1. The current price of a stock is $50. In 1 year, the price will be either $65 or $35. The annual risk-free rate is 10%. Find the price of a call option on the stock that has an exercise price of $55 and that expires in 1 year. (Hint: Use daily compounding)
2. The exercise price on one of Chrisardan Company’s call options is $20, its exercise value is $27, and its time value is $8. What are the option’s market value and the price of the stock?
Submit your answers in a Word document.
Text book: https://drive.google.com/file/d/1VYa3jR0L2irLOQbOo526_EMqZZtnjsKw/view?usp=sharing
Build a Model
a. Use the data given to calculate annual returns for | Goodman | Landry | Market | Index | 2015 | |||||||||
Data as given in the problem are shown below: | ||||||||||||||
Goodman Industries | Landry Incorporated | |||||||||||||
Year | Stock Price | Dividend | Includes Divs. | |||||||||||
2020 | $30.32 | $2.23 | $85.12 | $3.52 | 18,475.97 | |||||||||
2019 | $23.53 | $2.65 | $79.32 | $3.65 | 12,174.55 | |||||||||
2018 | $28.61 | $2.73 | $74.32 | $3.45 | 12,019.97 | |||||||||
2017 | $15.21 | $2.57 | $87.12 | $3.47 | 10,743.05 | |||||||||
2016 | $12.63 | $95.12 | $3.55 | 9,455.42 | ||||||||||
$13.21 | $2.25 | $84.25 | $3.25 | 8,163.96 | ||||||||||
We now calculate the rates of return for the two companies and the index: | ||||||||||||||
Kenneth D. Jackson: Change in stock price plus any dividends, divided by the previous stock price | ||||||||||||||
Average | ||||||||||||||
Note: To get the average, you could get the column sum and divide by 5, but you could also use the function wizard, fx. Click fx, then statistical, then Average, and then use the mouse to select the proper range. Do this for Goodman and then copy the cell for the other items. | ||||||||||||||
b. Calculate the standard deviation of the returns for Goodman, Landry, and the Market Index. (Hint: Use the sample standard deviation formula given in the chapter, which corresponds to the STDEV function in Excel.) | ||||||||||||||
Use the function wizard to calculate the standard deviations. | ||||||||||||||
Standard deviation of returns | ||||||||||||||
c. Construct a scatter diagram graph that shows Goodman’s and Landry’ returns on the vertical axis and the Market Index’s returns on the horizontal axis. | ||||||||||||||
It is easiest to make scatter diagrams with a data set that has the X-axis variable in the left column, so we reformat the returns data calculated above and show it just below. | ||||||||||||||
0.0% | ||||||||||||||
To make the graph, we first selected the range with the returns and the column heads, then clicked the chart wizard, then choose the scatter diagram without connected lines. That gave us the data points. We then used the drawing toolbar to make free-hand (“by eye”) regression lines, and changed the lines color and weights to match the dots. | ||||||||||||||
d. Estimate Goodman’s and Landry’s betas as the slopes of regression lines with stock returns on the vertical axis (y-axis) and market return on the horizontal axis (x-axis). (Hint: use Excel’s SLOPE function.) Are these betas consistent with your graph? | ||||||||||||||
Goodman’s beta | = | |||||||||||||
Landry’ beta = | ||||||||||||||
e. The risk-free rate on long-term Treasury bonds is 8.04%. Assume that the market risk premium is 6%. What is the expected return on the market? Now use the SML equation to calculate the two companies’ required returns. | ||||||||||||||
Market risk premium | 6.000% | |||||||||||||
Risk-free rate | 8.040% | |||||||||||||
Expected return on market = | + | |||||||||||||
Required return | ||||||||||||||
Goodman: | ||||||||||||||
Landry: | ||||||||||||||
f. If you formed a portfolio that consisted of 60% Goodman stock and 40% Landry stock, what would be its beta and its required return? | ||||||||||||||
The beta of a portfolio is simply a weighted average of the betas of the stocks in the portfolio, so this portfolio’s beta | ||||||||||||||
would be: | ||||||||||||||
Portfolio beta = | ||||||||||||||
g. Suppose an investor wants to include Goodman Industries’ stock in his or her portfolio. Stocks A, B, and C are currently in the portfolio, and their betas are | 0.769 | 0.985 | 1.423 | Stock A | Stock B | 20% | Stock C | |||||||
Beta | Portfolio Weight | |||||||||||||
30% | ||||||||||||||
100% | ||||||||||||||
Portfolio Beta = | ||||||||||||||
Required return on portfolio: | Risk-free rate + | Market Risk Premium * | * Beta |
Financial Options and Applications in Corporate Finance
CHAPTER 8
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Topics in Chapter
Financial Options Terminology
Option Price Relationships
Black-Scholes Option Pricing Model
Put-Call Parity
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The Big Picture: The Value of a Stock Option
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What is a financial option?
An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.
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What is the single most important
characteristic of an option?
It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset.
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Option Terminology (1 of 8)
Call option: An option to buy a specified number of shares of a security within some future period.
Put option: An option to sell a specified number of shares of a security within some future period.
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Option Terminology (2 of 8)
Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold.
Expiration date: The last date the option can be exercised.
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Option Terminology (3 of 8)
Exercise value: The value of a call option if it were exercised today =
Max[0, Current stock price – Strike price]
Note: The exercise value is zero if the stock price is less than the strike price.
Option price: The market price of the option contract.
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Option Terminology (4 of 8)
Time value: Option price minus the exercise value. It is the additional value because the option has remaining time until it expires.
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Option Terminology (5 of 8)
Writing a call option: For every new option, there is an investor who “writes” the option.
A writer creates the contract, sells it to another investor, and must fulfill the option contract if it is exercised.
For example, the writer of a call must be prepared to sell a share of stock to the investor who owns the call.
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Option Terminology (6 of 8)
Covered option: A call option written against stock held in an investor’s portfolio.
Naked (uncovered) option: An option written without the stock to back it up.
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Option Terminology (7 of 8)
In-the-money call: A call whose strike price is less than the current price of the underlying stock.
Out-of-the-money call: A call option whose strike price exceeds the current stock price.
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Option Terminology (8 of 8)
LEAPS: Long-term Equity AnticiPation Securities that are similar to conventional options except that they are long-term options with maturities of up to 2 ½ years.
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Consider the following data:
Strike price = $25.
Stock Price Call Option Price
$25 $3.00
30 7.50
35 12.00
40 16.50
45 21.00
50 25.50
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Exercise Value of Option
Price of
stock (a) Strike
price (b) Exercise value
of option (a)–(b)
$25.00 $25.00 $0.00
30.00 25.00 5.00
35.00 25.00 10.00
40.00 25.00 15.00
45.00 25.00 20.00
50.00 25.00 25.00
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Market Price of Option
Price of
stock (a) Strike
price (b) Exer.
val. (c) Mkt. Price
of opt. (d)
$25.00 $25.00 $0.00 $3.00
30.00 25.00 5.00 7.50
35.00 25.00 10.00 12.00
40.00 25.00 15.00 16.50
45.00 25.00 20.00 21.00
50.00 25.00 25.00 25.50
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Time Value of Option
Price of
stock (a) Strike
price (b) Exer.
Val. (c) Mkt. P of
opt. (d) Time value
(d) – (c)
$25.00 $25.00 $0.00 $3.00 $3.00
30.00 25.00 5.00 7.50 2.50
35.00 25.00 10.00 12.00 2.00
40.00 25.00 15.00 16.50 1.50
45.00 25.00 20.00 21.00 1.00
50.00 25.00 25.00 25.50 0.50
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Call Time Value Diagram
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Option Time Value Versus Exercise Value
The time value, which is the option price less its exercise value, declines as the stock price increases.
This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.
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The Binomial Model
Stock assumptions:
Current price: P = $27
In next 6 months, stock can either
Go up by factor of 1.41
Go down by factor of 0.71
Call option assumptions
Expires in t = 6 months = 0.5 years
Exercise price: X = $25
Risk-free rate: rRF = 6%
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Binomial Payoffs at Call’s Expiration
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Create portfolio by writing 1 option and buying Ns shares of stock.
Portfolio payoffs:
Stock is up: Ns(P)(u) − Cu
Stock is down: Ns(P)(d) − Cd
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The Hedge Portfolio with a Riskless Payoff
Set payoffs for up and down equal, solve for number of shares:
Ns= (Cu − Cd) / P(u − d)
In our example:
Ns= ($13.07 − $0) / $27(1.41 − 0.71)
Ns=0.6915
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Riskless Portfolio’s Payoffs at Call’s Expiration: $13.26
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Riskless payoffs earn the risk-free rate of return.
Find PV of riskless payoff. Discount at risk-free rate compounded daily.
N = 0.5(365)
I/YR = 6/365
PMT = 0
FV = −$13.26 (because we want to know how much we would want now to give up the FV)
PV = $12.87
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Alternatively, use the PV formula
(daily compounding).
PV = $13.26 / (1 + 0.06/365)365*0.5
= $12.87
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The Value of the Call Option
Because the portfolio is riskless:
VPortfolio = PV of riskless payoff
By definition, the value of the portfolio is:
VPortfolio = Ns(P) − VC
Equating these and rearranging, we get the value of the call:
VC = Ns(P) − PV of riskless payoff
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Value of Call
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
VC = 0.6915($27)
− $13.26 / (1 + 0.06/365)365*0.5
= $18.67 − $12.87
= $5.80
(VC = $5.81 if no rounding in any intermediate steps.)
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Portfolio Replicating the Call Option
From the previous slide we have:
VC = Ns(P) − Payoff / (1 + rRF/365)365*t
The right side of the equation is the same as creating a portfolio by buying Ns shares of stock and borrowing an amount equal to the present value of the hedge portfolio’s riskless payoff (which must be repaid).
The payoffs of the replicating portfolio are the same as the option’s payoffs.
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Replicating Portfolio Payoffs: Amount Borrowed and Repaid
Amount borrowed:
PV of payoff = $12.87
Repayment due to borrowing this amount:
Repayment = $12.87 (1 + rRF/365)365*t
Repayment = $13.26
Notice that this is the same as the payoff of the hedge portfolio.
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Replicating Portfolio Net Payoffs
Stock up:
Value of stock = 0.6915($38.07) =$26.33
Repayment of borrowing = $13.26
Net portfolio payoff = $13.07
Stock down:
Value of stock = 0.6915($19.17) =$13.26
Repayment of borrowing = $13.26
Net portfolio payoff = $0
Notice that the replicating portfolio’s payoffs exactly equal those of the option.
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Replicating Portfolios and Arbitrage
The payoffs of the replicating portfolio exactly equal those of the call option.
Cost of replicating portfolio
= Ns(P) − Amount borrowed
= 0.6915($27) − $12.87
= $18.67 − $12.87
= $5.80
If the call option’s price is not the same as the cost of the replicating portfolio, then there will be an opportunity for arbitrage.
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Arbitrage Example
Suppose the option sells for $6.
You can write option, receiving $6.
Create replicating portfolio for $5.80, netting $6.00 −$5.80 = $0.20.
Arbitrage:
You invested none of your own money.
You have no risk (the replicating portfolio’s payoffs exactly equal the payoffs you will owe because you wrote the option.
You have cash ($0.20) in your pocket.
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Arbitrage and Equilibrium Prices
If you could make a sure arbitrage profit, you would want to repeat it (and so would other investors).
With so many trying to write (sell) options, the extra “supply” would drive the option’s price down until it reached $5.80 and there were no more arbitrage profits available.
The opposite would occur if the option sold for less than $5.80.
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Multi-Period Binomial Pricing
If you divided time into smaller periods and allowed the stock price to go up or down each period, you would have a more reasonable outcome of possible stock prices when the option expires.
This type of problem can be solved with a binomial lattice.
As time periods get smaller, the binomial option price converges to the Black-Scholes price, which we discuss in later slides.
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Assumptions of the
Black-Scholes Option Pricing Model
The stock underlying the call option provides no dividends during the call option’s life.
There are no transactions costs for the sale/purchase of either the stock or the option.
Risk-free rate, rRF, is known and constant during the option’s life.
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Assumptions
Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate.
No penalty for short selling and sellers receive immediately full cash proceeds at today’s price.
Call option can be exercised only on its expiration date.
Security trading takes place in continuous time, and stock prices move randomly in continuous time.
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What are the three equations that make up the OPM?
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What is the value of the following call option according to the OPM?
Assume:
P = $27
X = $25
rRF = 6%
t = 0.5 years
σ = 0.49
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First, find d1.
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Second, find d2.
d2 = d1
d2 = 0.4819
d2 = 0.1355
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Third, find N(d1) and N(d2)
N(d1) = N(0.4819) = 0.6851
N(d2) = N(0.1355) = 0.5539
Note: Values obtained from Excel using NORMSDIST function. For example:
N(d1) = NORMSDIST(0.4819)
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Fourth, find value of option.
VC = $27(0.6851) – $25 e-(0.06)(0.5) (0.5539)
= $18.4977 – $13.4383
= $5.06
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What impact do the following parameters have on a call option’s value?
Current stock price: Call option value increases as the current stock price increases.
Strike price: As the exercise price increases, a call option’s value decreases.
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Impact on Call Value (1 of 2)
Option period: As the expiration date is lengthened, a call option’s value increases.
Longer time to expiration increases probability of very high stock price, which has big payoff.
Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price.
Risk-free rate: Call option’s value tends to increase as rRF increases (reduces the PV of the exercise price).
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Impact on Call Value (2 of 2)
Stock return variance: Option value increases with variance of the underlying stock.
Higher variance increases probability of very high stock price, which has big payoff.
Also increases the probability of a very low stock price, but payoff is zero for any price below the strike price.
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Put Options
A put option gives its holder the right to sell a share of stock at a specified stock on or before a particular date.
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Put-Call Parity
Portfolio 1:
Put option,
Share of stock, P
Portfolio 2:
Call option, VC
PV of exercise price, X
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Portfolio Payoffs at Expiration Date T for PT
CHAPTER 7
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Topics in Chapter
Features of common stock
Valuing common stock
Dividend growth model
Free cash flow valuation model
Market multiples
Preferred stock
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Corporate Valuation and Stock Valuation
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Common Stock: Owners, Directors, and Managers
Represents ownership.
Ownership implies control.
Stockholders elect directors.
Directors hire management.
Since managers are “agents” of shareholders, their goal should be: Maximize stock price.
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Classified Stock
Classified stock has special provisions for each class, usually involving voting rights and dividend rights.
Usually named Class A, Class B, etc.
New shares in IPO sometimes have voting restrictions but full dividend rights.
Founders’ shares usually have voting rights but dividend restrictions.
Standard & Poor’s no longer allows new additions to its indices to have classified stock.
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Tracking Stock
The dividends of tracking stock are tied to a particular division, rather than the company as a whole.
Investors can separately value the divisions.
Its easier to compensate division managers with the tracking stock.
But tracking stock usually has no voting rights, and the financial disclosure for the division is not as regulated as for the company.
Very few companies have tracking stock.
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Different Approaches for Valuing Common Stock
Free cash flow model
Constant growth
Nonconstant growth
Dividend growth model
Constant growth
Nonconstant growth
Using the multiples of comparable firms
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The Free Cash Flow Valuation Model: FCF and WACC
Free cash flow (FCF) is:
The cash flow available for distribution to all of a company’s investors.
Generated by a company’s operations.
The weighted average cost of capital (WACC) is:
The overall rate of return required by all of the company’s investors.
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Value of Operations (Vop)
The PV of expected future FCF, discounted at the WACC, is the value of a company’s operations (Vop):
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Sources of Value
Value of operations
Nonoperating assets
Short-term investments and other marketable securities
Ownership of non-controlling interest in another company
Value of nonoperating assets usually is very close to figure that is reported on balance sheets.
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Claims on Corporate Value
Debtholders have first claim.
Preferred stockholders have the next claim.
Any remaining value belongs to stockholders.
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Total Corporate Value: Sources and Claims
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Value of operations= PV of FCF discounted
at WACC
Conceptually correct, but how do you find the present value of an infinite stream?
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Suppose FCFs are expected to grow at a constant rate, gL, starting at t=1, and continue forever. What happens to FCF?
What is the value of operations if FCFs grow at a constant rate? See next slide.
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Value of operations in terms of FCF1 and gL:
We can multiply and divide by (1+gL), for a reason that will soon be clear, as shown on the next slide.
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Rewritten value of operations:
We can group , as shown on the next slide.
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Value of operations with grouped terms:
We can group the terms, as shown on the next slide.
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Value of operations if FCF grows
at a constant rate:
What happens toif t gets large? It depends on the size of gL relative to WACC. See next slide.
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What happens to as t gets large?
If gL < WACC: Then < 1.
If gL ≥ WACC: Then ≥ 1.
What happens to the value of operations if gL ≥ WACC? See next slide.
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What happens to the value of operations
if gL ≥ WACC?
Vop = (Big) + (Bigger) + (Even Bigger) + …+ (Really big!) = Infinity! So g can’t be greater than or equal to WACC!
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What happens to the value of operations
if gL ≤ WACC?
Vop = (Small) + (Smaller) + (Even smaller) + …+ FCF0 (Really small!) = ?
All the terms get smaller and smaller, but what happens to the sum? See next slide
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What is the sum of an infinite number of factors that get smaller at a geometric rate?
Consider this example. The first row is t. The second row is a number that is less than 1 that is compounded to the power of t. The third row is the cumulative sum.
t 1 2 3 4 . . . ∞
(1/2)t 1/2 1/4 1/8 1/16 1/∞ ≈ 0
Σ(1/2)t 1/2 3/4 7/8 15/16 ≈ 1
This sum converges to 1. Similarly, converges (although not to 1). See next slide.
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Constant Growth Formula for Value of Operations: gL begins at Time 1
If FCF are expected to grow at a constant rate of gL from Time 1 and afterwards, and gL
The higher ROIC causes a big increase in Vop,0.
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Impact of Lower Capital Requirements
No Change Improve CR
g0,1 10% 10%
g1,2 8% 8%
g2,3 5% 5%
g3,4 5% 5%
gL 5% 5%
OP 4.5% 4.5%
CR 56.0% 51.0%
ROIC 8.0% 8.8%
Vop,0 $958 $1,191
WACC 9.00% 9.00%
Lower capital requirements increases the ROIC.
ROIC of 8.8% > 8.26%
The higher ROIC causes an increase in Vop,0.
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Impact of Simultaneous Improvements in OP and CR
No Change Improve OP and CR
g0,1 10% 10%
g1,2 8% 8%
g2,3 5% 5%
g3,4 5% 5%
gL 5% 5%
OP 4.5% 5.5%
CR 56.0% 51.0%
ROIC 8.0% 10.8%
Vop,0 $958 $1,756
WACC 9.00% 9.00%
The ROIC is much higher due to the improvements in operations.
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Impact of Simultaneous Improvements in Growth, OP, and CR
No Change Improve All
g0,1 10% 11%
g1,2 8% 9%
g2,3 5% 6%
g3,4 5% 6%
gL 5% 6%
OP 4.5% 5.5%
CR 56.0% 51.0%
ROIC 8.0% 10.8%
Vop,0 $958 $2,008
WACC 9.00% 9.00%
The ROIC is much higher due to the improvements in operations.
With a higher ROIC, growth adds substantial value.
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Summary: Value of operations for previous combinations of ROIC and gL
ROIC ROIC ROIC ROIC ROIC
8.0% 8.8% 9.8% 10.8%
gL 5% $958 $1,191 $1,523 $1,756
gL 6% $933 $1,247 $1,694 $2,008
The ROIC is much higher due to the improvements in operations.
With a higher ROIC, growth adds substantial value.
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Are volatile stock prices consistent with rational pricing?
The previous slide shows that small changes in ROIC and growth cause large changes in value.
Similarly, small changes in the cost of capital (WACC), perhaps due to changes in risk or interest rates, cause large changes in value.
As new information arrives, investors continually update their estimates of operating profitability, capital requirements, growth, risk, and interest rates.
If stock prices aren’t volatile, then this means there isn’t a good flow of information.
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Value of dividend-paying stock = PV of dividends discounted at required return
Conceptually correct, but how do you find the present value of an infinite stream?
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Suppose dividends are expected to grow at a constant rate, gL, forever.
D1 = D0(1 + gL)1
D2 = D0(1 + gL)2
Dt = D0(1 + gL)t
What is the present value of a constant growth Dt when discounted at the stock’s required return, rs? See next slide.
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Present Value of a Constant Growth Dividend
What happens to as t gets bigger?
If gL
So gL must be less than rs for the constant growth model to be applicable!!
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Required rate of return: beta = 1.2, rRF = 7%,
and RPM = 5%.
Use the SML to calculate rs:
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Estimated Intrinsic Stock Value:
D0 = $2.00, rs = 13%, gL = 6%
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Expected Stock Price in 1 Year
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Expected Dividend Yield and Capital
Gains Yield (Year 1)
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Total Year 1 Return
Total return = Dividend yield + Capital gains yield.
Total return = 7% + 6% = 13%.
Total return = 13% = rs.
For constant growth stock:
Capital gains yield = 6% = gL.
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Rearrange model to rate of return form:
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Nonconstant Growth Stock
Nonconstant growth of 30% for Year 0 to Year 1, 25% for Year 1 to Year 2, 15% for Year 2 to Year 3, and then long-run constant gL = 6%.
Can no longer use constant growth model.
However, growth becomes constant after 3 years.
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Steps to Estimate Current Stock Value
Forecast dividends for nonconstant period, which ends at horizon date after which growth is constant at gL.
Find horizon value, which is PV of dividends beyond horizon date discounted back to horizon date
Horizon value = =
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Steps to Estimate Current Stock Price (Continued)
Find PV of each dividend in the forecast period.
Find PV of horizon value.
Sum PV of dividends and PV of horizon value.
Result is estimated current stock value.
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Example of Estimating Current Stock Value (D0 = $2.00, rs = 13%)
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Expected Dividend Yield and Capital
Gains Yield (t = 0)
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Expected Dividend Yield and Capital Gains Yield (after t = 3)
During nonconstant growth, dividend yield and capital gains yield are not constant.
If current growth is greater than g, current capital gains yield is greater than g.
After t = 3, gL = constant = 6%, so the
capital gains yield = 6%.
Because rs = 13%, after t = 3 dividend
yield = 13% – 6% = 7%.
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Using Stock Price Multiples to Estimate
Stock Price
Analysts often use the P/E multiple (the price per share divided by the earnings per share).
Example:
Estimate the average P/E ratio of comparable firms. This is the P/E multiple.
Multiply this average P/E ratio by the expected earnings of the company to estimate its stock price.
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Using Entity Multiples
The entity value (V) is:
the market value of equity (# shares of stock multiplied by the price per share)
plus the value of debt.
Pick a measure, such as EBITDA, Sales, Customers, Eyeballs, etc.
Calculate the average entity ratio for a sample of comparable firms. For example,
V/EBITDA
V/Customers
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Using Entity Multiples (Continued)
Find the entity value of the firm in question. For example,
Multiply the firm’s sales by the V/Sales multiple.
Multiply the firm’s # of customers by the V/Customers ratio
The result is the firm’s total value.
Subtract the firm’s debt to get the total value of its equity.
Divide by the number of shares to calculate the price per share.
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Problems with Market Multiple Methods
It is often hard to find comparable firms.
The average ratio for the sample of comparable firms often has a wide range.
For example, the average P/E ratio might be 20, but the range could be from 10 to 50. How do you know whether your firm should be compared to the low, average, or high performers?
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Comparing the FCF Model and Dividend
Growth Model
Can apply FCF model in more situations:
Privately held companies
Divisions of companies
Companies that pay zero (or very low) dividends
FCF model requires forecasted financial statements to estimate FCF
Takes more effort than just forecasting dividends, but…
Provides more insights into value drivers.
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Preferred Stock
Hybrid security.
Similar to bonds in that preferred stockholders receive a fixed dividend which must be paid before dividends can be paid on common stock.
However, unlike bonds, preferred stock dividends can be omitted without fear of pushing the firm into bankruptcy.
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Value of Preferred Stock
(Dividend = $2.10; rps = 7%)
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Risk and Return
CHAPTER 6
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Topics in Chapter
Basic return and risk concepts
Stand-alone risk
Portfolio (market) risk
Risk and return: CAPM/SML
Market equilibrium and market efficiency
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Determinants of Intrinsic Value: The Cost of Equity
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What are investment returns?
Investment returns measure the financial results of an investment.
Returns may be historical or prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
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An investment costs $1,000 and is sold after
1 year for $1,060.
Dollar return:
$ Received$ Invested
$1,060 $1,000$60.
Percentage return:
$ Return/$ Invested
$60/$1,0000.066%.
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What is investment risk?
Investment risk is exposure to the chance of earning less than expected.
The greater the chance of a return far below the expected return, the greater the risk.
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Scenarios and Returns for the 10-Year Zero Coupon T-bond Over the Next Year
Scenario Probability Return
Worst Case 0.10 −14%
Poor Case 0.20 −4%
Most Likely 0.40 6%
Good Case 0.20 16%
Best Case 0.10 26%
1.00
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Discrete Probability Distribution for Scenarios
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Example of a Continuous Probability Distribution
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Calculate the expected rate of return (r ̂ ) on the bond for the next year.
= 0.10(-14%) + 0.20(-4%) + 0.40(6%)
+ 0.20(16%) + 0.10(26%)
= 6%
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Use Excel to Calculate the Expected Value of a Discrete Distribution
= SUMPRODUCT(Probabilities,Returns)
SUMPRODUCT:
Multiplies each value in the first array (the range of cells with probabilities) by its corresponding value in the second array (the range of cells with returns).
Sums the products.
This is identical to the formula on the previous slide.
See Ch06 Mini Case.xlsx
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Consider these probability distributions for two investments. Which riskier? Why?
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Stand-Alone Risk: Standard Deviation
Stand-alone risk is the risk of each asset held by itself.
Standard deviation measures the dispersion of possible outcomes.
For a single asset:
Stand-alone risk = Standard deviation
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Variance (σ2) and Standard Deviation (σ) for Discrete Probabilities
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Standard Deviation of the Bond’s Return During the Next Year
σ2 = 0.10 (-0.14 – 0.06)2
+ 0.20 (-0.04 – 0.06)2
+ 0.40 (0.06 – 0.06)2
+ 0.20 (0.16 – 0.06)2
+ 0.10 (0.26 – 0.06)2
σ2 = 0.0120
σ =
σ = 0.1095 = 10.95%
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Use Excel to Calculate the Variance and Standard Deviation of a Discrete Distribution
= SUMPRODUCT(Probabilities,Returns− ,Returns−)
SUMPRODUCT:
Multiplies each value in the first array (the range of cells with probabilities) by its corresponding value in the second array (the range of cells with returns less the expected return) and by the third array (which is identical to the second array).
Sums the products; the result is variance.
Take the square root of the variance to get the standard deviation. See Ch06 Mini Case.xlsx
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Understanding the Standard Deviation
If the returns are normally distributed:
Outcome will be more than 1 σ away from about 31.74% ≈ 32% of the time:
16% of the time below −σ
16% of the time above +σ.
If = 6% and σ =10.95% ≈ 11%:
16% of the time return <−5% = 6% − 11%
16% of the time return > 17% = 6% + 11
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Useful in Comparing Investments
Investments with bigger standard deviations have more risk.
High risk doesn’t mean you should reject the investment, but:
You should know the risk before investing
You should expect a higher return as compensation for bearing the risk.
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Using Historical Data to Estimate Risk
Analysts often use discrete outcomes to analyze risk for projects; see Chapter 11.
But for investments, most analysts normally use historical data rather than discrete forecasts to estimate an investment’s risk unless it is a very special situation.
Most analysts use:
48 to 60 months of monthly data, or
52 weeks of weekly data, or
Shorter period using daily data.
Use annual returns here for sake of simplicity.
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Formulas for a Sample of T Historical Returns
Tedious to calculate by hand, easy in Excel. See next slide.
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Excel Functions a Sample of T Historical Returns
Suppose “SampleData” is the cell range with the T historical returns.
=AVERAGE(SampleData)
=STDEV(SampleData)
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Historical Data for Stock Returns
Year Market Blandy Gourmange
1 30% 26% 47%
2 7 15 −54
3 18 −14 15
4 −22 −15 7
5 −14 2 −28
6 10 −18 40
7 26 42 17
8 −10 30 −23
9 −3 −32 −4
10 38 28 75
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Average and Standard Deviations for
Stand-Alone Investments
Use formulas shown previously (tedious) or use Excel (easy)
What is Blandy’s stand-alone risk?
Note: analysts often use past risk as a predictor of future risk, but past returns are not a good prediction of future returns.
Market Blandy Gourmange
Average return 8.0% 6.4% 9.2%
Standard deviation 20.1% 25.2% 38.6%
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How risky is Blandy stock?
Assumptions:
Returns are normally distributed.
σ is 25.2%
Expected return is about 6.4%.
16% of the time (approximately), return will be:
< −18.8% (6.4%−25.2% = −18.8)
> 32.6% (6.4%+25.2% = 32.6)
Stocks are very risky!
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Portfolio Returns
The percentage of a portfolio’s value that is invested in Stock i is denoted by the “weight” wi. Notice that the sum of all the weights must equal 1.
With n stocks in the portfolio, its return each year will be:
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Example: 2-Stock Portfolio
Form a portfolio by selling 25% of the Blandy stock and investing it in the higher-risk Gourmange stock.
The portfolio return each year will be:
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Historical Data for Stocks and Portfolio Returns
Year Blandy Gourmange Portfolio of Blandy
and Gourmange
1 26% 47% 31.3%
2 15 −54 −2.3
3 −14 15 −6.8
4 −15 7 −9.5
5 2 −28 −5.5
6 −18 40 −3.5
7 42 17 35.8
8 30 −23 16.8
9 −32 −4 −25.0
10 28 75 39.8
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Portfolio Historical Average and
Standard Deviation
The portfolio’s average return is the weighted average of the stocks’ average returns.
The portfolio’s standard deviation is less than either stock’s σ!
What explains this?
Blandy Gourmange Portfolio
Average return 6.4% 9.2% 7.1%
Standard deviation 25.2% 38.6% 22.2%
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How closely do the returns follow one another?
Notice that the returns don’t move in perfect lock-step: Sometimes one is up and the other is down.
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Correlation Coefficient (ρi,j)
Loosely speaking, the correlation (r) coefficient measures the tendency of two variables to move together.
Estimating ri,j with historical data is tedious:
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Excel Functions to Estimate
the Correlation Coefficient (ρi,j)
“Stocki” and “Stockj” are the cell ranges with historical returns for Stocks i and j.
Est. ρi,j = Rij =Correl(Stocki,Stockj)
Correlation between Blandy (B) and Gourmange (G):
Est. ρB,G = 0.11
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2-Stock Portfolios
r = −1
2 stocks can be combined to form a riskless portfolio: σp = 0.
r = +1
Risk is not “reduced”
σp is just the weighted average of the 2 stocks’ standard deviations.
−1 < r < −1
Risk is reduced but not eliminated.
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Adding Stocks to a Portfolio
What would happen to the risk of an average 1-stock portfolio as more randomly selected stocks were added?
sp would decrease because the added stocks would not be perfectly correlated.
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Risk vs. Number of Stocks in Portfolio
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Stand-alone risk = Market risk + Diversifiable risk
Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification.
Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification.
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Conclusions
As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.
sp falls very slowly after about 40 stocks are included. The lower limit for sp is about 20% = sM .
By forming well-diversified portfolios, investors can eliminate about half the risk of owning a single stock.
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Can an investor holding one stock earn a return commensurate with its risk?
No. Rational investors will minimize risk by holding portfolios.
Investors bear only market risk, so prices and returns reflect the amount of market risk an individual stock brings to a portfolio, not the stand-alone risk of individual stock.
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Market Risk Due to an Individual Stock
How do you measure the amount of market risk that an individual stock brings to a well-diversified portfolio?
William Sharpe developed the Capital Asset Pricing Model (CAPM) to answer this question.
And the answer is….. See next slide.
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Market Risk as Defined by the CAPM
Define:
wi is the percent of the portfolio invested in Stock i.
σM is the standard deviation of the market index.
ri,M is the correlation between Stock i and the market.
The contribution of Stock i to the standard deviation of a well-diversified portfolio (σp) is:
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Market Risk and Beta (1 of 2)
The beta of Stock i (bi) is defined:
The contribution of Stock i to the standard deviation of a well-diversified portfolio (σp) is:
Given the standard deviation of the market and the percent of the portfolio invested in Stock i, beta measures the impact of Stock i on σp.
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Market Risk and Beta (2 of 2)
So Stock i contributes more risk (has a higher beta) if
its correlation with the market, , is larger and/or
its stand-alone risk, , is larger
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Required Return and Risk: General Concept
Investors require a return for time (for tying their funds up in the investment).
rRF, the risk-free rate
Investors require a return for risk, which is the extra return above the risk-free rate that investors require to induce them to invest in Stock i.
RPi, the risk premium of Stock i.
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Required Return and Risk: The CAPM
RPM is the market risk premium. It is the extra return above the risk-free rate that that investors require to invest in the overall stock market:
RPM = rM − rRF.
The CAPM defines the risk premium for Stock i as:
RPi = bi (RPM)
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The Security Market Line: Relating Risk and Required Return (1 of 2)
The Security Market Line (SML) puts the pieces together, showing how to determine the return required for bearing a stock’s risk:
SML: ri = rRF + (RPM)bi
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The Security Market Line: Relating Risk and Required Return
Risk depends on beta:
The part of σp due to Stock i = wi σM bi
Required return depends on beta:
ri = rRF + (RPM) bi
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Correlation Between Blandy and the Market
Using the formula for correlation or the Excel function =CORREL, Blandy’s correlation with the market (ρB,M) is:
ρB,M = 0.481
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Beta for Blandy
Use the previously calculated standard deviations for Blandy and the market to estimate Blandy’s beta:
b = ρB,M (σB/σM)
b = 0.481(.252/.201) = 0.60
The average beta is equal to 1.0, so Blandy’s stock contributes less risk to a well-diversified portfolio than does the average stock.
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Required Return for Blandy
Inputs:
rRF = 4% (given)
RPM = 5%(given)
b = 0.60 (estimated)
ri = rRF + bi (RPM)
ri = 4% + 0.60(5%) = 7%
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Comparing Risk and Return for Different Stocks
The beta of an average stock is 1.0; Gourmange’s beta is 1.3. How do their required returns compare with Blandy’s?
rAvg_company = 4%+ 1.0 (5%) = 9%
rG = 4%+ 1.3 (5%) = 10.5%
rB = 4%+ 0.6 (5%) = 7%
Blandy’s stock contributes less risk to a well-diversified portfolio than do Gourmange or the average stock, so Blandy’s investors require a lower rate of return.
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Using a Regression to Estimate Beta
Run a regression with returns on the stock plotted on the Y-axis and returns on the market portfolio plotted on the X-axis.
The slope of the regression line is equal to the stock’s beta coefficient.
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Excel: Plot Trendline Right on Chart
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Estimated Beta from Regression
The trendline is plotted on the previous slide, including the regression equation.
y = 0.6027x + 0.0158
b = Slope = 0.6027 (same as before)
Easier way—use the Excel SLOPE function.
b =SLOPE(y_values,x_values)
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Web Sites for Beta
http://finance.yahoo.com
Enter the ticker symbol for a “Stock Quote”, such as IBM or Dell, then click GO.
When the quote comes up, select Key Statistics from panel on left.
www.valueline.com
Enter a ticker symbol at the top of the page (registration is free).
Most stocks have betas in the range of 0.5 to 1.5.
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Impact on SML of Increase in Risk-Free Rate
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Impact on SML of Increase in Risk Aversion
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Calculate the weights for a portfolio with $1.4 million in Blandy and $0.6 million in Gourmange.
Find the weights:
wB = $1.4/($1.4+$0.6) = 70%
wG = $0.6/($1.4+$0.6) = 30%
The portfolio beta is the weighted average of the stocks’ betas:
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Calculate the portfolio beta.
bp = 0.7(bBlandy) + 0.3(bGour.)
= 0.7(0.60) + 0.3(1.30)
= 0.81.
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What is the Required Return on the Portfolio?
(1) Use SML:
rp = rRF + bp (RPM)
= 4.0% + 0.81%(5%) = 8.05%.
(2)Use fact that rp =
rp= 0.7(7.0%) + 0.3(10.5%)
= 8.05%.
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Portfolio Performance Evaluation
Relative to SML (1 of 2)
Portfolio Manager Portfolio Manager
JJ CC
Portfolio beta 0.7 1.4
Risk-free rate 4% 4%
Market risk premium 5% 5%
Portfolio required return 7.5% 11.0%
Portfolio actual return 8.5% 9.5%
Over/under performance +1.0% -1.5%
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Portfolio Performance Evaluation
Relative to SML (2 of 2)
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Intrinsic Values and Market Prices
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What is required for the market to be in equilibrium?
The market price of a security must equal the security’s intrinsic value (intrinsic value reflects the size, timing, and risk of the future cash flows).
Market price = Intrinsic value
The expected return a security must equal its required return (which reflects the security’s risk).
= r
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How is equilibrium established?
If the market price is below the intrinsic value (or if the expected return is above the required return), then the security is a “bargain.”
Buy orders will exceed sell orders, bidding up the market price (which also drives down the expected return, given no change in the asset’s cash flows).
“Profitable” trading (i.e., earning a return greater than justified by risk) will continue until the market price is equal to the intrinsic value.
The opposite occurs if the market price is above the intrinsic value.
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Efficient Market Hypothesis (EMH):
It’s all about the info.
The EMH asserts that when new information arrives, prices move to the new equilibrium price very, very quickly because:
There are many really smart analysts looking for mispriced securities.
New information is available to most professional traders almost instantly.
When mispricing occurs (due to new info or inefficient markets), analysts have billions of dollars to use in taking advantage of the mispricing– which then quickly eliminates the mispricing.
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Implications of Efficient Market Hypothesis (EMH)
Stocks are normally in equilibrium.
One cannot “beat the market” by consistently earning a return higher than is justified by a stock’s risk.
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Testing the EMH
Choose a trading strategy and implement it over a large sample.
Pick an asset pricing model, like the CAPM, and measure the required return of the strategy’s investments.
Measure the actual return.
Actual > required? Reject EMH.
Notice that this is a “joint” test of the EMH and the particular asset model– if the test rejects the EMH, it could be that the asset pricing model is wrong.
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Weak-form EMH
Current prices already reflect all the information “contained” in past prices, so you cannot earn excess returns with strategies based on past prices.
Example strategy: Invest in stocks that have declined below their previous 52-week low.
This is a type of “technical” analysis.
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Weak-form EMH: Empirical Evidence
Most empirical evidence supports weak-form EMH because very few trading strategies consistently earn in excess of the CAPM prediction.
Two exceptions with small excess returns:
Short-term momentum
Long-term reversals
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Semistrong-form EMH
Current prices already reflect all publicly available information, so you cannot earn excess returns with strategies based on information from financial statements or other public sources.
Example strategy: Invest in stocks with past 3-year annual earnings growth greater than 10% and a ratio of R&D to sales greater than 10%.
This is a type of “fundamental” analysis.
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Semistrong-form EMH: Empirical Evidence
Most empirical evidence supports the semistrong-form EMH.
In fact, the vast majority of portfolio managers do not consistently have returns in excess of CAPM predictions.
Two exceptions that earn excess returns:
Small companies
Companies with high book-to-market ratios
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Strong-form EMH
All information, even inside information, is embedded in stock prices so you cannot earn excess returns ever.
Not true—excess returns can be gained by trading on the basis of insider information.
Illegal! Go to jail now!
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Market Bubbles and Market Efficiency
Market bubbles:
Prices climb rapidly to heights that would have been considered extremely unlikely before the run-up.
Trading volume is unusually high.
Many new investors (or speculators?) eagerly enter the market.
Prices suddenly fall precipitously.
What does this imply about the EMH?
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Bubbles are hard to puncture.
If there is a bubble, why don’t traders take positions that make big profits when the bubble bursts?
It is hard to recognize a bubble until after it bursts—then it seems obvious!
Trading strategies expose traders to possible big negative cash flows if the bubble is slow to burst.
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Market Efficiency: The Bottom Line
For most stocks, for most of the time, it is generally safe to assume that the market is reasonably efficient.
Many investors have given up trying to beat the market, which helps explain the popularity of index funds.
However, bubbles do occur infrequently.
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
The CAPM: The Bottom Line
Empirical tests of CAPM have statistical problems that make empirical verification or rejection virtually impossible.
Most corporations use the CAPM to determine their stock’s required return.
Most researchers use multi-factor models to identify the portion of a stock’s return that remains unexplained after accounting for the model’s factors.
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Has the CAPM been completely confirmed
or refuted?
No. The statistical tests have problems that make empirical verification or rejection virtually impossible.
Investors’ required returns are based on future risk, but betas are calculated with historical data.
Investors may be concerned about both stand-alone and market risk.
© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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