Combining two uncorrelated signals, Sharpes add in quadrature

Question

You have two trading signals. Alone, one runs at Sharpe $0.8$ , the other at Sharpe $0.6$ . Their returns are uncorrelated.

Optimally combined, what Sharpe can the pair achieve? What if they were highly correlated instead?

Accepted Answer

When you blend uncorrelated return streams with optimal weights, the squared Sharpes add. The combined Sharpe is S{	ext{combined}} = \sqrt{S1^2 + S2^2} = \sqrt{0.8^2 + 0.6^2} = \sqrt{0.64 + 0.36} = \sqrt{1.0} = 1.0. Two ordinary signals become a materially better one, Sharpe 1.0 beats either standalone. The reason is diversification: combining independent edges keeps the returns adding while the risks only partially reinforce (variances add, but the cross term is zero because they're uncorrelated), so the return-to-risk ratio improves. The intuition generalizes: n uncorrelated signals of equ…

Combining two uncorrelated signals, Sharpes add in quadrature

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