financing) for corporations. Do you agree with that statement? Explain.
· 13.5 Your firm will have a fixed interest expense for the next 10 years. You recently found out that the marginal income tax rate for the firm will change from 30 percent to 40 percent next year. Describe how the change will affect the cash flow available to investors.
· 13.6 Describe why it is not usually appropriate to use the coupon rate on a firm’s bonds to estimate the pretax cost of debt for the firm.
· 13.7 Maltese Falcone, Inc., has not checked its weighted average cost of capital for four years. Firm management claims that since Maltese has not had to raise capital for new projects in four years, they should not have to worry about their current weighted average cost of capital. They argue that they have essentially locked in their cost of capital. Critique management’s statements.
· 13.8 Ten years ago, the Edson Water Company issued preferred stock at a price equal to the par value of $100. If the dividend yield on that issue was 12 percent, explain why the firm’s current cost of preferred capital is not likely to equal 12 percent.
· 13.9 Discuss under what circumstances you might be able to use a model that assumes constant growth in dividends to calculate the current cost of equity capital for a firm.
· 13.10 Your boss just finished computing your firm’s weighted average cost of capital. He is relieved because he says that he can now use that cost of capital to evaluate all projects that the firm is considering for the next four years. Evaluate that statement.
Questions and Problems
· 13.2 WACC: What is the weighted average cost of capital?
· 13.3 Taxes and the cost of debt: How are taxes accounted for when we calculate the cost of debt?
· 13.4 Cost of common stock: List and describe each of the three methods used to calculate the cost of common stock.
· 13.5 Cost of common stock: Whitewall Tire Co. just paid an annual dividend of $1.60 on its common shares. If Whitewall is expected to increase its annual dividend by 2 percent per year into the foreseeable future and the current price of Whitewall’s common shares is $11.66, what is the cost of common stock for Whitewall?
· 13.6 Cost of common stock: Seerex Wok Co. is expected to pay a dividend of $1.10 one year from today on its common shares. That dividend is expected to increase by 5 percent every year thereafter. If the price of Seerex is $13.75, what is Seerex’s cost of common stock?
· 13.7 Cost of common stock: Two-Stage Rocket paid an annual dividend of $1.25 yesterday, and it is commonly known that the firm’s management expects to increase its dividend by 8 percent for the next two years and by 2 percent thereafter. If the current price of Two-Stage’s common stock is $17.80, what is the cost of common equity capital for the firm?
· 13.8 Cost of preferred stock: Fjord Luxury Liners has preferred shares outstanding that pay an annual dividend equal to $15 per year. If the current price of Fjord preferred shares is $107.14, what is the after-tax cost of preferred stock for Fjord?
· 13.9 Cost of preferred stock: Kresler Autos has preferred shares outstanding that pay annual dividends of $12, and the current price of the shares is $80. What is the after-tax cost of new preferred shares for Kresler if the flotation (issuance) costs for preferred are 5 percent?
· 13.10 WACC: Describe the alternatives to using a firm’s WACC as a discount rate when evaluating a project.
· 13.11 WACC for a firm: Capital Co. has a capital structure, based on current market values, that consists of 50 percent debt, 10 percent preferred stock, and 40 percent common stock. If the returns required by investors are 8 percent, 10 percent, and 15 percent for the debt, preferred stock, and common stock, respectively, what is Capital’s after-tax WACC? Assume that the firm’s marginal tax rate is 40 percent.
13.12 WACC: What are direct out-of-pocket costs?
· 13.14 Finance balance sheet: Explain why the cost of capital for a firm is equal to the expected rate of return to the investors in the firm.
· 13.15 Current cost of a bond: You know that the after-tax cost of debt capital for Bubbles Champagne is 7 percent. If the firm has only one issue of five-year bonds outstanding, what is the current price of the bonds if the coupon rate on those bonds is 10 percent? Assume the bonds make semiannual coupon payments and the marginal tax rate is 30 percent.
· 13.16 Current cost of a bond: Perpetual Ltd. has issued bonds that never require the principal amount to be repaid to investors. Correspondingly, Perpetual must make interest payments into the infinite future. If the bondholders receive annual payments of $75 and the current price of the bonds is $882.35, what is the after-tax cost of this debt for Perpetual if the firm is in the 40 percent marginal tax rate?
· 13.17 Current cost of a bond: You are analyzing the cost of debt for a firm. You know that the firm’s 14-year maturity, 8.5 percent coupon bonds are selling at a price of $823.48. The bonds pay interest semiannually. If these bonds are the only debt outstanding, what is the after-tax cost of debt for this firm if it has a 30 percent marginal and average tax rate?
· 13.18 Taxes and the cost of debt: Holding all other things constant, does a decrease in the marginal tax rate for a firm provide incentive for the firm to increase or decrease its use of debt?
· 13.19 Cost of debt for a firm: You are analyzing the after-tax cost of debt for a firm. You know that the firm’s 12-year maturity, 9.5 percent semi-annual coupon bonds are selling at a price of $1,200. If these bonds are the only debt outstanding for the firm, what is the after-tax cost of debt for this firm if it has a marginal tax rate of 34 percent? What if the bonds are selling at par?
· 13.20 Cost of common stock: Underestimated Inc.’s common shares currently sell for $36 each. The firm’s management believes that its shares should really sell for $54 each. If the firm just paid an annual dividend of $2 per share and management expects those dividends to increase by 8 percent per year forever (and this is common knowledge to the market), what is the current cost of common equity for the firm and what does management believe is a more appropriate cost of common equity for the firm?
· 13.21 Cost of common stock: Write out the general equation for the price of the stock for a firm that will grow dividends very rapidly at a constant rate for the four years after the next dividend is paid and will grow dividends thereafter at a constant, but lower rate. Discuss the problems in estimating the cost of equity capital for such a stock.
· 13.22 Cost of common stock: You have calculated the cost of common stock using all three methods described in the chapter. Unfortunately, all three methods have yielded different answers. Describe which answer (if any) is most appropriate.
· 13.23 WACC for a firm: The managers of a firm financed entirely with common stock are evaluating two distinct projects. The first project has a large amount of unsystematic risk and a small amount of systematic risk. The second project has a small amount of unsystematic risk and a large amount of systematic risk. Which project, if taken, is more likely to increase the firm’s cost of capital?
· 13.24 WACC for a firm: The Imaginary Products Co. currently has debt with a market value of $300 million outstanding. The debt consists of 9 percent coupon bonds (semiannual coupon payments) which have a maturity of 15 years and are currently priced at $1,440.03 per bond. The firm also has an issue of 2 million preferred shares outstanding with a market price of $12.00. The preferred shares pay an annual dividend of $1.20. Imaginary also has 14 million shares of common stock outstanding with a price of $20.00 per share. The firm is expected to pay a $2.20 common dividend one year from today, and that dividend is expected to increase by 5 percent per year forever. If Imaginary is subject to a 40 percent marginal tax rate, then what is the firm’s weighted average cost of capital?
· 13.25 Choosing a discount rate: For the Imaginary Products firm in Problem 13.24, calculate the appropriate cost of capital for a new project that is financed with the same proportion of debt, preferred shares, and common shares as the firm’s current capital structure. Also assume that the project has the same degree of systematic risk as the average project that the firm is currently undertaking (the project is also in the same general industry as the firm’s current line of business).
· 13.26 Choosing a discount rate: If a firm anticipates financing a project with a capital mix different than its current capital structure, describe in realistic terms how the firm is subjecting itself to a calculation error if its historical WACC is used to evaluate the project.
· 13.28 You are an external financial analyst evaluating the merits of a stock. Since you are using a dividend discount model approach to evaluate a cost of equity capital, you need to estimate the dividend growth rate for the firm in the future. Describe how you might go about doing this.
· 13.29 You know that the return of Momentum Cyclicals common shares is 1.6 times as sensitive to macroeconomic information as the return of the market. If the risk-free rate of return is 4 percent and market risk premium is 6 percent, what is Momentum Cyclicals’s cost of common equity capital?
· 13.30 In your analysis of the cost of capital for a common stock, you calculate a cost of capital using a dividend discount model that is much lower than the calculation for the cost of capital using the CAPM model. Explain a possible source for the discrepancy.
· 13.31 RetRyder Hand Trucks has a preferred share issue outstanding that pays a dividend of $1.30 per year. The current cost of preferred equity for RetRyder is 9 percent. If RetRyder issues additional preferred shares that pay exactly the same dividend and the investment banker retains 8 percent of the sale price, what is the cost of the new preferred shares for RetRyder?
· 13.32 Enigma Corporation’s management believes that the firm’s cost of capital (WACC) is too high because the firm has been too secretive with the market concerning its operations. Evaluate that statement.
· 13.33 Discuss what valuable information would be lost if you decided to use book values in order to calculate the cost of each of the capital components within a firm’s capital structure.
· 13.34 Hurricane Corporation is financed with debt, preferred equity, and common equity with market values of $20 million, $10 million, and $30 million, respectively. The betas for the debt, preferred stock, and common stock are 0.2, 0.5, and 1.1, respectively. If the risk-free rate is 3.95 percent, the market risk premium is 6.01 percent, and Hurricane’s average and marginal tax rates are both 30 percent, what is the company’s weighted average cost of capital?
· 13.35 You are working as an intern at Coral Gables Products, a privately owned manufacturing company. Shortly after you read
Chapter 13
in this book, you got into a discussion with the Chief Financial Officer (CFO) at Coral Gables about weighted average cost of capital calculations. She pointed out that, just as the beta of the assets of a firm equals a weighted average of the betas for the individual assets, as shown in Equation 7.11:
· 13.36 The CFO described in Problem 13.35 asks you to estimate the beta for Coral Gables’s common stock. Since the common stock is not publicly traded, you do not have the data necessary to estimate the beta using regression analysis. However, you have found a company with publicly traded stock that has operations which are exactly like those at Coral Gables. Using stock returns for this pure-play comparable firm, you estimate the beta for the comparable company’s stock to be 1.06. The market value of that company’s common equity is $45 million, and it has one debt issue outstanding with a market value of $15 million and an annual pretax cost of 4.85 percent. The comparable company has no preferred stock.
35. If the risk-free rate is 3.95 percent and the market risk premium is 6.01 percent, what is the beta of the assets of the comparable company?
35. If the total market value of Coral Gables’ financing consists of 35 percent debt and 65 percent equity (this is what the CFO estimates the market values to be) and the pretax cost of its debt is 5.45 percent, what is the beta for Coral Gables’s common stock?
1. 13.37 Estimate the weighted average cost of capital for Coral Gables using your estimated beta and the information in the problem statement in Problem 13.36. Assume that the average and marginal tax rates for Coral Gables are both 25 percent.
37. Expected market return.
37. Rate of return required by stockholders.
37. Cost of retained earnings plus dividends.
37. Risk the company incurs when financing.
1. 13.39
Dot.Com
has determined that it could issue $1,000 face value bonds with an 8 percent coupon paid semiannually and a five-year maturity at $900 per bond. If
Dot.Com
‘s marginal tax rate is 38 percent, its after-tax cost of debt is closest to:
38. 6.2 percent.
38. 6.4 percent.
38. 6.6 percent.
38. 6.8 percent.
1. 13.40 Morgan Insurance Ltd. issued a fixed-rate perpetual preferred stock three years ago and placed it privately with institutional investors. The stock was issued at $25.00 per share with a $1.75 dividend. If the company were to issue preferred stock today, the yield would be 6.5 percent. The stock’s current value is:
39. $25.00.
39. $26.92.
39. $37.31.
39. $40.18.
1. 13.41 The Gearing Company has an after-tax cost of debt capital of 4 percent, a cost of preferred stock of 8 percent, a cost of equity capital of 10 percent, and a weighted average cost of capital of 7 percent. Gearing intends to maintain its current capital structure as it raises additional capital. In making its capital-budgeting decisions for the average-risk project, the relevant cost of capital is:
40. 4 percent.
40. 7 percent.
40. 8 percent.
40. 10 percent.
1. 13.42 Suppose the cost of capital of the Gadget Company is 10 percent. If Gadget has a capital structure that is 50 percent debt and 50 percent equity, its before-tax cost of debt is 5 percent, and its marginal tax rate is 20 percent, then its cost of equity capital is closest to:
41. 10 percent.
41. 12 percent.
41. 14 percent.
41. 16 percent.
Sample Test Problems
· 13.1 The Balanced, Inc., has three different product lines. Its least risky product line has a beta of 1.7, while its middle-risk product line has a beta of 1.8, and its most risky product line has a beta of 2.1. The market value of the assets invested in these lines is $1 billion for the least risky line, $3 billion for the middle-risk line, and $7 billion for the riskiest product line. What is the beta of the assets of The Balanced, Inc.? (Hint: see problem 13.35 on page 439.)
· 13.2 Ellwood Corp. has a five-year bond issue outstanding with a coupon rate of 10 percent and a price of $1,039.56. If the bonds pay coupons semiannually, what is the pretax cost of the debt and what is the after-tax cost of the debt? Assume the marginal tax rate for the firm is 40 percent.
· 13.3 Miron’s Copper Corp. management expects its common stock dividends to grow 1.5 percent per year for the indefinite future. The firm’s shares are currently selling for $18.45, and the firm just paid a dividend of $3.00 yesterday. What is the cost of common stock for this firm?
· 13.4 Micah’s Time Portals has a preferred stock issue outstanding that pays an annual dividend of $2.50 per year and is currently selling for $27.78 a share. What is the cost of preferred stock for this firm?
· 13.5 The Old Time New Age Co. has a portfolio of projects with a beta of 1.25. The firm is currently evaluating a new project that involves a new product in a new competitive market. Briefly discuss what adjustment Old Time New Age might make to its 1.25 beta in order to evaluate this new project.
Surveys of capital budgeting practices at major public firms in the United States indicate that a large percentage (possibly as high as 80 percent) of firms use the cost of capital for a firm or a division in capital budgeting calculations. For a discussion of this evidence, see the article titled “Best Practices in Estimating the Cost of Capital: Survey and Synthesis,” by R. F. Bruner, K. M. Eades, R. S. Harris, and R. C. Higgins, which was published in the Spring/Summer 1998 issue of Financial Practice and Education.
The total expected cash flows at Boeing also include cash flows from projects that the firm is expected to undertake in the future, or what are often referred to as growth opportunities. This idea is discussed in detail in later chapters. For our immediate purposes, we will assume that these cash flows are expected to equal $0.
We will discuss how firms finance their assets in more detail in
Chapters 15
and
16
. For the time being, we will simply assume that a firm uses some combination of debt and equity. Here we use the term debt in the broadest sense to refer to all liabilities, including liabilities on which the firm does not pay interest, such as accounts payable. As is common practice, we focus only on long-term interest-bearing debt, such as bank loans and bonds, in the cost of capital calculations. The reason for this is discussed in the next section.
As we will discuss in Section 13.2, if markets are efficient, the prices we observe in the markets will reflect the true costs of the different securities that the firm has outstanding.
We are ignoring the effect of taxes on the cost of debt financing for the time being. This effect is discussed in detail in Section 13.2 and explicitly incorporated into subsequent calculations.
These types of costs are incurred by firms whenever they raise capital. We only show how to include them in the cost of bond financing and, later, in estimating the cost of preferred stock, but they should also be included in calculations of the costs of capital from other sources, such as bank loans and common equity.
There is a limit on the total amount of home loan interest payments that you can deduct when you calculate your taxable income. For instance, in 2011 you could deduct interest payments on loans with a total face value of $1,100,000 ($1,000,000 mortgage plus $100,000 home equity loan).
We use the term “risk-free” here to refer to assets that have no default risk. Investors in the assets can still face interest rate risk as described in
Chapter 8
.
Some firms issue a type of stock that has an equity claim on only part of their business. If a project is similar to the part of the business for which “tracking stock” like this has been sold, the returns on the tracking stock can be used to estimate the beta for the common stock used to finance a project.Order an Essay Now & Get These Features For Free:
