Sharpe ratio: formula, a worked S&P 500 example and the limits

The Sharpe ratio is the average return an investment earned above a risk-free rate, divided by the standard deviation of that excess return. It measures how much extra return each unit of volatility bought. Over the ten years from 2016 to 2025, the S&P 500 earned an average of 13.50 percentage points a year more than three-month Treasury bills, with a standard deviation of 15.45 points, which gives a Sharpe ratio of 0.87.

The result moves with the risk-free rate chosen, the return frequency, the way monthly figures are annualized and the dates covered. The US stock market as a whole scored 0.82 over the same ten years measured with monthly data, and 1.34 over the three years to December 2025.

The Sharpe ratio formula, term by term

The excess return. Each period's return minus that period's risk-free return, subtracted period by period. The three-month bill paid 0.04% on average in 2021 and 5.28% in 2023, so a single ten-year average rate would misstate the excess return in both years. Morningstar, which publishes Sharpe ratios for funds, uses the three-month Treasury bill as the risk-free benchmark for US portfolios and subtracts it month by month. The bill's rate is published daily by the Federal Reserve and was 4.08% on September 25, 2026.

The average. An arithmetic mean of the excess returns, the sum divided by the number of periods. A compounded (geometric) average comes out lower.

The standard deviation. How widely the excess returns spread around their own average. The usual estimate is the sample standard deviation, which divides the sum of squared deviations by the number of periods minus one. In Excel, with ten years of excess returns in cells C1 to C10, the whole ratio is =AVERAGE(C1:C10)/STDEV(C1:C10), the same two-function formula William Sharpe gave for a spreadsheet in 1994.

Because the risk-free return has little variance of its own, the standard deviation of the excess return is close to the standard deviation of the fund's return. In the example below they differ by 0.15 points.

Worked example: the S&P 500 from 2016 to 2025

The data are the annual total returns of the S&P 500 with dividends reinvested and the average three-month Treasury bill rate for each year, from the historical returns dataset maintained at NYU Stern, whose bill series comes from the Federal Reserve's H.15 release. An S&P 500 index fund would have earned these returns minus its expense ratio.

1. Average excess return. The ten excess returns add up to 134.96. Divided by 10, the average is 13.50 percentage points a year.
2. Deviations from the average. Subtract 13.50 from each year. The largest are 2022 at -33.63, 2018 at -19.70 and 2019 at +15.60.
3. Squared deviations. Squaring and adding all ten gives 2,149.5. The single year 2022 contributes 1,130.9 of that, more than half.
4. Variance and standard deviation. 2,149.5 divided by 9 (ten years minus one) is 238.8. Its square root is 15.45.
5. Sharpe ratio. 13.50 divided by 15.45 is 0.87.

In step three, one bad year supplied most of the measured risk, so the ratio for 2016 to 2025 depends heavily on whether 2022 sits inside the window. Drop 2022 and use 2016 to 2021 plus 2023 to 2025, and the ratio rises to about 1.6.

Annualizing a monthly Sharpe ratio with the square root of 12

Most published Sharpe ratios are computed from monthly returns and then converted to an annual figure. The conversion multiplies the monthly ratio by the square root of 12, about 3.46. The logic is that the average return grows in proportion to time while the standard deviation grows with the square root of time, so the ratio grows with the square root of time as well. Morningstar annualizes this way, and Sharpe recommended measuring over short periods such as months and then annualizing, because monthly data contain twelve times as many observations.

Applied to the US stock market over the same ten years, using the 120 monthly excess returns of all NYSE, NYSE American and Nasdaq stocks from the Kenneth French data library:

• Average monthly excess return: 1.079%
• Standard deviation of monthly excess returns: 4.543%
• Monthly Sharpe ratio: 1.079 ÷ 4.543 = 0.238
• Annualized: 0.238 × 3.464 = 0.82

The monthly route gives 0.82 on the whole US market, against 0.87 for the annual route on the S&P 500, with different indexes and different return intervals.

The square-root rule holds exactly only when returns are independent from one month to the next. When they are serially correlated, the correct multiplier changes. For returns that follow a first-order autoregressive process, the annual Sharpe ratio is 2.88 times the monthly ratio when the monthly autocorrelation is +20%, and 4.17 times when it is -20%, against 3.46 for independent returns. Positive autocorrelation makes volatility grow faster than the square root of time, so multiplying by 3.46 overstates the annual ratio. The US market's first-order monthly autocorrelation over 2016 to 2025 was -0.13, inside the range of plus or minus 0.18 that chance alone produces 95% of the time with 120 observations, so the standard conversion is a reasonable choice for this series.

How precise is a measured Sharpe ratio?

A historical Sharpe ratio is an estimate, and it carries a standard error. When returns are independent and identically distributed, the standard error of a Sharpe ratio estimated from T observations is approximately the square root of (1 + SR² ÷ 2) ÷ T, where SR is the ratio measured at the same frequency as the data. The formula comes from Andrew Lo's 2002 paper "The Statistics of Sharpe Ratios" in the Financial Analysts Journal.

For the S&P 500 example, with SR = 0.87 and T = 10 years:

• SR² ÷ 2 = 0.38
• (1 + 0.38) ÷ 10 = 0.138
• Standard error = square root of 0.138 = 0.37
• 95% confidence interval = 0.87 ± 1.96 × 0.37, from 0.14 to 1.60

With ten annual observations, a true ratio of 0.3 and one of 1.5 both sit inside that interval. In Lo's own sample, the Vanguard 500 Index fund had an annual Sharpe ratio of 0.85 with a 95% interval of 0.33 to 1.36, even with monthly data.

The t-statistic for the average excess return equals the historical Sharpe ratio times the square root of the number of observations, a relationship Sharpe spelled out in 1994. For the example, 0.87 × √10 gives 2.76.

Precision falls further when many strategies are tried and only the best is reported. Among 1,000 independent backtests of strategies with no edge, each run on ten years of monthly data, the best one is expected to show an annualized Sharpe ratio of about 1.0 by chance alone, using the expected-maximum formula behind the deflated Sharpe ratio. David Bailey and Marcos López de Prado published that measure in the Journal of Portfolio Management in 2014. It adjusts the hurdle for the number of trials and for skewed and fat-tailed returns.

What is a good Sharpe ratio?

A widely repeated scale calls a ratio between 1 and 1.99 good, 2 to 2.99 very good and 3 or more outstanding, and treats readings above 2 as a possible sign of leverage. Measured long-run results sit well below that scale:

Berkshire Hathaway's 0.79 was the highest of any US stock or mutual fund with at least 40 years of history over 1976 to 2017. It came from an average excess return of 18.6% a year on volatility of 23.5%, produced by leverage of about 1.7 to 1 applied to cheap, safe, high-quality stocks, the same traits described in our article on factor investing.

The same US market that produced 0.82 over ten years produced 1.34 over the three years to December 2025, because those three years had strong returns and no drawdown as deep as 2022's. Morningstar's headline figure uses the most recent 36 months, so a fund page can show a ratio above 1 after three strong years, as the whole US market did at the end of 2025.

A comparison holds only between funds measured over the same dates, with the same risk-free rate and the same return frequency, and ideally within one category. Adding assets can raise return and still lower the ratio: in our article on the 60/40 portfolio, adding commodities, private equity or real estate to the mix raised returns and raised risk by more.

William Sharpe's 1966 and 1994 papers

William F. Sharpe introduced the measure in "Mutual Fund Performance", published in the Journal of Business in January 1966, and named it the reward-to-variability ratio. By the early 1990s textbooks called it the Sharpe index or the Sharpe measure and Morningstar called it the Sharpe ratio, the term Sharpe himself adopted in 1994. He shared the 1990 Nobel prize in economic sciences with Harry Markowitz and Merton Miller for pioneering work in financial economics.

His 1994 article in the Journal of Portfolio Management, "The Sharpe Ratio", restated the measure more generally.

Ex-ante and ex-post Sharpe ratios

The ex-ante Sharpe ratio uses expected returns and a predicted standard deviation, and it is the version that belongs in a decision about the future. The ex-post Sharpe ratio uses realized returns over a historical period, as in the worked example. Practitioners compute the ex-post version and use it as a forecast of the ex-ante one, an assumption Sharpe described as subject to serious question when the historical figures are unadjusted.

A benchmark in place of the risk-free rate

In 1966 the comparison asset was a riskless security. In 1994 Sharpe defined the ratio on any differential return, the fund's return minus a benchmark's, including a benchmark portfolio with the same investment style as the fund. When the benchmark is an index, the same calculation is usually called the information ratio.

Leverage leaves the ratio unchanged

Borrowing at the risk-free rate to double a position doubles both the excess return and its standard deviation, so the ratio stays the same. Sharpe called this scale independence. When borrowing costs more than the risk-free rate, the extra interest comes out of the numerator and the ratio falls.

The ratio ignores correlation

Picking the fund with the highest Sharpe ratio is sound when it will be the only risky holding, or when the candidates have similar correlations with everything else the investor owns. When correlations differ, a fund with a lower ratio can improve the whole portfolio more, because it diversifies better.

Where the Sharpe ratio misleads

The ratio summarizes returns with two numbers, the mean and the standard deviation. That is a complete description only when returns are roughly normal and independent over time, which Sharpe acknowledged in 1994 by setting other moments of the distribution outside the scope of the measure.

Option selling and fat left tails

A strategy that sells out-of-the-money options collects a small premium most months and suffers a large loss occasionally, a fat left tail in its return distribution. Until the loss arrives, its returns look steady and its Sharpe ratio looks high. The strategy that maximizes an expected Sharpe ratio without any skill sells out-of-the-money calls and out-of-the-money puts in uneven amounts, which produces regular modest profits and a large exposure to extreme negative events. Integral Investment Management, a Dallas hedge fund, promoted what it called the highest Sharpe ratio in the industry in 1998. Its returns appear to have come in part from short positions in out-of-the-money puts on US stock indexes, and the Art Institute of Chicago lost nearly 43 million dollars invested with it.

Smoothed returns from illiquid assets

When a fund holds assets that rarely trade, its monthly values come from appraisals, models or stale prices. Reported returns then move more smoothly than the underlying economic returns, volatility is understated and the Sharpe ratio is overstated. Across 908 hedge funds in the TASS database, the degree of smoothing varied widely from one strategy style to another. The smoothing shows up as positive autocorrelation in monthly returns, which is the case where the square root of 12 fails. In Lo's sample, a mortgage-backed securities fund had an annualized Sharpe ratio of 4.03 using the standard conversion and 2.44 once serial correlation was accounted for, an overstatement of 65%. A convertible and option arbitrage fund fell from 4.35 to 2.99. A risk arbitrage fund with negative autocorrelation rose from 3.25 to 3.83.

A Sharpe ratio computed on a hedge fund's gross returns ignores the management and incentive fees an investor pays, which our article on hedge fund fees works through, so fund comparisons should use net returns.

Negative Sharpe ratios

When the average excess return is below zero, more volatility makes the ratio less negative. Two funds that each lagged Treasury bills by 2 points a year score -0.40 with 5% volatility and -0.10 with 20% volatility. Ranked by Sharpe ratio, the more volatile fund comes first.

Sortino, Treynor, information and Calmar ratios compared

Each alternative keeps the idea of return per unit of risk and changes what counts as risk.

Sortino ratio. Frank Sortino's measure replaces the standard deviation with downside deviation, so years above the target add nothing to risk. With the Treasury bill as the target in the S&P 500 example, only 2018 (-6.20) and 2022 (-20.13) fall short. Their squares, 38.44 and 405.22, add up to 443.66. Divided by all ten years that is 44.37, and its square root is 6.66. The Sortino ratio is 13.50 ÷ 6.66 = 2.03, more than twice the Sharpe ratio, because the strong years of 2019 and 2021 no longer count as risk.

Treynor ratio. Jack Treynor divided the excess return by beta, the sensitivity to the market, which suits a fund that is one part of a diversified portfolio. Berkshire Hathaway earned 18.6% a year above bills with a market beta of 0.69, a Treynor ratio of 18.6 ÷ 0.69 = 27.0%. The market by definition has a beta of 1, so its Treynor ratio equals its 7.5% excess return.

Information ratio. The excess return over a benchmark divided by tracking error, the standard deviation of that difference. It measures a manager's active decisions after stripping out the market. Adjusted for market exposure, Berkshire's information ratio was 0.64.

Calmar ratio. It divides the annualized return by the largest peak-to-trough loss, often measured over 36 months, and is used mostly to evaluate commodity trading advisers and hedge funds. For the US stock market over 2016 to 2025, the compound annual return of 14.72% divided by the 24.84% fall between the end of December 2021 and the end of September 2022 gives 0.59. Over the 36 months to December 2025, with a 23.02% annual return and a largest drawdown of 9.12%, it was 2.52.

Questions about Sharpe ratio values

Is 0.5 a good Sharpe ratio?

Over long periods it is about what a broad stock index has delivered. The S&P 500 had an annual Sharpe ratio of about 0.43 from 1928 to 2025, and the median US equity mutual fund with at least ten years of history had 0.39 over 1976 to 2017. A fund that holds 0.5 over ten years or more has done better than most. Over a three-year window the comparison changes: the whole US stock market scored 1.34 in the three years to December 2025.

What does a Sharpe ratio of 2.5 mean?

The average excess return was 2.5 times the standard deviation of returns. For an annual figure, a fund with 10% volatility would have earned 25 percentage points a year above Treasury bills. Few funds hold that level for long: the highest Sharpe ratio among US equity mutual funds with at least 30 years of history over 1976 to 2017 was 0.93. A reading of 2.5 over a long period calls for a look at how the returns were produced, including option selling, leverage, illiquid holdings and the length of the track record.

Can a Sharpe ratio be negative?

Yes. It is negative whenever the investment returned less than the risk-free rate over the period measured. A negative ratio is hard to rank: of two funds that each lagged Treasury bills by 2 points, the one with 20% volatility scores -0.10 and the one with 5% volatility scores -0.40, so the riskier fund looks better.

Is a higher Sharpe ratio always better?

Between two funds of the same type with similar correlations to the rest of a portfolio, and measured over the same period with the same risk-free rate, the higher ratio shows more excess return per unit of volatility. Across different strategies the comparison breaks down, because the ratio ignores skewness, fat tails, smoothed prices and correlation with other holdings.