4
$\begingroup$

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 ?

$\endgroup$
2
$\begingroup$

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.

You can do the rest of the math.

$\endgroup$
  • 1
    $\begingroup$ Looks reasonable, but where do these equations come from? $\endgroup$ – Alex C May 16 '18 at 23:48
  • 2
    $\begingroup$ 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 $\sqrt{\mu^{\top} \Sigma^{-1}\mu}$. $\endgroup$ – shabbychef May 17 '18 at 5:30
  • $\begingroup$ 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. $\endgroup$ – steveo'america May 17 '18 at 18:50

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.