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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.

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  • $\begingroup$ 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, 2016 at 2:21

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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$\rho=\frac{\beta_A \beta_B\sigma_M^2}{\sigma_A \sigma_B}$

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