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Suppose we have the following information on stocks $X$, $Y$, and $Z$:

  1. Expected Returns: $E(R_X)=10\%$, $E(R_Y)=12\%$.
  2. Standard Deviations: $\sigma_X=10\%$, $\sigma_Y=15\%$, $\sigma_Z=10\%$
  3. Pairwise Correlations: $\rho_{XY}=0$, $\rho_{XZ}=0$, $\rho_{YZ}=0.5$.

Assume that the CAPM holds and that the market portfolio consists of the above three stocks weighted equally. Find the expected return of Stock Z.

Attempt: We can first get $\sigma_M$ by using the formula for the variance of a three-asset portfolio. Then, from there, we can solve for the $\beta$ for each stock using $\beta=\frac{\text{Cov}(R_i,R_M)}{\sigma_M^2}$. However, I'm not sure how to compute for the correlation between the stock return and the market return.

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To compute the correlation between the stock return (let us say $R_X$) and the market return $R_M$, you just write:

$\rho_{R_X,R_M} = \frac{Cov(R_X,R_M) }{\sigma_{R_X}\sigma_{R_M}} = \frac{Cov(R_X, \frac{1}{3}R_X + \frac{1}{3}R_Y + \frac{1}{3}R_Z ) }{\sigma_{R_X}\sigma_{R_M}} $

Once you get the market variance and the $\beta$, you just write the CAPM formula for $X$ and $Y$:

$E[R_X] = R_{rf} + \beta_{X,M}(E[R_M] - R_{rf})$

$E[R_Y] = R_{rf} + \beta_{Y,M}(E[R_M] - R_{rf})$

And then you go two équations with two unknows variables ($R_{rf}$ and $E[R_M]$). You solve it and finally you easily get $E[R_Z]$ from $E[R_M]$

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