Bus5Sbf Statistics | Computation Of Assessment Answers

1. Calculate returns for these three series in Excel or any software of your choice using the transformation: r  = 100*ln(P  / P ) and perform the Jarque-Berra test of normally distributed returns for each of Boeing and GD. What do you infer about the distribution of the two stock returns series? Describe also the risk and average return relationship in each of the two stocks.

 

2. Test a hypothesis that the average return on GD stock is different from 2.8%. Which test statistic would you choose to perform this hypothesis test and why? Also, specify the distribution of the test statistic under the null hypothesis.

 

3. Before investing in one of the two stocks, you rst want to compare risk associated with each of the two stocks. Perform an appropriate hypothesis test using 5% signicance level and interpret your results.

 

4. Besides, you want to determine whether both stocks have same population average return. Perform an appropriate hypothesis test using information in your sample of  60 observations on returns. Report your ndings and also mention which stock will you prefer and why?

 

5. Compute excess return on your preferred stock as y  = r  - r    and excess market return as x  = r  - r   and perform the following tasks.

 

a.    Estimate the CAPM using linear regression by regressing the excess return on your preferred stock (y ) on excess market return (x ) and properly report your regression results.

 

b.    Interpret the estimated CAPM beta-coecient in terms of the stock's riskiness in comparison with the market.

 

c.    Interpret the value of R .

 

d.    Interpret 95% condence interval for the slope coecient. 

 

6.  Using the condence interval approach to hypothesis testing, perform the hypothesis test to determine whether your preferred stock is a neutral stock. 

 

7. One of the assumptions of ordinary least squares (OLS) method is; normally distributed error term in the model. Perform an appropriate hypothesis test to determine whether it is plausible to assume normally distributed errors.

The calculations in this question involve computation of return, risk- return relationship and the Jarque- Berra test of normality. The returns for the S&P, Boeing, DG and Treasury notes stocks are produced in the excel output. Returns measure the performance of the stocks in the market (Jordan, Stephen , Randolph, & Bradford, 2010).

The relationship between risk and return can be established by finding the variance of the returns. Risk is the variance of the returns (Meigs, Walter, & Robert , 1970). From the output, it is established that the stock with a higher average return has lower risk and that with a lower average return has a higher risk. This is an indication that highly risky stocks have lower returns.

	Average Return	Risk
Boeing	1.26	33.91
GD	1.58	20.15

A Jarque Berra test of normality is used for testing whether a variable is normally distributed or not (Frankfort-Nachias, 2015). The results of the Jarque-Berra test are outlined below for the two stocks Boeing and GD.

C.  Jarque- Berra  Test Statistics and The P Value
			JB Test Statistics	P Value
Boeing	9.833333	0.772545	7.596689773	0.022408
GD	9.833333	0.058949	0.579665126	0.748389

 

The p value of the test is less than 0.05 for the Boeing. This is an indication that the returns are normally distributed (Stuart A., 1999). The p value is more than 0.05 for the GD stocks. This is an indication that the returns of GD are not normally distributed (Stuart A., 1999).

2.Hypothesis testing of Singe Population Mean

This question is about a hypothesis test that the average return on GD stock is difference from 2.8 %. This is a test for mean. Since the sample size is large (i. e more than 30), the most suitable test is the Z- test. The following hypothesis is tested.

H0:  The average return on GD is equal to 2.8

H1: The average return on GD is not equal to 2.8

This is a two tailed test. The Z score value of the test is -2.0924 and the P value is 0.018. The Value is less than the alpha value. We reject the null hypothesis (Knight, 2000). We conclude that there is sufficient evidence to prove that the average return on the GD is not equal to 2.8.

3.F-test: Hypothesis testing to compare equality of variance in both stocks

This question is about investigating the difference in the risks associated with the Boeing and GD stocks. Risk is the variance of return (Tim, 2005). Therefore, this test involves comparing the variances of the returns of the two stocks. This is done using F- test for two sample variances.

The following hypothesis is tested;

H0: There is no difference in the risk associated with the two stocks

H1: There is difference in the risk associated with the two stocks.

After running the test in excel, the following output is obtained;

	Boeing	GD
Mean	1.256454	1.57727
Variance	33.91273	20.14665
Observations	59	59
Df	58	58
F	1.683294
P(F<=f) one-tail	0.024794
F Critical one-tail	1.545768

From the output, the p value is 0024794. The p value is less than the alpha value= 0.05. We reject the null hypothesis. We conclude that there is sufficient evidence to prove that there is a difference in the risk associated with the two stocks.

4.Hypothesis testing – Comparing Two Population Means

This question is about comparison of means of two populations, Boeing and GD. Since the population size is large i. e more than 30, we use a single factor ANOVA (Ana, Jose, & Jorge, 2003). The following hypothesis is tested.

H0: There is no significant difference in the average returns.

H1: There is significant difference in the average returns.

After running a single factor ANOVA test, the following out is obtained;

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	3.036223	1	3.036223	0.112329	0.738113	3.922879
Within Groups	3135.444	116	27.02969
Total	3138.48	117

From the output above, the p value is 0.738113. This is more than the alpha value=0.05. We fail to reject the null hypothesis. We conclude that there is sufficient evidence to prove that there is no significant difference in the average returns.  

5.Computing excess returns

This question is investigating about the Capital Asset Pricing Model (CAPM). CAPM is one of the models used in the capital markets to determine returns on the market and returns on an individual stock (David & David, 2000). In this case, CAPM has been estimated using the excess return on Boeing stock (Yt) and the excess market return (Xt).

CAPM is a linear equation hence it can be estimated using the linear regression equation (Frank & Harrell, 2001). The beta for CAPM is 0.1090. This is the risk premium of the market (Irving, 1950). The R square is 0.01190. This implies that the sample data explains 1.190% of the population (Krishnamoorthy, 2005). This is an indication that the data is not good enough for making inferences about the population (Lind, 2008).

The confidence interval for the CPAM is (-5.530, 2.178). This implies that for any given sample, we are 95% confidence that the beta of CAPM will fall in the interval (-5.530, 2.178) (Suhov & Kelbert, 2005).

Regression Statistics
Multiple R	0.109092451
R Square	0.011901163
Adjusted R Square	-0.00504799
Standard Error	5.845686384
Observations	60

ANOVA
	df	SS	MS		F		Significance F
Regression	1	24.28354251	24.28354		0.710626		0.402699
Residual	59	2016.150909	34.17205
Total	60	2040.434451
	Coefficients	Standard Error	t Stat	P-value		Lower 95%		Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0	#N/A	#N/A	#N/A		#N/A		#N/A	#N/A	#N/A
Xt	-1.585894049	1.881280962	-0.84299	0.402641		-5.35033		2.17854	-5.35033	2.17854

6.Confidence Interval approach to a hypothesis test

This question is testing the confidence interval approach to hypothesis testing. The following hypothesis is tested;

H0: Boeing stock is not a neutral stock

H1: Boeing stock is a neutral stock

The following values are used for calculation of the test statistics;

Xbar=	1.256454
Standard deviation =	5.823464
sample size=	59
Standard Error=	0.75815
df=	58
The 95% Confidence interval, the t critical =				2.001717

After running the test, the p value is found to be 0.00368. this is less than the alpha value. We reject the null hypothesis. We conclude that statistically, there is sufficient evidence to prove that Boeing stock is a neutral stock (Sid, Anandi, & Robert, 2007).

7.Testing assumption of normally distributed errors

This question is testing on the assumption of normality. One of the assumptions of normality is that the error terms are normally distributed (David & David, 2000). The following hypothesis is tested;

H0: The error terms are normally distributed.

H1: The error terms are not normally distributed.

This test can be conducted by running a summary test. For the error terms to be normally distributed, the mean and the median should be equal. Secondly, the kurtosis should be equal to zero or lose to zero (Frankfort-Nachias, 2015). From the output below, the conditions for normality are not met. Therefore, we conclude that there is sufficient evidence to prove that the error terms are not normally distributed (David & David, 2000

Summary Statistics		Xt
Yt
		0.007666667
Mean	-0.865703331	0.052215674
Standard Error	0.750793914	-0.041
Median	-0.299915303	#N/A
Mode	#N/A	0.404460874
Standard Deviation	5.815624652	0.163588599
Sample Variance	33.8214901	-0.915897031
Kurtosis	1.389686511	0.276006434
Skewness	-0.598862051	1.599
Range	31.02872928	-0.665
Minimum	-20.80176586	0.934
Maximum	10.22696342	0.46
Sum	-51.94219986	60
Count	60

References

Ana, M., Jose, G. B., & Jorge, A. L. (2003). Stochastic Models: Symposium on Probability and Stochastic Processes .

David, J. S., & David, S. (2000). Handbook of parametric and nonparametric statistical Procedures.

Frank, E., & Harrell, J. (2001). Regression Modelling Strategies: Models, Logistic Regression, and Survival Analysis.

Frankfort-Nachias, C. &.-G. (2015). Social Statistics for a diverse society. Thousand Oaks, CA: Sage Publications.

Irving, J. G. (1950). Probability and the Weighing Evidence.

Jordan, Stephen , A. R., Randolph, W. W., & Bradford, D. (2010). Fundamentals of corporate finance. Boston: McGraw-Hill Irwin.

Knight, K. (2000). Mathematical Statistics- Volume in Texts in Statistical Scence Series. Chapman and Hall.

Krishnamoorthy, K. (2005). Handbook of Statistical Distributions with Applications.

Lind, D. A. (2008). Statistical Techniques in Business & . Boston.: McGraw-Hill Irwin.

Meigs, Walter, B., & Robert , F. (1970). Financial Accounting. McGraw-Hill Book Company.

Sid, M., Anandi, P. S., & Robert, A. C. (2007). Practicing Financial Planning for Proffesionals . Rochester Hills Publishing.

Stuart A., O. K. (1999). Kendall’s Advanced Theory of Statistics: Volume 2A- Classical Inference & the linear Model.

Suhov, Y., & Kelbert, M. (2005). Probability and Statisics by exasmple. basic probability and statistics.

Tim, S. (2005). Mastering Statistical Process Control: A handbook for Performance Improvement Using Cases.