Great year, bad Sharpe? A note on the joint distribution of performance and risk-adjusted return

Returns distributions are heavy-tailed across asset classes. In this note, I examine the implications of this stylized fact for the joint statistics of performance and risk-adjusted return. Using both synthetic and real data, I show that the Sharpe ratio does not increase monotonically with performance: in a sample of price trajectories, the largest Sharpe ratios are associated with suboptimal mean returns; conversely, the best performance never corresponds to the largest Sharpe ratios. This counter-intuitive effect is unrelated to the risk-return tradeoff familiar from portfolio theory: it is a consequence of asymptotic correlations between the sample mean and sample standard deviation of heavy-tailed variables (which are absent in the Gaussian case). In addition to its very large sample noise, the non-monotonic association of the Sharpe ratio with performance puts into question its status as the gold standard of investment quality.