Monthly returns have a Sharpe ratio of 0.3 and first order autocorrelation of 0.3 with an AR(1) structure. What is the correctly annualised Sharpe ratio?

Monthly returns have a Sharpe ratio of 0.3 and first order autocorrelation of 0.3 with an AR(1) structure. What is the correctly annualised Sharpe ratio?

Approach: Annualising scales the mean by 12 and the standard deviation by the square root of the variance of a twelve month sum. Write that variance with the autocovariance terms included.

0.78. The variance of a twelve month sum is sigma^2*(12 + 2*sum over k from 1 to 11 of (12 - k)*rho^k). With rho = 0.3 the sum is 4.531, so the bracket is 12 + 9.06 = 21.06 rather than 12 and the correct scaling factor is 12/sqrt(21.06) = 2.615 in place of sqrt(12) = 3.464. The annualised Sharpe ratio is 0.3*2.615 = 0.78, which is 75% of the 1.04 the naive convention reports. Return smoothing from stale or model marks is the usual source of positive autocorrelation, so the correction bites hardest on credit, private holdings and anything priced off a matrix rather than a trade. Negative autocorrelation, as in a fast reversal book, pushes the factor above sqrt(12).

Follow-up: What autocorrelation would make the correction raise an annualised Sharpe ratio of 1.0 to 1.2?

Key concepts: autocorrelation, sharpe ratio, annualisation, return smoothing.