Solution
A stock can be identified as underperformed stock or out performed stock based on the value of alpha of the regression equation. On performing the regression equation of the stocks of Microsoft from January 1998 to December 2008, the following solution was observed.
SUMMARY OUTPUT 

Regression Statistics 

Multiple R 
0.584179 

R Square 
0.341265 

Adjusted R Square 
0.336197 

Standard Error 
0.089101 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.534676 
0.534676 
67.34784 
1.94E13 

Residual 
130 
1.032074 
0.007939 

Total 
131 
1.56675 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.008773 
0.007755 
1.131246 
0.260034 
0.00657 
0.024116 
0.00657 
0.024116 
rmrf 
1.320997 
0.160968 
8.206573 
1.94E13 
1.002541 
1.639454 
1.002541 
1.639454 
Table 1: Output of regression equation of the stocks of Microsoft
(Source: created by author)
The value of alpha of this regression equation is 0.008773, which is greater than 0. Since the value of alpha greater than zero indicates that the stock outperforms consistently, it could be interpreted that the stocks of Microsoft outperforms consistently.
Evaluation of the claim that Microsoft is an aggressive stock
Solution
A stock is said to be an “aggressive stock” when the beta value of the stock is greater than one as the variation of the stock is more. The stock is said to be a defensive stock when the beta value of the stock is less than one as the variation of the stock is less. In the regression equation of the stocks of Microsoft from January 1998 to December 2008, it was seen that the value of beta was 1.320997, which is greater than one (Seber and Lee 2012). Thus, it can be interpreted that the stocks of Microsoft (a Tech stock) is an aggressive stock as the beta value of this stock is greater than one.
Evaluation of the claim that the stocks of MobilExxon are a defensive stock
Solution
A stock is said to be a defensive stock if the beta value of the stock is less than one which indicates that the variation of the stock is less. On performing regression equation of the stocks of MobilExxon (xom), the following result had been found.
SUMMARY OUTPUT 

Regression Statistics 

Multiple R 
0.376656 

R Square 
0.14187 

Adjusted R Square 
0.135269 

Standard Error 
0.049673 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.053029 
0.053029 
21.49221 
8.53E06 

Residual 
130 
0.320757 
0.002467 

Total 
131 
0.373786 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.010556 
0.004323 
2.441511 
0.015971 
0.002002 
0.019109 
0.002002 
0.019109 
rmrf 
0.416019 
0.089737 
4.63597 
8.53E06 
0.238485 
0.593554 
0.238485 
0.593554 
Table 2: Output of Regression equation of the stocks of xom
(Source: created by author)
The value of beta coefficient of this regression equation is 0.416019, which is greater than one. This suggests that the variation of the stock is high (Cameron and Trivedi 2013). Thus, the claim that the stocks of MobilExxon (xom) are defensive is not true. This is because the value of the stocks is not less than one.
The table is filled as per the results of regression equation at 95% confidence interval for the stocks given.
SUMMARY OUTPUT (dis) 

Regression Statistics 

Multiple R 
0.538186 

R Square 
0.289644 

Adjusted R Square 
0.28418 

Standard Error 
0.068417 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.248122 
0.248122 
53.00681 
2.83E11 

Residual 
130 
0.608522 
0.004681 

Total 
131 
0.856644 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.001526 
0.005955 
0.256295 
0.798128 
0.01026 
0.013307 
0.01026 
0.013307 
rm  rf 
0.899889 
0.123601 
7.280578 
2.83E11 
0.655358 
1.144419 
0.655358 
1.144419 
Table 3: Regression output table of “dis”
(Source: created by author)
SUMMARY OUTPUT (ge) 

Regression Statistics 

Multiple R 
0.623297 

R Square 
0.388499 

Adjusted R Square 
0.383795 

Standard Error 
0.054897 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.248906 
0.248906 
82.5917 
1.45E15 

Residual 
130 
0.391781 
0.003014 

Total 
131 
0.640687 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.001509 
0.004778 
0.315749 
0.7527 
0.00794 
0.010962 
0.00794 
0.010962 
rm  rf 
0.90131 
0.099176 
9.087997 
1.45E15 
0.705103 
1.097518 
0.705103 
1.097518 
Table 4: Regression output table of “ge”
(Source: created by author)
SUMMARY OUTPUT (gm) 

Regression Statistics 

Multiple R 
0.479893 

R Square 
0.230298 

Adjusted R Square 
0.224377 

Standard Error 
0.112137 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.489115 
0.489115 
38.89646 
5.8E09 

Residual 
130 
1.634723 
0.012575 

Total 
131 
2.123838 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.00887 
0.00976 
0.90923 
0.364914 
0.02818 
0.010435 
0.02818 
0.010435 
rm  rf 
1.263461 
0.202585 
6.236702 
5.8E09 
0.862671 
1.664251 
0.862671 
1.664251 
Table 5: Regression output table of “gm”
(Source: created by author)
SUMMARY OUTPUT (ibm) 

Regression Statistics 

Multiple R 
0.636171 

R Square 
0.404714 

Adjusted R Square 
0.400135 

Standard Error 
0.070081 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.43408 
0.43408 
88.38246 
2.47E16 

Residual 
130 
0.63848 
0.004911 

Total 
131 
1.07256 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.008527 
0.0061 
1.397893 
0.164526 
0.00354 
0.020595 
0.00354 
0.020595 
rm  rf 
1.190259 
0.126607 
9.401195 
2.47E16 
0.939782 
1.440736 
0.939782 
1.440736 
Table 6: Regression output table of “ibm”
(Source: created by author)
SUMMARY OUTPUT (xom) 

Regression Statistics 

Multiple R 
0.376656 

R Square 
0.14187 

Adjusted R Square 
0.135269 

Standard Error 
0.049673 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.053029 
0.053029 
21.49221 
8.53E06 

Residual 
130 
0.320757 
0.002467 

Total 
131 
0.373786 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.010556 
0.004323 
2.441511 
0.015971 
0.002002 
0.019109 
0.002002 
0.019109 
rm  rf 
0.416019 
0.089737 
4.63597 
8.53E06 
0.238485 
0.593554 
0.238485 
0.593554 
Table 7: Regression output table of “xom”
(Source: created by author)
SUMMARY OUTPUT (msft) 

Regression Statistics 

Multiple R 
0.584179 

R Square 
0.341265 

Adjusted R Square 
0.336197 

Standard Error 
0.089101 

Observations 
132 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.534676 
0.534676 
67.34784 
1.94E13 

Residual 
130 
1.032074 
0.007939 

Total 
131 
1.56675 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Lower 95.0% 
Upper 95.0% 
Intercept 
0.008773 
0.007755 
1.131246 
0.260034 
0.00657 
0.024116 
0.00657 
0.024116 
rm  rf 
1.320997 
0.160968 
8.206573 
1.94E13 
1.002541 
1.639454 
1.002541 
1.639454 
Table 8: Regression output table of “msft”
(Source: created by author)
The above finding would help to frame the table given below:

β^ 
95% Confidence Interval for β 



Lower bound 
Upper bound 
Microsoft 
1.3189 
1.002541 
1.639454 
GE 
0.8993 
0.705103 
1.097518 
GM 
1.2614 
0.862671 
1.664251 
IBM 
1.1882 
0.939782 
1.440736 
Disney 
0.8978 
0.655358 
1.144419 
XOM 
0.4140 
0.238485 
0.593554 
Table 9: results of the estimation
(Source: created by author)
Interpretation of 95% confidence interval of GE, GM, Disney in terms of their risk profile
Solution
The beta value of “GE” was found to be 0.8993. The lower bound of 95% confidence interval was found to be 0.705103 while the upper bound was found to be 1.097518. The value of beta was found to be less than one (Kleinbaum et al. 2013). This can be interpreted that there is less deviation in the stocks of GE, which shows that the stocks of GE is defensive. Also, the confidence interval of this stock have the lower bound in the defensive region and it eventually changes to aggressive region in the upper bound.
The beta value of the stocks of GM was found to be 1.2614. The value of the lower bound of 95% confidence interval is 0.8626 and the value of the upper bound is 1.664251. Since the beta value of the stock is greater than one, the stocks of GM are aggressive (Montgomery et al. 2015).
The beta value of the stocks of Disney was 0.8978, which is less than one. The lower bound of the 95% confidence interval is 0.655358 while the value of upper bound is 1.144419. The stocks of Disney are defensive as the value of beta is less than one. This indicates that there is less variation among the stocks of Disney.
Choice of three stocks from the above
Solution
The investor seeks three stocks that have well diversified risk profiles. This indicates that the stocks must be aggressive and beta value of the chosen three stocks must be greater than one. This is because aggressive stocks indicate greater variation in the stocks which also indicates that the risk profiles are well diversified. From the above calculations it was seen Microsoft, GM and IBM are the three stocks that would be suitable for the investor. This is because the beta values of these three stocks are greater than one and they are aggressive stocks.
Explanation of expected excess returns of the stocks
Solution
The regression equation shows that the stocks of “dis” have the beta coefficient of 1.007 of the market premium. This shows that the stocks of “dis” would be influenced positively by the factor of 1.007 by market premium. The coefficient of size premium and value premium is given by 0.00041 and 0.002923 respectively. This indicates that the size premium would influence the stocks of “dis” negatively by a factor 0.00041 and value premium would influence if positively by a factor 0.002923.
The regression equation of “ge” shows that the value of coefficients of market premium, size premium and value premium are 0.9567, 0.00564, 0.00209. This shows that market premium influences the stocks of “ge” positively by a factor 0.9567 while the other two factors influence the stocks negatively by a factor of 0.00564 and 0.00209 respectively (Draper and Smith 2014).
The regression equation of “gm” shows that the value of coefficients of market premium, size premium and value premium are 1.57, 0.00082 and 0.008574 respectively. This shows that market premium and value premium influence the stocks of “gm” positively by a factor of 1.57 and 0.008574 respectively. The factor of size premium influences the stocks of “gm” negatively by a factor of 0.00082.
The regression equation of “ibm” shows that the value of coefficients of market premium, size premium and value premium are 1.1493, 0.00219, 0.00267. This shows that the factor of market premium influences the stocks of “ibm” positively by a factor of 1.1493 while size premium and value premium influence the stocks negatively by a factor of 0.00219 and 0.00267 respectively.
The regression equation of “msft” shows that the value of coefficients of market premium, size premium and value premium are 1.035, 0.00354 and 0.01084. This shows that the market premium influences the shares of “msft” positively by a factor of 1.035 while the size premium and the value premium influences the stocks negatively by a factor of 0.00354 and 0.01084 respectively (Draper and Smith 2014).
The regression equation of “xom” shows that the value of coefficients of market premium, size premium and value premium are 0.5490, 0.00175, and 0.002786. This shows that the market premium and value premium influences the stocks of “xom” positively by a factor of 0.5490 and 0.002786 respectively while the factor of size premium influence the stocks of “xom” negatively by a factor of 0.00175.
Testing of hypothesis using ftest for “dis”
Solution
Ftest was done for the variable “dis” to test the claim that “dis” is unrelated to size and value premium in context of the coefficients of FamaFrench model. The hypothesis of this test is as follows:
H_{0} : “dis” is unrelated to size premium
H_{1} : “dis” is related to size premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


dis 
smb 
Mean 
0.001378697 
0.308864 
Variance 
0.006539264 
16.59104 
Observations 
132 
132 
df 
131 
131 
F 
0.000394144 

P(F<=f) onetail 
0 

F Critical onetail 
0.749410102 

Table 10: ftest of “dis”and “smb”
(Source: created by author)
Here, the value of F is less than the critical value of F; i.e. 0.000394144 < 0.749410102. Since, the f value of the test is less than the critical F value, the “null hypothesis” is accepted and it can be interpreted that “dis” is unrelated to size premium.
The hypothesis of this Ftest is as follows:
H_{0} : “dis” is unrelated to value premium
H_{1} : “dis” is related to value premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


dis 
hml 
Mean 
0.001378697 
0.360606 
Variance 
0.006539264 
14.18226 
Observations 
132 
132 
df 
131 
131 
F 
0.000461088 

P(F<=f) onetail 
0 

F Critical onetail 
0.749410102 

Table 11: ftest of “dis”and “hml”
(Source: created by author)
The table shows that the F value of the test is 0.000461088, which is less than the critical value of F of the test 0.749410102. It shows that the “null hypothesis” is accepted and “dis” is unrelated to value premium.
Hypothesis test of “xom” using ftest
Solution
Hypothesis test would be done in order to test the claim that the size and value premium do not affect the stocks of “xom” in context of the coefficients of FamaFrench model. Ftest would be used in this case. The following hypothesis would be given for this test.
H_{0} : “xom” is unrelated to value premium
H_{1} : “xom” is related to value premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


xom 
hml 
Mean 
0.010488 
0.360606 
Variance 
0.002853 
14.18226 
Observations 
132 
132 
df 
131 
131 
F 
0.000201 

P(F<=f) onetail 
0 

F Critical onetail 
0.74941 

Table 12: ftest of “xom” and “hml”
(Source: created by author)
The table shows that the f value of the test is less than the critical value of the test; i.e. 0.000201 < 0.74941. This leads to the acceptance of “null hypothesis” and it can be interpreted that “xom” is unrelated to value premium.
The following hypothesis would be used to test the relationship between stocks of “xom” and the size premium.
H_{0} : “xom” is unrelated to size premium
H_{1} : “xom” is related to size premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


xom 
smb 
Mean 
0.010488 
0.308864 
Variance 
0.002853 
16.59104 
Observations 
132 
132 
df 
131 
131 
F 
0.000172 

P(F<=f) onetail 
0 

F Critical onetail 
0.74941 

Table 13: ftest of “xom” and “smb”
(Source: created by author)
Ftest shows that the f value of the test is 0.000172 and the critical value of the test is 0.74941 (Sen and Srivastava 2012). This shows that the F value of the test is less than the critical value. The “null hypothesis” is accepted in this case and “xom” is unrelated to size premium”.
Hypothesis test of “msft” to test the claim that it has same sensitivity to the size premium and value premium in context of the coefficients of FamaFrench model
Solution
Ftest would be used to test the claim that it has same sensitivity to the size premium and value premium in context of the coefficients of FamaFrench model. The hypothesis of the test is as follows:
H_{0} : “msft” is unrelated to size premium
H_{1} : “msft” is related to size premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


msft 
smb 
Mean 
0.008557 
0.308864 
Variance 
0.01196 
16.59104 
Observations 
132 
132 
df 
131 
131 
F 
0.000721 

P(F<=f) onetail 
0 

F Critical onetail 
0.74941 

Table 14: ftest of “msft” and “smb”
(Source: created by author)
The f value of the test is 0.000721 which is less than the critical value of the test 0.74941. This leads to the acceptance of “null hypothesis” and “msft” is unrelated to size premium.
The hypothesis of the test is as follows:
H_{0} : “msft” is unrelated to value premium
H_{1} : “msft” is related to value premium
The table below provides the result of this ftest:
FTest TwoSample for Variances 


msft 
hml 
Mean 
0.008557 
0.360606 
Variance 
0.01196 
14.18226 
Observations 
132 
132 
df 
131 
131 
F 
0.000843 

P(F<=f) onetail 
0 

F Critical onetail 
0.74941 

Table 15: ftest of “msft” and “hml”
(Source: created by author)
The f value of the test is 0.000843 and the critical value of the test is 0.74941 (Sanderson and Windmeijer 2016). The f value of the test is less than the critical value of the test. This leads to acceptance of “null hypothesis” and “msft” is unrelated to value premium.
Cameron, A.C. and Trivedi, P.K., 2013. Regression analysis of count data (Vol. 53). Cambridge university press.
Draper, N.R. and Smith, H., 2014. Applied regression analysis. John Wiley & Sons.
Kleinbaum, D.G., Kupper, L.L., Nizam, A. and Rosenberg, E.S., 2013. Applied regression analysis and other multivariable methods. Nelson Education.
Montgomery, D.C., Peck, E.A. and Vining, G.G., 2015. Introduction to linear regression analysis. John Wiley & Sons.
Sanderson, E. and Windmeijer, F., 2016. A weak instrument Ftest in linear IV models with multiple endogenous variables. Journal of Econometrics, 190(2), pp.212221.
Seber, G.A. and Lee, A.J., 2012. Linear regression analysis (Vol. 936). John Wiley & Sons.
Sen, A. and Srivastava, M., 2012. Regression analysis: theory, methods, and applications. Springer Science & Business Media.
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