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?