How long to tell a Sharpe 1.5 from a Sharpe 1.0?

Question

Manager A has posted an annualized Sharpe of $1.5$ over five years; manager B, $1.0$ over five years, on an independent book.

Can you conclude A is genuinely better than B? How many years would it take to distinguish them?

Accepted Answer

The standard errors. For i.i.d. normal returns the Sharpe estimate has \operatorname{SE}(\widehat{SR}) \approx \sqrt{(1 + 	frac12 SR^2)/T}. At T = 5: \operatorname{SE}A \approx \sqrt{	frac{1 + 1.125}{5}} = \sqrt{0.425} \approx 0.65, \qquad \operatorname{SE}B \approx \sqrt{	frac{1 + 0.5}{5}} = \sqrt{0.30} \approx 0.55. The difference. With independent books the standard error of the difference combines in quadrature: \operatorname{SE}{\Delta} = \sqrt{\operatorname{SE}A^2 + \operatorname{SE}B^2} \approx \sqrt{0.425 + 0.30} = \sqrt{0.725} \approx 0.85. The observed gap is 1.5 - 1.0 = 0.5, so …

How long to tell a Sharpe 1.5 from a Sharpe 1.0?

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