The Sharpe and Sortino ratios
How the Sharpe ratio and Sortino ratio are calculated, a worked example of downside deviation, and why both can mislead on short or cherry-picked samples.
Part of the Statistics for Investors and Traders track on Agenticks. About 12 minutes, written for a intermediate reader.
Read first: Volatility and standard deviation, Distributions and fat tails in stock returns.
Last reviewed .
Risk notice. Backtested and hypothetical results are simulated: no trades were placed, the rules were applied to past data with the benefit of hindsight, and they may not reflect liquidity, slippage or every cost. Past results, real or simulated, do not predict future results.
Key takeaways
- Sharpe ratio = (return - risk-free rate) / standard deviation of returns. It treats upside and downside swings alike.
- The Sortino ratio divides by downside deviation instead, so strategies whose swings are mostly gains score higher.
- With a 10% return, a 4% risk-free rate, 15% volatility and 8% downside deviation, Sharpe is 0.40 and Sortino is 0.75.
- Both ratios depend on the period measured, assume roughly symmetric returns and can be inflated by testing many variations.
A 12% return sounds better than an 8% return, until you learn the 12% came with stomach-turning swings and the 8% barely wobbled. Risk-adjusted measures try to put return and risk into one number. The two most common are the Sharpe ratio and the Sortino ratio. William Sharpe proposed his ratio in 1966 as a "reward-to-variability" measure and set out its modern form in 1994. Frank Sortino and Lee Price later described a version that measures risk only by returns below a minimum acceptable return, on the reasoning that swings above that target are not what investors mean by risk.
Same top, different bottom
Sharpe ratio = (return - risk-free rate) / standard deviation of returns. Sortino ratio = (return - risk-free rate or target) / downside deviation. The top is the excess return, what was earned above a near-riskless alternative such as short-term government bills. Sharpe divides it by all the volatility; Sortino divides it by the volatility of the shortfalls only.
Worked example: Sharpe and Sortino from annual figures
A hypothetical strategy returned 10% a year with 15% annual volatility and 8% downside deviation, while the risk-free rate was 4%. Illustrative assumptions.
| Annual return | 10% |
|---|---|
| Risk-free rate | 4% |
| Excess return (10% - 4%) | 6% |
| Sharpe ratio (6 / 15) | 0.40 |
| Sortino ratio (6 / 8) | 0.75 |
| Same 6% excess return | Sharpe 0.40, Sortino 0.75 |
Hypothetical figures, not results from any real strategy. The calculator below uses the same formulas and defaults.
Downside deviation works like standard deviation, except only the shortfalls count. Take four monthly returns of +5%, -3%, +6% and -4%, with a target of 0%. Treat every month at or above the target as 0 and keep the shortfalls: 0, -3, 0, -4. Square them (0, 9, 0, 16), add them (25), divide by the number of months, all four of them (6.25), and take the square root: 2.5%. The ordinary sample standard deviation of the same four months is about 5.23%, because it also counts the big up months as risk. So with the same average return of 1% a month, the monthly Sortino ratio (1 / 2.5 = 0.40) is more than twice the monthly Sharpe ratio (1 / 5.23 = 0.191). When most of a strategy's swings are upside, Sortino gives it credit for that. When the swings are mostly on the downside, it gives no such credit.
Sharpe and Sortino calculator
Enter annual figures. The defaults are the hypothetical strategy above, assumptions for illustration rather than typical returns for any market.
Worked example: the higher return is not always the better ratio
Two hypothetical strategies, measured over the same years, with a 4% risk-free rate. Illustrative assumptions.
| Strategy A return and volatility | 15% and 25% |
|---|---|
| Strategy A Sharpe ratio ((15 - 4) / 25 = 11 / 25) | 0.44 |
| Strategy B return and volatility | 9% and 10% |
| Strategy B Sharpe ratio ((9 - 4) / 10 = 5 / 10) | 0.50 |
| Return earned per unit of volatility | B is higher, even though A returned more |
A higher ratio says the return came with less bumpiness. It does not say which strategy suits a given person.
A made more money in this example, and it also swung two and a half times as hard to do it. The ratio asks how much return each strategy delivered for each unit of volatility it put you through, and on that measure B did better. Whether A's extra return was worth its swings is a judgement the ratio cannot make, which is why it works best as one input among several.
Ratios computed from monthly or daily returns are usually scaled to a year by multiplying by the square root of the number of periods: about 3.464 for monthly data and about 15.87 for daily data. The four months above give an annualized Sharpe ratio of about 0.191 x 3.464 = 0.66 and an annualized Sortino ratio of 0.40 x 3.464 = 1.39. This step quietly assumes that each period's return is unrelated to the last, the same assumption behind the square root of time in the volatility lesson.
Both ratios are useful for comparing like with like. Both are also easy to misread. They assume returns are roughly symmetric. A strategy that collects small gains most months and takes a rare large loss, such as one that sells insurance-like options, can show a high Sharpe ratio for years before the loss arrives. Standard deviation barely sees a fat tail that has not happened yet. They depend on the period. Sharpe himself noted that the ratio is not independent of the time period over which it is measured. The same strategy measured from a market low looks better than when measured from a high. They are estimates from a sample. Andrew Lo showed that when monthly returns are correlated with each other, simply multiplying by the square root of 12 can overstate an annual Sharpe ratio, by as much as 65% in his hedge fund example. With few returns the estimate is noisy, as the sample size lesson explains. They can be gamed. Smoothing reported returns, picking a flattering start date, or testing hundreds of variations and reporting the best all inflate the number. The deflated Sharpe ratio of David Bailey and Marcos Lopez de Prado adjusts for how many strategies were tried and for returns that are not normally distributed. The overfitting lesson covers the trial-and-error side.
- Sharpe ratio
- Divides excess return by the standard deviation of all returns, Penalizes a big up month as much as a big down month
- Sortino ratio
- Counts only returns below a target as risk, Uses downside deviation in the bottom of the fraction
- Both
- Subtracts the risk-free rate or a target from the return, Can look excellent before a rare large loss shows up
A ratio is a summary, not a verdict
Read a Sharpe or Sortino ratio next to the drawdown, the number of trades, the period it covers and how many variations were tried. A high ratio on a short, hand-picked sample is weaker evidence than a modest one on a long, honest sample.
You can read a Sharpe ratio honestly
You can now compute the Sharpe and Sortino ratios, build a downside deviation by hand, and name the ways both ratios can flatter a strategy.
Check your understanding
- A strategy returned 12% with 20% volatility while the risk-free rate was 2%. What is its Sharpe ratio?
- 0.50. Excess return is 12% - 2% = 10%, and 10 / 20 = 0.50.
- Two strategies have the same Sharpe ratio. A's big swings are mostly up months; B's are mostly down months. Which has the higher Sortino ratio?
- A. Sortino counts only shortfalls as risk. A's swings are mostly upside, so its downside deviation is smaller.
- Why might a strategy that sells insurance-like options show a high Sharpe ratio for years and then lose heavily?
- Its rare large loss had not happened yet, so its standard deviation understated its real risk. Small steady gains give a low standard deviation until the fat-tail event arrives.
- What does the deflated Sharpe ratio adjust for?
- How many strategy variations were tried, and returns that are not normally distributed. Testing many variations and keeping the best inflates the Sharpe ratio by luck alone. The deflated version corrects for that.
Common questions
- What is a good Sharpe ratio?
- There is no universal cut-off. The ratio depends on the period, the data frequency and how many strategies were tried before this one was picked, so it is most useful for comparing similar strategies measured the same way over the same period.
- What is the difference between the Sharpe and Sortino ratios?
- Both divide excess return by a measure of risk. The Sharpe ratio uses the standard deviation of all returns; the Sortino ratio uses downside deviation, which counts only returns below a target, so it does not penalize large up moves.
Terms defined in this lesson
Continue
- Previous: Drawdown math and recovery
- Next: Regression basics: beta and R squared
- Volatility and standard deviation
- Distributions and fat tails in stock returns
- Sample size and why ten trades mean nothing
- Overfitting and curve-fitting (the cardinal sin)
- Statistics for Investors and Traders track
- All Agenticks lessons
- Trading glossary
Sources
- Sharpe, W. F. (1994). The Sharpe ratio. The Journal of Portfolio Management, 21(1), 49-58.
- Sortino, F. A., & Price, L. N. (1994). Performance measurement in a downside risk framework. The Journal of Investing, 3(3), 59-64.
- Lo, A. W. (2002). The statistics of Sharpe ratios. Financial Analysts Journal, 58(4), 36-52.
- Bailey, D. H., & Lopez de Prado, M. (2014). The deflated Sharpe ratio: Correcting for selection bias, backtest overfitting, and non-normality. The Journal of Portfolio Management, 40(5), 94-107.
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