Two strategies are run on separate histories and report Sharpe ratios of 1.0 and 0.5. How many years are needed before that gap is two standard errors wide, and what changes if they are run on the same period with a return correlation of 0.5?

Two strategies are run on separate histories and report Sharpe ratios of 1.0 and 0.5. How many years are needed before that gap is two standard errors wide, and what changes if they are run on the same period with a return correlation of 0.5?

Approach: The difference of two independent estimates has variance equal to the sum of the variances. Then redo it with a covariance term for the paired case.

32 years. Each annualised Sharpe estimate has a standard error near 1/sqrt(T) in years, so an independent difference has standard error sqrt(2/T). Requiring 0.5 = 2*sqrt(2/T) gives sqrt(2/T) = 0.25 and T = 32 years of sample for each. Run on the same window with a return correlation of 0.5, the variance of the difference carries a factor (1 - rho) = 0.5, the standard error becomes sqrt(1/T) and the requirement halves to 16 years. That is the whole argument for comparing managers on a common period and testing the difference series directly rather than differencing two separately estimated ratios. Any ranking built on three years of data separates nothing inside half a Sharpe point.

Follow-up: How would you test whether a manager's Sharpe ratio has changed between the first and second half of a track record?

Key concepts: sharpe ratio, standard error, return correlation, sample length.