How Return Correlation Limits Sharpe Gains from Combining Strategies
Summary
The note explains why combining two trading strategies does not guarantee a higher Sharpe ratio. It treats strategy return streams as assets in a mean-variance portfolio and shows that the achievable Sharpe depends on their expected returns, volatilities, and correlation. With the example Sharpe ratios of one and one-half, the answer identifies a correlation equal to the ratio of the smaller Sharpe to the larger as the point with no diversification benefit.
For independent returns, the optimal Sharpe combines geometrically, producing only a modest improvement in the example. Lower or negative correlation can provide greater diversification, while the answer’s limiting negative-correlation case yields an unbounded theoretical Sharpe. That result is a mathematical edge case rather than a practical promise; the discussion does not address estimation error, trading costs, leverage constraints, or whether the return streams remain stable out of sample.
Key ideas
- A higher combined Sharpe is not guaranteed by adding strategies with different individual Sharpes.
- The diversification benefit depends on the correlation between strategy return streams.
- For independent returns, the optimal Sharpe combines through a geometric relationship.
- Strong negative correlation can raise the theoretical combined Sharpe substantially.
- The analysis omits practical constraints and uncertainty in estimated returns and correlations.
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Full text
# Method to combine trading signals to achieve higher sharpe
# Method to combine trading signals to achieve higher sharpe
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.
## Answer by steveo'america (score 5)
https://quant.stackexchange.com/a/37824
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.