Omega Ratio

The Omega ratio is a relative measure of the likelihood of achieving a given return, such as a minimum acceptable return (MAR) or a target return. The higher the omega value, the greater the probability that a given return will be met or exceeded. Omega represents a ratio of the cumulative probability of an investment’s outcome above an investor’s defined return level (a threshold level), to the cumulative probability of an investment’s outcome below an investor’s threshold level. The omega concept divides expected returns into two parts – gains and losses, or returns above the expected rate (the upside) and those below it (the downside). Therefore, in simple terms, consider omega as the ratio of upside returns (good) relative to downside returns (bad).

Omega Ratio – The Omega Ratio is a measure of performance that doesn’t assume a normal distribution of returns.

Omega Ratio

Where

r is the threshold return, and

F is cumulative density function of returns.

There are several ways to estimate the risk of not achieving a given return, but most of them assume that returns are normally distributed. However, as stated above, investment returns are not normally distributed, as they tend to be skewed or “fat-tailed” (i.e., there are more extreme returns than implied by the theoretical normal distribution). The omega calculations are important as they use the actual return distribution rather than a theoretical normal distribution. Thus the omega ratio and its components more accurately reflect the historical experience of the investment being measured.

Since omega considers all information available from an investment’s historical return data, it can be used to rank potential investments in a manner specific to the investor’s threshold level. However, the omega decisions are not static for at least two reasons:

1. As return information is updated, the probability distribution will change and omega must be updated.

2. As an investor’s threshold level changes, the rankings among comparative investments may change.

Therefore, omega allows investors to visualize the trade-off between risk and return at different threshold levels for various investment choices. Note that when the threshold is set to the mean of the distribution, the omega ratio is equal to 1.

Figure 13a and 13b highlights the omega ratios for two different threshold levels (a 0% return and a 10% return) for the S&P 500 Index and a range of funds (F1 to F9). Therefore, if an investor’s minimum acceptable return (MAR) is 10% rather than 0%, the omega rankings among the investments will change. For example, for a threshold return above 0%, F1 has the highest omega ratio followed by F2 and F3, but for a threshold return above 10%, F8 has the highest omega ratio followed by F9.

Figure 13a: Omega Ratio 0%
Figure 13b: Omega Ratio 10%

A key question to consider is how the Omega Ratio compares when ranking Omega Ratio relative to more commonly used Statistics.Table 3 summarizes the return data for Fund A and a mix of asset class benchmarks. The table is sorted on Sharpe Ratio, and the Omega Ratio has a 1% monthly return threshold. When looking at the rankings, Bonds appear to be a good choice when looking at Sharpe Ratio, however the Bonds have the lowest Omega Ratio at the 1% monthly return threshold. Fund A provides the highest Sharpe and Omega ratios and has the smallest drawdown.

Table 3: Omega and Common Investment Statistics

Because the Sharpe ratio is calculated from return data that has been averaged or annualized, the resulting ranking of the investments do not include higher levels of information specific to the shape of the distribution of the underlying return data. Therefore, it is reasonable to conclude that the observed differences in rankings are due to the higher levels of information contained in the Omega calculations. In effect, Omega as a risk-adjusted measure provides investors with additional information to better understand the risk/reward characteristics encapsulated within an investment’s historic returns.

Figure 14 illustrates the omega ratios for five different investments (Funds A-E) for a 12-month holding period. The two selected thresholds are a minimum acceptable return (MAR) of 2% (dashed purple line) and a target return of 8% (solid purple line). If a 2% downside is an investor’s primary concern, then Figure 14 shows that there is considerable difference between the funds, as Fund A (the red line) has a much better chance of exceeding the downside (i.e. it has a higher omega ratio at 2%).

However, there is less difference between the funds as far as earning a target return of 8%. This means that choosing between the funds should be based more on the downside risk than on the expected return. The 8% target is close to the crossover point of all the funds. The Omega ratio is a useful investment tool because it can be used in a compact way to show how different investment options relate to a target return and to a MAR.

Figure 14: Omega Ratio - Take II

Figure 14 used a 12-month holding period, which is appropriate for short-term expectations. However, Figure 15 uses an investment horizon of 5 years (60-month holding period), and shows that downside risk is less of a consideration for this period. The main decision is the target return. Once again, it is essential to consider the specific time period when analyzing investment returns.

Figure 15: Omega Ratio - Take III

Investors who are required to select and monitor investment managers should develop a basic understanding of investment statistics. Quantitative tools can provide you with good insight that you can use in your qualitative interviews with managers and when monitoring your investments.