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:

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


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

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.


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