# Portfolio Allocation given Sharpe Ratio

If there are two portfolios with sharpe ratios of 1.2 and 0.5, what would be the allocation rationale.

If the correlation between portfolios is:

$a. 0$

$b. 0.8$

$c.-0.8$

I see there is a diversification benefit in the c case, but is there a way to decide weights without more information ?

The optimal Sharpe you can achieve, by the Markowitz portfolio, is $$\sqrt{\frac{1}{1-\rho^2} \left( 1.2^2 - 2 \rho (1.2) (0.5) + 0.5^2 \right)}.$$ The optimal portfolio is $$\frac{1}{1-\rho^2} \begin{bmatrix} 1 & -\rho \\ -\rho & 1 \end{bmatrix} \begin{bmatrix} 1.2\\ 0.5 \end{bmatrix},$$ where $\rho$ is the correlation of the assets.
• Rescale the assets to unit risk. Their returns are then their Sharpe ratios, and their covariance is the correlation matrix, with 1 on the diagonal and $\rho$ on the off diagonal. If you invert that covariance matrix you get the $1 / (1-\rho^2)$ times the matrix term. So the lower equation is just $\Sigma^{-1} \mu$ on the rescaled returns. The units are 'risk units'; a real portfolio would be positively rescaled to the desired risk level. The top equation is simply $\sqrt{\mu^{\top} \Sigma^{-1}\mu}$. – shabbychef May 17 '18 at 5:30
• BTW, you can confirm that the minimum optimal Sharpe is achieved when $\rho = 0.5/1.2$, in which case the optimal Sharpe is 1.2, i.e. there is zero diversification benefit. – steveo'america May 17 '18 at 18:50