Statistics for Business and Finance - Hypothesis Testing, CAPM Model and Regression Analysis

This article covers hypothesis testing, CAPM model and regression analysis for stocks and index returns. It includes normal distribution test, t-test, risk comparison, beta significance test and more.

Statistics for Business and Finance
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1) Returns for the given three series (Stocks and index) are represented below.
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Normal distribution test: Jarque- Berra Test
Relevant test statistic: Jarque- Berra (JB) statistic
Formula for JB stat computation
JB statistic for Boeing Returns is higher than critical value and hence, null hypothesis would be
rejected. Therefore, it can be concluded that variable (Boeing Returns) is not normally
distributed. Further, JB statistic for General Dynamics Returns is also higher than critical value
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and hence, null hypothesis would be rejected. Therefore, it can be concluded that variable
(General Dynamics Returns) is not normally distributed.
2) Hypothesis test for the claim that average return on GD stock differs from 2.8%.
The test is the relevant test to check the claimof hypothesis isindicated below .
t statistic ¿ 1.56−2.8
( 4.45
√ ( 60 ) ) =−2.40987
Hypothesis test Two tailed
Degree of freedom ¿ 60−1=59
The p value 0.0191
Significance level 0.05
Observation( p value≪ Significancelevel ) ¿ 0.0191≪0.05
Result Reject null hypothesis∧¿
accept alternative hypothesis
Conclusion Average return onGD stock differs ¿ 2.8 % .
3) Comparison of associated risk for each of the given two stocks
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The p value (two tailed) =2*0.1073=0.2146
Significance level 0.05
Observation (p value> Significance level) = 0.2146<<0.05
Result Cannot reject null hypothesis and
cannot accept alternative hypothesis
Conclusion There is no significant difference present in the
associated risk for each of the given stock returns
(BA and GD)
4) Hypothesis testing for deciding whether the two stocks have same population average return
or not
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The p value (two tailed) 0.9452
Significance level 0.05
Observation (p value> Significance level) = 0.9452>0.05
Result Cannot reject null hypothesis and
cannot accept alternative hypothesis
Conclusion There is no significant difference present in the
returns for each of the given stock returns (BA and
GD)
The above hypotheses tests clearly refer to no significant difference existing in the population
return and risk related to GD and Boeing monthly stock movements. As a result, no conclusion
can be drawn by referring to the hypothesis results derived above. Hence, the decision needs to
be made on the basis of the sample data. Clearly, GD stock emerges as the preferred stock based
on the sample data since it offers higher average monthly returns during the considering period
and has a lower risk.
(5) In order to estimate the CAPM model for the preferred stock, the excess returns need to be
firstly calculated by deducting the treasury returns (proxy for risk free returns) from the monthly
returns computed in part 1. For running the CAPM model, the independent variable of choice is
the excess monthly returns on S&P index while the dependent variable is the excess stock
returns.
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The CAPM model can be expressed using the equation illustrated as follows.
(c) R2 = 0.440
(d) The 95% confidence interval in relation to beta is (0.647,1.192).
(6) The hypothesis test for stock beta significance is illustrated below.
In the above hypothesis, the value hypothesised is zero. To decide on whether null hypothesis
would be rejected or not, the key consideration is if the relevant confidence interval contains the
hypothesised value or not. The relevant confidence interval for beta has been already been
computed and contains 1 which would imply that null hypothesis rejection is not permissible in
this case.
(7) Normal distribution test: Jarque- Berra Test
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Relevant test statistic: Jarque- Berra (JB) statistic
Formula for JB stat computation
The computes JB statistic for the residual error has been computed in the excel and has been
arrived at 19.21. Considering the above value, it can be concluded that this does not exceed the
critical value of JB statistic and hence the assumption about residuals being normally distributed
is correct.
PART B: Interpretation
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(1) Based on the inferential statistical technique, it is correct to conclude that the mean stock
returns for the two stocks in this case i.e. Boeing and General Dynamics do not follow a normal
distribution. Hence, the underlying distribution would be non-normal.
For the given stocks, the risk and returns summary needs to be provided. The mean returns on
the Boeing stock is 1.23% which is lesser than the corresponding returns on GD stock which is
1.30%. The indicator of risk is standard deviation which is lower for GD as comparison to
Boeing monthly returns.
(2) The T distribution is the relevant distribution with regards to the null hypothesis. This is
relevant as the standard deviation of population is not known. Also, the average GD stock returns
tend to be differing from the hypothesized value 2.8%.
(3) The GD and Boeing monthly stock returns do not have difference in relation to the
underlying risk captured by the variances of the monthly stock returns.
(4) The GD and Boeing monthly stock returns do not have difference in relation to the
underlying returns captured by the average returns of the monthly stock returns.
(5) (b) Beta estimated in accordance with the CAPM approach for GD is 0.92. This I
representative that excess market returns change by 1 % would reflect a corresponding change of
0.92% in the GD excess stock returns on an average.
(c) The interpretation of the R2 or coefficient of determination can be carried out as shown
below. The market (i.e. S&P index) related excess returns is capable of offering explanation in
regards to 44% of the movements witnessed in the excess stock returns while the remaining
variation is unexplained.
(d) It can be predicted with 95% probability that the beta of the GD stock would be within the
interval marked by the boundaries marked by 0.647 and 1.192.
(6) The confidence interval based testing of hypothesis implies that it can be concluded that
stock is neutral as the beta does not deviate significantly from 1.
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(7) A minimum requirement of linear regression is that the residuals or error terms should be
normally distributed which is not fulfilled in this case as evidence from the normality related
hypothesis test with regards to the residuals of the CAPM model.
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