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?


1 Answer 1


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:

$$\text{Cov}(R_A, R_B) = \beta_A\beta_B\text{Cov}(R_M, R_M) = \beta_A \beta_B\sigma_M^2$$

  • 1
    $\begingroup$ It is true that in general "we do not know what the correlation is between two assets just from their covariance with a third asset", but here as you can see here the situation is much more restricted: The two assets are a linear combination of the third asset and random terms $\epsilon_A, \epsilon_B$ orthogonal to each other and to $R_m$. Basically the first two assets are multiples of the third plus noise $\endgroup$
    – Alex C
    Jan 28, 2017 at 3:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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