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A post hoc test on the Sharpe ratio

Steven Pav

Abstract

We describe a post hoc test for the Sharpe ratio, analogous to Tukey's test for pairwise equality of means. The test can be applied after rejection of the hypothesis that all population Signal-Noise ratios are equal. The test is applicable under a simple correlation structure among asset returns. Simulations indicate the test maintains nominal type I rate under a wide range of conditions and is moderately powerful under reasonable alternatives.

A post hoc test on the Sharpe ratio

Abstract

We describe a post hoc test for the Sharpe ratio, analogous to Tukey's test for pairwise equality of means. The test can be applied after rejection of the hypothesis that all population Signal-Noise ratios are equal. The test is applicable under a simple correlation structure among asset returns. Simulations indicate the test maintains nominal type I rate under a wide range of conditions and is moderately powerful under reasonable alternatives.

Paper Structure

This paper contains 18 sections, 13 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The test
  3. Bonferroni Cutoff:
  4. Arbitrary correlation structure:
  5. Examples
  6. Simulations under the null
  7. Basic Simulations
  8. Varying ${{n}_{}}$ and ${{k}_{}}$:
  9. Varying $\rho$:
  10. Feasible Estimator, Varying $\rho$:
  11. Feasible Estimator, Misspecified Model, Varying $\rho$:
  12. Simulations under the alternative
  13. Under the alternative, one good:
  14. Under the alternative, half good:
  15. Real Assets
  16. ...and 3 more sections

Figures (10)

  • Figure 1: The quantiles of the range of Sharpe ratio from 5,000 simulations are plotted against a transformed Tukey distribution, $\sqrt{(1-\rho)/{{n}_{}}\xspace}\, {q}^{}_{{\cdot},{{{k}_{}}\xspace},{\infty}}\xspace$. The points show little deviation from the plotted $y=x$ line.
  • Figure 2: The computed p-values from 5,000 simulations are plotted against a uniform law, visually confirming that the p-values are nearly uniform. Simulations use the exact $\rho$ to compute the p-values via ptukey. We transform the p-values and plot $\left| 2p - 1 \right|\xspace$ in log-log space to emphasize the tails. The points show little deviation from the plotted $y=x$ line.
  • Figure 3: The empirical type I rate at the nominal 0.05 level is plotted against the number of days in each simulation, for different values of ${{k}_{}}\xspace$. Simulations use the exact $\rho$ to perform the hypothesis test. The two facets show rejection rates under the $df={{n}_{}}\xspace-1$ and $df=\infty$ cutoffs. When using the $df=\infty$ cutoff, the test is anti-conservative for the "large ${{k}_{}}$, small ${{n}_{}}$" case, but the nominal rate is nearly achieved for the $df={{n}_{}}\xspace-1$ cutoff.
  • Figure 4: The empirical type I rate at the nominal 0.05 level is plotted against the correlation, $\rho$. Simulations use the exact $\rho$ to perform the hypothesis test, and the $df={{n}_{}}\xspace-1$ cutoff.
  • Figure 5: The empirical type I rate at the nominal 0.05 level is plotted against the correlation, $\rho$. Simulations use an estimated$\rho$ to perform the hypothesis test, and the $df={{n}_{}}\xspace-1$ cutoff.
  • ...and 5 more figures