A strategy returns 1.2% a month with a 4% monthly standard deviation. What is its annualised Sharpe ratio under the usual convention, and under what condition does that number overstate the truth?

A strategy returns 1.2% a month with a 4% monthly standard deviation. What is its annualised Sharpe ratio under the usual convention, and under what condition does that number overstate the truth?

Approach: Form the monthly ratio, then ask what has to be true of the returns for the square root of time scaling to be exact.

1.04. The monthly Sharpe ratio is 1.2/4 = 0.3 and the square root of time convention multiplies by sqrt(12) = 3.464, giving 1.039. The scaling holds when monthly returns are independent and identically distributed, since the mean of a sum of twelve grows with 12 while the standard deviation grows only with sqrt(12). Positive autocorrelation breaks it, because the variance of the twelve month sum then exceeds twelve monthly variances and the honest annual figure is lower than the scaled one. Smoothed marks on illiquid holdings, systematic option selling and any book with serially correlated P&L fail the condition in the same direction, so the reported Sharpe ratio is biased upward exactly where the strategy is hardest to exit.

Follow-up: What does square root of time scaling do to a Sharpe ratio when returns are negatively autocorrelated?

Key concepts: sharpe ratio, square root of time, independence, autocorrelation.