Comparing Sharpes with unequal track lengths

Question

Manager A has an annualized Sharpe of 1.8 over 8 years; Manager B has an annualized Sharpe of 0.8 over 6 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 (note the different track lengths). With \operatorname{SE}(SR) \approx \sqrt{(1 + 	frac{1}{2}SR^2)/T}: \operatorname{SE}A = \sqrt{\frac{1 + 1.62}{8}} = \sqrt{0.3275} \approx 0.57,\qquad \operatorname{SE}B = \sqrt{\frac{1 + 0.32}{6}} = \sqrt{0.22} \approx 0.47. Standard error of the difference. \operatorname{SE}{	ext{diff}} = \sqrt{0.3275 + 0.22} = \sqrt{0.5475} \approx 0.74. Test the gap. z = \frac{1.8 - 0.8}{0.74} \approx 1.35. Verdict. A z of 1.35 is below 2, so the one-point gap is not significant, even though A has both the higher Sharpe and the longer record. Un…

Comparing Sharpes with unequal track lengths

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