There are a few thread with the question of methods to combine different trading signals/strategies. But is there any method that can ensure that by combining two signals, we can achieve a better Sharpe ratio?

For example, I have two simple signals one with a Sharpe of 1 and the other with a Sharpe of 0.5. I would like to generate a new one using these two signals to get better Sharpe. I've tried boosting or simple linear combinations of the two but didn't get better results. Any suggestions will be appreciated.


There is no guarantee you can improve the Sharpe in this case, depending on the correlation of the returns streams. For the two asset case (you can model your strategies as assets and take a linear combination of them), if the correlation of the two assets is equal to the ratio of Sharpes (smaller to larger), there is zero diversification benefit.

For example, in your case, re-lever the assets to have unit volatility, so they have expected returns of 1 and 0.5. The covariance matrix is then $$ \Sigma = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \\ \end{bmatrix} $$ The optimal achievable Sharpe is that of the Markowitz portfolio and has value equal to $\sqrt{\mu^{\top} \Sigma^{-1} \mu}$. In your case this will equal $$ SR=\sqrt{\begin{bmatrix}1 & 0.5\end{bmatrix}% \left(\frac{1}{1 - 0.5^2}% \begin{bmatrix} 1 & -0.5 \\ -0.5 & 1 \end{bmatrix} \right) \begin{bmatrix} 1\\0.5 \end{bmatrix}} = 1 $$ If, however, your returns streams are negatively correlated, an infinitely high Sharpe is possible--replace the 0.5 in the $\Sigma^{-1}$ (but not the $\mu$) with a $\rho$ near -1.

When your returns streams are independent, $\rho=0$, Sharpes 'add' geometrically, so you would expect an optimal achievable Sharpe of $\sqrt{1 + \frac{1}{4}}$, which is only a modest improvement over the 1, and you might not notice the improvement.


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