Sharpe Ratio

Formula

r_f,period = (1 + r_f,annual) ^ (1/P) − 1
excess_t   = R_t − r_f,period

Sharpe = mean(excess) / std(excess, ddof=1) · sqrt(P)

P = 252 by default, 242 for China A-shares, 365 for a crypto-style calendar.
std = 0 → the ratio is undefined (returned as NaN).

Conventions that change the number

• The performance calculator uses this compounding convention, sample standard deviation, and the same sqrt(P) annualisation. The glossary function is written to match it.
• Some published Sharpes subtract r_f/P, or divide by the population standard deviation, or leave the ratio un-annualised. Those are different statistics. Quote P and the risk-free treatment next to the number.
• Downside-only volatility is the Sortino ratio. Active return over tracking error is the information ratio.

Pitfalls

• It is a sample moment. A short window, or a window that happens to contain one regime, moves the ratio a lot. It does not carry a promise about the next window.
• Volatility is not the risk you may care about. Two series with the same Sharpe can have very different drawdowns and tail losses. See also VaR and expected shortfall.
• Illiquid and smoothed returns look better. A series that only updates occasionally understates standard deviation and overstates Sharpe. The same is true of a backtest that ignores costs.
• Annualising assumes i.i.d. periods. Strong autocorrelation makes sqrt(P) the wrong scaling. The function still multiplies by sqrt(P); the assumption is yours to check.

Python reference implementation

Requires numpy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download sharpe-ratio.py · All snippets

import numpy as np

def sharpe_ratio(returns, rf_annual=0.0, periods=252) -> float:
    """Annualised Sharpe ratio.

    rf_annual is compounded to a per-period rate: (1 + rf_annual) ** (1/periods) - 1.
    Volatility is the sample standard deviation of excess returns. Zero volatility
    returns NaN. This matches /tools/performance-analytics/.
    """
    r = np.asarray(returns, dtype=float)
    r = r[np.isfinite(r)]
    if r.size < 2:
        raise ValueError("need at least two finite returns")
    if periods <= 0:
        raise ValueError("periods must be positive")
    rf = (1.0 + rf_annual) ** (1.0 / periods) - 1.0
    excess = r - rf
    sd = float(excess.std(ddof=1))
    if sd == 0.0:
        return float("nan")
    return float(excess.mean() / sd * np.sqrt(periods))

FAQ

Why compound the risk-free rate instead of dividing by 252?

An annual rate of 2 percent is not 2/252 per day under compound interest. The per-period rate that compounds back to 2 percent is (1.02)^(1/252) − 1, which is slightly less than 2/252. The difference is small at low rates and is the convention the performance calculator on this site uses.

Is a Sharpe ratio of 1 good?

It means the sample's mean excess return equalled its annualised volatility. Whether that is attractive depends on the sample, the costs that were left out, and the drawdowns. The ratio is not a threshold for taking a position.