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Deriving Maximum Sharpe Portfolio Weights up to a Scale Factor

Article Quant Q&A · Author: Felipe Teti
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Summary

The document concerns maximizing a portfolio's Sharpe ratio when expected excess payoffs and the covariance matrix of asset payoffs are given. It writes the objective as expected excess payoff divided by portfolio volatility, then differentiates it with respect to the weight vector. The response treats scalar terms in the first-order condition as constants and rearranges the condition to show that the portfolio weights are proportional to the inverse covariance matrix applied to expected excess payoffs.

This establishes the direction of the optimal weight vector, while a multiplicative scale remains undetermined because scaling all weights leaves the ratio unchanged. The brief answer does not derive a particular normalization or address constraints such as budget, leverage, or short-sale limits. Its result therefore applies to the unconstrained formulation shown and relies on the covariance matrix being invertible.

Key ideas

  • The Sharpe objective depends on expected excess payoff and portfolio variance.
  • The first-order condition identifies weights proportional to inverse covariance times expected excess payoff.
  • The scale factor is not determined by a scale-invariant Sharpe objective.
  • Additional portfolio constraints can change the solution and are not handled in the derivation.

Tags

Portfolio constructionStatisticsRisk management
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# Optimal Portfolio Formulation


# Optimal Portfolio Formulation












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 by steveo'america (score 1, accepted)

https://quant.stackexchange.com/a/61915

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.