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

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