Risk metrics beyond the Sharpe ratio: Treynor, Calmar, Omega, Ulcer Index and more
The Sharpe ratio is the best-known metric for return per unit of risk, but it is not the only one. Each of the other risk metrics answers a slightly different question: how much return was there per unit of market risk (Treynor)? Per percent of the deepest decline (Calmar)? How do gains compare with losses (Omega, gain/loss)? How painful were the declines overall (Ulcer Index)? This article works through all of them with one example and shows when each one makes sense.
The shared example
All the metrics use the same series of twelve monthly returns from a mixed portfolio. The risk-free rate is 2% a year, and the beta against the market is 0.9.
Treynor ratio: return per unit of market risk
Treynor = (R − r_f) / β- R
- portfolio return p.a.
- β
- beta against the market, see beta and alpha
The Treynor ratio, named after Jack Treynor, divides the excess return not by the total volatility but by beta, i.e. only by the market risk. In the example: (7.44 − 2) / 0.9 = 6.05. It suits a portfolio that is only one building block within a large, well-diversified total portfolio: there, only the contribution to market risk counts, because the remaining risk is offset by the other building blocks. For a single portfolio that makes up your entire wealth, the Sharpe ratio is more appropriate. With a beta close to zero, the Treynor ratio becomes uselessly large.
Calmar ratio: return per unit of the deepest decline
Calmar = annual return / |maximum drawdown|- maximum drawdown
- largest decline from a peak, see maximum drawdown
The Calmar ratio comes from the world of managed futures (introduced by Terry Young) and was originally calculated over the last 36 months. In the example: 7.44 / 4.10 = 1.82. It answers a question investors ask intuitively: how many years of return set against the worst loss? Its weakness: it depends on a single event and on the length of the period (see below). Related variants such as the Sterling ratio divide by the average of several large drawdowns instead of only the largest.
Ulcer Index: how deep and how long
Ulcer Index = √(mean of the squared drawdowns)
Martin ratio = (R − r_f) / Ulcer Index- drawdown
- percentage distance from the previous peak in each period (0 at the peak)
The Ulcer Index of Martin and McCann (1989) measures not only the lowest point but every phase below the peak: deep declines count heavily because they are squared, and long periods under water count because of the many periods involved. It thus sums up the “ulcer” an investor experiences when looking at their portfolio. In the example, the portfolio is below its peak in seven of twelve months; the Ulcer Index is 1.93%, and the Martin ratio is (7.44 − 2) / 1.93 = 2.83.
| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Drawdown in % | 0 | −1.5 | 0 | −4.1 | −2.37 | −1.79 | −3.95 | −0.59 | 0 | −0.8 | 0 | 0 |
Omega and gain/loss: gains against losses
Ω(L) = Σ max(rₜ − L, 0) / Σ max(L − rₜ, 0)
Gain/loss = mean of the positive returns / |mean of the negative returns|- L
- minimum acceptable return (threshold), often 0 or the risk-free rate per period
The Omega ratio, introduced by Con Keating and William Shadwick and described in detail alongside other metrics in Bacon (2008), divides the sum of all gains above a threshold by the sum of all losses below it. It uses the whole distribution, not just the mean and standard deviation. Skewness and fat tails are therefore taken into account (see skewness and kurtosis). In the example, with a threshold of 0: 16.1 / 8.6 = 1.87. A value above 1 means that the gains outweigh the losses.
The gain/loss ratio, by contrast, compares the average winning month with the average losing month: 2.01 / 2.15 = 0.94. Here it is below 1, even though the portfolio made a clear gain, because there were twice as many winning months as losing months. Gain/loss therefore describes the size of the swings, Omega the overall result. Together with the share of positive months, the two give a good picture. In the example, two thirds of the months were positive; that more than makes up for the slightly smaller winning months. In asset pricing theory, Bernardo and Ledoit (2000) used a related gain-loss ratio to derive price bounds for assets in incomplete markets.
M²: return at the same level of risk
M² = Sharpe ratio × σ_benchmark + r_f- σ_benchmark
- volatility of the benchmark index
Modigliani and Modigliani (1997) translated the Sharpe ratio into a return: what return would the portfolio have delivered if it had been brought to the volatility of the benchmark index through leverage or by adding money market holdings? With a Sharpe ratio of (7.44 − 2) / 8.05 = 0.68 (more precisely 0.677) and an index volatility of 15%, the result is 0.677 × 15 + 2 = 12.15%. The advantage: percentages can be compared directly with the index return, whereas Sharpe ratios remain abstract.
Upside and downside capture
Upside capture = geometric mean of the portfolio in months with a rising index / geometric mean of the index in those months- Downside capture
- the same for months with a falling index
Capture ratios describe how a fund behaves relative to its index in rising and falling phases. A defensive fund has both values below 100%; it is good if the downside value is clearly lower than the upside value. Like all metrics of this kind, it needs many months to be meaningful.
Which metric for what?
| Metric | Measures | Well suited for |
|---|---|---|
| Sharpe | return per unit of total volatility | whole portfolios, comparing strategies |
| Sortino | return per unit of downside volatility | investments with asymmetric returns |
| Treynor | return per unit of market risk (beta) | building blocks of a large portfolio |
| Information ratio | excess return per unit of deviation from the index | active funds against their benchmark index |
| Calmar | return per unit of maximum drawdown | strategies focused on limiting losses |
| Ulcer / Martin | depth and duration of declines | investors who cope badly with long loss periods |
| Omega | gains relative to losses across the whole distribution | hedge funds, options strategies |
| M² | return at the same risk as the index | an easy-to-understand comparison with the index |
The Sortino ratio and the information ratio have their own articles. Together with upside and downside capture (how much of the index's gains and losses a portfolio participates in), these metrics cover the usual questions.
For advanced readers: what distorts the metrics
- Period and drawdown: the expected maximum drawdown grows with the length of the period. Magdon-Ismail et al. (2004) show for a Brownian motion that in the long run it grows logarithmically with time when the drift is positive, with the square root of time when there is no drift, and linearly when the drift is negative. Calmar ratios from three years and from thirty years are therefore not comparable.
- Data frequency: with monthly data, you do not see drawdowns within a month; with daily data, they turn out deeper. The Ulcer Index and Calmar ratio therefore depend on the frequency.
- Threshold for Omega: Ω depends heavily on the chosen minimum return. Always use the same threshold for comparisons.
- Beta estimate for Treynor: if the beta is uncertain or close to zero, the Treynor ratio fluctuates wildly. It only makes sense with stable, clearly positive betas.
- Few events: Calmar and Ulcer often hinge on a single crisis. A period covering several cycles, or a simulation, gives more meaningful results.
How Portfolio Lab calculates it
In the Backtest, Portfolio Lab calculates the Calmar ratio as the CAGR divided by the absolute value of the maximum drawdown over the entire selected period (not just 36 months). The gain/loss ratio is the average of the positive period returns divided by the absolute value of the average of the negative ones. With a benchmark, the Treynor ratio (annualised mean excess return divided by beta), M² (Sharpe ratio of the excess returns times the volatility of the benchmark's excess returns, plus the mean interest rate, annualised) and upside and downside capture via geometric means are added. Portfolio Lab does not currently show Omega or the Ulcer Index; however, the drawdown chart and the table of the largest declines show their depth and duration. Details are in the metrics methodology.
Frequently asked questions
What is a good Calmar ratio?
Over three years, values above 1 are often considered good, and above 3 very good. Over longer periods, Calmar ratios come out lower because the maximum drawdown grows over time.
What is the difference between the Sharpe and Treynor ratios?
Sharpe divides the excess return by the total volatility, Treynor only by the market risk, beta. Treynor suits building blocks of a broadly diversified total portfolio.
What does the Ulcer Index measure?
The depth and duration of all declines from the peak: the square root of the mean of the squared drawdowns. Long periods under water increase it just as much as deep slumps do.
What is the Omega ratio?
The sum of the gains above a minimum return divided by the sum of the losses below it. It takes the whole distribution into account, including skewness and fat tails.
Which risk metric is the best?
None on its own. Sharpe and Sortino for return per unit of volatility, Calmar and the Ulcer Index for declines, the information ratio for active funds. Together they give a complete picture.
Why is my gain/loss ratio below 1 even though I made money?
Because it only compares the average size of winning and losing months. If there were many more winning months than losing months, it can be below 1 while the result is still positive.
Over what period should I compare metrics?
Over the same period with the same data frequency, ideally covering at least one full market cycle including a crisis. Metrics from a few calm years almost always overestimate quality.
Literature
- Bacon, Carl R. (2008): Practical Portfolio Performance Measurement and Attribution. Wiley, 2nd edition.
- Bernardo, Antonio E.; Ledoit, Olivier (2000): Gain, Loss, and Asset Pricing. Journal of Political Economy 108 (1), pp. 144–172. doi:10.1086/262114
- Magdon-Ismail, Malik; Atiya, Amir F.; Pratap, Amrit; Abu-Mostafa, Yaser S. (2004): On the maximum drawdown of a Brownian motion. Journal of Applied Probability 41 (1), pp. 147–161. doi:10.1239/jap/1077134674
- Martin, Peter G.; McCann, Byron B. (1989): The Investor's Guide to Fidelity Funds. Wiley.
- Modigliani, Franco; Modigliani, Leah (1997): Risk-Adjusted Performance. The Journal of Portfolio Management 23 (2), pp. 45–54. doi:10.3905/jpm.23.2.45
This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.