Probabilistic Sharpe Ratio (PSR) and Backtest Overfitting

Introduction

The Sharpe ratio is one of the most abused statistics in quantitative finance. Two strategies can have the same Sharpe ratio even if one is estimated from a short, skewed, fat-tailed sample and the other from a long, well-behaved history. A raw Sharpe number says nothing about statistical confidence, non-normality, or multiple testing.

The Probabilistic Sharpe Ratio (PSR) estimates the probability that an observed Sharpe ratio exceeds a benchmark \(SR^*\), while adjusting for skewness and kurtosis:

Here, \(T\) is the sample length, \(\gamma_3\) is skewness, \(\gamma_4\) is kurtosis, and \(\Phi\) is the standard normal CDF. The denominator inflates uncertainty when returns are asymmetric or fat-tailed — exactly what classical Sharpe ignores.

Python Implementation

import numpy as np
import pandas as pd
from scipy.stats import skew, kurtosis, norm

def probabilistic_sharpe_ratio(returns, sr_benchmark=0.0, periods_per_year=252):
    r = pd.Series(returns).dropna()
    sr_hat = np.sqrt(periods_per_year) * r.mean() / r.std(ddof=1)

    T  = len(r)
    g3 = skew(r, bias=False)
    g4 = kurtosis(r, fisher=False, bias=False)  # Pearson kurtosis

    numerator   = (sr_hat - sr_benchmark) * np.sqrt(T - 1)
    denominator = np.sqrt(1 - g3 * sr_hat + ((g4 - 1) / 4.0) * sr_hat**2)
    z = numerator / denominator

    return {
        "sharpe":   sr_hat,
        "psr":      norm.cdf(z),
        "skew":     g3,
        "kurtosis": g4,
        "z_score":  z
    }

# Example
np.random.seed(42)
rets = np.random.normal(0.0005, 0.01, 500)
print(probabilistic_sharpe_ratio(rets, sr_benchmark=1.0))

Ranking strategies by PSR instead of raw Sharpe forces the strategy to "earn" its Sharpe under a stronger evidentiary standard. It penalizes short histories, punishes ugly tail behavior, and gives you a way to compare estimated skill against a benchmark such as \(SR^* = 1\). Standard Sharpe is descriptive. PSR is inferential.