Question 1

Problems 1-6 on pages 511-512 of Ruppert/Matteson.

  1. Would you reject the null hypothesis that alpha is zero for any of the seven stocks? Why or why not?
  2. Use model ( 17.20 ) to estimate the expected excess return for all seven stocks. Compare these results to using the sample means of the excess returns to estimate these parameters. Assume for the remainder of this lab that all alphas are zero. (Note: Because of this assumption, one might consider reestimating the betas and the residuals with a no-intercept model. However, since the estimated alphas were close to zero, forcing the alphas to be ex- actly zero will not change the estimates of the betas or the residuals by much. Therefore, for simplicity, do not reestimate.)
  3. Compute the correlation matrix of the residuals. Do any of the residual correlations seem large? Could you suggest a reason why the large correlations might be large? (Information about the companies in this data set is available at Yahoo Finance and other Internet sites.
  4. Use model ( 17.20) to estimate the covariance matrix of the excess returns for the seven companies.
  5. What percentage of the excess return variance for UTX is due to the market?
  6. An analyst predicts that the expected excess return on the market next year will be \(4 \% .\) Assume that the betas estimated here using data from \(2004-2006\) are suitable as estimates of next year’s betas. Estimate the expected excess returns for the seven stocks for next year.

In our solution we require running the following code to extract the data and fit our model.

1. \(\mathbf{Solution.}\qquad\) We extract the p-values for the intercept \(\alpha\) from each fitted regression model and check if any are significant.

Table 1: \(p\)-values for \(\alpha_j\)
GM_AC F_AC UTX_AC CAT_AC MRK_AC PFE_AC IBM_AC
0.421 0.143 0.389 0.234 0.912 0.409 0.312

From Table 1 we see that none of the \(p\)-values of the \(\alpha_j\) intercepts are significant so we fail to reject the null that any of the \(\alpha_j = 0\).

   

2. \(\mathbf{Solution.}\qquad\) Below we estimate expected excess returns for each of the 7 stocks using our fitted model and compare these estimates to the sample means of stock excess returns. We see that both estimates differ.

Table 2: Expected Excess Returns from fitted model
GM_AC F_AC UTX_AC CAT_AC MRK_AC PFE_AC IBM_AC
0.023 0.024 0.019 0.027 0.015 0.018 0.016
Table 3: Sample Means of Stock Excess Returns
GM_AC F_AC UTX_AC CAT_AC MRK_AC PFE_AC IBM_AC
-0.045 -0.074 0.049 0.084 0.007 -0.022 -0.016

   

3. \(\mathbf{Solution.}\qquad\) From the correlation matrix of residuals below, we see that GM and Ford are the most highly correlated with correlation equal to 0.51. This makes sense since they are both American automotive companies.

Correlation matrix of residuals from fitted regression model.

Figure 1: Correlation matrix of residuals from fitted regression model.

   

4. \(\mathbf{Solution.}\qquad\) To estimate the covariance matrix of returns, we use the following formulas, \[\begin{aligned} \sigma_{j}^{2} &=\beta_{j}^{2} \sigma_{M}^{2}+\sigma_{\epsilon, j}^{2} \\ \sigma_{j j^{\prime}} &=\beta_{j} \beta_{j^{\prime}} \sigma_{M}^{2} \text { for } j \neq j^{\prime} \end{aligned} \]

Table 4: Estimated Covariance Matrix of Excess Stock Returns
GM_AC F_AC UTX_AC CAT_AC MRK_AC PFE_AC IBM_AC
GM_AC 5.512 0.686 0.541 0.781 0.416 0.526 0.455
F_AC 0.686 3.673 0.557 0.804 0.428 0.542 0.468
UTX_AC 0.541 0.557 1.230 0.634 0.338 0.427 0.370
CAT_AC 0.781 0.804 0.634 2.441 0.487 0.617 0.534
MRK_AC 0.416 0.428 0.338 0.487 3.329 0.328 0.284
PFE_AC 0.526 0.542 0.427 0.617 0.328 2.004 0.359
IBM_AC 0.455 0.468 0.370 0.534 0.284 0.359 0.992

   

5. \(\mathbf{Solution.}\qquad\) Below we compute the percentage of the excess return variance for UTX that is due to the market.

## [1] 0.357

   

6. \(\mathbf{Solution.}\qquad\) Below we estimate the expected excess returns for the seven stocks for next year assuming that the excess return on the market next year will be 4%.

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 4.81 4.95 3.91 5.64    3  3.8 3.28

Question 2

Exercise 11 on page 514 of Ruppert/Matteson, with the following adjustments.

Suppose there are three risky assets with the following betas and \(\sigma_{\epsilon_{j}}^{2}\) when regressed on the market portfolio. \[ \begin{array}{c|c|c} j & \beta_{j} & \sigma_{\epsilon_{j}}^{2} \\ \hline 1 & 0.8 & 0.012 \\ 2 & 0.9 & 0.025 \\ 3 & 0.7 & 0.015 \end{array} \]

Assume \(\epsilon_{1}, \epsilon_{2},\) and \(\epsilon_{3}\) are uncorrelated. Suppose also that the variance of \(R_{M}-\mu_{f}\) is 0.02

  1. What is the beta of an equally weighted portfolio of these three assets?
  2. What is the variance of the excess return on the equally weighted portfolio?
  3. What proportion of the total risk of asset 1 is due to market risk?

(a) \(\mathbf{Solution.}\qquad\) Using the CAPM regression model with no intercept, the excess returns of asset \(j\) are defined as, \[\begin{equation} R_{j}-\mu_{f}=\beta_{j}\left(R_{M}-\mu_{f}\right)+\varepsilon_{j}\label{eq:1} \end{equation}\]

Then we can define our equally-weighted portfolio as, \[ R_{p}=\frac{1}{3}\sum_{j=1}^{3}R_{j} \]

Applying the CAPM formula (\ref{eq:1}) to \(R_{p}\) we have, \[\begin{align} R_{p}-\mu_{F} & =\beta_{p}\left(R_{M}-\mu_{f}\right)+\varepsilon_{p}\label{eq:2} \end{align}\]

where, \[ \beta_{p}=\frac{1}{3}(0.8+0.9+0.7)=\clyx(\frac{1}{3}(0.8+0.9+0.7)) \]

   

(b) \(\mathbf{Solution.}\qquad\) We take the variance of equation (\ref{eq:2}), \[\begin{align*} \textrm{Var}\left(R_{p}-\mu_{f}\right) & =\textrm{Var}\left(\beta_{p}\left(R_{M}-\mu_{f}\right)+\varepsilon_{p}\right)\\ & =\beta_{p}^{2}\textrm{Var}\left(R_{M}-\mu_{f}\right)+\textrm{Var}\left(\varepsilon_{p}\right)\\ & =(0.8)^{2}(0.02)+\left(\frac{1}{3}\right)^{2}(0.012+0.025+0.015)\\ & =\clyx((0.8)^{2}(0.02))+\clyx(\left(\frac{1}{3}\right)^{2}(0.012+0.025+0.015))\\ & \approx\clyx((0.8)^{2}(0.02)+\left(\frac{1}{3}\right)^{2}(0.012+0.025+0.015)) \end{align*}\]

   

(c) \(\mathbf{Solution.}\qquad\) The proportion of risk for asset \(1\) is calculated as, \[\begin{align*} \frac{\beta_{1}^{2}\sigma_{M}^{2}}{\beta_{1}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon_{1}}^{2}} & =\frac{(0.8)^{2}(0.02)}{(0.8)^{2}(0.02)+0.012}\\ & =\clyx(\frac{(0.8)^{2}(0.02)}{(0.8)^{2}(0.02)+0.012}) \end{align*}\]

   

Question 3

Consider time series model of \[ X_{n}=U \delta_n + \epsilon_n \]

where \(\delta_{1}, \delta_{2}, \ldots\) are iid rvs with mean 1 and variance of \(1, \epsilon_{1}, \epsilon_{2}, \ldots\) are iid rvs with mean 0 and variance of \(\frac{1}{4}\) and are independent of \(\left\{\delta_{i}\right\}_{i=1}^{\infty},\) and \(U\) is Uniform \((0,1)\) rv which is independent of \(\left\{\epsilon_{i}\right\}_{i=1}^{\infty}\) and \(\left\{\delta_{i}\right\}_{i=1}^{\infty}\)

  1. Derive the mean function and auto-covariance function of the time series \(\left\{X_{n}\right\}_{n=1}^{\infty}\) and validate that this time series is weakly stationary.
  2. Show that the time series \(\left\{X_{n}\right\}_{n=1}^{\infty}\) is not ergodic. Hint. Consider the auto-correlation, and show that it does not satisfy a necessary property for ergodicity.

(a) \(\mathbf{Solution.}\qquad\) First we calculate the mean function, \[\begin{align*} \mu_{X}(n) & =\mathbb{E}\left[X_{n}\right]\\ & =\mathbb{E}\left[U\delta_{n}+\varepsilon_{n}\right]\\ & =\mathbb{E}[U]\mathbb{E}\left[\delta_{n}\right]+\mathbb{E}\left[\varepsilon_{n}\right]\\ & =\frac{1}{2}(1)+0=\frac{1}{2} \end{align*}\]

Next, we calculate the auto-covariance function \[\begin{align} \Sigma_{X}(m,n) & =\textrm{Cov}\left(X_{m},X_{n}\right)\nonumber \\ & =\textrm{Cov}\left(U\delta_{m}+\varepsilon_{m},U\delta_{n}+\varepsilon_{n}\right)\nonumber \\ & =\textrm{Cov}\left(U\delta_{m},U\delta_{n}\right)+\underbrace{\textrm{Cov}\left(U\delta_{m},\varepsilon_{n}\right)}_{=0}+\underbrace{\textrm{Cov}\left(\varepsilon_{m},U\delta_{n}\right)}_{=0}+\textrm{Cov}\left(\varepsilon_{m},\varepsilon_{n}\right)\nonumber \\ & =\textrm{Cov}\left(U\delta_{m},U\delta_{n}\right)+\textrm{Cov}\left(\varepsilon_{m},\varepsilon_{n}\right)\label{eq:3} \end{align}\]

where, \[ \textrm{Cov}\left(\varepsilon_{m},\varepsilon_{n}\right)=\begin{cases} \frac{1}{4} & \text{if }m=n\\ 0 & \text{if }m\neq n \end{cases} \]

and, \[\begin{align*} \textrm{Cov}\left(U\delta_{m},U\delta_{n}\right) & =\mathbb{E}\left[U^{2}\delta_{m}\delta_{n}\right]-\mathbb{E}\left[U\delta_{m}\right]\mathbb{E}\left[U\delta_{n}\right]\\ & =\mathbb{E}\left[U^{2}\right]\mathbb{E}\left[\delta_{m}\delta_{n}\right]-\left(\frac{1}{2}\right)^{2}\\ & =\frac{1}{3}\mathbb{E}\left[\delta_{m}\delta_{n}\right]-\frac{1}{4} \end{align*}\]

where, \[ \mathbb{E}\left[\delta_{m}\delta_{n}\right]=\begin{cases} 2 & \text{if }m=n\\ 1 & \text{if }m\neq n \end{cases} \]

Then equation (\ref{eq:3}) becomes, \[ \Sigma_{X}(m,n)=\begin{cases} \frac{1}{3}(2)-\frac{1}{4}+\frac{1}{4}=\frac{2}{3} & \text{if }m=n\\ \frac{1}{3}(2)-\frac{1}{4}=\frac{1}{12} & \text{if }m\neq n \end{cases} \]

We know that a weakly stationary time series is both,

  • mean stationary: constant mean
  • covariance stationary: covariance between two observations only depends on their time lag.

This holds true by the \(\mu_{X}\left(n\right)\) and \(\Sigma_{X}(m,n)\) we computed above.

   

(b) \(\mathbf{Solution.}\qquad\) To compute the auto-correlation function, we compute, \(\Sigma_{X}(m,m)=\operatorname{Cov}\left(X_{m},X_{m}\right),\) which we know from part a) is equal to \(\frac{2}{3}\). Thus, \[ \rho_{X}\left(h\right)=\begin{cases} \frac{2/3}{2/3}=1 & \text{if }h=0\\ \frac{1/12}{2/3}=\frac{1}{8} & \text{if }h\neq0 \end{cases} \]

The auto-correlation function shows that the correlation between past and future events will always be equal to \(1/8\). This does not satisfy the definition of an ergodic time-series since past and future events should be approximately independent and thus have \(0\) correlation.