# The G-Omega Ratio: A New and More Conservative Performance Metric than Omega

The Omega ratio http://en.wikipedia.org/wiki/Omega_ratio is a relatively new performance metric invented by Keating and Shadwick. It was designed to be a better way to capture all moments of the distribution to give a fair accounting of the upside versus the downside risk that is superior to the Sharpe Ratio. From a semantic perspective it does truly capture the notion of “reward versus risk” in terms that most investors are familiar with. The Omega uses a threshold or some level of required return to calculate the upside above the threshold versus the downside below the threshold. My own testing shows that the Omega is in fact competitive or superior to the Sharpe ratio in many applications. It is often better for use in ranking assets, strategies or portfolio managers- especially those that have non-normal distributions. Winton Capital provides an interesting white paper on the topic http://www.performance-measurement.org/Winton2003.pdf of using the Omega for performance evaluation.

The calculation of the Omega statistic is actually quite simple–don’t let the fancy notation deter you. The calcuation boils down to a few simple steps:

1) identify a threshold, T– lets say it is zero for this example

2) calculate the excess returns > T, or take the sum of the returns >0 in this example minus the threshold (subtract zero for all returns above zero and take the sum)

3) calculate the excess returns< T, or take the sum of the returns <0 in this example minus the threshold (subtract zero for all returns below zero and take the sum)

4) take the sum in step 2 divided by -1* the sum in step 3……….this is the Omega Ratio

The conventional Omega Ratio suffers from a few minor flaws. First, it is skewed by outliers because it simply takes the ratio of the sum above versus the sum below the threshold. Any extreme observations will lead to an Omega Ratio that does not correctly capture the true nature of the distribution. Second, the use of a sum measurement fails to account for the true ratio of upside versus downside return potential in the absence of a difference in frequency between the two. That is, assuming that the frequency of returns above the threshold versus below were to change dramatically, what would be the effective upside versus downside. Third, the Omega does not account for the impact of compounding returns–it simply takes the sum of returns versus the threshold. This does not address the fact that compounding requires a low variance around returns to comparably match the sum or arithmetic mean of returns.

The “G-Omega” is a simple modification to the above ratio that addresses these concerns. The simple difference is that the G-Omega uses the geometric average of returns above the threshold versus the geometric average of returns below the threshold instead of the sum subject to a maximum value of 10 and a minimum value of 0. This ratio is purely focused on the evolution of upside compounding potential versus downside compounding potential. The G-Omega is indifferent to the frequency of returns above the threshold versus the frequency below the threshold. It is also less sensitive to outliers since it is taking the average of several values. The G-Omega can potentially have a value of less than 1 even when there are mainly positive observations versus negative observations. This is because the average returns below the threshold can exceed the average returns above the threshold when tail risk is high. From this perspective, the G-Omega can provide a very conservative view of an asset, strategy or portfolio. It is potentially a nice complement to the original Omega metric.

Thanks for the link to that paper. Did not know the Omega ratio…

But I can’t match your formula explanation to the formula in the paper…

The paper draws a Cumulative Distribution Function and expresses omega as the ratio of

sorry – fat finger error! finishing:

omega as the ratio between 1) the surface above the CDF for x>T and 2) the surface below the CDF for x<T (using integration of the CDF in the formula).

It does not add returns as you mention…

One of the consequences is that outliers would not have such a great impact: since by definition the CDF value for an outlier is close to 0 or 1…

More to the point, the CDF DOES take frequency in consideration by its very nature.

Not sure if you or I missed the point of the Omega ratio but one of us sure did!

hi oh, neither of us missed the point 🙂 thanks for highlighting what was not mentioned which was that this is a simplified reduction of the omega calculation to make it accessible. what i was trying to say is that frequency is a source of estimation error the g-omega does not take frequency into account. outliers have an effect equivalent to what you would expect based on the reduced formula.

best

david

David’s simple steps to calculate Omega (using ratios of sums of returns) is equivalent to the formal Omega formula (as in the original paper). I simulated various sets of returns and the two calculations always produce the same results. But what is the intuition behind these being equivalent? On the face of it, they seem different. If there is no easy intuitive explanation of the link between the two, then what is the formal derivation from one to the other?

hi george, thank you and you are correct. the derivation is something that i can post at some point should i find the time. in general all we want to know is the area above the threshold versus the area below adjusted for probability. that is the link.

best

david

You can also do Upper Partial Moment / Lower Partial Moment

The formal definition of the Omega Ratio is a ratio of integrals of the CDF between different limits (above and below the MAR). For empirical returns, you can simply use a ratio of sums of the returns above and below the MAR. Remember, an integral is simply a sum to the limit. I can provide a formal mathematical derivation

The two approaches give the same result for a large number of results . There’s an excel spreadsheet at http://investexcel.net/219/calculate-the-omega-ratio-with-excel/ to help you calculate the Omega Ratio

Oddly enough, this method seems to be quite volatile to the selected time frame. For example, on the SPY, a lookback period of 100 produced risk adjusted results just slightly better than buy and hold. A 400 period lookback increased total returns by 400% over the 100 period test. However, a 200 period lookback produced a total 25% loss over the period tested.