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

TL;DR

This paper investigates how heavy-tailed return distributions affect the joint behavior of absolute performance and the Sharpe ratio, revealing that top-performing investments often do not have the highest risk-adjusted returns due to statistical properties.

Contribution

It demonstrates that heavy tails cause a non-monotonic relationship between performance and Sharpe ratio, challenging the reliability of the Sharpe ratio as a measure of investment quality.

Findings

Best in-sample performance does not imply best Sharpe ratio.

Heavy tails induce asymptotic correlations affecting metrics.

Sharpe ratio's reliability as a performance metric is questioned.

Abstract

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.