Skip to content
CyclepediaThe trading encyclopedia
ITENESPT-BRFRDE
Learning path Silver Repeatable method

Sharpe ratio

Mean differential return per unit of standard deviation, conditional on the benchmark, frequency, window and annualisation convention.

On this page

Who this is for — Readers comparing return with variability who want to avoid rankings built from incompatible series, benchmarks or annualisation methods.

The Sharpe ratio expresses mean differential return per unit of standard deviation of that same differential. William F. Sharpe called the original measure the reward-to-variability ratio and in 1994 set out a general form based on the difference between a fund and its relevant benchmark.

For periodic observations:

Dₜ = Rₚ,ₜ − Rᵦ,ₜ;   SR = mean(Dₜ) / sd(Dₜ)

When the benchmark is a duration- and currency-consistent risk-free asset, the numerator is commonly called excess return. If the benchmark is risky, the measure concerns differential return and should not automatically be labelled risk-free excess return.

Sharpe ratio: excess return per unit of volatility Numerator and denominator must share frequency, period and conventions Sharpe ratio: excess return per unit of volatility Numerator and denominator must share frequency, period and conventions SR = mean(rₚ − r_f) / sd(rₚ − r_f) Excess return Portfolio return minus arisk-free rate aligned to theperiod. Volatility Standard deviation of the sameobserved excess returns. Annualisation Scaling depends on assumptionsabout the series and actualfrequency. Interpretation No universal cut-off replacesdistribution, sample andcomparison. Cyclepedia · educational diagram: state conventions, period and data
The ratio compresses mean and dispersion into one number; benchmark, sample and distribution shape remain necessary for interpretation.

Ex ante and ex post

An ex-ante Sharpe ratio uses expected differential return and risk. An ex-post ratio uses the observed sample mean and standard deviation. The latter describes history and does not automatically become a forecast.

A report should state period and observations; frequency and calendar; currency and FX; price or total return; gross or net basis and included costs; benchmark and source; sample or population standard deviation; annualisation; and whether the series is live, simulated or composite. A net daily five-year ratio and a gross monthly twelve-month ratio do not form a meaningful ranking.

Annualisation and autocorrelation

A widely used approximation is:

SR_annual ≈ SR_period × √m

It requires assumptions allowing means and variances to scale appropriately. With autocorrelation, smoothing, overlapping returns or changing volatility, multiplication by √12, √252 or √365 can distort the result.

Andrew Lo demonstrates that Sharpe-ratio statistics and annualisation depend on serial correlation. Higher frequency is not automatically more informative: stale prices and illiquid strategies can create an artificially smooth series.

Why there are no universal thresholds

Rules such as “below 1 poor, above 2 excellent” omit asset class and benchmark; frequency and window; leverage, liquidity and capacity; cost, fee and tax; selection bias and number of trials; skew, tails and drawdown; and estimation uncertainty.

A high ratio may arise from a short sample, smoothing, a payoff with frequent small gains and rare losses, or selection of the best among many trials. Negative Sharpe ratios also make simple “higher is always better” intuition problematic when leverage or benchmark changes. The ratio is a coordinate, not a universal grade.

Sharpe, Sortino and information ratio

Measure Numerator Denominator Question
Sharpe mean differential return standard deviation of the same differential How much mean return per unit of variability?
Sortino mean return above a target downside deviation relative to that target How much mean return per unit of target shortfall?
Information ratio mean active return tracking error How much relative return per unit of active risk?

The measures are not interchangeable. Changing denominator changes the definition of risk; changing benchmark changes the analysed series.

Example

A hypothetical strategy has mean monthly differential return of 0.40% and sample standard deviation of 2%:

SR_monthly = 0.40 / 2 = 0.20

Under square-root-of-time assumptions, SR_annual ≈ 0.20 × √12 ≈ 0.693. Serial correlation requires a calculation that includes covariance across periods. A gross series does not describe the net investor experience. The example illustrates the formula and does not classify 0.693 as acceptable or unacceptable.

Statistical and process limitations

Because Sharpe uses mean and standard deviation, it is sensitive to outliers, treats upside and downside dispersion symmetrically, may hide left-tail risk, depends on the benchmark, and is undefined when standard deviation is zero. It does not separate luck, factors, leverage and skill; correct data snooping or survivorship bias; or measure capacity, liquidity and drawdown.

Selecting one strategy from many backtests increases the chance of a favourable metric. Trial records, out-of-sample data, costs and uncertainty intervals are needed for interpretation.

Common error — Inferring Sharpe ratio from CAGR and maximum drawdown. Its denominator is the standard deviation of a periodic differential- return series; drawdown and volatility describe different properties.

Sources

  • William F. Sharpe, The Sharpe Ratio, The Journal of Portfolio Management (1994) — general formulation, ex-ante and ex-post versions and differential return.
  • William F. Sharpe, Mutual Fund Performance, The Journal of Business (1966) — original reward-to-variability measure.
  • Andrew W. Lo, The Statistics of Sharpe Ratios, Financial Analysts Journal (2002) — sampling distribution, autocorrelation and annualisation.
  • CFA Institute, Investment Risk and Performance — comparison of Sharpe, tracking error, information ratio, alpha and beta.
  • GIPS Standards, Handbook for Firms — disclosure of supplemental measures and identification of the risk-free rate.
Suggest an edit My proposals