When is a Sharpe gap finally significant? 15-year records

Question

Manager A has an annualized Sharpe of 1.5 over 15 years; Manager B has an annualized Sharpe of 0.3 over 15 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 = 15: \operatorname{SE}A = \sqrt{\frac{1 + 1.125}{15}} = \sqrt{0.1417} \approx 0.38,\qquad \operatorname{SE}B = \sqrt{\frac{1 + 0.045}{15}} = \sqrt{0.0697} \approx 0.26. Standard error of the difference. \operatorname{SE}{	ext{diff}} = \sqrt{0.1417 + 0.0697} = \sqrt{0.2113} \approx 0.46. Test the gap. z = \frac{1.5 - 0.3}{0.46} \approx 2.61. Verdict. A z of 2.61 is above 2, so this difference is statistically significant. Two things made it so: a long 15-year record (which shrank both error bars…

When is a Sharpe gap finally significant? 15-year records

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