Ex Ante Comparison of Maximum Sharpe Ratios and Incremental Variable Testing

A typical decision problem faced by investors is whether or not to include additional asset classes or return drivers into their portfolios. In the context of classical portfolio theory and in the presence of a riskless asset, the relevant criterion is a significant increase in the maximum Sharpe ratio (the Sharpe ratio of the tangency portfolio) achievable when enlarging the investment opportunity set. Although not all assumptions of classical portfolio theory are always satisfied by empirical return distributions, the Sharpe ratio is widely used in practice as a risk-adjusted performance measure and a benchmark. Most existing results on the distribution of the maximum Sharpe ratio depend on the assumption of multivariate normal return distributions. We generalize results from the literature to provide an analytical representation of the distribution of the difference between two maximum Sharpe ratios for much less restrictive distributional assumptions, both with and without short sales. For the long-only case, we provide conditions that allow us to check whether any local optima that may be encountered indeed correspond to the global optimum. Knowing the distribution of the difference enables us to test ex ante whether or not the inclusion of additional variables leads to a significant improvement in the maximum Sharpe ratio. High estimation uncertainty regarding means of asset returns makes it difficult to distinguish statistically between tangency portfolios at conventional significance levels