# Expected Return on Stock

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.

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 $$E[R_M]$$