Deriving Stock Correlation from Betas and Residual Independence
Summary
The document derives the correlation between two stocks under a single-market-factor setup. Each stock’s return is decomposed into a market-driven component, scaled by its beta, and a residual component. Given the assumption that residual returns are uncorrelated across the stocks, their covariance comes only from exposure to the shared market return. The covariance is therefore the product of the two betas and market variance.
Dividing this covariance by the product of the stocks’ volatilities gives the correlation formula. The exercise provides betas, individual volatilities, and market volatility, allowing a numerical answer to be calculated, though the accepted response stops at the formula. It does not complete the requested comparison of residual volatilities. The result depends on the single-factor decomposition and the stated zero residual covariance assumption.
Key ideas
- Stock returns are represented as a market component plus a residual component.
- If the residuals are uncorrelated, shared market exposure determines cross-stock covariance.
- The systematic covariance equals the product of both betas and the market variance.
- Correlation follows by dividing covariance by the product of the stocks’ volatilities.
- The derivation relies on the assumed factor model and residual independence.
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Full text
# Calculating the correlation of stock A with stock B
# Calculating the correlation of stock A with stock B
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.
## Answer by Alex C (score 2, accepted)
https://quant.stackexchange.com/a/29548
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$\rho=\frac{\beta_A \beta_B\sigma_M^2}{\sigma_A \sigma_B}$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.