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Fund of Hedge Funds Portfolio Optimization Using the Omega Ratio

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Fund of Hedge Funds Portfolio Optimization Using the Omega Ratio risk management / compliance LE Fund of Hedge Funds Portfolio IC T R Optimization Using the Omega Ratio A D E ® BY STEVE TOGHER, CAIA , clearly demonstrate how Omega quantitative measure should be an UR AND TambE BARsbaY captures these higher moments. It is end-all solution. an equation that adds to mean and Our objective here is to present a EAT variance, captures all of the higher method of “optimizing” a portfolio F t is interesting to ponder moment information in the return of hedge funds using the Omega how new and radical distribution, incorporates sensitivity ratio. For the purposes of our discus- theories and equations to return levels, and is intuitive and sion we will focus on a portfolio of Iin financial analysis become widely relatively easy to calculate. alternative investment managers accepted. It most likely is a function The body of literature about con- with nonnormal returns but, of of simplicity, elegance of design, structing and optimizing portfolios course, the underlying investments and ease of application. But it is of hedge funds continues to grow can be anything. We put optimizing even more interesting to observe and the alternatives are varied. In in quotes because our analysis is not how, once widely accepted, such most cases a comparison is made the sole input in determining the concepts can be misapplied. Perhaps to MVO, which suffers from similar optimized portfolio. two of the most common examples drawbacks as Sharpe when applied are the application of the Sharpe to hedge fund returns. DeSouza and Understanding the Omega Ratio ratio and mean variance optimiza- Gokcan (2004) defined an approach We believe that using the Omega tion (MVO) to hedge fund portfolio that is modeled on asset allocation ratio for this purpose is a rather intu- analysis. To be sure, both of these methodologies familiar in tradi- itive decision based on the abilities equations are elegant and beautiful tional investment management but and strengths of the measure. Those in design and both have changed takes into account the idiosyncrasies in agreement with the evidence that the way the world analyzes invest- of hedge fund investing as an asset more traditional measures such as ment options, but an overwhelming class. Amenc and Martellini (2002) the Sharpe ratio and other MVO- body of evidence suggests that they took the approach of evaluating type measures are inappropriate are not applicable to the asymmetri- out-of-sample performance of an for alternative investments should cal nature of hedge fund returns. improved estimator of the covari- appreciate Omega’s lack of assump- They don’t apply because they don’t ance structure of hedge fund index tions about distributions. accurately account for the impacts returns, focusing on its use for opti- In fact, it is from this lack of of the third and fourth moments mal portfolio selection. assumptions that Omega derives its of hedge fund return distributions, power. Because the measure takes which are used to calculate skew- Objective into account the exact distribution ness and kurtosis, respectively. The purpose of this paper is to build of the returns in question, it inher- That said, it is nearly impossible upon Omega’s intuitive nature and ently includes all of the information to review performance information discuss and review how Omega can encoded in the moments: mean, on a hedge fund without seeing the be used to optimize a fund of hedge variance, skewness, and kurtosis. fund’s Sharpe ratio on prominent funds portfolio. It is worth noting Admittedly, Omega is backward- display (assuming it is a compelling now, and we will expand on this looking and thus fallible, but it does number). Likewise the most popular later, that optimization through not have the added weakness of hedge fund databases and analyti- Omega is purely a quantitative assuming normality. In addition, cal programs all offer some type of pursuit and, as investment practi- as Cascon and Shadwick (2006) put MVO algorithm. tioners, we believe that while it is forth, if we compare investments Fortunately another elegant mea- an important part of multimanager that come from different affine fami- sure, the Omega ratio, is attaining hedge fund portfolio construction, lies of distributions, we then cannot widespread recognition and accep- it is not a final solution. And we use standard deviation as a “com- tance. Keating and Shadwick (2002) can find solace in this because no mon unit of risk,” which also leads 1 The Monitor The Voice of the Investment Management Consultants Association www.imca.org to a weakening of the arguments put Omega as an Optimizer able return, which can be anything forth by the Sharpe ratio and MVO, As posited by Favre-Bulle and Pache from zero to the risk-free rate to at least in the case of alternative (2003), a portfolio can be optimized virtually anything else. Optimizing investment manager analysis. using the Omega ratio at a speci- at a given threshold is rather simple The Omega ratio is defined by fied threshold which, depending and not unlike performing a tradi- Keating and Shadwick (2002) to be: on the investor, can be selected in tional MVO. Instead of minimizing a number of ways. In this case, the variance while maximizing return, threshold can be a minimum accept- one minimizes probability-weighted TABLE 1 Ordinary Risk Statistics ANNUaliZED StaNDARD DEViatioN ANNUaliZED RatE of RETURN Fund A .80% 14.60% where [a,b] is the interval of returns Fund B 0.97% 49.65% with a cumulative distribution Fund C 6.66% 33.4% function F(x), and r is the threshold Fund D 6.56% 13.88% return where the ratio is being evalu- Fund E 1.51% 16.11% ated. Clearly, a higher Omega ratio Fund F 6.45% 9.55% is better than a lower one, because it Fund G 1.69% 7.4% indicates either a greater probability- Fund H 3.94% 16.57% weighted gain or a smaller probabil- Fund I 0.38% 10.50% ity-weighted loss, or both, based on Fund J .80% 18.48% historical returns. Fund K 8.74% 19.57% FIGURE 1 Comparison of Omega Plots July / August 007 Risk Management / Compliance The Monitor 13 >> “PORTFOLIO OPTIMIZATION” CONTINUED interval but become less desirable loss and maximizes the probability- at other portions. Here, however, is weighted gain as calculated using As one can imagine, the where a somewhat more subjective the Omega ratio. decision would be made based on Our goal is to apply this proce- mechanics of the calculations are any number of factors that could dure in a similar fashion using the include such things as investor risk Omega ratio but across an entire quite straightforward while aversion and investor goals. range of threshold returns, thereby resulting in what we believe is the challenge lies in actually Steve Togher, CAIA®, is managing a rather robust optimization of director of alternative investments at sorts. We say “of sorts” because in defining and parsing out Fortigent LLC in New York, NY, where performing this procedure across he oversees the use of hedge funds and an entire range of thresholds it the best portfolios... other nontraditional investment solu- becomes somewhat challenging to tions. He earned a B.A. in computer define what a best portfolio would information systems and an M.B.A. in be, and thus we need to address finance, both from Baruch College. Con- other quantitative issues related to tact him at [email protected]. market factors impacting particular the upside and downside, as well as Tambe Barsbay is a quantitative strategies and more qualitative issues the value of the Omega ratio itself analyst of alternative investments with related to specific investor needs and in comparison to others across the Fortigent LLC and heads the Omega goals to ultimately come up with a entire range of thresholds. optimization programming team. He “best” portfolio. earned a B.S. cum laude with honors And so we can backtrack a bit Analysis in mathematical science from the now to examine in more detail the We will argue that the number New Jersey Institute of Technology quantitative aspect of choosing of funds being optimized is not and is pursuing an M.S. in mathemat- the best portfolio. Favre-Bulle and material to the study, provided the ics and statistics at Georgetown Univer- Pache (2003) did this at individual returns streams reflect nonnormal- sity. Contact him at tambe.barsbay@ thresholds by plotting efficient ity. Thus, we optimized a portfolio fortigent.com. frontiers of probability-weighted of 11 underlying single manager losses versus probability-weighted hedge funds with the ordinary risk References gains. We extend this by calculating statistics shown in table 1. Amenc, N., and L. Martellini. 2002. entire Omega ratio curves for large The optimization was run with Portfolio Optimization and Hedge numbers of portfolios. As one can very loose restrictions with regard Fund Style Allocation Decisions. The imagine, the mechanics of the cal- to minimums and maximums per Journal of Alternative Investments 5, no. culations are quite straightforward investment, a liberty that would not 2 (fall): 7–20. while the challenge lies in actually normally be taken in a real-world Cascon, A., and W. Shadwick. 2006. The defining and parsing out the best application, but it nonetheless is one CS Character and Limitations of the portfolios, a process that can be that allows us to produce meaning- Sharpe Ratio. The Journal of Investment loosened or tightened to produce ful results for illustrative purposes. Consulting 8, no. 1 (summer): 36–52. more or fewer results, respectively. Figure 1 shows an Omega plot DeSouza, C., and S. Gokcan.
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