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Question:
Discuss about the Econometrics for Stocks of Microsoft Outperforms.

1. Stocks of Microsoft outperforms or under performs in the market.

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 1.94E-13 Residual 130 1.032074 0.007939 Total 131 1.56675 Coefficients Standard Error t Stat P-value 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 1.94E-13 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.

2.

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.

3.

Evaluation of the claim that the stocks of Mobil-Exxon 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 Mobil-Exxon (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 8.53E-06 Residual 130 0.320757 0.002467 Total 131 0.373786 Coefficients Standard Error t Stat P-value 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 8.53E-06 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 Mobil-Exxon (xom) are defensive is not true. This is because the value of the stocks is not less than one.

4.

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 2.83E-11 Residual 130 0.608522 0.004681 Total 131 0.856644 Coefficients Standard Error t Stat P-value 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 2.83E-11 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 1.45E-15 Residual 130 0.391781 0.003014 Total 131 0.640687 Coefficients Standard Error t Stat P-value 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 1.45E-15 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 5.8E-09 Residual 130 1.634723 0.012575 Total 131 2.123838 Coefficients Standard Error t Stat P-value 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 5.8E-09 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 2.47E-16 Residual 130 0.63848 0.004911 Total 131 1.07256 Coefficients Standard Error t Stat P-value 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 2.47E-16 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 8.53E-06 Residual 130 0.320757 0.002467 Total 131 0.373786 Coefficients Standard Error t Stat P-value 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 8.53E-06 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 1.94E-13 Residual 130 1.032074 0.007939 Total 131 1.56675 Coefficients Standard Error t Stat P-value 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 1.94E-13 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)

5.

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.

6.

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.

Part II

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.

2.

Testing of hypothesis using f-test for “dis”

Solution

F-test 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 Fama-French model. The hypothesis of this test is as follows:

H0 : “dis” is unrelated to size premium

H1 : “dis” is related to size premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.749410102

Table 10: f-test 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 F-test is as follows:

H0 : “dis” is unrelated to value premium

H1 : “dis” is related to value premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.749410102

Table 11: f-test 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.

3.

Hypothesis test of “xom” using f-test

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 Fama-French model. F-test would be used in this case. The following hypothesis would be given for this test.

H0 : “xom” is unrelated to value premium

H1 : “xom” is related to value premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.74941

Table 12: f-test 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.

H0 : “xom” is unrelated to size premium

H1 : “xom” is related to size premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.74941

Table 13: f-test of “xom” and “smb”

(Source: created by author)

F-test 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”.

4.

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 Fama-French model

Solution

F-test 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 Fama-French model. The hypothesis of the test is as follows:

H0 : “msft” is unrelated to size premium

H1 : “msft” is related to size premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.74941

Table 14: f-test 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:

H0 : “msft” is unrelated to value premium

H1 : “msft” is related to value premium

The table below provides the result of this f-test:

 F-Test Two-Sample 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) one-tail 0 F Critical one-tail 0.74941

Table 15: f-test 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.

References

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 F-test in linear IV models with multiple endogenous variables. Journal of Econometrics, 190(2), pp.212-221.

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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