0
$\begingroup$

Background: This question is from Active Portfolio Management by Grinold and Kahn (Exercise 4 of Chapter 2- Consensus Expected Returns: The Capital Asset Pricing Model). I have no background in finance and am trying to work through this book on my own.

Question: Assume that residual returns are uncorrelated across stocks. Stock A has a beta of 1.15 and a volatility if 35 percent. Stock B has a beta of .95 and a volatility of 33 percent. If the market volatility is 20 percent, what is the correlation of stock A with stock B? Which stock has higher residual volatility?

Comments: 1. This question is not about the CAPM because we are assuming there are residual returns? 2. What is the exact definition of volatility being used here? 3. Is there a nice formula for the first part of the question? 4. I have a slight feeling for the second question since the beta determines how much of the volatility comes from the market. 5. The only formulas I think I need are $$ \beta_P = \frac {\operatorname{Cov}(r_P, r_M)}{\operatorname{Var}(r_M)}, \quad r_P = \beta_P + \theta_P, \quad \sigma_P^2= \beta_P^2\sigma_M^2 + \omega_P^2, $$ where $P$ is the portfolio, $M$ is the market and $r_P$ and $r_M$ are their respective excess returns, $\theta_P$ is the residual return, and $\omega_P$ is the residual variance of portfolio $P$ (i.e., the variance of $\theta_P$). Finally, just to be clear, $\operatorname{Var}(r_M) = \sigma_P^2$

Attempt at second part: Using the third formula above I get residua volatility for $A$ to be 35-23 = 12% and for $B$ to be 33- 19 = 14 %, so stock B has higher residual volatility.

$\endgroup$
  • $\begingroup$ I thought the third formula had in it variances, you seem to be applying it to volatilities when you do 35-23. Shouldn't it be $\sigma=\sqrt{35^2-23^2}$ ? $\endgroup$ – Alex C Aug 9 '16 at 2:21
2
$\begingroup$

The definition of correlation is $\rho=\frac{covar(r_A,r_B)}{\sigma_A \sigma_B}$.

We know $\sigma_A=0.35;\sigma_B=0.33$. What is the covariance?

Since the residuals are independent the covariance only comes from the systematic (market) movement in both stocks. A useful general formula is $covar(\beta_A X,\beta_B X)=\beta_A \beta_B var(X)$. (see https://en.wikipedia.org/wiki/Covariance#Properties ).

So putting it all together$\rho=\frac{\beta_A \beta_B\sigma_M^2}{\sigma_A \sigma_B}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.