Consider the equity prices of the United States companies Microsoft and Walmart for the period April 1990 to July 2004 ( \(T=172)\).

(a) Compute the continuously compounded (log) returns on Microsoft and Walmart.

(b) Compute the variance-covariance matrix of the returns on these two stocks. Verify that the covariance matrix of the returns is

\[ \left[\begin{array}{ll} 0.011332 & 0.002380 \\ 0.002380 & 0.005759 \end{array}\right] \]

where the diagonal elements are the variances of the individual asset returns and the off-diagonal elements are the covariances. Note that the off-diagonal elements are in fact identical because the covariance matrix is a symmetric matrix. n computing the elements of the covariance matrix use the biased form presented in Chapter 2 in which \(T\) is used in the denominator instead of \(T-1\).

(c) Use the expressions in (3.5) and (3.6) to verify that the minimum variance portfolio weights between these two assets are

\[ \begin{aligned} w_{1} & =\frac{\sigma_{2}^{2}-\sigma_{12}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2 \sigma_{12}}=\frac{0.005759-0.002380}{0.011332+0.005759-2 \times 0.002380}=0.274 \\ w_{2} & =1-w_{1}=1-0.274=0.726 \end{aligned} \]

(d) Using the computed weights in part (c), compute the return on the portfolio as well as its mean and variance (without any degrees of freedom adjustment).

(e) Estimate the regression equation

\[ r_{\text {Wmart }, t}=\beta_{0}+\beta_{1}\left(r_{\text {Wmart }, t}-r_{\text {Msoft }, t}\right)+u_{t} \]

where \(u_{t}\) is a disturbance term.

i. Interpret the estimate of \(\beta_{1}\) and discuss how it is related to the optimal portfolio weights computed in part (c).

ii. Interpret the estimate of \(\beta_{0}\).

iii. Compute the variance of the least squares residuals, without any degrees of freedom adjustment, and interpret the result.

(f) Using the results in part (e)
i. Construct a test of an equal weighted portfolio, \(w_{1}=w_{2}=0.5\).
ii. Construct a test of portfolio diversification.
(g) Repeat parts (a) to (f) for Exxon and GE.
(h) Repeat parts (a) to (f) for gold and IBM.