Bias and consistency of the maximum Sharpe ratio

We show that the maximum Sharpe ratio obtained via the Markowitz optimization procedure from a sample of returns on a number of risky assets is, under commonly satisfied assumptions, biased upwards for the population value. Thus investment advice, decisions and assessments based on the estimated Sharpe ratio will be overly optimistic. The bias in the estimator is shown theoretically and illustrated using a data set of Spiders and iShares. We obtain bounds on the difference between the sample maximum Sharpe ratio and its population counterpart and show that the sample estimator is consistent for the population value; thus the bias disappears asymptotically under some reasonable assumptions. However, the bias can be significant infinite samples and can persist even in very large samples. We demonstrate this with simulations based on portfolios formed from normally and t-distributed returns. As expected, the over-optimistic risk-return tradeoff predicted by the procedure is not reflected in corresponding good out-of-sample portfolio performance of the Spiders and iShares.