Sortino ratio and Calmar ratio: measuring downside risk
The Sortino ratio works like the Sharpe ratio, but only penalizes downside fluctuations. The Calmar ratio goes even further and divides the return by the largest loss. Both fit the question that really preoccupies investors better: how much return do I get for the risk of losing money?
Sortino ratio: the formula
Sortino ratio = (return p.a. − MAR) / downside deviation p.a.- MAR
- minimum acceptable return: the threshold below which a return counts as a loss (often 0% or the risk-free rate)
- downside deviation
- square root of the average of the squared shortfalls; months above the threshold count as 0
Downside deviation = √( (1 / n) × Σ min(r_t − MAR, 0)² ) × √12The Sortino ratio is higher than the Sharpe ratio because the good months do not increase the downside deviation. The more skewed the returns, the further apart the two are. That is why the Sortino ratio suits strategies with an asymmetric profile, such as trend following.
A common calculation mistake
Many spreadsheets and websites divide the sum of the squared shortfalls only by the number of negative months instead of by all months. This makes the downside deviation larger and the Sortino ratio smaller. In the example, there are four negative months:
Same Sharpe ratio, different risk
How much the shape of the returns matters is shown by three stylized strategies, each with an average monthly return of 1% over twelve months (minimum acceptable return and interest rate 0%):
| Strategy (monthly returns) | Sharpe p.a. | Sortino p.a. | Skewness |
|---|---|---|---|
| A: alternating +3% and −1% | 1.66 | 4.90 | 0.00 |
| B: −0.5% ten times, +8.5% twice | 0.99 | 7.59 | 1.79 |
| C: +1.9% ten times, −3.5% twice | 1.65 | 2.42 | −1.79 |
Figures from the table above, all three with an average monthly return of 1%. By Sharpe, A and C are level and B trails; by Sortino, B is clearly ahead and C far behind.
A and C have almost the same Sharpe ratio. But C earns steadily and loses rarely, and then a lot. That is the profile of selling insurance. Its Sortino ratio is halved as a result. B looks worst by Sharpe because the two big winning months inflate the standard deviation; by Sortino, B is clearly ahead. Which view is right depends on what you perceive as risk. For most investors, rare large losses are worse than rare large gains. That is exactly what the Sortino ratio is made for.
Where the idea comes from
In the early 1990s, Frank Sortino and his co-authors argued that investors understand risk not as fluctuation but as the danger of missing a goal. Sortino and van der Meer (1991) therefore proposed downside risk below a minimum acceptable return as a risk measure; Sortino and Price (1994) turned it into the metric that bears his name today. The minimum acceptable return is deliberately left to your choice: 0% for “no loss”, inflation for “preserving purchasing power”, or the return a withdrawal plan requires.
Behind the idea lies a general observation: returns are often not symmetrical. Option sellers earn steadily for a long time and rarely lose a lot (negative skewness); trend followers often lose a little and rarely win a lot (positive skewness). The standard deviation treats both the same; the downside deviation does not.
Calmar ratio: return versus drawdown
Calmar ratio = CAGR / |maximum drawdown|A Calmar ratio of 0.30 means: one year of average return covers just under a third of the worst decline. It was originally calculated over 36 months; over long backtests it depends heavily on a single crisis, see maximum drawdown. Related measures divide by the average of several large drawdowns (Sterling ratio) or by the square root of their squares (Burke ratio); Bacon (2008) brings these variants together.
In practice, the Calmar ratio answers a very tangible question: how many years of average return would the worst slump have cost? At 0.30, it is just over three years. For an investor who wants to withdraw money in the coming years, that is often more telling than any standard deviation. Its drawback is its dependence on the period: a Calmar ratio over ten years without a major crisis is hardly comparable with one over 30 years.
Gain/loss ratio and hit rate
Two simple additional figures help you understand a Sortino ratio. The hit rate is the share of positive periods; the gain/loss ratio is the average gain in positive periods divided by the average loss in negative ones. In the example, 8 of 12 months were positive (67%), averaging +1.69%, and the four negative ones averaged −2.50%.
For advanced readers: lower partial moments
The downside deviation is a special case of lower partial moments (LPM): the average of the shortfalls below the minimum acceptable return, raised to the power n. With n = 2 you get the downside variance, with n = 1 the average shortfall that underlies the Omega ratio, and with n = 0 the probability of missing the threshold. The higher n, the more heavily large losses count. The Kappa measures divide the excess return by the nth root of the LPM of order n. The Sortino ratio is Kappa 2.
LPM_n = (1 / T) × Σ max(MAR − r_t, 0)^nWhy should investors give downside risk particular weight? Research on the pricing of skewness provides a reason. Harvey and Siddique (2000) found that assets with unfavorable, systematic skewness have to offer a higher return:
“Systematic skewness is economically important and commands a risk premium”Their estimate of this premium averages 3.60% a year. So if you buy a strategy with a high Sharpe ratio but strongly negative skewness, you may only be getting your return as compensation for rare large losses. The Sortino ratio makes this partly visible, but does not capture it fully. For that, it is worth looking at skewness and kurtosis and value at risk.
Statistically, the Sortino ratio is harder to estimate than the Sharpe ratio: it relies only on the negative periods, i.e. on less data. With two or three years of monthly data, it rests on a handful of months and is correspondingly uncertain.
Which metric when?
| Metric | Divides by | Good for |
|---|---|---|
| Sharpe | Volatility | Comparing similar portfolios, optimization |
| Sortino | Downside fluctuation | Strategies with skewed returns |
| Calmar | Largest decline | Worst-case questions, withdrawal plans |
| Gain/loss ratio | Avg. losing month | How big are good months versus bad ones? |
None of these figures is the right one. Sharpe and Sortino describe the typical ups and downs, Calmar the single worst moment. Together with the maximum drawdown and other measures such as Omega and the Ulcer Index from the article risk metrics, they give a picture that no single metric can provide.
How Portfolio Lab calculates it
Portfolio Lab uses the risk-free return per period as the minimum acceptable return, i.e. the same time-varying money market rate as for the Sharpe ratio, so that the two metrics are comparable. The downside deviation is the square root of the mean of the squared negative excess returns over all periods, times √12 (or √252 for daily data); the numerator is the mean excess return times 12 or 252. The Calmar ratio is the CAGR divided by the absolute value of the maximum drawdown over the whole period, and the gain/loss ratio is the mean positive divided by the mean negative period value. Optimization can also maximize the Sortino ratio directly. Details in the metrics methodology.
Frequently asked questions
Which minimum acceptable return should I use?
0% for the question “loss or not”, the risk-free rate for a comparison with the Sharpe ratio. What matters is using the same threshold for all the portfolios you compare.
Why is my Sortino ratio much higher than my Sharpe ratio?
Because the good months do not count in the downside deviation. With symmetrical returns and a minimum acceptable return close to the average, the Sortino ratio is roughly 1.4 times the Sharpe ratio; with right-skewed returns, it is even higher.
What is the gain/loss ratio?
The average gain in positive months divided by the average loss in negative months. Values above 1 mean: good months are larger on average than bad ones.
What is a good Sortino ratio?
It is usually above the Sharpe ratio; over long periods, values of around 0.5 to 1 are good for broad portfolios. More important than a fixed value is a comparison under the same conditions.
Can I optimize a portfolio for the Sortino ratio?
Yes, Portfolio Lab's Optimization offers the objective “Max. Sortino ratio”. It favors mixes with few and small losing months. Like any optimization on historical data, it tends to fit itself to the past.
What is a good Calmar ratio?
For long-term equity portfolios with one major crisis in the period, 0.1 to 0.3 is usual. Values above 1 come almost exclusively from short periods without a deep slump.
What is the Omega ratio?
It divides the sum of gains above a threshold by the sum of losses below it, and so uses the whole distribution, not just the mean and the downside fluctuation. Values above 1 mean: more was earned above the threshold than was lost below it. More in the risk metrics article.
Literature
- Bacon, Carl R. (2008): Practical Portfolio Performance Measurement and Attribution. Wiley, 2nd edition.
- Harvey, Campbell R.; Siddique, Akhtar (2000): Conditional Skewness in Asset Pricing Tests. The Journal of Finance 55 (3), pp. 1263–1295. doi:10.1111/0022-1082.00247
- Sortino, Frank A.; van der Meer, Robert (1991): Downside risk. The Journal of Portfolio Management 17 (4), pp. 27–31. doi:10.3905/jpm.1991.409343
- Sortino, Frank A.; Price, Lee N. (1994): Performance Measurement in a Downside Risk Framework. The Journal of Investing 3 (3), pp. 59–64. doi:10.3905/joi.3.3.59
This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.