Is a Sharpe of 1.5 really better than a Sharpe of 0.9?

Question

Manager A has an annualized Sharpe of 1.5 over 5 years; Manager B has an annualized Sharpe of 0.9 over 5 years. They trade unrelated markets, so treat their returns as independent, roughly normal.

Is the difference in Sharpe ratios statistically significant at the 5% level?

Accepted Answer

Each Sharpe has its own error bar. Using the standard approximation, the standard error of an annualized Sharpe over T years is \operatorname{SE}(SR) \approx \sqrt{(1 + 	frac{1}{2}SR^2)/T}. Compute each: \operatorname{SE}A = \sqrt{\frac{1 + 	frac{1}{2}(1.5)^2}{5}} = \sqrt{0.425} \approx 0.65,\qquad \operatorname{SE}B = \sqrt{\frac{1 + 	frac{1}{2}(0.9)^2}{5}} = \sqrt{0.281} \approx 0.53. Standard error of the difference. For two independent estimates, the variances add, so the standard error of the gap is the square root of the sum of squared standard errors: \operatorname{SE}{	ext{diff}} …

Is a Sharpe of 1.5 really better than a Sharpe of 0.9?

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