If you are working in finance, you have almost surely heard of risk-reward ratios and probably used some of them to evaluate the performance of a stock, ETF, or any other investment strategy. Among the different alternatives, the most popular risk-reward ratio is the so-called Sharpe ratio, first introduced by William F. Sharpe in 1966.

It was originally termed *reward-to-variability ratio* and, consequently, it was defined as a quotient with reward in the numerator and variability in the denominator. Both concepts are computed from the returns distribution of an asset as follows:

*Reward*is computed as the average asset return.*Variability*is calculated as the standard deviation of the asset returns.

Usually, the costs of financing the position (or, equivalently, the opportunity cost of not investing) is taken into account by replacing the vanilla asset returns by excess asset returns above a pre-defined benchmark (customarily the risk-free return). Thus, leaving the Sharpe ratio definition as follows:

\[SR \triangleq \frac{\mu-r_b}{\sigma}\]

where \(\mu\) denotes the average asset return, \(r_b\), the average benchmark return, and \(\sigma\), the standard deviation of the excess returns.

The main drawback of the above definition is that only the first two statistical moments of the distribution are used. Consequently, it won’t capture all the useful information for some distributions (especially the asymmetric ones) and/or for investors attributing some sort of utility to higher order moments. To overcome this limitation, researchers have proposed many other risk-reward ratios: Calmar, Sortino, Sterling, Burke, Kappa, to cite some.

However, in this post, I will focus on some of today’s most celebrated ratios: the Omega ratio. The Omega ratio was introduced by Keating and Shadwick in 2002 in an article titled “A Universal Performance Measure”. It was pretentiously named “Omega” after the last letter in the Greek alphabet, as a reference to its ultimacy. But… did it really deserve the name?

## The original definition

Many sources, including the original article or Wikipedia, define the Omega ratio in terms of the cumulative return distribution, \(F(x)\), as follows:

\[\Omega (r)\triangleq {\frac {\int _{{r}}^{\infty }(1-F(x))\,dx}{\int _{{-\infty }}^{r}F(x)dx}}.\]

At this point, if you are familiar with other risk-reward ratios, you may have been surprised by the complexity of the definition. In its current form, it seems hard to intuitively explain what is going on in there. Those integrals of a cumulative distribution function (which is already the result of integrating a probability density function) are far from being intuitive. Let’s see what we can get after some mathematical manipulation.

## A more intuitive definition

The definition above can be transformed into the ratio of two option prices as follows:

\[\Omega (r) = \frac{E[\max(x-r,0)]}{E[\max(r-x,0)]} = \frac{e^{-r_f} E[\max(x-r,0)]}{e^{-r_f} E[\max(r-x,0)]} = \frac{C(r)}{P(r)},\]

where \(C(r)\) is the price of a European call option written on the asset and \(P(r)\) is the price of a European put option written on the asset, both for a strike price \(r\). The first step is a little tricky and I have ommited the details for the sake of conciseness. Still, if you are interested, the details can be found in the article ¨Omega as a performance measure¨ by Kazemi et al.

Interestingly, it can also be rewritten in terms of the put option price alone (i.e., using the put-call parity formula):

\[\Omega (r) = 1 + \frac{\mu – r}{P(r)} = 1 + \frac{\mu – r}{E[\max(r-x,0)]}.\]

This second formulation shows that, disregarding the constant 1, the Omega ratio can also be seen as a variation of the Sharpe ratio where the standard deviation is replaced by a one-sided measure of risk, the expected deviations below \(r\).

## So?

From what we have seen so far, we can conclude that **there is nothing significantly new in the definition of the Omega ratio**. It is basically a return-to-risk ratio using yet another measure of downside risk. Indeed, it is rather similar to another well-known ratio:

\[\text{Sortino ratio} = \frac{\mu – r}{\sqrt{E[\max(r-x,0)^2]}},\]

the only difference being that the quadratic mean in the denominator has been replaced by the arithmetic mean. In short, risk in Omega ratio is measured as the first order lower partial moment (LPM) of the returns whereas in Sortino ratio the second order LPM is used instead. Nothing fancy for this part.

Nonetheless, there is something about Omega ratio that is worth mentioning. If you paid careful attention, you may have noticed that since I first introduced the ratio, it was not defined as a simple statistic but as a function of a parameter, i.e., \(\Omega(r)\). The parameter \(r\) is the target return threshold and defines what is considered a gain (above the threshold) and a loss (below the threshold).

It can be thought of as a point where investor preferences can be inserted into the statistic, but the main contribution of this parameter is the paradigm shift it brings along: the Omega ratio is **no longer a single number** but a curve, the so-called **Omega curve** or function. Then, its main appeal over the Sharpe ratio is that it allows to plot rankings for varying levels of risk aversion by varying the threshold.

Along with this very same line, recent results (e.g. “Risk aversion vs. the Omega ratio: Consistency results” by Balder and Schweizer) have shown that **Omega ratio is consistent with second order stochastic dominance**. In plain words, this means that the preferences expressed by the Omega ratio (again, and some other related ratios) are consistent with those of a non-satiated and risk-averse investor, no matter the specific shape of the utility function.

Drop a line below if you want to know more and see you in my next post!