  Returns distributions are heavy-tailed across asset classes. In this note, I
examine the implications of this well-known stylized fact for the joint
statistics of performance (absolute return) and Sharpe ratio (risk-adjusted
return). Using both synthetic and real data, I show that, all other things
being equal, the investments with the best in-sample performance are never
associated with the best in-sample Sharpe ratios (and vice versa). This
counter-intuitive effect is unrelated to the risk-return tradeoff familiar from
portfolio theory: it is, rather, a consequence of asymptotic correlations
between the sample mean and sample standard deviation of heavy-tailed
variables. In addition to its large sample noise, this non-monotonic
association of the Sharpe ratio with performance puts into question its status
as the gold standard metric of investment quality.
