2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Yeah, that´s it... Thanks, man. $\endgroup$ – Felipe Teti Mar 23 at 18:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.