# Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?

Suppose the consumer Solves $$\max -e^{-\gamma W}$$ where $$W=X^T D -X^Tp R_f$$ where $$X$$ is the vector invested in a risky asset and $$D\sim N(E[D],\Sigma^2_D)$$ and $$R=\sim N(E[R],\Sigma^2_R)$$. Then $${ X=(\gamma \Sigma_R)^{-1}(E[R-R_f])}$$. Is this formula correct?

My reasoning is as follows: $$e^{-\gamma W}=e^{-\gamma X(E[D]-p R_f)+\frac{1}{2}\gamma^2 X \Sigma X^T}$$ Hence $$X=(\gamma \Sigma_D)^{-1}(E[D-p R_f])$$ Hence $$X=(\gamma \Sigma_R)^{-1}(E[R-R_f])$$

Here $$\Sigma_D$$ and $$\Sigma_R$$ refer to variance vector for dividend and returns.

• Hey, can you please explain what $p$ is, and what's the difference betwen $R$ and $R_f$? Where does $R$ enter the formula in the first place? Oct 13, 2022 at 6:16
• The optimal solution, though, looks good at first sight :) Oct 13, 2022 at 7:50
• @Kermittfrog $R_f$ is risk free rate and $R=D/p$ is return on risky assets. Oct 18, 2022 at 16:42