Solutions to Assignment 4
1.
Here are the expected returns and the standard deviations of returns for the three stocks:
SUNW 
USW 
WMT 

exp. Ret 
0.07438 
0.025103 
0.028305 
stdev. 
0.128537 
0.061013 
0.089955 
Here is the variancecovariance matrix:
SUNW 
USW 
WMT 

SUNW 
0.016522 
0.001715 
0.002278 
USW 
0.001715 
0.003723 
0.001707 
WMT 
0.002278 
0.001707 
0.008092 
The tables below show the expected returns and the variances for the portfolios that can be constructed by taking the stocks, two at a time. The chart below graphs the combination lines.
Portfolios with SUNW and USW 
Portfolios with SUNW and WMT 

Proportion in SUNW 
USW 
Exp ret 
var. 
stdev. 
Proportion in SUNW 
WMT 
Exp ret 
var. 
stdev. 

0 
1 
0.025103 
0.003723 
0.061013 
0 
1 
0.028305 
0.008092 
0.089955 

0.1 
0.9 
0.030031 
0.003489 
0.05907 
0.1 
0.9 
0.032913 
0.00713 
0.084437 

0.2 
0.8 
0.034958 
0.003592 
0.059935 
0.2 
0.8 
0.03752 
0.006569 
0.081046 

0.3 
0.7 
0.039886 
0.004031 
0.063493 
0.3 
0.7 
0.042128 
0.006409 
0.080054 

0.4 
0.6 
0.044814 
0.004807 
0.069332 
0.4 
0.6 
0.046735 
0.00665 
0.081546 

0.5 
0.5 
0.049742 
0.005919 
0.076933 
0.5 
0.5 
0.051343 
0.007292 
0.085395 

0.6 
0.4 
0.054669 
0.007367 
0.08583 
0.6 
0.4 
0.05595 
0.008336 
0.091301 

0.7 
0.3 
0.059597 
0.009151 
0.095662 
0.7 
0.3 
0.060558 
0.009781 
0.098897 

0.8 
0.2 
0.064525 
0.011272 
0.106169 
0.8 
0.2 
0.065165 
0.011627 
0.107826 

0.9 
0.1 
0.069453 
0.013729 
0.117169 
0.9 
0.1 
0.069773 
0.013874 
0.117786 

1 
0 
0.07438 
0.016522 
0.128537 
1 
0 
0.07438 
0.016522 
0.128537 
Portfolios with USW and WMT 

Proportion in USW 
WMT 
Exp ret 
var. 
stdev. 
0 
1 
0.028305 
0.008092 
0.089955 
0.1 
0.9 
0.027985 
0.006899 
0.083059 
0.2 
0.8 
0.027665 
0.005874 
0.076641 
0.3 
0.7 
0.027345 
0.005017 
0.07083 
0.4 
0.6 
0.027024 
0.004328 
0.065787 
0.5 
0.5 
0.026704 
0.003807 
0.061701 
0.6 
0.4 
0.026384 
0.003454 
0.058771 
0.7 
0.3 
0.026064 
0.003269 
0.057177 
0.8 
0.2 
0.025743 
0.003252 
0.057029 
0.9 
0.1 
0.025423 
0.003403 
0.058339 
1 
0 
0.025103 
0.003723 
0.061013 
a. If I had not yet computed the portfolio variances for different portfolio combinations,
and I wanted to minimize portfolio variance to the greatest extent,
I would choose USW and WMT, because they have the smallest pairwise covariance, and their own variances are also small.
b. You can decrease the standard deviation of returns of a portfolio consisting of two stocks below the standard deviations of returns of either stock by itself by choosing two stocks that have low correlation and picking the appropriate combination of both stocks. The portfolio standard deviation can be raised beyond the standard deviation of returns of either asset simply by weighting one stock sufficiently heavily.
2.
Regression of R_{IBM} on R_{S&P500} using return data for 19791983
Regression Statistics 

Multiple R 
0.557787248 

R Square 
0.311126614 

Adjusted R Square 
0.299249487 

Standard Error 
0.046555226 

Observations 
60 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.056775717 
0.056775717 
26.19544319 
3.65702E06 

Residual 
58 
0.125708566 
0.002167389 

Total 
59 
0.182484283 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.000579407 
0.006338236 
0.091414564 
0.927478186 
0.013266756 
0.012107942 
R_{IBM} 
0.718914152 
0.140463717 
5.118148414 
3.65702E06 
0.437745684 
1.00008262 
Parts 1., 2.,
The numbers under the rubric "Coefficients" are the most important ones. The statistical model that the regression assumes is:
R_{IBM,t} = intercept + slope(R_{S&P500}) + error.
The estimated value of the intercept is given in the row labelled "Intercept" as 0.000579407. The estimated value of the slope is below that, and is 0.7189. This is the IBM’s beta estimate, and it means that during the sample period, a 1% increase in the market return meant that, on average, IBM’s return increased by 0.72%.
The number under the "standard error" column for that row is 0.14; how do we interpret this? Suppose we could draw lots of 60observation samples from the assumed underlying normal distribution that generated IBM’s and the market’s returns over that period. Now, if we ran regressions using that data, we would get many beta estimates. If we plotted the frequency distribution for those beta estimates, we would get a normal distribution centered around 0.72 with a standard deviation of 0.14. We can now interpret the last two numbers in the same row as well. What they tell us is that we can say with 95% confidence that the true value of IBM’s beta over that period was between 0.438 and 1.000.
To make more sense of the intercept, we need to make use of the CAPM. The CAPM says that
E(R_{IBM}) = R_{f} + b_{IBM}(E(R_{m})R_{f}) = (1 b_{IBM})R_{f} + E(R_{m})
If we take the expectation of the regression model above, we get
E(R_{IBM}) = intercept + E(R_{m})
Comparing the two models, we see that according to the CAPM, the intercept should equal (1 b_{IBM})R_{f}; i.e. if the average return on IBM over that period relative to its market risk was as predicted by the CAPM, the estimated intercept should have been (1 b_{IBM})R_{f}. This also means that we can use the value of the intercept less (1 b_{IBM})R_{f} as a measure of how the stock performed over that period. If we look at average oneyear Tbill rates during that period, they were around 9.5%. Since we are using monthly returns, the monthly rate is 0.759%. Using that estimate, the CAPM suggests that the intercept should have been (10.72)(.0075995) = 0.002126, whereas the actual estimate was only 0.00058. Clearly the stock underperformed the market, after adjustment for the beta risk of IBM. However, if we look at the 95% confidence limits for IBM, we see that the the CAPM value is within the limits derived from the regressoin. Hence we cannot reject the hypothesis that there was no underperformance.
The R^{2} of the regression is the proportion of the variance of the return on IBM that can be explained by movements in the market portfolio. Hence, in our case, 31.11% of the variance of R_{IBM }could be explained by market movements.
Part 3. The beta estimate can be improved in several ways. One would be to combine the estimated beta with the a priori estimate of one, since the average beta of all assets in the market is, by definition, one. Another method would be to take into account the financial leverage and operating leverage of IBM during the period, since financial and operating leverages are important determinants of stock betas.
Regression using return data for 19941998
Regression Statistics 

Multiple R 
0.571233482 

R Square 
0.326307691 

Adjusted R Square 
0.314692306 

Standard Error 
0.069882609 

Observations 
60 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
0.137192994 
0.137192994 
28.09271504 
1.87524E06 

Residual 
58 
0.283247584 
0.004883579 

Total 
59 
0.420440578 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.01232967 
0.009997292 
1.233300966 
0.222438497 
0.007682069 
0.03234141 
R_{IBM} 
1.205925099 
0.227522042 
5.300256129 
1.87524E06 
0.750490591 
1.661359608 
Part 4. The beta estimate provided by Yahoo as of Friday, March 24 was 1.06. This is different from 1.20. However, it is interesting to note that 1.06 is within the 95% confidence limits. Furthermore, if we took an average of 1.20 and the a priori estimate of one, we would get a beta estimate closer to 1.06. Yahoo also uses data for the last five years to estimate its beta; hence the difference is probably simply due to a slightly different dataset.
Part 5., and 6.
The beta for IBM has increased from the early 80s to the late 90s. The reason for this is most likely that IBM has changed from being a staid, utilitylike company that mainly leased out mainframes to a company that is very much a technologybased "New Economy" company that does very well when the economy is doing well, but has much significantly prospects when the economy is not doing well.