Sharpe, Sortino and Calmar Can Each Call the Same Two Portfolios Equally Good. Omega Ratio Is Built to Disagree
7 min read
Loading...
7 min read
Two portfolios can post the exact same average return and the exact same volatility, which means they also post the exact same Sharpe ratio. As far as that one number is concerned, they are identical investments. They are not. One of them loses a little money most months and makes it all back in one huge month. The other makes a little money most months and gives it all up in one huge month. Same Sharpe ratio, opposite portfolios to actually hold. The Omega ratio is the number built to tell them apart.
The Sharpe ratio divides excess return by standard deviation, where every swing, up or down, counts as risk. The Sortino ratio divides excess return by downside deviation, where only the swings below a target count. The Calmar ratio divides annualized return by the single worst drawdown. All three do the same two things before they divide anything: they boil an entire history of returns down to one number for reward, and one number for risk.
The Omega ratio skips that boiling-down step. Instead of picking one number to stand in for risk, it takes every return in the history, above and below a threshold you choose, and weighs the whole shape of the gains against the whole shape of the losses.
Omega(threshold) = (Sum of every return above the threshold, minus the threshold) ÷ (Sum of the threshold, minus every return below it)
A return that lands 10 percentage points above the threshold contributes 10 to the top. A return that lands 3 points below it contributes 3 to the bottom. Nothing gets squared, averaged into a single deviation, or thrown away before the division happens. That is the real difference from Sharpe, Sortino and Calmar: each of them summarizes the distribution first and divides second. Omega divides the distribution itself.
Con Keating and William F. Shadwick introduced the Omega ratio in their 2002 paper "A Universal Performance Measure," arguing that mean-variance measures like the Sharpe ratio quietly assume returns are shaped like a symmetric bell curve, and that this assumption throws away real information whenever a portfolio's actual returns are not shaped that way. Because Omega uses every individual return in the sample rather than two summary statistics, it responds to skew, whether the unusually large moves in the history tend to be gains or losses, in a way a standard deviation cannot, since a standard deviation treats a large gain and a large loss of the same size as identical amounts of risk.
Picture two hypothetical portfolios, each with ten months of returns.
Portfolio A posts +2% in nine of those months and -12% in the tenth. Portfolio B posts -0.8% in nine of those months and +13.2% in the tenth.
| Portfolio A | Portfolio B | |
|---|---|---|
| Nine months | +2% each | -0.8% each |
| One month | -12% | +13.2% |
| Average monthly return | 0.6% | 0.6% |
| Standard deviation | 4.2% | 4.2% |
| Sharpe ratio (0% target) | 0.14 | 0.14 |
| Omega ratio (0% threshold) | 1.50 | 1.83 |
The average and the spread were chosen to land on exactly the same number for both portfolios, so the Sharpe ratio comes out identical: 0.6 divided by 4.2 is 0.14 either way. As far as the Sharpe ratio can tell, these are the same portfolio.
The Omega ratio disagrees. At a 0% threshold, Portfolio A's nine gaining months add up to 18 percentage points of upside (2% times 9), against a single 12-point loss: 18 ÷ 12 = 1.50. Portfolio B's one gaining month alone is worth 13.2 points, against nine small losses totaling 7.2 points (0.8% times 9): 13.2 ÷ 7.2 = 1.83.
The Omega ratio needs a threshold before it means anything, the same way the Sortino ratio needs a target. Set it at 0%, and a portfolio's Omega ratio tells you how its gains stack up against its losses. Set it somewhere else, and the number moves, sometimes by a lot, because months that counted as gains a moment ago can flip to counting as shortfalls.
There is one threshold that always produces the same answer regardless of a portfolio's shape: set it exactly at the portfolio's own average return, and the Omega ratio comes out to exactly 1.0, for any distribution. Recomputing Portfolios A and B above at a 0.6% threshold instead of 0%: Portfolio A's gains above 0.6% total 12.6 points, and its one loss below 0.6% is also 12.6 points once measured from that higher line. Portfolio B works out to the identical 12.6 against 12.6. Both come out to exactly 1.0, because the amount every return sits above the average always balances the amount every return sits below it, by definition of what an average is. That is precisely why a threshold set at the average tells you nothing about shape, and why practitioners set the Omega ratio's threshold at 0% or at the risk-free rate instead, since only a threshold away from the average lets the shape of the gains and losses actually show up in the number.
An Omega ratio above 1.0 at a given threshold means gains outweigh losses relative to that line; below 1.0 means losses dominate. There is no fixed, universally agreed cutoff for what counts as a strong Omega ratio the way some ratios carry rough conventional bands, since the number's scale depends heavily on which threshold was chosen.
That last point matters more for the Omega ratio than for most risk-adjusted measures: two Omega ratios are only comparable if they were computed at the exact same threshold. A 1.83 measured at a 0% threshold and a 1.2 measured at a 2% threshold are not ranked against each other by size alone, they answered two different questions.
NetWort's Portfolio Health card already shows your Sharpe ratio, calculated from your actual transaction history, next to your volatility and max drawdown. It does not currently compute a full Omega ratio from that history.
What NetWort does compute, on every individual asset's own page, is a Gain/Pain ratio: your sum of gaining periods divided by your sum of losing periods, the exact same calculation as an Omega ratio set at a 0% threshold. You can see it for yourself on the BTC asset page, or on any holding, under "Worth the Risk," alongside that same asset's Sharpe and Sortino ratios. A Gain/Pain ratio well above 1.0 next to a middling Sharpe ratio is the same signal this whole article is about: a portfolio or asset whose good periods are doing more work than its raw volatility number alone would suggest.