'Omega Alpha' is a new approach which efficiently identifies sources of risk-adjusted return, provides easily managed strategies for delivering absolute return and is data rather than model driven. It generalizes and consistently out-performs traditional regression alpha. Omega Alpha is produced by optimization using a new class of statistics (Omega Metrics) which are especially well-suited to finance. This article demonstrates the impact of Omega Alpha in an example of a back-test of a European equity manager. It also includes an example, from a study by Con Keating and Claudio Semadeni, of the same approach applied to a fixed income fund. We conclude with a brief discussion of the reasons for its success.

Every investor is faced with the same problem: construct a portfolio of assets balancing risk tolerance and reward needs. The assets may be shares, bonds, mutual funds or hedge funds. Decisions must be made on the basis of past performance coupled to views about the near and longer term future.

A simple measure of success is out-of-sample absolute return-appropriately adjusted for risk. Adjusting for risk is, of course, a difficult issue. What one can say with certainty is that if all returns were normally distributed the Sharpe ratio would be the ideal measure of risk-adjusted return. We can also say with certainty that financial returns are far from normal, especially returns from hedge funds.

The assumption of normality implies symmetry in up and downside returns. The goal of hedge fund managers, however, is to produce asymmetric return with maximal upside and this is the basis of their appeal to investors. The use of variance as a risk measure is clearly at odds with this goal. Variance produced by out-performance is welcome. Fat tails are desirable as long as they occur on the right side of the mean.

Omega functions [Box 1], introduced in 2002 ('A Universal Performance Measure' C. Keating and W.F. Shadwick J.Performance Measurement 6 no 2, 2002 pp 59-84) provide a new mathematical tool kit ('Omega Functions', A. Cascon, C. Keating and W.F. Shadwick, The Finance Development Centre 2003) for the study of returns distributions. These new tools led in turn to a new approach to risk-adjusted return and with it a new approach to the general portfolio problem or the search for 'alpha'. This is based on new statistics ('Omega Functions and Omega Metrics', A. Cascon and W.F. Shadwick The Finance Development Centre 2004) that arise naturally through the use of the Omega function of a distribution as a proxy for the distribution itself [Box]. These statistics (which we have called Omega Metrics or Omega scores) generalise Sharpe ratios by taking into account departures from normality such as fat tails and asymmetry in a way that is both consistent with financial goals and mathematically well-founded. As aresult, portfolios optimised on the basis of their Omega score can be expected to outperform those constructed by mean/variance tools. Extensive tests of the performance of such portfolios across asset classes and over different market conditions bears this out.

This note provides an introduction to our new approach illustrated by examples of its application to the construction of long-short funds by hedging long equity and fixed income funds with a short position in a benchmark index. This is followed by a brief discussion of the reasons for the success of this extension of standard financial engineering.

The two classical ingredients of hedge fund strategies are short selling and leverage. The contradictions inherent in combining the traditional mean/variance approach to risk adjusted return with hedge fund strategies are apparent in the simplest case. -hedging to reduce the variability in returns. (After which leverage may or may not be applied.)

Consider the case of a long equity manager who often, even if not always, beats his benchmark both in rising and falling markets. Such a manager has stock picking and/or market timing skills and conventional finance theory tells us how to identify the 'alpha' that he produces.

Standard finance theory makes a plausible assumption, namely that the part of the return that we wish to isolate is the part that is uncorrelated with the benchmark.

From this assumption, basic algebra and statistics tell us that we can use linear regression to identify the manager's 'beta' with respect to the market. Adding to the manager's returns a short position of beta times the index leaves us with the manager's alpha (on average) in a minimum variance investment. If we make the further assumption that variance is the correct measure of risk we have done an optimal job of hedging the market exposure and extracting the manager's alpha.

But there is an obvious problem with this approach, which is the implicit assumption of symmetry between up and downside returns. If the manager's returns rise at beta times those of the market as the market rises, the model behind this strategy tells us that they must also fall at beta times those of the market when the market declines. This is of course in contradiction to the observation about the manager's skill that we started with. The assumption that the return we really want to extract is the component uncorrelated with the market is at odds with market timing or stock picking skill and the alpha which has been identified by using this model is sub-optimal in general. Likewise, while reducing variance on the downside is clearly a good thing, it is also clear that doing this without consideration of the impact on the upside is also sub-optimal.

The use of Omega Metrics to optimize a portfolio of manager and benchmark provides a robust approach to this problem ('From Omega to Portable Alpha, W.F. Shadwick, The Finance Development Centre 2004). Omega scores [Box1]-the numbers produced by the application of Omega Metrics-are designed to capture the features of returns distributions which are financially meaningful. They do this by rewarding fat tails on the upside and penalising them on the downside. The Omega scores also reward the mean of the distribution both on its size and on the degree of concentration around it.

Omega scores are produced by applying financially meaningful and mathematically well-founded principles, but the real test is how they perform out of sample in producing risk adjusted return. The examples below show that there is much to be gained by applying this approach.

We apply this approach first to a European Equity fund. Like most equity funds, this one struggled somewhat in the severe downturn followingthe collapse of the equity bubble. However, as Figure 1 shows, its Omega score was consistently above that of the index. We compare the performance that would have been produced from January 2000 to August 2004 by the regression alpha and the Omega alpha for this fund, using the benchmark MSCI Europe index. In both cases, three year rolling windows of monthly returns were used to provide quarterly portfolio weights which were held without update for 3 months at which point the optimisation was repeated with the most recent three years of returns. No leverage was used. The 56 month out of sample period provides a good test of these approaches both through the downturn, the 2003 recovery and most of the 'difficult' first 3 quarters of 2004. The evolution of $1 invested with each of the three funds in January 2000 is shown in Figure 2.

It is clear from Figure 4 that in terms of absolute return the regression alpha improves on the fund's performance but is beaten handily by the Omega Alpha fund. One of the reasons for this is that although the fund's beta was positive during the entire out of sample period, by the fourth quarter of 2003 the market upturn had moved the Omega Alpha fund to a small positive position in the index. This is another example of the cost of the assumption that the part of the manager's return we want to capture is the part that is uncorrelated with the index. In this case the 'risk' that was being reduced was the risk of a rising market and the regression alpha, as it was designed to do, reduced this risk- losing most of the benefit of the 2003 rally in the process.

Table 1 shows the standard performance figures for the 56 month out of sample period. As one would expect, given that it should produce the minimum variance combination of the Equity fund and the index, the regression alpha has lower variance than the Omega Alpha. It also has a higher Sharpe ratio although one can see that the Omega Alpha has lower downside risk and its additional variance arises on the upside ( as the range of monthly returns indicates.) The Omega scores on the other hand, reflect what common sense would say about the relative risks of the regression and Omega funds.

The benefits of our approach are not limited to equity funds nor to periods which include severe market downturns. The next example illustrates the Omega Alpha which can be extracted from a bond fund during a period where fixed income managers experienced excellent market conditions.(C. Keating and C. Semadeni, 2004)

The portfolio ingredients in this case are a Fixed Income Fund and its benchmark, the FTSE All Bond Index over the period from June 1999 to July 2002. Again, as Figure 3 shows, using our Omega score criterion, the fund consistently out-performed its benchmark. The analysis was conducted as above, using three year rolling windows with quarterly updates providing an out of sample period from August 2003 to the end of June 2004.

The absolute return performance with no leverage and with 2 times leverage is shown in Figures 4 and 5. As one would expect, given the market conditions for fixed income funds over the out of sample period, absolute return for the hedged funds is lower than that of the fund itself. On a risk adjusted basis however (see Table 2) it is clear that a great deal has been gained by standard measures such as Sharpe ratio or monthly drawdown, as well as in terms of the Omega score.

Table 2 also makes clear the cost of the variance minimization which drives the regression alpha. While the Omega Alpha portfolio has both higher variance and a higher monthly drawdown than the regression hedge, the increase of drawdown by 19 basis points has been accompanied by an increase in the maximum monthly return of 60 basis points and an increase in the annualized mean return of 216 basis points. Clearly the assumption of symmetry between up and down side is a costly one in this case.

The significant shift of variance to the upside achieved by the Omega Alpha portfolio means that a great deal of leverage could have been applied while still maintaining downside risk-by any of the standard measures- lower than that of the original fund. Gearing the Omega Alpha fund by a factor of 3.5 would produce an annual return in excess of 15% with variance lower than that of the original fund.

Optimising on Omega score is an obvious strategy for construction of any portfolio. By modifying the weight given to downside versus upside and to mean versus tail behaviour, the Omega scores can also be tailored to match investor risk appetite from highly conservative capital preservation to aggressive growth goals.

Omega scores provide a new approach to the identification and extraction of alpha as the examples from equity and fixed income funds illustrate. Numerous other examples show that the Omega Alpha obtained by optimising a position in a fund and a market index is an excellent source of easily managed, low cost, alpha. The key ingredients are simple: a fund and a benchmark index in which long or short positions can be taken. If the fund regularly out-performs the index (as judged by their Omega scores) then an Omega optimal position long the fund and short (or occasionally long) the index will produce returns that are adjusted for risk in a meaningful way.

This approach presents the opportunity to produce a new generation of traditional and synthetic investment products, such as long short index trading strategies, absolute return funds from mutual funds, and synthetic funds of hedge funds from investable indices.

In all of these cases, the common sense goals of risk adjustment are met more successfully by Omega Metrics than by the standard mean variance toolkit, as would be expected in a world where normality of returns is grossly at odds with observed behaviour.

Omega functions provide an alternative representation of a probability distribution. They are based on the intuitive observation (C. Keating and W.F. Shadwick, 'A Universal Performance Measure', J Performance Measurement 6 no 2, 2002 pp 59-84) that the quality of a bet on a return above a threshold r can be measured by the ratio of the probability weighted gains to probability weighted losses, both measured relative to r. This leads to the formulation of the Omega function for a distribution F as the ratio of the expected value, with respect to F, of a 'virtual call' with strike r to the value of a 'virtual put' with strike r: Omega(r)= E(max(x-r,0))/E(max(r-x,0)). (A. Cascon, C. Keating and W.F. Shadwick, 'Omega Functions', The Finance Development Centre 2003) In other words, the value of the Omega function of F at r is the ratio of the value of all of the upside relative to r to all of the downside relative to r.

As r is allowed to vary over the support of the distribution, the resulting function turns out to be mathematically equivalent to the distribution itself. An immediate consequence of this surprising fact is that the Omega function of F may be used as a proxy for the distribution. This has many advantages over conventional approaches. For one thing, Omega functions are immediately informative about the relative weights of up and downside tails and left or right bias in a distribution. These properties are evident in the shape of the graph of the Omega function, or more conveniently, the graph of the logarithm of the Omega function. Figure B1 shows log(Omega) for two CTA fundsas a function of monthly return over a 46 month data set.

The naive approach to ranking funds by their Omega functions is to fix a threshold return (such as the risk free rate or a target or benchmark mean) and use the value of the Omega function at that threshold as the rank.

The Omega functions for these funds illustrate the (insurmountable) problem with this approach. CTA1 has more upside than CTA2 but also has more downside and a lower mean return. Which is the better investment based on the available data?

The answer to this question requires a more sophisticated approach-one which takes into account all of the information in the Omega function. We require real-valued functionals of Omega functions which, like Sharpe ratios in the case of normally distributed returns, allow us to rank distributions. Unlike Sharpe ratios, Sortino ratios or other mean-variance measures, they should take fat tails and left or right asymmetry (whether it arises from variance or whatever other cause) into account.

The requirements of a functional on Omega functions are therefore very simple. It should:

- penalise fat tails below the mean
- reward fat tails above the mean
- reward higher means on 'quality'.

The first two features are self-explanatory. They provide for a moment-free assessment of the positive or negative bias in a fund's returns distribution. In the case of two mirror image distributions such as buying and selling the same lottery ticket, the score on fat tails would be negative for the sell and positive for the buy.

By 'quality' in the third we mean, for example, that a manager whose returns are almost uniformly distributed around his average return would score lower than one who achieved the same average but with less dispersion around it.

The Omega F2 rankings introduced in ("Omega Functions and Omega Metrics-Hedging with CTAs, A. Cascon and W.F. Shadwick The Finance Development Centre 2004") satisfy these requirements. Like a Sharpe ratio, they allow us to rank two or more distributions. Applied to normal distributions, the Omega F2 score produces the same rankings as the Sharpe ratio.

Figure B2 is a time series record of the Omega scores for two CTA funds over the last 36 of 46 months in a data sample. This record shows the stable separation of the scores of the two funds. It also shows that CTA1 is not significantly different from the average CTA fund over the same period. The persistence of such relationships out of sample is a key to the success of optimization on the basis of Omega scores.

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## Commentary

## Issue 136

## Could a Severe Equities Sell-off Prolong Economic Expansion?

## Erik Norland, Senior Economist and Executive Director, CME Group