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

Paper: arXiv 2302.08829

Authors: Matteo Smerlak

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.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The note provides a highly insightful, counter-intuitive theoretical and empirical examination of the joint distribution between total return and the Sharpe ratio under heavy-tailed distributions. It effectively highlights a structural flaw in using the Sharpe ratio as a primary ranking metric, combining solid mathematical derivations of asymptotic sample statistics with empirical validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze joint distribution of performance & Sharpe ratio in heavy-tailed returns] --> B[Data Inputs: Synthetic & Real Multi-Asset Return Data]
    B --> C[Methodology: Evaluate sample mean vs. sample standard deviation correlations]
    C --> D[Computational Analysis: Model asymptotic behavior under heavy-tailed distributions]
    D --> E[Key Finding 1: Top in-sample performers NEVER have the best Sharpe ratios]
    D --> F[Key Finding 2: Non-monotonic association driven by asymptotic correlations, not risk-return tradeoff]
    E & F --> G[Outcome: Questions Sharpe ratio as the gold standard metric due to sample noise]