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backtest_statistics

Concept Overview

Turns a return or equity series into the handful of statistics a strategy is actually judged on: annualised Sharpe, information ratio, the drawdown and time-under-water profile, average holding period, bet concentration, and the multiple-testing corrections — probabilistic and deflated Sharpe — that say whether a Sharpe is real. Those corrections are why this module exists rather than a two-line Sharpe helper: AFML Chapter 14’s point is that a Sharpe reported without the number of trials behind it is uninterpretable.

When to Use

Reach for it after a backtest run, at model-selection time, and again in production monitoring. Use deflated_sharpe_ratio whenever the strategy is the survivor of a search — a grid, a parameter sweep, a family of variants — and pass the trial count honestly; sharpe_ratio alone flatters every one of them. Note that drawdown_and_time_under_water consumes a timestamped equity curve, not a return vector, and that every annualisation constant must match your bar frequency.

Mathematical Foundations

Sharpe Ratio

$\mathrm{SR}=\frac{\mu-r_f}{\sigma}\sqrt{n}$

where $\mu$ and $\sigma$ are the mean and standard deviation of the per-bar returns, $r_f$ the per-bar risk-free rate, and $n$ the number of bars per year (entries_per_year) — the annualisation constant must match your bar frequency.

Information Ratio

$\mathrm{IR}=\frac{\mu-r_b}{\sigma_{(r-r_b)}}$

where $r_b$ is the benchmark return and $\sigma_{(r-r_b)}$ the tracking error, i.e. the standard deviation of the excess return series.

Probabilistic Sharpe Ratio

$\mathrm{PSR}(\mathrm{SR}^*)=Z\left[\frac{(\widehat{\mathrm{SR}}-\mathrm{SR}^*)\sqrt{T-1}}{\sqrt{1-\hat\gamma_3\widehat{\mathrm{SR}}+\frac{\hat\gamma_4-1}{4}\widehat{\mathrm{SR}}^2}}\right]$

where $Z[\cdot]$ is the standard normal CDF, $\widehat{\mathrm{SR}}$ the observed (non-annualised) Sharpe ratio, $\mathrm{SR}^*$ the benchmark being tested against, $T$ the number of returns, and $\hat\gamma_3,\hat\gamma_4$ the sample skewness and kurtosis. Non-normal returns lower the confidence a given Sharpe deserves.

Deflated Sharpe Ratio

$\mathrm{DSR}=\mathrm{PSR}(\mathrm{SR}_0),\qquad \mathrm{SR}_0=\sqrt{V[\{\widehat{\mathrm{SR}}_n\}]}\left((1-\gamma)Z^{-1}\!\left[1-\tfrac{1}{N}\right]+\gamma Z^{-1}\!\left[1-\tfrac{e^{-1}}{N}\right]\right)$

where $N$ is the number of strategy variants you tried, $V[\{\widehat{\mathrm{SR}}_n\}]$ the variance of their Sharpe ratios, $\gamma\approx0.5772$ the Euler-Mascheroni constant, and $Z^{-1}$ the normal quantile function. $\mathrm{SR}_0$ is the Sharpe you would expect the best of $N$ independent worthless strategies to post, so DSR is the PSR measured against that bar instead of against zero. deflated_sharpe_ratio accepts either the raw $\{\widehat{\mathrm{SR}}_n\}$ or the $(\text{sd}, N)$ pair via estimates_param.

Usage Examples

Rust

Compute Sharpe and drawdown

use chrono::{Duration, NaiveDateTime};
use openquant::backtest_statistics::{drawdown_and_time_under_water, sharpe_ratio};

let returns = vec![0.01, -0.005, 0.007, -0.002, 0.003];
let sharpe = sharpe_ratio(&returns, 252.0, 0.0);

// Drawdown and time-under-water are computed on a *timestamped equity curve*,
// not on the return series: the function needs the timestamps to measure how
// long each high-water mark went un-recovered.
let t0 = NaiveDateTime::parse_from_str("2024-01-02 00:00:00", "%Y-%m-%d %H:%M:%S")?;
let mut equity = 1.0;
let curve: Vec<(NaiveDateTime, f64)> = returns
    .iter()
    .enumerate()
    .map(|(i, r)| {
        equity *= 1.0 + r;
        (t0 + Duration::days(i as i64), equity)
    })
    .collect();

// dollars = false reports each drawdown as a fraction of its high-water mark.
let (drawdowns, time_under_water) = drawdown_and_time_under_water(&curve, false);
println!("sharpe={sharpe:.3} drawdowns={drawdowns:?} tuw={time_under_water:?}");

API Reference

Python API

backtest_stats.sharpe_ratio
backtest_stats.information_ratio
backtest_stats.probabilistic_sharpe_ratio
backtest_stats.deflated_sharpe_ratio
backtest_stats.minimum_track_record_length
backtest_stats.timing_of_flattening_and_flips
backtest_stats.average_holding_period
backtest_stats.bets_concentration
backtest_stats.all_bets_concentration
backtest_stats.drawdown_and_time_under_water

Rust API

sharpe_ratio
deflated_sharpe_ratio
probabilistic_sharpe_ratio
drawdown_and_time_under_water
average_holding_period

Risk Notes and Caveats

Use annualization constants consistent with your bar frequency.
Deflated Sharpe is useful when strategy mining many variants.