# Calculating R* in a two-asset world

In chapter 5 of John Cochrane's Asset pricing, we derive a state-space interpretation of the mean variance frontier by defining $$R^*$$ and $$R^{e*}$$. A little forward, we have this formulation: $$R^* = \frac{1'E(RR')^{-1}R}{1'E(RR')^{-1}1}$$ How can we write down the R* for a two asset economy with a risky asset (mean = $$\mu$$, variance = $$\sigma^2$$)?

This is my work:

$$R=[R_f \space \space R]'$$. We have:

$$R^* =\frac{[1 \space \space 1] {\begin{bmatrix} E(R_f^2) & E(R_fR)\\ E(RR_f) & E(R^2) \end{bmatrix}}^{-1} [R_F \space R]'}{[1 \space \space 1] {\begin{bmatrix} E(R_f^2) & E(R_fR)\\ E(RR_f) & E(R^2) \end{bmatrix}}^{-1} [1 \space 1]'}$$

taking expectations:

$$=\frac{[1 \space \space 1] {\begin{bmatrix} R_f^2 & R_f\mu\\ R_f\mu & \mu^2 + \sigma^2 \end{bmatrix}}^{-1} [R_F \space R]'}{[1 \space \space 1] {\begin{bmatrix} R_f^2 & R_f\mu\\ R_f\mu & \mu^2 + \sigma^2 \end{bmatrix}}^{-1} [1 \space 1]'}$$

Do the inverse:

$$=\frac{[1 \space \space 1] {\begin{bmatrix} \mu^2 + \sigma^2 & -R_f\mu\\ -R_f\mu & R_f^2 \end{bmatrix}} [R_F \space R]'}{[1 \space \space 1] {\begin{bmatrix} \mu^2 + \sigma^2 & -R_f\mu\\ -R_f\mu & R_f^2 \end{bmatrix}} [1 \space 1]'}$$

Multiply:

$$=\frac{[\mu^2 + \sigma^2 -R_f\mu \space \space \space \space \space -R_f\mu + R_f^2] [R_f \space \space R]'}{[\mu^2 + \sigma^2 -R_f\mu \space \space \space \space \space -R_f\mu + R_f^2] [1 \space \space 1]'}$$

Multiply: $$=\frac{R_f(\mu^2 + \sigma^2 -R_f\mu)+R(-R_f\mu + R_f^2 )}{\mu^2 + \sigma^2 -2R_f\mu + R_f^2}$$

So we were able to write down the $$R^*$$ as a linear coefficient of R:

$$R^*=AR + B$$

This is my work, but I have no reference to check if this is indeed correct. Especially because still I feel I have not grasped the state-space interpretation fully. Would be grateful if someone confirms my work/or suggest better solutions.