From Alpha to Omega: The Omega Measure

by Dr. Fred Novometsky, Ph.D
J.P. Morgan Investment Analytics & Consulting

The remarkable financial innovation that has taken place in the past 20 years has transformed the investment management process as well as the analysis of realized investment performance. There are many illustrations of the problems facing today’s institutional investors, including alpha transport, market neutral long / short investing, target date funds, liability driven investing and enhanced equity indexation. The assessment of expected tail losses, the analysis of derivative security exposures, the estimation of value-at-risk and the application of multi-factor performance attribution models are examples of some of the tools that investment performance analysts are using to respond to the world of investing.

Traditional measures of risk-adjusted returns such as Sharpe ratio, Treynor ratio and information ratio have been questioned by academics and practitioners as to their effectiveness in stressful markets such as the recent credit turbulence. New measures are being advocated that better assess downside risk and upside potential relative to target returns. A zero target might be appropriate for an absolute return investor seeking nominal capital preservation. An endowment or foundation might select an appropriate inflation rate for real capital preservation. A risk-free rate might serve the needs of an investor targeting minimum opportunity costs. The investment fund of a defined benefit pension plan might choose the actuarial rate to protect actuarial funding of the plan. Finally, a moving target such as a benchmark is the choice of a relative performance oriented investor.

This article introduces the omega measure developed by Shadwick and Keating (2002) and Cascon et al. (2003). This measure is the ratio of the average realized reward or return in excess of a given target return to the average realized loss relative to the same target return. The mathematical concepts behind the omega are somewhat involved, but you can get a sense of the measure and its estimation from historical realized returns.

Illustration of the Omega Measure

Exhibit 1 illustrates the computation of the omega measure for the Russell 3000 index using the quarterly returns between December 31, 2001 and December 31, 2004.

Exhibit 1 – Omega Measure Calculation for a 5% Annual Target Return

The annual return target is assumed to be 5%, which when decompounded is equivalent to a 1.2272% quarterly return. The column labeled Downside contains a simple binary indicator. The value is 1 if the corresponding quarterly return of the index is less than the quarterly target return, and is 0 otherwise. The column labeled Upside contains another simple binary indicator. The value is 1 if the corresponding quarterly return of the Russell 3000 is greater than the target return, and is 0 otherwise. The column labeled LPM is used to estimate the average realized loss below the target return. The abbreviation stands for lower partial moment. The values in this column are the product of the Downside indicator and the target return minus the index return. The column labeled UPM is used to estimate the average realized return above the target return, and this abbreviation stands for upper partial moment. The values in this column are the product of the Upside indicator and the index return minus the target return. The row labeled Average contains the average values for the lower and upper partial moments. The omega measure is the average upper partial moment divided by the average  lower partial moment.

The omega measure for the 5% annual target return is 1.1186. All things being equal, if you increase the target return, you should expect a lower UPM, a higher LPM and a lower omega measure. Exhibit 2 shows the computation of the omega measure for a 10% annual target return. An increase in the target return results in a reduction of UPM from 3.77% to 3.18% with an increase in LPM from 3.37% to 3.97%. The resultant omega measure is 0.8023.

Exhibit 2 – Omega Measure Calculation for a 10% Annual Target Return

Omega Measure of Representative Asset Classes

It is instructive to compare the omega measure of representative asset classes for equity assets, fixed income assets and alternative investments. The omega measure is computed using quarterly returns for the time period between June 30, 2000 and June 30, 2009. The annual target return ranges from no loss (0%) up to 10%.

In Exhibit 3, we summarize the omega measures for four (4) equity asset classes – the Russell 3000 index, the MSCI World index, the MSCI World ex US index and MSCI Emerging Markets index.

Exhibit 3 – Omega Measure for Equity Assets

As shown, the MSCI Emerging Markets index has the best omega measure for all target returns while the Russell 3000 index has the worst. Note that the omega measure declines uniformly for all the indices, and relative to scale of the vertical axis, are relatively uniform across target returns.

Exhibit 4 presents the omega measures for fixed income assets using the Bar Cap US index, the Bar Cap World index, the Bar Cap World ex US index, and the Bar Cap Asia Pacific index. At lower target returns, the Bar Cap US dominates all of the other indices. After a 5% target return, the other indices have better loss-adjusted rewards. The range from 1 to 9 is fairly dramatic.

Exhibit 4 – Omega Measure for Fixed Income Assets

Exhibit 5 illustrates the omega measure for alternative investments. These are represented by the NAREIT composite real estate index, the HedgeFund.net (HFN) fund of funds index, the HFN CTA managed futures index, the Dow Jones UBS commodities index, the Cambridge Associates private equity index, and the NACREIF timberland index. The high omega measures for CTA Managed Futures and Timberland are dramatic and help make the case for their inclusion in investor portfolios as described in a recent article by Abrams in Pensions & Investments.

Exhibit 5 – Omega Measure for Alternative Assets

Conclusion

It can be seen that the omega measure can be a useful tool for rating the risk-adjusted performance of asset classes. The same could be said for assessing the loss-adjusted upside potential of investment managers. It is the evaluation of the total portfolio for institutional investors that provides some startling results. In Exhibit 6, we present the omega measure for three different types of U.S. institutional investors – corporate pensions, public pensions, and endowments.

Exhibit 6 – Omega Measure for Institutional Investors

The average asset allocation for each of these investors is used to derive quarterly portfolio returns that are used to evaluate the omega measure. A special case, namely the Yale endowment, is included in the analysis. The pioneering use of private equity, hedge funds, managed futures and natural resources, coupled with substantially reduced exposure to traditional asset classes, results in an overwhelmingly superior omega measure. Note the similarity between the average corporate pension plan and endowment.

This article has introduced the omega measure, a relatively new tool for evaluating investment performance results that take into account an investor’s desired return target.Sample numerical computations of the measure are provided. The graphical illustrations show how the omega measure can be used to evaluate asset classes, institutional investor portfolios, and domestic market asset allocation decisions. Harlow (1991) and Kane and Bartholomew (2003) show that downside risk measures and omega measures can be used for designing portfolios in a post modern-portfolio theory world. The reader interested in learning more about omega measures should consult the papers by Bacmann and Scholz (2003) and Kazemi et al. (2003).

To view the next article, Multiple Asset Class Return Comparison, click here.

References

Abrams, R., R. Shaduri, et al. (2009). How managed futures can help portfolio, June 15. Pensions & Investments. New York.

Bacmann, J. F. and S. Scholz (2003). "Alternative Performance Measures for Hedge Funds." AIMA Journal June 2003.

Cascon, A., C. Keating, et al. (2003). The Omega Function. London, UK, The Finance Development Centre.

Harlow, W. V. (1991). "Asset Allocation in a Downside Risk Framework." Financial Analysts Journal 47: 28-40.

Kane, S. J. and M. C. Martholomew-Biggs (2003). Optimizing Omega. Hatfield, UK, School of Physics Astronomy and Mathematics, University of Hertfordshire.

Kazemi, H., T. Schneeweis, et al. (2003). Omega as a Performance Measure. Amherst, MA, CISDM, University of Massachusetts.

Scherer, B. (2007). Portfolio Construction and Risk Budgeting, Third Edition. London, UK, Risk Books.

Shadwick, W. and C. Keating (2002). "A Universal Performance Measure." Journal of Performance Measurement Spring 2002: 59-84.