Sharpe ratio calculator

Two ways to use the Sharpe ratio calculator

From a return series. Enter one return per period in percent, oldest first, such as twelve monthly returns from a fund factsheet or a spreadsheet column. The calculator finds the mean and the sample standard deviation itself and shows the table of deviations.

From mean, SD and risk-free rate. If you already have the average return and the standard deviation for one period, switch “Start from” and enter them directly. They are used as entered; nothing is recomputed.

In both modes the risk-free rate is entered as an annual rate and divided by the number of periods in a year (3% a year is 0.25% a month). Some analysts compound instead, (1 + Rf)^1/12 − 1; at ordinary interest rates the two differ in the third decimal place of the ratio.

The Sharpe ratio formula

S = (R̄p − Rf) ÷ σp

R̄p is the mean return of the portfolio over the period, Rf the risk-free return for the same period and σp the standard deviation of the portfolio’s returns. The numerator is the excess return; the denominator is the risk taken to earn it. William Sharpe set out the ratio in this excess-return form in “The Sharpe Ratio” (Journal of Portfolio Management, 1994).

The standard deviation is where most of the work is. This calculator uses the sample standard deviation, which divides the sum of squared deviations by n − 1:

σp = √( Σ(r − r̄)² ÷ (n − 1) )

Strictly, Sharpe’s definition uses the standard deviation of the excess returns r − Rf. With a constant risk-free rate, as here, subtracting it from every return does not change the spread, so the two are the same number.

Annualizing a Sharpe ratio

A mean return grows in proportion to time, while a standard deviation grows with the square root of time when returns are independent. So a per-period Sharpe ratio is annualized by multiplying by the square root of the number of periods in a year:

S(annual) = S(period) × √k k = 252 daily, 52 weekly, 12 monthly, 1 annual

Worked example: twelve monthly returns

The calculator starts with these monthly returns (%) and a 3% annual risk-free rate:

2.1, −1.3, 3.4, 0.8, −0.5, 1.9, 2.7, −2.2, 1.5, 0.6, 3.1, −0.4

1. Mean: the returns sum to 11.7, so r̄ = 11.7 ÷ 12 = 0.975% a month.
2. Deviations from the mean run from −3.175 to 2.425; their squares sum to 35.2625 (%²).
3. Sample SD: √(35.2625 ÷ 11) = √3.2057 = 1.7904% a month.
4. Risk-free rate per month: 3% ÷ 12 = 0.25%.
5. Monthly Sharpe ratio: (0.975 − 0.25) ÷ 1.7904 = 0.725 ÷ 1.7904 = 0.4049.
6. Annualized: 0.4049 × √12 = 0.4049 × 3.4641 = 1.403.

The same answer comes from annualizing the inputs first: a mean of 0.975 × 12 = 11.7% a year, an SD of 1.7904 × √12 = 6.2023% a year, and (11.7 − 3) ÷ 6.2023 = 1.403.

Reading a Sharpe ratio

A higher ratio means more excess return for each unit of volatility. The number is only useful next to another Sharpe ratio worked out the same way: the same period of history, the same return frequency, the same risk-free rate and the same annualizing factor. A fund measured over a calm three years will usually show a higher ratio than the same fund measured over a period that includes a crash.

There is no universal “good” value. Thresholds such as 1 or 2 that appear in articles are conventions, and they do not transfer between asset classes or between daily and monthly data. A Sharpe ratio is a way to rank alternatives, not a pass mark.

Limits of the Sharpe ratio

• Non-normal returns. The standard deviation treats a large gain and a large loss alike and says nothing about skew or fat tails. Strategies that sell options can show a high Sharpe ratio until the rare large loss arrives.
• Estimation error in σ. Twelve or even sixty returns give a noisy estimate of the standard deviation, and the ratio inherits that noise. Small differences between two Sharpe ratios are rarely meaningful.
• Autocorrelation. The √k rule assumes returns are independent. Smoothed returns, common in private assets and some hedge funds, understate the SD and overstate the annualized ratio.

For the risk of a mix of assets rather than a single return series, see the portfolio standard deviation calculator, and for how the standard deviation is used across finance, the guide to standard deviation in finance.