The current financial crisis confirms the limits of standard quantitative approaches for portfolio optimization. The failures of these approaches contributed to the growing interest for advanced measures such as the Omega ratio. 

What is the Omega Ratio?

The Omega ratio is the probability-adjusted ratio of gains to losses relative to a given threshold return r. The higher Omega the better: a high Omega means that there is more density return on the right of the threshold return than on the left side:

The Omega ratio uses all the information contained within the historical (or simulated) returns series of the financial instrument. It is especially adapted to asymmetric distributions where risk is not captured by the sole volatility.
As pointed out by Kazemi, Schneeweis, Gupta (2003), the Omega ratio can be formulated as a call-put ratio: it is the ratio of the call price to the put price for the chosen threshold.

How does it differ from the Sharpe and Sortino ratios?

On the one hand, the Sharpe and Sortino ratios are computed based on the sole knowledge of the two first moments of the distribution. The Sharpe ratio measures the excess return to volatility ratio while the Sortino ratio estimates the excess return to “bad” volatility ratio (also called semi-volatility or downside volatility, which is a truncated version of the second-order moment).
On the other hand, the Omega ratio takes into account all the moments of the distribution (mean return, volatility, skewness, kurtosis and higher moments).  As a consequence, it is valid for nonnormal returns and suitable for the asymmetric nature of hedge fund returns for instance.

When and how did the Omega ratio start to gain in popularity?

The Omega ratio was introduced by Keating and Shadwick in 2002 (“A universal Performance Measure” - The Finance Development Centrer, London ). Since then, it has become increasingly popular, partly because it is very intuitive and easy to compute. The Omega ratio derives it power from its universality: it takes into account all the moments of the distribution, hence it is valid to deal with nonnormal returns. As opposed to the Sharpe ratio, it is suitable for the asymmetric nature of hedge fund returns.

Since 2007, the failures of standard approaches based on static second order moments contributed to the growing interest for advanced measures such as the Omega ratio. Indeed, using Omega as a risk measure to construct a fund of hedge fund reduces the extreme negative risk of the portfolio: it selects the assets with the lowest density below the threshold return without altering the upside part of the distribution.

How has the application of the Sharpe ratio been misapplied in Funds of Hedge Fund management and how does the Omega ratio overcomes that?

Mean-variance optimization (introduced by the Nobel prize H. Markowitz) is equivalent to finding the portfolio with optimal Sharpe ratio for a given target return. This optimization approach is suitable for (close to) normally distributed returns. But the Sharpe ratio explains only part of the risk for asymmetric returns and can even be misleading, as it assumes that all the risk is explained by the 2 first moments of the distribution.

As a consequence, monitoring and optimizing the Sharpe ratio in a static framework is not sufficient to determine the shape of the whole distribution of returns (fat tails, asymmetry or peakedness for instance), especially when dealing with hedge funds.

As explained before, the Omega ratio takes into account all the moments of the distribution of returns. It might be tempting to formulate an optimization problem consisting in maximizing the Omega ratio given some business constraints, but this approach has a major drawback: the Omega ratio is not convex and leads to unstable and dangerous solutions. The best choice in practise is not always to implement the exact formulation of the optimization problem to be solved. To overcome this issue, it is possible to solve a robustified convex formulation of the Omega ratio optimization problem.