Fin 320 Advanced Financial Analysis
Homework Assignments
Spring 2000
Prof. P.V. Viswanath

Solutions to Assignment 4

1.

Here are the expected returns and the standard deviations of returns for the three stocks:

Here is the variance-covariance matrix:

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.

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.

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 60-observation 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 one-year T-bill 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 (1-0.72)(.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 R² 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.

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, utility-like company that mainly leased out mainframes to a company that is very much a technology-based "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.