When CAPM Implies Covariance from Market Betas
Summary
The document examines whether the single-index model determines the covariance between two stocks from their market betas and the market variance. It explains the textbook formula by expressing each stock’s excess return as a market-linked component plus an idiosyncratic residual. If the residuals are uncorrelated with each other and with the market, only the shared market component contributes to the stocks’ covariance.
Under those assumptions, covariance equals the product of the two betas and market variance; the exercise’s inputs illustrate the calculation. The question’s counterexample shows why covariance with a third variable alone cannot generally determine pairwise covariance. The result therefore depends on the single-index or CAPM residual assumptions, especially zero cross-asset residual covariance. It should not be treated as a general identity for arbitrary assets.
Key ideas
- In a single-index model, each asset return is decomposed into market exposure and an idiosyncratic residual.
- If residuals are mutually uncorrelated and uncorrelated with the market, pairwise covariance comes from the shared market component.
- Under those assumptions, the covariance is the product of the assets’ betas and market variance.
- Covariances with a third asset do not determine pairwise covariance without additional model assumptions.
Tags
Full text
# Using CAPM to find correlation of two assets with each other
# Using CAPM to find correlation of two assets with each other
I stumpled upon an exercise in an investments book:
> The data below describe a three-stock financial market that satisfies the single-index model.
```
Stock Capitalization Beta Mean Excess Return Standard Deviation
A $3,000 1.0 10% 40%
B $1,940 0.2 2% 30%
C $1,360 1.7 17% 50%
```
> The standard deviation of the market-index portfolio is 25%. a. What is the mean excess return of the index portfolio? b. What is the covariance between stock A and stock B ?
With the solution to the second question given as:
> $Cov(R_A, R_B) = \beta_A \beta_B \sigma_M^2 = 1 * 0.2 * .25^2 = .0125$
This translates to $\beta_A \beta_B \sigma_M^2 = \frac{Cov(A,M)}{\sigma_M^2}*\frac{Cov(B,M)}{\sigma_M^2}*\sigma_M^2 = \frac{Cov(A,M)Cov(B,M)}{\sigma_M^2}$
However, I could not derive this formula, and mathematically, we do not know what the correlation is between two assets just from their covariance with a third asset (except that we can give upper and lower bounds, in some cases).
For example, if $A,B$ i.i.d., and $M := A+B$, then
$Cov(A,B) = 0$ by construction, but $Cov(A,M) = Cov(B,M) = Cov(A,A+B) = Cov(A,A)+Cov(A,B) = Var(A)$.
Am I missing something? Are the assumptions of CAPM playing into this?
Are the sample solutions incorrect?
## Answer by Comp_Warrior (score 4, accepted)
https://quant.stackexchange.com/a/32124
The solution provided can be derived using the CAPM. For asset $A$ you have:
$$R_A-R_f = \alpha_A +\beta_A(R_M-R_f)+\epsilon_A$$
Similarly for asset B:
$$R_B-R_f = \alpha_B +\beta_B(R_M-R_f)+\epsilon_B$$
Calculate the covariance:
$$\text{Cov}(R_A, R_B) = \text{Cov}(\beta_AR_M, \beta_BR_M)$$
Here I have dispensed with all the constant terms, and also used the usual CAPM assumption that $\epsilon$ represents idiosyncratic volatility, so $\text{Cov}(\epsilon_A, \epsilon_B) = 0$, $\text{Cov}(R_M, \epsilon_A) = 0$ and $\text{Cov}(R_M, \epsilon_B) = 0$. So we have:
$$\text{Cov}(R_A, R_B) = \beta_A\beta_B\text{Cov}(R_M, R_M) = \beta_A \beta_B\sigma_M^2$$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.