A Sharpe of 2.0 versus 1.0 over four years: real gap?

Question

Strategy A has an annualized Sharpe of 2.0 over 4 years; Strategy B has an annualized Sharpe of 1.0 over 4 years. They are independent, roughly normal.

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

Accepted Answer

Each Sharpe's error bar. With \operatorname{SE}(SR) \approx \sqrt{(1 + 	frac{1}{2}SR^2)/T} and T = 4: \operatorname{SE}A = \sqrt{\frac{1 + 2}{4}} = \sqrt{0.75} \approx 0.87,\qquad \operatorname{SE}B = \sqrt{\frac{1 + 0.5}{4}} = \sqrt{0.375} \approx 0.61. Standard error of the difference. Variances add for independent estimates: \operatorname{SE}{	ext{diff}} = \sqrt{0.75 + 0.375} = \sqrt{1.125} \approx 1.06. Test the gap. z = \frac{2.0 - 1.0}{1.06} \approx 0.94. Verdict. A z of 0.94 is well under 2, so a full one-point Sharpe gap over four years is not statistically significant. Notice that …

A Sharpe of 2.0 versus 1.0 over four years: real gap?

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