Some Improved Estimators of Maximum Squared Sharpe Ratio

By assuming multivariate normal distribution of excess returns, we find that the sample maximum squared Sharpe ratio (MSR) has a significant upward bias. We then construct estimators for MSR based on Bayes estimation and unbiased estimation of the squared slope of the asymptote to the minimum variance frontier (ψ2). While the often used unbiased estimator may lead to unreasonable negative estimates in the case of finite sample, Bayes estimators will never produce negative values as long as the prior is bounded below by zero although it has a larger bias. We also design a mixed estimator by combining the Bayes estimator with the unbiased estimator. We show by simulation that the new mixed estimator performs as good as the unbiased estimator in terms of bias and root mean square errors and it is always positive. The mixed estimators are particularly useful in trend analysis when MSR is very low, for example, during crisis or depression time. While negative or zero estimates from unbiased estimator are not admissible, Bayes and mixed estimators can provide more information.