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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 ?

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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.

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    $\begingroup$ Looks reasonable, but where do these equations come from? $\endgroup$
    – Alex C
    Commented May 16, 2018 at 23:48
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    $\begingroup$ 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
    Commented May 17, 2018 at 5:30
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    $\begingroup$ 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
    Commented May 17, 2018 at 18:50

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