How Risk-Adjusted Returns works

Scope

Reports standard risk-adjusted performance metrics on a daily simple-returns series, optionally with a benchmark column. Annualized via √252 trading days.

Input format

date,strategy,benchmark
2024-01-02,0.0012,0.0008
2024-01-03,-0.0005,-0.0002
...

The benchmark column is optional. If present it enables beta / alpha / tracking error / information ratio.

Formulas

Sharpe (annualized): (mean(excess) / stdev(excess)) × √252 where excess = r - rf_daily.

Sortino (annualized): (mean(excess) / downside_stdev) × √252. Downside stdev uses only negative excess returns: √(Σ min(0, r-rf)² / N).

Omega: Σ max(0, r_excess) / Σ max(0, -r_excess). Threshold = 0 (above rf). Returns positive infinity if there are no negative excess returns.

Calmar: ann_return / max_drawdown. Max drawdown computed on cumulative wealth starting at 1.

Beta: cov(r, b) / var(b).

Alpha (CAPM, annualized): R_p - R_f - β · (R_b - R_f) where R_p and R_b are the annualized returns of strategy and benchmark, R_f is the annualized risk-free rate.

Tracking error (annualized): stdev(r - b) × √252.

Information ratio: (mean(r - b) × √252) / tracking_error.

Skewness: standardized third moment.

Excess kurtosis: standardized fourth moment − 3.

Assumptions + limitations

1. Daily frequency. Annualization constant 252 assumes daily bars on a trading calendar. For weekly/monthly data, adjust externally.
2. Simple returns. Not log returns. For strategies reported in log returns, convert before upload.
3. Independent observations. Serial correlation inflates the apparent Sharpe. For strategies with material autocorrelation, expect the annualized number to overstate live realisation by 10–30%.
4. Normal-ish tails assumed for some intuitions. Skew + kurtosis flag deviation; Sharpe itself does not penalize non-normality. Use Sortino + Calmar + kurtosis together.
5. Risk-free rate is a constant annual percentage; de-annualized to daily for the subtraction. A time-varying rf requires preprocessing.
6. Benchmark alignment. The strategy and benchmark columns are expected to share the same dates. Missing or mis-aligned rows produce biased beta.
7. Sample size. Minimum 20 observations enforced. Stable estimates typically require 2+ years of daily data.

References

• Sharpe, W. F. (1966). "Mutual Fund Performance." Journal of Business 39(1).
• Sortino, F. A., & van der Meer, R. (1991). "Downside Risk." Journal of Portfolio Management 17(4).
• Young, T. W. (1991). "Calmar Ratio: A Smoother Tool." Futures (magazine).
• Keating, C., & Shadwick, W. F. (2002). "A Universal Performance Measure." Journal of Performance Measurement 6(3).
• Bailey, D. H., & Lopez de Prado, M. (2014). "The Deflated Sharpe Ratio." Journal of Portfolio Management 40(5).

Connects to

• Backtest Overfitting Score — after Sharpe looks good, check whether it survives PBO and Deflated Sharpe.
• Fractional Kelly Sizer — once risk metrics are calibrated, size accordingly.

Changelog

• 2026-04-20 — Initial release.