A strategy has a true annual Sharpe ratio of 0.5. Using the standard error of an annualised Sharpe estimate over T years of returns, how many years are needed to reject a zero Sharpe at the 5% level one-sided with 80% power, and does sampling daily instead of monthly help?

A strategy has a true annual Sharpe ratio of 0.5. Using the standard error of an annualised Sharpe estimate over T years of returns, how many years are needed to reject a zero Sharpe at the 5% level one-sided with 80% power, and does sampling daily instead of monthly help?

Approach: Use the asymptotic standard error of the Sharpe estimate, then require the true value to clear the critical value by the amount that leaves 80% in the rejection region.

28. The annualised Sharpe estimate has approximate standard error sqrt((1 + SR^2/2)/T), which at SR = 0.5 is sqrt(1.125/T). A one-sided 5% test with 80% power needs the true value to sit z_{0.95} + z_{0.80} standard errors above zero, which is 1.645 + 0.84 = 2.485. So 0.5 must reach 2.485*sqrt(1.125/T), giving sqrt(T) of 5.27 and T of 27.8, so 28 years. Sampling daily rather than monthly does not help, because the standard error depends on the calendar length T and not on the number of observations inside it; finer sampling estimates the mean no better. A two-year backtest reporting a Sharpe of 0.5 therefore carries a t-statistic near 0.7, and if that backtest was selected from hundreds of candidates the honest evidence is weaker still.

Follow-up: How does the required T change when the returns have an autocorrelation of 0.2 at lag one month?

Key concepts: Sharpe ratio, statistical power, standard error, sampling frequency.