## Line Chart of Price Indexes

The report discusses about the volatility and risk market, return and market return that is observed as previous studies. The report also focuses to refer the method of evaluating the company’s price indexes through its market price movements. The price indexes for Boeing (BA) and International Business Machines (IBM) has been selected for representing. However, the historical price data of the individual values from 1^{st} February 2010 and 31^{st} July 2015 cannot illustrate the proper results. Hence, the historical index prices of S&P500 index and the 10 years’ US Treasury Bill are included in the valuation process for computing the beneficial results. S&P500 index shows the market summarisation and the 10 years’ US Treasury Bill indicates the risk-free return of the market values.

The trend of the market prices for a definite time-period depicts market prices trend. The first line chart includes all the three types of trend lines of price indexes. The second, third and fourth line charts include the line charts independently. The price indexes of in the line charts are:

The IBM price index has grown and then got reduction within this period of 1^{st} December 2010 to 31^{st} May 2016. It could be accomplished from the above line charts that the price indexes of both S&P500 and Boeing (BA) have gradually increased from 01/12/2010 to 31/05/2016. The S&P500 index has significantly higher price indexes than Boeing (BA) price indexes and IBM price indexes. Generally, S&P500 has price index greater than IBM price index. However, from June 2014 both are getting almost similar.

Calculation of returns of S&P500, Boeing and IBM:

The calculation of returns of S&P500, Boeing (BA) and IBM are calculated in excel file and spread sheet number 2.

Summary Statistics:

The average return of Boeing (BA) is more than average returns of IBM (1.01398826>0.071483586). The risk is normally defined by standard deviation of returns of close rates of price indexes. The risk in terms of standard deviation indicates that Boeing (BA) return is more volatile than IBM return as 5.98765036917879>5.98765036917879. The range of Boeing (BA) return is 30.8197292766617 whereas the range of IBM return is 28.8655719892539. Therefore, fluctuation is lesser in case of IBM return than Boeing (BA) return. The maximum value of Boeing (BA) return is 12.2869634151351 and the minimum value is (-18.5327658615267) whereas the minimum and maximum values of IBM returns are (-14.3826499212464) and (14.4829220680075).

Jerque-Bera test is carried out for testing the normality of BA and IBM price indexes.

## Calculation of Return Prices of Each Prices

The Jerque-Bera test statistic (JB) is-

Firstly, the JB test statistics of Boeing (BA) (16.7353) and IBM price indexes (15.0915) are calculated. For two-tail Chi-square tests, Boeing (BA) and IBM price returns failed to attain normality at 95% confidence limit. The Chi-square value for 2 degrees of freedom and 5% level of significance is 5.99. Calculated JB values for both the price returns are greater than chi-square critical. Hence, both the price indexes are not normally distributed.

A one-sample t-test elaborates whether the average price return of Boeing (BA) return is at least 3% or not. The t-statistic is – We reject the null hypothesis of average price return greater than or equal to 0.03 as T_{0.05 }< T_{cric }(1.3249<1.9977), at 5% level of significance. Hence, it can be said that the average price index of Boeing (BA) return is not at least 3%.

The riskiness of returns of two price indexes could be more significantly compared by F-test of two samples variances. The F-test for comparing the riskiness of the returns of price indexes of IBM and Boeing (BA) are incorporated here. The F-statistic is calculated as

Hypotheses:

The risk associated with each of the two price returns is compared by F-statistic. The calculated F-statistics is 1.404554937. For Boeing (BA) and IBM price returns, p-value of the F-statistic is found is 0.088449703. It is greater than 0.05. The null hypothesis is rejected at 5% level of significance. Hence, it could be concluded that level of volatility of the two price returns for the particular period are unequal to each other.

The average return is shown by the average of returns of the price indexes. Hence, for comparing the mean return of Boeing (BA) and IBM price indexes, two sample z-test (for unequal samples) and two sample t-test (for equal samples of unequal variances) could be executed on the calculated returns of the Boeing (BA) and IBM price indexes.

Hypotheses:

Null hypothesis (H_{0}): μ_{BA} = μ_{IBM}

Alternative hypothesis (H_{A}): μ_{BA} ≠ μ_{IBM}

Z-test:

The z-statistic is given as z.

For comparing the average returns of each of the two investing price indexes, a z-test is applied. The variances are known for each of the price returns (BA=35.85195694 and IBM=25.52549281). The calculated z-statistic is 0.96991968. The p-value for two-tail z-statistic is 0.33208653 (>0.05). Hence, we can reject the null hypothesis of equality of averages of returns of two price indexes at 5% level of significance. The returns of these two indexes are not equal.

## Summary Statistics

Two sample t-test for unequal variances:

The two-sample t-statistic is given by the statistic,

The t-test assuming equal variances of BA and IBM price returns provide the t-statistic 0.96991968. The p-value of the two-tail t-test is 0.333974621. The level of significance is 5%, which is lesser than calculated p-value. Therefore, we accept the null hypothesis of equality of averages of both the price returns.

Inference:

According to the price return averages and price return standard deviations (risk), equality is established. Therefore, we cannot draw definite conclusion to choose any one price indexes between BA and IBM. Hence, we continue with both of them. Subsequently, we are willing to price excess return, excess market return and CAPM of both the price returns. With the help of these, we can discover the volatility of both the price returns. The preferable price index would be detected after that.

The excess return and excess market returns are calculated in the spreadsheet 4 of excel file.

The Capital Asset Pricing Model (CAPM) is known as CAPM that is one of the fundamental models in the financial field. The CAPM illustrates variability in the rate of return (r_{t}) as a function of the rate of return on a market portfolio (r_{M,t}) involving all publicly traded price indexes. Specifically, the bête (slope) measures the sensitivity of variation of given return of security in the entire market. Value of beta defines whether the price return is a defensive, a neutral price index or an aggressive price index. Together with an intercept (β_{0}) and an error term (u_{t}) in the model, we have a simple linear regression model –

The regression models are –

BA excess return = 0.40176 + (1.1193)*Excess Market Return

IBM excess return = (-1.12349) + 0.7204*Excess Market Return

The calculated beta-value for Boeing (BA) excess return and Excess market return is 1.1193. The calculated beta-value for IBM excess return and Excess market return is 0.7204. The calculated beta-values define that BA price returns have volatility level of 111.93% of the market, whereas the volatility level of IBM compared to the market is 72.04%. Therefore, it can be said that Boeing (BA) is highly volatile than IBM. Hence, Boeing (BA) return is considered more profitable than IBM price indexes.

The simple linear regression describe that the values of R^{2} of BA and IBM are 0.40482754 and 0.2379855. The R^{2} measures the linear association of the dependent variable with the independent variable. Therefore, from the values of R^{2} of the two price returns it can be concluded that BA excess return (40.48%) is more related than IBM (23.79%) with excess market return.

Confidence Interval of IBM Price returns:

- For Boeing price returns, slope (β
_{1}) = 11931665, Standard Error = 0.170989, d.f. = 64, t-value = 6.546119. Hence, the 95% confidence interval for the slope coefficient would be (0.777621689, 1.4610116079). - For IBM price returns, slope (β
_{1}) = 72041148, Standard Error = 0.16241145, d.f. = 64, t-value = 4.43571844. Hence, the 95% confidence interval for the slope coefficient would be (0.395858, 1.0449648).

The testing of aggressiveness of the price returns needs the following hypothesis:

Null hypothesis (H_{0}): β_{1} = 1

Alternative hypothesis (H_{1}): β_{1} < 1

For both the price returns, the p-values are positive t-value and equal degrees of freedom 64. The 95% confidence intervals for beta values of both Boeing (BA) and IBM price returns are (0.777621689, 1.4610116079) and (0.395858, 1.0449648). The confidence intervals near to 0 indicates more neutral nature for price exchange return. The confidence intervals of t-statistics demonstrate that IBM price return is more neutral.

The method of ordinary least squares (OLS) helps to establish the normality with diagrams. The error terms in the model are graphically referred in normal probability plot. The distribution of residual values is symmetric with respect to the market return values. It indicates that the error terms are not following normal distributions for IBM price returns.

Besides the Jerque-Bera test of the residual values of IBM excess price returns, indicate that JB test statistic is 18.21556 for IBM price indexes because of predicted excess value. For chi-square distribution of 2 degrees of freedom at 5% level of significance, the chi-square statistic is 5.99. Therefore, we accept the normality of residual values of IBM price indexes.