<h2>CHAPTER 1</h2><p><b>Asset Allocation for Hedge Fund Strategies: How to Better Manage Tail Risk</p><p>Arjan Berkelaar, Adam Kobor, and Roy Kouwenberg</p><br><p>ABSTRACT</b></p><p>Most approaches to risk budgeting are based on tracking error and value at risk(VaR). In addition, the return streams from any investment process are usuallyassumed to be serially uncorrelated and normally distributed. This assumption,however, does not necessarily hold in reality. In this chapter, we consider tworelatively new risk measures that are better suited to deal with nonnormal andserially correlated return streams and that are superior to tracking error (orvolatility) and value at risk. We show how these measures can be used indetermining an optimal risk allocation, allowing investors to better manage thetail and drawdown risks in their portfolios. By better managing these risks,investors can achieve superior risk-adjusted returns.</p><br><p><b>INTRODUCTION</b></p><p>Many institutional investors are searching for sources of diversification andreturn-enhancing strategies in order to improve the performance of theirportfolios. An area where many of them hope to achieve superior risk-adjustedreturns is hedge funds. Shifting asset allocations toward hedge funds is not aguarantee of success, however. Unlike equity and bond markets that compensateinvestors with a positive risk premium over the long term, returns fromselecting hedge fund managers are conditional on skill. To be successful inpicking hedge funds, a strong risk management process and a disciplinedinvestment approach are required.</p><p>Most approaches to risk and asset allocation, both in practice and in theacademic literature, are based on standard deviation and value at risk. Inaddition, the return streams are usually assumed to be normally distributed.This assumption is quite convenient, allowing investors to use the well-knownmean-variance workhorse to derive optimal allocations. We refer interestedreaders to Berkelaar et al. (2006) for a risk budgeting framework wheninvestment returns are normally distributed. In the case of hedge funds,however, the normal distribution fails to adequately describe the returndistribution. Basing risk allocations on mean-variance optimization may resultin a considerable misallocation of risk that could result in suboptimalportfolios and lower investment returns.</p><p>In this chapter, we consider conditional value at risk (CVaR)—a relativelynew risk measure that is better suited to deal with nonnormal return streams.The advantage of this risk measure is that it is easy to use and allows fornumerical tractability. In this chapter, we derive optimal portfolios for mean-CVaRinvestors and compare results with those of a mean-variance investor.Others have also studied the impact of skewness and fat tails on optimalportfolios. Krokhmal et al. (2003) consider a portfolio of individual hedgefunds and study the performance of various risk constraints, including CVaR andconditional drawdown at risk (CDaR), with in-sample and out-of-sample tests.Amin and Kat (2003) study the optimal allocation among stocks, bonds, and hedgefunds in a mean-variance-skewness optimization framework. Kouwenberg (2003)studies the added value of investment in individual hedge funds for investorswith passive stock and bond portfolios, taking into account the nonnormality ofthe return distribution.</p><p>We use the Hedge Fund Research, Inc. (HFRI) indexes for several hedge fundstrategies to determine optimal allocations. The period for the historical timeseries is January 1990 to December 2007. We show that investors who manage tailrisk by basing their portfolios on mean-CVaR optimization should be able toproduce superior returns on their hedge fund portfolio. Using historical returnsfor several hedge fund strategies, the incremental return could be as much as100 to 200 basis points (bps). We also present results based on a forward-lookingsimulation model for hedge fund returns with more modest assumptionsabout expected returns. In this case, the incremental return is about 40 to 60bps.</p><p>This chapter is organized as follows. The second section of this chapterdiscusses the CVaR measure. We also show to what extent risk could beunderestimated by assuming that returns are serially uncorrelated and normallydistributed. This chapter's third section discusses a framework that can beapplied to develop forward-looking return scenarios for a wide range of hedgefund strategies. In our simulation approach, we separate the systematic market(or beta) exposures prevalent in hedge fund returns from an alpha component thatcan be attributed to skill. Our factor model is kept simple, and we onlydecompose the hedge fund returns into an equiity beta and fixed-income beta. Thesensitivities of hedge fund returns to equity and fixed-income returns are timevarying. We use a Kalman filtering approach to capture the time-varying natureof the beta factors. To simulate realistic return scenarios for equities andbonds, we use regime switching models. By capturing both normal and stressfulperiods, regime switching models can describe the empirically observed skewnessand kurtosis in stock and bond returns. By linking the return on hedge funds tothe returns on public asset classes, they will also exhibit skewed and fat-taileddistributions. In the fourth section of this chapter, we compare theoptimal asset allocations based on mean-variance optimization with the optimalasset allocations based on mean-CVaR optimization. First, we study the resultsbased on the historical return distribution, and then we present optimalallocations derived using our forward-looking simulation framework. The finalsection summarizes our conclusions.</p><br><p><b>DOWNSIDE RISK AND RISK ALLOCATION</b></p><p>We assume that the investor wants to maximize the expected return acrossdifferent hedge fund strategies subject to a constraint on the maximum allowablerisk. This portfolio optimization problem can be formulated mathematically as</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)</p><p>where w is the vector of portfolio weights, μ<i>'w</i> equals the expectedreturn of the portfolio, <i>R(w)</i> indicates the risk measure used to measureportfolio risk, and <i>R</i><sub>max</sub> is the overall risk budget. Weassume that the portfolio weights are nonnegative and sum to 100 percent.</p><p>Most approaches to risk budgeting are based on volatility and value at risk(VaR) and rely on the assumption that returns are serially uncorrelated andnormally distributed. While these two assumptions are convenient and allow forsimple formulas for the optimal allocation of risk, they may not be realisticfor hedge fund returns.</p><p>We consider several hedge fund indexes from HFRI. <b>Table 1.1</b> shows theskewness, kurtosis, and serial correlation for various hedge fund strategiesbased on monthly historical returns from January 1990 to December 2007. As thenumbers in <b>Table 1.1</b> show, the assumption that hedge fund returns arenormally distributed and serially uncorrelated is violated in many cases. Thishas been observed by several others in the literature. Typically, however, theseviolations are dealt with through ad hoc adjustments, e.g., to the returns or byinflating historical volatilities to make the resulting return streams closer tothat of a normal distributed and independent and identically distributed(i.i.d.) variable.</p><p>In the presence of positive serial correlation, volatilities tend beunderestimated and Sharpe ratios overestimated. When returns are seriallycorrelated, we can no longer annualize monthly volatilities by multiplying withthe ?12. An extensive discussion and some analytical results on the effect ofserial correlation on Sharpe ratios can be found in Lo (2002).</p><p>In this chapter we are concerned with investment processes that may generatereturn streams that have outliers (indicated, for example, by skewness and tailsthat are fatter than for a normal distribution). In these situations, volatilityis no longer a meaningful indicator of risk, and using it to determine anoptimal hedge fund portfolio may result in a misallocation of risk. Value atrisk is often advocated in the investment industry as a risk measure that issuperior to volatility as it captures potential downside risk. Value at risk hasseveral weaknesses, however, that make it a poor risk measure for the purpose ofrisk budgeting.</p><p>These weaknesses are widely documented in the literature [see, e.g., Artzner etal. (1999)]. Probably, the most severe drawback is that VaR, in general, is nota coherent risk measure in a sense defined by Artzner et al. (1999). Value atrisk does not, in general, satisfy the property of subadditivity defined byArtzner et al. (1999). This means that diversification may, in fact, increaserisk. This is a serious shortcoming when we try to allocate risk. Basak andShapiro (2001) show that the optimal portfolio that maximizes expected utilityover wealth subject to a constraint on VaR results in adverse investmentbehavior, and may, e.g., increase extreme losses in the tail. Finally, VaR is,in general, a nonconvex and nonsmooth function. This can lead to multipleoptima, which creates problems for portfolio optimization.</p><p>In this chapter we use CVaR. Basak and Shapiro (2001) also consider an investorthat maximizes expected utility over wealth subject to a constraint on CVaR.They show that the optimal portfolio is much better behaved compared to aninvestor that constrains the VaR of the portfolio. Conditional VaR is aconditional tail expectation. In general, CVaR is the weighted average of VaRand losses exceeding VaR. As a result, CVaR always exceeds VaR and is useful inquantifying risks beyond VaR, taking into account both the likelihood and themagnitude of possible losses. Conditional VaR should be considered superior toVaR for quantifying downside risks. Unlike VaR, it is easily verified that CVaRis a coherent risk measure and consistent with risk-aversion (second-orderstochastic dominance). In addition, CVaR is a smooth and convex function, makingit better suited for the purpose of portfolio optimization. Finally, Rockafellerand Uryasev (2000; 2002) have shown that mean-CVaR optimization can beformulated as a linear programming problem that can be solved easily even in thecase of nonnormal return distributions. (The linear programming formulation isprovided in Appendix A.)</p><p>We are interested in determining the optimal risk allocation from portfoliooptimization problem (1) with <i>R (W) CVaR</i> = -E(r' w | r' w ≤VaR<sub>α</sub>). In the special case when returns are normallydistributed with mean μ and volatility σ, the α% CVaR over aninvestment horizon <i>T</i> can be calculated as</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>where [xi], = <i>N</i><sup>-1</sup> (α) and where <i>N</i>(·) denotesthe cumulative normal distribution function and ψ(·) denotes the normaldensity function. For example, the 95 percent CVaR for one year for a portfoliowith annual mean and volatility μ and σ, is given by -μ + 2.06σ (in contrast, the 95 percent VaR equals -μ + 1.64 σ).</p><p>For a time series of returns of length <i>N</i>, the α% CVaR can becalculated as</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>The confidence level α effectively determines how much weight the investor putson tail risk. When α -> 0, the α% CVaR will be equal to the averagereturn. When α v 1, the α% CVaR will be equal to the worst-casereturn.</p><p>As mentioned above, a portfolio optimization with CVaR constraints can bereformulated as a linear programming problem. The linear programming formulationallows an investor to maximize the expected portfolio return subject to CVaRconstraints while simultaneously calculating the VaR of the optimal portfolio.Linear programming problems can be solved easily with most commerciallyavailable optimization packages. <b>Table 1.2</b> shows the monthly 99 percentVaR and CVaR for several hedge fund strategies for historical monthly returnsfrom January 1990 to December 2007.</p><p>To get a better sense of how much tail risk could be underestimated by using anormal distribution for hedge fund returns, we calculated the ratio of the 99percent CVaR based on the historical distribution to the 99 percent CVaR basedon a normal distribution with a mean and volatility equal to the historical meanand volatility. <b>Figure 1.1</b> shows this ratio on the vertical axis versusthe Jarque-Bera statistic (on a logarithmic scale) on the horizontal axis.Clearly, the more returns deviate from a normal distribution, the more risk isbeing underestimated by a normal distribution.</p><p>A nice property of mean-CVaR optimization models is that when returns arenormally distributed and serially uncorrelated, the optimal risk allocations areidentical to those derived from mean-variance optimization. In other words,mean-variance analysis is a special case of the mean-CVaR optimization.</p><p>Some investors may shun the use of risk measures such as CVaR for fear that itwill require a huge investment in new technology to upgrade their current risksystems that rely heavily on VaR. We believe that CVaR will eventually take overthe popularity of VaR in the investment industry. While many banks andinstitutional investors have heavily invested in VaR technology in the lastseveral years, upgrading existing VaR-based risk systems to CVaR is relativelycheap. The CVaR for a portfolio can be calculated as the average of several VaRnumbers for the portfolio at different confidence levels. If risk systems canspit out VaR numbers for a portfolio at different confidence levels, it isrelatively easy to calculate (or approximate) the CVaR of the portfolio.</p><br><p><b>MODELING HEDGE FUND RETURN DISTRIBUTIONS</p><p>The Simulation Framework</b></p><p>When constructing an optimal portfolio across a wide range of hedge fundstrategies, several approaches can be taken. The simplest approach is to run anoptimization based on historical returns. By now, we indeed have a reasonablylong history of hedge fund returns, and we will present historical optimizationresults in the section Optimal Hedge Fund Allocations. The historical approachis naïve however. Historical returns are not necessarily representative goingforward. As the hedge fund industry grows and matures, it is reasonable toexpect lower alphas in the future due to, e.g., increased competition. Inaddition, historical hedge fund data potentially suffers from several biases,such as survivorship bias or backfill bias. <i>Survivorship bias</i> means thathedge funds that run out of business may disappear from the database.<i>Backfill bias</i>, on the other hand, refers to the fact that hedge fundscan voluntarily report their performance to databases and typically reporthistorical performance after having had a good run.</p><p>We need a more rigorous approach to determine optimal hedge fund portfolios. Ourmain considerations are the following: While hedge funds follow activestrategies, their performance cannot be entirely attributed to skill (i.e.,alpha, or uncorrelated excess returns). Several studies have shown that asignificant portion of the returns on various hedge fund strategies can beexplained by the returns on stocks and bonds. For each hedge fund strategy, webuild a simple factor model with both a beta component linked to the performanceof public asset classes and an alpha component that is independent from the betafactors.</p><p>The sensitivity of hedge fund returns relative to the returns on public assetclasses is not constant over time. Style drifts and changing bets, among others,explain why it would be unrealistic to assume constant betas exposures. We applya Kalman filter methodology to capture this time-varying behavior. We model thereturn of hedge fund strategy <i>h</i> in period <i>t</i> as follows:[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where α<sub>h.t</sub>is the time-varying alpha component of strategy <i>h</i> and the other twocomponents represent the systematic beta components. This means that theperformance in every period can be attributed partially to the performance ofequity and fixed-income markets as well as to the skill and success of the fundmanager.</p><p>To properly quantify tail risk, we need to model the returns on fixed income andequities in a realistic fashion. We use regime switching models to estimate andsimulate the return processes of these two asset classes. Regime switchingmodels do a reasonable job at fitting the empirically observed skewness andkurtosis in asset returns. By linking hedge fund returns to the returns onpublic asset classes, they will also inherit the regime switching property, andthus, potentially exhibit asymmetric and leptokurtic distributions.</p><p><i>(Continues...)</i>