Optimal Portfolio Formulation

Question

I'm currently studying Luenberg's Article "Projection Pricing" (Jrl of Optimization Theory and Applications, Vol. 109, No. 1, pp. 1–25, April 2001) and there is a claim that I can't prove. In brief, I'm trying to find the maximum Sharpe Ratio portfolio, so: $$ \text{maximize } \frac{\omega^T(\bar{y}-R_f p)}{\sqrt{\omega^TV \omega}} $$ Where $\omega$ is the weights vector, $\bar{y}$ expected payoff of the assets, $p$ is the price vector, $V$ the covariance Matrix and $R_f = 1+r_f$, $r_f$ is the riskfree rate. And he claims that the solution is: $$ \omega = \gamma (\bar{y}-R_f p)^T V^{-1} $$ I tried to use the first Kuhn-Tucker condition:

$$ \mathcal{L}(\omega) = \frac{\omega^T(\bar{y}-R_f p)}{\sqrt{\omega^TV \omega}} $$

$$ \frac{\partial\mathcal{L(\omega)}}{\partial\omega} = 0 \Rightarrow \frac{(\bar{y}-R_f p)}{\sqrt{\omega^TV \omega}} - \frac{\omega^T(\bar{y}-R_f p)}{\sqrt{(\omega^T V \omega)^3}} V \omega =0 $$ But no success from there...

Maybe I'm not seeing some kind of manipulation or the problem is missing some constraints.

Does anyone know how to prove it?

Thanks.

Answer 1

Treat scalars as annoying constants to be dealt with later and solve the KKT conditions up to that scaling. You already showed that $$ \frac{\bar{y} - R_fp}{c_1} - \frac{c_2}{c_1^3}V\omega = 0. $$ This would suffice to prove the identity of $\omega$ up to scaling.