An Alternative Performance Measure · the distributional properties of this portfolio present an excellent basis for evaluating the performance of these funds. The underlying notion

An Alternative Performance Measure ∗

Alexandre Hocquard, Nicolas Papageorgiou, Bruno Remillard,

HEC Montreal

October 7, 2009

Abstract

In this paper, we present a new alternative performance measure (APM) whichevaluates not only for the marginal distribution of a given fund but also its’ dependence(correlation) with a reference portfolio. This performance measure is of particular valuein assessing hedge fund return as the latter are selected not only for yield enhancementbut also for their diversification benefits. The methodology is adapted from Americanoption theory and based on earlier work of Dybvig (1988), Kat and Palaro (2005)and Papageorgiou et al. (2008). It offers a unique metric for hedge fund performanceevaluation.

∗Corresponding author: Nicolas Papageorgiou, Finance Department, HEC Montreal, 3000 Cote Sainte-Catherine, Montreal,QC, H3T 2A7, Canada. All the authors are at HEC Montreal and can be reached [email protected]. This work is supported in part by the Fonds pour la formation de chercheurset l’aide a la recherche du Gouvernement du Quebec, by the Natural Sciences and Engineering ResearchCouncil of Canada and by Desjardins Global Asset Management.

1

1 Introduction

The impressive growth of the hedge fund industry, with all its’ well publicized success storiesand spectacular failures, has naturally led to an increased scrutiny of fund managers andof their investment strategies. Unfortunately, the performance measures proposed in thefinancial literature so far have proven to be ineffective at providing a true picture of truehedge funds performance. Assumptions about the underlying distribution of hedge fundreturns and/or about their systematic risk exposures are the most common drawbacks ofthe performance measures proposed to date. Efforts by several authors to adapt traditionalperformance measures in order to account for the irregularities of hedge fund returns haveprovided limited improvements. This includes deriving Sharpe ratios that account for highermoments (Mahdavi (2004) and Lo (2002)) and factor models that incorporate non-linearities(i.e. Fung and Hsieh (2001), Agarwal and Naik (2004)). Although these improvements offersome added insight into hedge fund performance, they are still unreliable and cannot accom-modate the diversity of hedge fund strategies. This paper re-examines the issue of hedgefund performance from a very different angle. We distance ourselves from the traditionalapproaches inherited from the mutual fund literature, and propose a much more robust andflexible performance evaluation tool that finds its roots in the option pricing literature. Wepresent a new performance measure which builds on earlier research by Dybvig (1988) andPapageorgiou et al. (2008). We no longer concern ourselves with the actual time-series, butrather with the statistical properties of the fund over a period of time. This performancemeasures provides a unbiased and complete picture of hedge fund performance since it con-siders the entire distribution of returns as well as the funds correlation with other assets.

The regulatory freedom that hedge fund managers enjoy coupled the asymmetricincentive fee structure, naturally leads to hedge fund strategies that exhibit non-normal re-turn distributions. As a result, Bernardo and Ledoit (2000) demonstrate that traditionalperformance measures, such as the Sharpe ratio, lead to results that are severely misleading.Grinblatt and Titman (1989) show how managers can generate positively skewed distribu-tion by selling call options on the index, thereby improving their Jensen’a alpha. For certainhedge fund strategies, the resulting skewed distributions occur naturally as a consequenceof the type of risk exposures taken by the manager. In particular, Mitchell and Pulvino(2001) show that the resulting payoff structure of merger arbitrage funds resembles sellinguncovered index put options and Fung and Hsieh (2001) find that systematic trend-followersessentially produce the payoff diagram of a lookback straddle. For other funds, however, theskewed distribution might also result from managers attempting to ”game” the performancemeasures. If managers are aware of the bias that a non-normal payoff structures induce inthese measures they will have incentives to select strategies that maximize a given perfor-mance measure rather than choosing an investment strategy that is optimal for the investor.In a recent paper, Goetzmann et al. (2006) demonstrate how to manipulate widespread, tra-

2

ditional performance measures and propose a manipulation-proof metric. However, whilethey provide a theoretical demonstration of how to ”game” a given performance measure,little is known on whether or not the implementation of such strategies would be successful.

Given the failings of the market model of Sharpe (1964), Lintner (1965) and Mossin(1966) and its related performance measures to provide a viable benchmark, researchers haveturned towards Sharpe (1992) style-based analysis to extract information about hedge fundreturns. The underlying idea is to try and separate the returns that are due to systematicexposure to risk factors (beta returns) from those that are due to manager skill (alpha re-turns). Early papers by Fung and Hsieh (1997), Liang (1999), Ackermann et al. (1999), andAgarwal and Naik (2000) obtain low and very variable R-squares. Certain hedge fund charac-teristics explain some of the difficulties in identifying a well specified linear model. The use offinancial derivatives, leverage, dynamic trading strategies and the asymmetric performancefee structures are some of the most obvious sources of non-linearities in hedge fund returns.Papers by Fung and Hsieh (2001), Agarwal and Naik (2004), Mitchell and Pulvino (2001),Lambert et al. (2009), Diez de los Rios and Garcia (2005) improve the explanatory powerof factor models by including risk premia that attempt to account for these non-linearities.Nonetheless, the heterogeneity of hedge funds both within and across styles makes it verydifficult to find a general factor model that is well-specified and spans the different strategies.

This paper proposes a novel approach to hedge fund performance evaluation. Byfocusing on the distribution of returns rather than on the actual time-series, and by utilizingtools from option pricing theory, the Alternative Performance Measure (APM) can be usedto directly compare funds that possess very different characteristics. Furthermore, the APMevaluates not only the marginal distribution of a given fund but also considers its’ correlationwith other assets, thereby incorporating the diversification benefits of the fund into theperformance measure.

The paper will be structured as follows. Section 2 explains the intuition underlyingthe APM, while section 3 details the implementation of the model. Section 4 describes thedifferent performance measures used in this study. Section 5 presents the numerical resultsand Section 6 concludes.

2 The Alternative Performance Measure

The Alternative Performance Measure addresses the many dimensions that make the per-formance evaluation of alternative assets so challenging. Hedge fund investors are not onlyconcerned with risk/return properties of the fund, but increasingly expect low correlationwith their portfolio of traditional assets. The performance measure must account not onlyfor the stand-alone properties of the specific fund but also its’ diversification benefits. Theperformance evaluation of the fund must encapsulate all the benefits that the fund offers to

3

the investor.

This differs from traditional performance measures which focus purely on the risk/returncharacteristics of the fund. Our approach also dissociates itself from the factor model ap-proach, which aims to identify a relationship between the time-series of the fund and somesystematic risk factor(s). We will no longer concern ourselves with the actual time-seriesof returns. Instead, we will consider the wider picture: we will evaluate the distributionalproperties of the fund.

The Alternative Performance Measure finds its roots in the option pricing literature.The most effective way to compare options that have different characteristics (maturity, strikeprice, underlying characteristics, exotic patterns...) is to price the options and derive thefair market value for each one. The option with the most valuable features should obtain thehighest price. Over the last twenty years, academics and practitioners alike have developedmany techniques to price and hedge American options, exotic options, or even real options.The literature has proposed many innovations to overcome the simplifying assumptions ofBlack and Scholes framework, such as the family of GARCH models, stochastic volatilitymodels, and Markov chains.

A further novelty with APM is that the option payoff that we seek to price is writtenon the distributional properties of the underlying security. The option is basically the payoffthat transforms the statistical properties of the underlying security into the statistical prop-erties of the hedge fund that is being evaluated. In reality, what concerns the investor is notthe strategy itself but rather the ultimate payoff of the strategy (i.e. his terminal wealth). Ahedge fund should be evaluated on his capacity to deliver two specific characteristics. First,the ability to generate end-of-month returns that satisfy criteria regarding excess returns,volatility, skewness, kurtosis (the marginal distribution), and second the ability to respecta specific dependence property with the investor’s portfolio. A hedge fund investment, likeany alternative investment,should be viewed as a payoff structure that delivers those proper-ties (marginal return distribution and dependence) at the end of a predefined horizon. Theidea of evaluating the distributional characteristics of a fund or asset was first put forth byDybvig (1988) with the Payoff Distribution Model and later extended to a bi-variate settingby Kat and Palaro (2005). The bivariate Payoff Distribution Model represents an inter-esting contribution to the performance evaluation and asset pricing literature, however theimplementation is quite challenging. Papageorgiou et al. (2008) propose a methodology forcalculating and replicating the payoff function that overcomes several of the inconsistenciesin Kat and Palaro (2005).

Of course, in order to price any payoff function, we need to identify the underlyingsecurity on which the payoff is written. This usually is a trivial decision, however in thecase of hedge funds, there is no representative investible index on which we can base ouranalysis (unlike, for example, the S&P 500 for equity Mutual funds). Therefore, we will needto derive a tradable benchmark portfolio that captures the typical beta exposures for the

4

hedge fund industry. Although researchers have been unsuccessful at deriving multi-factormodels that explain hedge fund returns on an individual level, Hasanhodzic and Lo (2007)show that on an aggregate basis it is feasible to specify a model that captures the typicalexposures of the hedge fund industry. Although the time-series properties of this multi-factorbenchmark might be quite different to those of many hedge fund strategies, we believe thatthe distributional properties of this portfolio present an excellent basis for evaluating theperformance of these funds. The underlying notion is that hedge fund returns are generatedby exposure to these beta factors, however the dynamic nature of hedge fund investingdistorts the time-series properties of the beta returns. In effect, these distortions have thesame impact as entering into a specific derivative contract on the underlying benchmarkportfolio. We will therefore evaluate each fund by pricing the payoff that it generates withrespect to this benchmark portfolio. This will allow for a direct comparison across fundsthat have potentially very different properties.

The APM prices the payoff by deriving, using the properties of benchmark portfolioand the investor’s portfolio, the cheapest dynamic trading strategy that produces the samepayoff as the hedge fund. By pricing this bi-variate payoff, we obtain a performance measurethat perfectly reflects the entire benefits of the hedge fund investment. In addition, if theunderlying security or benchmark portfolio is tradable, we can derive the hedging strategythat replicates the hedge fund’s payoff function.

3 Implementation of the APM

The implementation of the APM can be broken down into three separate stages, specificallythe modeling of the distributions and dependence structures, the calculation of the payofffunction, and finally the selection of the hedging/dynamic replicating strategy.

3.1 Modeling the returns

There are three marginal distributions (hedge fund, benchmark portfolio and investor port-folio) and two dependence structures (hedge fund/investor portfolio and benchmark/investorportfolios) that need to be modeled. Let us recall that the APM evaluates the hedge fundby generating it’s statistical properties (marginal distribution and correlation with the in-vestor’s portfolio) from the joint distributional properties of the investor portfolio and thebenchmark portfolio. More generally, this implies that given two risky assets S(1) and S(2),it is possible to “reproduce” the statistical properties of the joint return distribution of assetS(1) and a third asset S(3). In our case, asset S(1) is the investor portfolio, asset S(2) is abenchmark portfolio and asset S(3) is the hedge fund. In order to provide a robust solutionin this framework, we propose the following approach to model the returns of three assets.

5

3.1.1 Joint returns of hedge fund and investor’s portfolio

The first step consists in estimating the monthly distribution of the hedge fund returns aswell as the dependence between the hedge fund and the investor’s portfolio. There are noparticular restrictions regarding the choice of the distribution for the hedge fund or the copulafunction1 between the hedge fund and the investor’s portfolio, (C1,3). For the distribution ofthe hedge fund, we use the empirical cumulative density function of the selected fund. Fora robust estimation of the copula function C1,3, it is preferable to use normalized ranks, i.e.if Ri1 represents the rank of Yi among Y1, . . . , Yn and if Ri2 represents the rank of Zi amongZ1, . . . , Zn, with Rij = 1 for the smallest observations, then set

Ui =Ri1

n + 1, Vi =

Ri2

n + 1, i = 1, . . . , n.

To estimate θ one aims to maximize the pseudo-log-likelihood∑i=1

log cθ(Ui, Vi),

as suggested in Genest et al. (1995). For example, if the copula is the Gaussian copulawith correlation ρ, the pseudo-likelihood estimator for ρ yields the famous van der Waerdencoefficient defined to be the correlation between the pairs {Φ−1(Ui), Φ

−1(Vi); i = 1, . . . , n}.For other families that can be indexed by Kentall’s tau, e.g., Clayton, Frank and Gumbelfamilies, one could estimate the parameter by inversion of the sample Kendall’s tau. See,e.g., Genest et al. (2006).

Finally, to test for goodness-of-fit, one can use Cramer-von Mises type statistics forthe empirical copula or for the Rosenblatt’s transform. The latter could be the best choicegiven that ∂

∂uC1,3(u, v) needed to be calculated for the evaluation of the payoff function.

These tests are described in Genest et al. (2007) and in view of their results, we recommend

to use the test statistic S(B)n .

3.1.2 Joint returns of investor’s and benchmark portfolios

Next, we model the joint monthly distribution of the investor’s portfolio and the bench-mark portfolio using bivariate Gaussian mixtures. This family of mixtures provides uswith the required flexibility to appropriately model the joint distribution of the two port-folios. Other possibilities for modeling the joint distribution include certain multi-variateGARCH processes (see Hafner and Rombouts (2007)), or Markov Switching models. UnlikePapageorgiou et al. (2008), our goal is not to perform a study on the replication properties of

1Copula functions are used to model the dependence as they offer greater flexibility (such as tail-dependence) then traditional correlation measures

6

the model, we do not need to model the joint daily distribution of the two portfolios. Notethat the fact that Gaussian Mixtures have known aggregation properties is an importantfeature. This means that if we chose to implement a daily hedging strategy that replicatesthe monthly payoff, the distribution of the joint-monthly returns of the investor portfolio andthe benchmark portfolio would be consistent with the joint-distribution of the daily returns.

In order to choose the optimal number of regimes, we first need to estimate theparameters of the model, and then provide a goodness-of-fit test to evaluate whether agreater number of regimes is required. The estimation method is based on the EM algorithmof (Dempster et al., 1977) and is presented in Appendix B.1 for the bivariate case. A newgoodness-of-fit test, described in Appendix B.3, is then performed to assess the suitabilityand to select the number of mixture regimes m. The proposed test, based on the work inGenest et al. (2007), uses the Rosenblatt’s transform.

3.2 Calculating the payoff function

The payoff function will allow us to transform the joint distribution of the investor’s portfolioand the benchmark portfolio into the bivariate distributions of the hedge fund and theinvestor’s portfolio. In Papageorgiou et al. (2008), the payoff’s return function g is shown tobe calculable using the bivariate distribution function F1,2 between the investor’s portfolioand the benchmark portfolio, the marginal distribution function of the targeted fund F3, and

the copulas C1,3 describing the dependence structure between the joints returns(R

(1)0,T , R

(3)0,T

)of the reference portfolio and the targeted fund. The exact expression for g is given by

g(x, y) = Q{x, P

(R

(2)0,T ≤ y|R(1)

0,T = x)}

, (1)

where Q(x, α) is the order α quantile of the conditional law of R(3)0,T given R

(1)0,T = x, i.e., for

any α ∈ (0, 1), q(x, α) satisfies

P{

R(3)0,T ≤ Q(x, α)|R(1)

0,T = x}

= α.

Using properties of copulas, e.g. Nelsen (1999), the conditional distributions can be expressedin terms of the margins and the associated copulas.

In our methodology, since the monthly returns(R

(1)0,T , R

(2)0,T

)are modeled by a Gaus-

sian mixtures with parameters (πk)mk=1, (μk)

mk=1 and (Ak)

mk=1, the conditional distributions

can be expressed as follows

P(R

(2)0,T ≤ y|R(1)

0,T = x)

=

m∑k=1

πk(x)φ{y; μk(x), σ2}

7

where πk(x), μk(x) and σ2 are given by formulas (2) and (3).

πk(x1) =πkφ(x1; μk1, σ

2k1)∑m

j=1 πjφ(x1; μj1, σ2j1)

(2)

andμk(x1) = αk + βkx1, σ2

k = σ2k(1 − ρ2

k). (3)

3.2.1 Univariate Payoff

As detailed in Amin and Kat (2003), the univariate payoff function can be rewriten to re-lax the correlation feature and focus only on the marginal distribution of the targeted fundF (3). The authors show that given an underlying asset S(2) with monthly returns R(2) anda targeted distribution to deliver F (3), it is possible to “evaluate” the statistical propertiesof the returns at time T (end of month).

There exists a function g such that the distribution of g(R(2)

)is the same as the

distribution F (3). This payoff’s return function g is easily shown to be calculable using thedistribution function F (2) of the underlying asset and the marginal distribution function ofthe targeted distribution F (3).

The exact expression for g is given by

g(x) = Q{P

(R(2) ≤ x

)}, ; ∀x ∈ R (4)

where Q(α) is the order α quantile of the distribution F (3).An other notation for g is:

g(x) =(F (3)

(F (2)(x)

))−1, ; ∀x ∈ R (5)

Only marginal distributions are needed in a univariate pricing model, which can bechosen as a gaussian mixture distribution.

These payoff functions g and g falls in the same category of more classical knownpayoffs such as put and call options except than instead of being written on the underlyingprice, they are written on the underlying monthly return. This implies a more adaptedpayoff to integrate the whole risk profile of the underlying returns density for g, and for gthe dependance structure with the investor’s portfolio also. We will mainly focus on thederivation of the bivariate function g since the Alternative Performance Measure tries toevaluate the benefits of adding an alternative investment to a given portfolio. In order to

8

evaluate the payoff value, one need to develop a pricing and hedging algorithm which isconsistent with a discrete time hedging.

3.3 Pricing the Payoff

Having solved for the payoff function g, we need to price the payoff function. In order toachieve this, we implement the hedging approach proposed in Papageorgiou et al. (2008).Note that there is no ”risk-neutral” evaluation involved in our approach and that all calcu-lations are carried out under the objective probability measure. The complete descriptionof the hedging and pricing is provided in Appendix C. The algorithm provides us with theprice V0 and the hedging strategy ϕ. In effect, the idea is to solve for the portfolio (V0, ϕ)such as to minimize the expected square hedging error

E[β2

T {VT (V0, ϕ) − CT}2] ,

where βT is the discount factor and CT = 100 exp{g

(R1

0,T , R20,T

)}is the payoff at maturity.

Note that the price V0 represents the premium at time t = 0 for a payoff equivalentto an investment in fund S(3). That is, over a number of months, the payoff delivers boththe marginal distribution F (3) and dependence structure C1,3.

• If V0 > 0: the option is more expensive than the fund, indicating that there is more tothe strategy than systematic dynamic exposures to readily available market premia.

• If V0 < 0: the option is less expensive than the fund, indicating that the fund is annot an efficient investment vehicle. Investors can obtain better returns (for the samehigher moments and correlation with their portfolio) by dynamically replicating thepayoff of the option.

This methodology delivers a fair price for the fund investment opportunity. V0 is aperformance measure that allows us to compare investments in various funds by rankingthem in terms of price. This measure is specific for a given investor because of its propertyof reflecting the dependence structure with a given reference portfolio.

4 Constructing the benchmark and Investor portfolios

In this section we will examine how the components of the two tradable portfolios are selected,and show a couple of illustrations of how exactly the model works.

9

4.1 The benchmark portfolio

The choice of the benchmark portfolio is a crucial step in evaluating the performance. Un-fortunately there are no representative investible indices for the hedge fund industry, sowe attempt to identify a multi-factor model, using investible securities, that best explainsthe returns for the HFR Fund Weighted Composite Index. The model must capture thetypical exposures of hedge fund managers in a variety of asset classes, including equity,bonds, commodities and foreign exchange. The model is selected such as to maximize theadjusted R-square over the sample period. We are not particularly concerned with issues ofmulti-collinearity as we are not interested in testing the significance of any specific factornor the significance of the resulting alpha coefficient, but rather we want a model that iswell-specified and characterizes the aggregate hedge fund exposures. The selected modelsatisfies:

HFRt = αt +n∑

i=1

βi,tXi + εt (6)

Where HFR is the the HFRI Fund Weighted Composite Index monthly return series,Xi the i − th factor monthly return series and εt the error term. The α and β coefficientsare estimated through a classical unconstrained linear regression routine. The factors thatare selected for the benchmark portfolio are the S&P 500, Nasdaq 100, MSCI Europe AsiaFar East (EAFE), MSCI Emerging Markets, Goldman Sachs Commodity Index, Dollar SpotIndex, Lehman US Aggregate Bond Index and the spread between the Russell 2000 and S&P500. (1 − ∑n

i=1 βi) is invested in a US Libor-1 month contract to ensure financing.

All indices are total returns indices (assuming a full reinvestment of the paid dividendsin the index). All monthly data series are extracted from the Bloomberg Database. Notethat the Lehman US Aggregate index is now available under Barclays license. The portfolioweights for the sample period and the two sub-samples are presented in table 3. The adjustedR-square is around 85%, with equity making up close to 50% of the overall exposure of theHFR Fund Weighted Composite Index.

4.2 The investor’s portfolio

The investor’s portfolio is a completely exogenous choice, and is dictated by the holdingsof the investor. To illustrate our methodology and its dependance valuation properties, weselect the investor portfolio so as to resemble a typical conservative US pension fund, with anallocation of 60% in the S&P 500 Total Return index and 40% in the Lehman US AggregateBond index.

10

4.3 Example of Payoff function

To illustrate our methodology, we will present both the univariate payoff function g andthe bivariate function g for two HFR indices using the selected benchmark portfolio andinvestor’s portfolio. The two indices are the HFR Fund Weighted Composite Index and theHFR Short Biais Index from January 1990 to December 2008. These two indices are selecteddue to their very different characteristics, thereby enabling us to demonstrate the flexibilityof the methodology. We will look at the marginal distributions of both funds, the univariateand bi-variate payoff functions and illustrate the impact of the correlation dimension on theprice of the payoff.

Table provides the descriptive statistics for the two indices as well as for the bench-mark

Table 1: Monthly Statistics for 1990 − 2008

Fund Mean Std.Dev. Skew. Kurt. Min. Max. Corr.HFRI Fund Weighted Composite 0.98% 2.05% -0.79 5.90 -8.70% 7.65% 71.77%HFRI Short Bias 0.25% 5.61% 0.18 5.30 -21.21% 22.84% -66.65%Benchmark Portfolio 0.35% 1.88% -1.22 6.63 -9.07% 4.65% 78.65%Investor’s Portfolio 0.62% 2.60% -0.92 5.55 -11.99% 7.67% 100.00%

The gaussian mixture estimation are presented in the appendix in table 14. Univariatedistributions are inferred from the bivariate parameters. The marginal inverse densities ofthe HFR indices are computed using their empirical cumulative density function.

4.3.1 Univariate payoff

Figures 1 and 2 plot the end-of-month payoff functions for both HFR indices against themonthly returns of the benchmark portfolio. Basically, these plots tell us the required returnfor the HFR index for any given return level of the benchmark portfolio.

The replicated payoff kernel density can be mapped with both the HFR Indices targetand the benchmark portfolio used in the hedging process.

We observe in these payoff structures that the HFR Short Biais Index exhibits morevolatility than the HFR Fund Weighted Composite Index (the range of possible outcomes isfar greater). The steep slope on the negative outcome returns of the HFR Fund WeightedComposite Index illustrates the important negative skewness of the index. The replicatedpayoffs illustrate the ability of the methodology to distort the original properties of the

11

−4% −2% 0% 2% 4% 6% 8%−10%

−8%

−6%

−4%

−2%

0%

2%

4%

6%

8%

Monthly Benchmark Portfolio Return

HF

RI F

W P

ayo

ff M

on

thly

Ret

urn

Figure 1: HFR Fund Weighted Payoff g

−4% −2% 0% 2% 4% 6%−40%

−30%

−20%

−10%

0%

10%

20%

30%

Monthly Benchmark Portfolio Return

HF

RI S

ho

rt B

iais

Mo

nth

ly R

etu

rn

Figure 2: HFR Short Bias Payoff g

benchmark portfolio returns in order to generate the targeted marginal distribution of theselected Hedge Fund Index by dynamically trading the benchmark portfolio.

12

−20% −15% −10% −5% 0% 5% 10% 15%0

5

10

15

20

25

30

Monthly Return

HFRI FWReplication PayoffBenchmark Portfolio

Figure 3: HFRI Fund Weighted replicated payoff (univariate)

−40% −30% −20% −10% 0% 10% 20% 30% 40%0

5

10

15

20

25

Monthly Return

HFRI Short Biais

Replication Payoff

Benchmark Portfolio

Figure 4: HFRI Short Biais replicated payoff (univariate)

4.3.2 Bivariate payoff

The bi-variate end-of-month payoff function for the HFR Fund Weighted Composite Indexand HFR Short Bias Index are presented in figure 5 and 6. The payoff are expressed inreturns relative to the end-of-month returns of the benchmark and investor portfolios.

13

Figure 5: HFRI Fund Weighted Payoff g

Figure 6: HFRI Short Bias Payoff g

14

4.3.3 Pricing the payoffs

In table 2 we present the price (premia) associated with both the univariate and bivariatepayoffs of the two HFR indices. The results for all the HFR indices ar presented in theresults section.

Table 2: APM values: Univariate and Bivariate premia

Fund Univariate BivariateHFRI Fund Weighted Composite 0.590% 0.450%HFRI Short Bias -0.061% 0.552%

For the HFR Fund Weighted Composite Index, the univariate payoff value is 0.59%,signifying that there is a 59 bps monthly positive premium in investing in the HFR FundWeighted Composite Index rather than trying to replicate its statistical properties througha hedging protocol using the selected benchmark portfolio. The APM value (bi-variate) is0.450%, implying a 45 bps monthly positive premium in investing in the HFR Fund WeightedComposite Index rather than trying to replicate its statistical properties and its dependancestructure with a reference investor’s portfolio through a hedging protocol using the selectedbenchmark portfolio. The difference in the cost of the uni-variate and bi-variate payoff is14 bps per month. It is interesting that accounting for the correlation actually reduces thepremium related to the strategy, indicating that the HFRI fund weighted index does notoffer much value in terms of diversification to the investor’s portfolio..

The pricing of the univariate and bi-variate payoffs for the HFR Short Bias indexhighlights the importance of accounting for correlation. The univariate payoff value is -0.061%, signifying that there is a negative 6.1 bps monthly premium for HFR Short BiasIndex, whereas the APM value (bi-variate) is 0.552%, implying a 55.2 bps monthly posi-tive premium in investing in the HFR Short Bias Index. By only considering the marginaldistribution of the Short Fund Index, we greatly underestimate the value of this strategy,specifically relating to it’s low (negative) correlation with our investor’s portfolio (or in otherwords, its diversification benefits).

15

5 Review of the performance measures

In order to highlight the effectiveness and uniqueness of the APM, we will contrast theresults to those obtained using traditional performance measures, specifically the Sharperatio, the Omega ratio and a regression based alpha. We select these measures becausethey are standards in the industry, have well known characteristics (and drawbacks) andthereby provide excellent comparative measures for our analysis. A quick review of howtheses performance measures are calculated is provided below, and then we provide someevidence, using simulated distributions, of the limitations of these measures.

5.1 The traditional performance measures

5.1.1 The Sharpe Ratio (SR)

The Sharpe ratio introduced by Sharpe (1966) is the most commonly used ratio in the financeindustry. The main attraction of this measure is that it is easy to calculate and interpret.The main assumption is that any asset class can be fully characterized in terms of risk-returnrelationship by the expected excess return and the variance of the asset class. Sharpe ratio isa measure of the mean excess-return per unit of risk, in a world where the return distributionof assets is Gaussian and risk is therefore symmetric.

The Sharpe ratio (SR) can be expressed as:

SR =

(E

[R(3)

] − Rf

)σR(3)

(7)

where E[R(3)

]is the expected return of the fund’s return R(3), Rf the risk free rate and

σR(3) the standard deviation of fund’s return R(3).

By construction the Sharpe ratio is neither sensitive to higher moments of the marginalreturn distribution R(3), nor to dependence structure characteristics with a investor’s port-folio S(1).

5.1.2 The Omega Ratio (Ω)

The Omega ratio introduced by Keating and Shadwick (2002) was developed to relax hy-pothesis of gaussian distribution from investment returns made by the Sharpe ratio. Infact, it is a well accepted fact of empirical finance and practical finance that returns are notdistributed normally. In addition to the first and the second moment, higher moments arerequired for a complete description of sources of risk. This measure leads to a full charac-terization of the risk reward properties of the distribution by measuring the overall impact

16

of all moments. It also incorporates the beneficial impact of gains as well as the detrimentaleffect of losses, given any individuals loss threshold.

Omega ratio (Ω) can be expressed as:

Ω(L) =

∫ b

L[1 − F3(x)] dx∫ L

aF3(x)dx

(8)

where x is the one period fund’s return R(3), L a threshold selected by the investor (couldbe Rf), and (a, b) the upper and lower bounds of the fund’s return distribution F3.

Omega could also be written:

Ω(L) =E [max(x − L, 0)]

E [max(L − x, 0)](9)

By construction the Omega ratio is sensitive to higher moments of the marginalreturn distribution R(3), but not affected by the dependence structure characteristics with areference portfolio S(1).

5.1.3 Multi-Factor Alpha

The alpha is estimated by regressing the excess retun of the HFRI indices against the eight-market factors that were selected for the benchmark portfolio. The factors are the S&P 500,Nasdaq 100, MSCI Europe Asia Far East (EAFE), MSCI Emerging Markets, Goldman SachsCommodity Index, Dollar Spot Index, Lehman US Aggregate Bond Index and the spreadbetween the Russell 2000 and S&P 500.

HFRj,t − Rft = αj +

n∑i=1

βji,t(Xi,t − Rft) + εj,t (10)

With Rf the US Libor 1-month risk free rate. The αj measures the excess returnthat cannot be attributed to systematic exposure to the different risk factors.

5.2 Comparison between performance measures

In this section, we will attempt to demonstrate, using ”fictitious” funds, the appeal of theAPM relative to other more established performance measures. This will allow us to highlightthe some of the drawbacks of traditional performance measures, and demonstrate that theyprovide an incomplete picture regarding the true performance of alternative investments. Webasically want to demonstrate, using a very simple framework, some of the well-documented

17

limitations of the traditional performance measures. In order to achieve this, we will rankseveral funds with different characteristics using three of the performance measures, specif-ically the Sharpe ratio, the Ω ratio and the APM. We cannot use the multi-factor alpha inthis example as we will be simply making assumptions about the distribution of the funds’returns not about the actual time-series. The marginal distribution of the funds’ returnswill be characterized by a Johnson distribution which allows us to assume varying levels ofasymmetry. We will also assume that the correlation between investor’s portfolio and thefund is characterized by a Gaussian copula. We will vary the mean (μ), volatility (σ), theskewness (sk) of the funds and the correlation (ρ) between the hedge fund and the investor’sportfolio. The Johnson distribution excess kurtosis is fixed to 2.

Specifically, these are the annualized parameters that can vary:

- μ can take values 5% and 10%.

- σ can take values 6% and 14%.

- sk can take values −1, 0 and 1

- ρ can take values −0.5, 0 and 0.5.

We will evaluate each measure for all combination of the parameters μ, σ, sk and ρ.This leads to 36 different funds. The threshold will be the risk free rate Rf with a fixedvalue of 5% per year. For the APM, the benchmark portfolio will be obtained by performingthe regression analysis on the HFR Composite Index.

Table 4 presents here the ranks of each fund for the Sharpe Ratio. As expected, thefunds with the best Sharpe Ratio are those with the highest mean and lowest volatility. Thismeasure does not account for higher moments, and therefore does not differentiate betweenfunds that offer different levels of skewness. This leads to potentially seriously flawed rank-ing, as negative skewness mis-evaluation for investments that appear to be non-risky in angaussian framework but with a very large potential downside risk (negatively skewed funds).This measure do not allocate value for diversification, which is a drawback for alternativeinvestment purpose.

The Omega ratio provides a more detailed ranking, as it makes no assumption aboutthe nature of the funds’s distribution. Table 5 presents the ranks of each fund using theOmega Ratio. Omega ratio allows us to differentiate investments with downside risk andupside potential, with a full characterization of the marginal distribution. As expected, thetop ranking fund is the one with the best mean-variance trade-off and positive skewness.Then ranking goes down along the skewness, as downside risk increases. Interestingly, when

18

we look at ranks 22 to 34, we note that for a given mean, it is not necessarily the fund withthe lowest volatility that has the best ranking. Occasionally, we observe that a fund with apositive skewness but a larger volatility is better than the fund with zero skewness and lowervolatility. Nevertheless, ranks are constant along correlation with the reference portfolio,and the benefits from diversification are not reflected with the Omega ratio.

The APM should now incorporate an extra dimension into the rankings, resultingfrom the complete characterization of the marginal distribution of the fund’s return and thedependence structure between the investor’s portfolio and the fund. Table 6 presents theranks of each fund using the APM. As expected, APM rankings are affected by all parame-ters, μ, σ, sk and ρ. In this particular setting for the investor’s reference portfolio and thebenchmark portfolio, the results are quite intuitive. Similar to the Omega ratio, all thingsbeing equal elsewhere, the APM gives a higher ranking to funds that exhibit positive skew-ness. Rankings are, however, also differentiates the rankings with respect to correlation withthe investor’s portfolio. The APM attributes greater value funds that have low correlationwith the investor’s portfolio. The APM demonstrates a greater excellent results for rankinga large variety of different funds, providing a much complete picture than the Sharpe andOmega ratios.

6 Empirical Results

Now that we have demonstrated the potential benefits of using a multidimensional perfor-mance measure, we apply this evaluation on HFRI indices. The empirical results will beorganized as follows. First we will evaluate each HFR index over the entire sample period(1990 − 2008). We rank the hedge fund indices using all 4 performance measures, and an-alyze the relative rankings. Next, we break the sample into two sub-samples, specifically(1990 − 1999) and (1999 − 2008) and undertake the same analysis. Finally, for robustness,we remove the final year of the sample (2008) and re-examine the latter sub-sample in orderto control for the impact of the recent financial crisis.

6.1 HFRI Index descriptions

The HFRI Monthly Indices are a series of benchmarks designed to reflect hedge fund industryperformance by constructing equally weighted composites of constituent funds, as reportedby the hedge fund managers listed within HFR Database. The HFRI range in breadth fromthe industry-level view of the HFRI Fund Weighted Composite Index, which encompassesover 2000 funds, to the increasingly specific-level of the sub-strategy classifications. In thispaper, we will focus on the 22 indices for which we have a sufficiently long data series. All

19

the information regarding the construction and criteria for inclusion in the different indicescan be found on the Hedge Fund Research website.2

6.2 Entire Sample Period 1990 − 2008

6.2.1 Descriptive Statistics

For each index, a description of their marginal distribution and dependence structure param-eters is presented in Table 8 and Table 9 respectively. The performance varies significantlyacross the hedge fund indices. The highest returns are attained by the HFRI QuantitativeDirectional, the HFRI Macro and HFRI Emerging Markets indices, with monthly returnsnear or above 120 bps per month. The weakest returns over the entire sample are generatedby the Short Bias Strategy with monthly returns of 25 bps. The Short bias also exhibitsthe greatest volatility of all the HFR indices. With the exception of the Macro strategiesand the Short bias funds, all the HFRI indixes exhibit negative skewness over the sampleperiod. This is consistent with evidence in the literature (Agarwal and Naik (2004), Huberand Kaiser (2004), Gupta and Liang (2005) and Jaeger and Wagner (2005)). The correlationbetween the HFRI indices and the investor’s portfolio (last column) is surprisingly consistentacross strategies. With the exception of the Short bias strategy (-66.65%), Defensive Fundof Funds (12.5%) and Equity Market Neutral (24%), the correlations are all in the 50%-70%range.

6.2.2 Ranking of HFR indices

The rankings according the four performance measures are presented in table 10. As wecan observe from the Spearman Rank correlation statistic between the four performancemeasures in table 11, the Omega measure and Sharpe ratio provide a quasi-identical rankingacross the different hedge fund indices. The rankings are however significantly different whenthey are based on the multi-factor regression alpha or on the APM.

The Sharpe and Omega measures, are both risk adjusted measures that do not at-tempt to dissociate systematic risk from idiosyncratic risk exposures, and therefore it is notterribly surprising that they generate a similar classification. We could have expected theOmega measure to punish the funds that exhibited negative skewness, however since almostall the funds are negatively skewed the impact on the ranking of accounting for the highermoments is limited. On a risk-adjusted basis, the Merger Arbitrage and Macro indices pro-vide the best performance. The Short bias index is the only one to have a negative excess

2www.hedgefundresearch.com

20

return, and the two layers of fees clearly impact the performance of the Fund of Fund indices.

The ranking based on the alpha and APM measures differ substantially from those ofthe other two performance measures. Let us recall that alpha will represent the excess returnthat cannot be explained by systematic exposure to the eight selected market factors, andthe APM represents the excess return with respect to the payoff that replicates the statisti-cal properties of the fund (marginal distribution and dependence) based on the benchmarkportfolio. All the alphas are positive and significant at the 99% confidence level, with theexception of the Fixed Income/Convertible Bond and Short Bias indices (these are conse-quently the two lowest ranking indices). The top strategy based on alpha is the EmergingMarkets Index which provides an excess return of 85 bps per month. This is closely followedby the two Macro and the strategies which generate excess returns of 79 and 69 bps permonth respectively. The Quantitative Directional Index, which is similar in style to a sys-tematic Macro, is right behind with 68 bps per month. Merger Arbitrage, which was rankedfirst with both the Omega and Sharpe ratios, drops to twelfth based on alpha (37 bps), indi-cating that much of the return from this strategy is attributable to systematic risk exposures.

The ranking from the APM distinguishes itself in a number of ways. Generally, theranking is closer to that of obtained using alpha rather than Omega or Sharpe. This isnot surprising as the APM and the alpha are rely on the same systematic risk factors inthe performance evaluation process. The results between the two measures are also easilycomparable as they are both quoted in terms of excess return. Let us recall than when wecalculate the alpha, we allow for the weight of each factor to be determined by the uncon-strained linear regression for each index, whereas for the APM, the weights are determinedby the HFRI Fund Weighted Composite. For the APM, we are not concerned with findingthe model with the best fit for each index, but rather determine how much of their returndistribution can be explained by a dynamic exposure to a static portfolio of market factors.One might expect that because we are not allowing the weights to be decided by the specificindices, the excess return would necessarily be higher, however this is not the case. In fact,there is no distinct pattern in the magnitude of the excess returns when we compare thealpha to the APM.

The most marking contrast relate to the Macro strategies and the Short Bias strat-egy. The APM attributes a excess return of 400 bps less than the alpha for the two HFRIMacro indices. Macro strategies are generally quite dynamic, and have been shown byFung and Hsieh (2001) to have payoffs that resemble straddles. The flexibility of the APMclearly does a better job of identifying and evaluating this dynamic exposure to the marketfactors. The other index whose performance evaluation changes dramatically is the ShortBias index. Since the APM replicates the dependence structure of the index with the in-

21

vestor’s portfolio, considerable importance is placed on the low (and in this case negative)correlation with the the stock/bond portfolio. In effect, the APM attributes one of thehighest rankings to this strategy over the entire sample period. It is interesting to note thatall the indices exhibit positive excess return, providing evidence that over the last 20 yearshedge funds have offered more than repackaged beta returns.

6.3 Sub-Sample Period: 1990 − 1999


For each index, a description of their marginal distribution and dependence structure pa-rameters is presented in Table 16 and Table 17 respectively. The highest returns are attainedby the HFRI Quantitative Directional (201 bps per month), the HFRI Equity Hedge (189bps per month) and HFRI Macro (167 bps per month). The weakest returns over the en-tire sample are generated by the Short Bias Strategy with an average monthly loss of 29bps. The Short bias also exhibits the greatest volatility of all the HFR indices. As for theentire sample period, most funds exhibit negative skewness over the first sub-sample pe-riod. The correlation between the HFRI indices and the investor’s portfolio (last column) isbetween 30% and 70% for all strategies, with the exception of the Short bias strategy (-65%).


The rankings according the four performance measures are presented in table 18. Onceagian, the Spearman Rank correlation statistic between the four performance measures intable 11 clearly demonstrate that the performance measures can be seperated into 2 classes.The Omega measure and Sharpe ratio provide a quasi-identical ranking across the differenthedge fund indices (SRC = 0.971) and the alpha and APM present correlation of 0.888.

The Sharpe and Omega rank the Merger Arbitrage, Event Driven and Equity hedgeas the top three strategies. On the other end of the spectrum, we find the Short Bias, Emerg-ing Market and Fund of Fund indices. When we account for the systematic risk exposureof the funds, the alpha attributes the best ranking to the Macro index (which was also themost common strategy in the 90s), followed by the Emerging Markets and Equity Hedge.It is perhaps a bit surprising to find the Emerging Markets in the top performers given theAsian crisis in the latter part of the period, however there was also an important run-up inprices prior to the crisis. The Equity hedge, which are generally directional equity strategies,tend to perform well in markets that are trending. The lowest alphas are attributed to the

22

Equity Market Neutral and to the Fund of Fund indices.

The HFRI indices that present the statistical properties that are the most costly to re-produce are the Quantitative Directional, Macro and Emerging Market strategies. Similarlyto the alpha rankings, the least interesting properties are generated by the Fund of Fund in-dices and the Equity Market Neutral strategies. Regardless of the performance measure, thetwo layers of fees on the Fund of Funds clearly impacts their performance. Notwithstandingtheir poor performance relative to their peers, Fund of Funds still seem to offer value to in-vestors as they offer positive excess return on the basis of both the alpha and APM measures.

6.4 Sub-Sample Period: 1999 − 2008


For each index, a description of their marginal distribution and dependence structure param-eters is presented in Table 24 and Table 25 respectively. Contrary to the first sub-sample,the highest returns are attained by the HFRI Short Bias index and the HFRI QuantitativeDirectional. This is not surprisingly given the significant drawdowns in both 2000 and 2008.The weakest returns over the second sub-sample are generated by the Fixed-Income (Con-vertible Abitrage) and the FOF indices (with the exception of the Defensive FOF index).The Short bias again exhibits the greatest volatility of all the HFR indices and once againmost funds exhibit negative skewness over the sub-sample period. The correlation betweenthe HFRI indices and the investor’s portfolio (last column) is more variable across the strate-gies. The Short bias offers a strong negative correlation (-67%), however Market Neutral,the Defensive FOF, and the Macro indices offer low correlations.


The rankings according the four performance measures are presented in table 26. Over thesecond sub-sample there is less disparity across the four measures, with the APM presentinga Spearman Rank correlation of approximately 0.8 with each of the three other measures(table 27).

The Sharpe and Omega rank the Defensive Fund of Fund and two Macro indices as thebest strategies. It is interesting to note that six strategies, specifically the FOF Diversified,FOF Conservative, FOF Diversified, FOF Composite, Fixed-Income/Convertible Arbitrageand Fixed Income/Corporate, all have negative excess returns over the second sub-sample.In contrast with the results from the first sub-sample, the alpha is only significant at the 99%

23

confidence level for 5 strategies, specifically the Systematic Diversified Macro, the EmergingMarkets, the Quantitative Directional, Event Driven and Merger Arbitrage. Negative alphas(although statistically not significant) are attributed to the Short Bias strategy and to thetwo Fixed-Income Strategies. It is interesting to contrast the adjusted R-squares for thedifferent strategies over the two sub-samples (tables 21 and table 29). We note these aregenerally higher in the second period, indicating a greater dependence on market factor ex-posures as a source of returns in the later period. The most significant increases in adjustedR-square are for the Relative Value (Fixed-Income and Multi-Strategy) whereas the onlysignificant decreases concern the Macro strategies.

The HFRI indices that present the most compelling statistical properties are a mixedbag of what we identified with the Sharpe, Omega and alpha measures. The SystematicDiversified Macro provides the return distribution that is the most costly to reproduce, fol-lowed by the Emerging Market and Distressed Debt strategies. Six indices have statisticalproperties that could be generated more efficiently using a dynamic trading strategy on thebenchmark portfolio. These include the two Fixed-Income strategies, the Multi-Strategyindex, and three of the Fund of Fund indices. Generally, the excess returns generated bythe HFRI indices are much lower than in the first sub-period. This is largely due to thesignificant increase in competition in the hedge fund industry, with an ever increasing num-ber of funds undertaking similar strategies. The APM measure helps us clearly identify thestrategies that are simply re-packaging beta (be it dynamic or static) from those are trulygenerating alpha.

6.5 Sub-Sample Period 1999 − 2007: Pre-crisis

In order to eliminate the impact of the recent financial crisis, we exclude 2008 from our anal-ysis of the second sub-sample. A description of the marginal distribution and dependencestructure parameters is presented on tables 32 and 33. The rankings according the fourperformance measures are presented in table 34. The results indicate that the results for thesecond sub-period are not solely driven by the 2008 crisis. The Fixed-Income strategies werethe most badly impacted by the crisis, due to a rise in defaults and a lack of liquidity inthe bond and money markets, and the Short Bias funds were understandably the ones thatprofited the most from the great de-leveraging of the financial institutions. In fact when weeliminate 2008, the Sharpe ratio of all the funds increases with the sole exception of the ShortBias strategy. The impact on the alpha and APM measures when we eliminate 2008 is lessuniform. Since these measures rely on market factors, what is important is not the absoluteperformance of the strategy but rather it’s relative performance with respect to the marketfactors. If we consider the APM results, for certain strategies the performance improves

24

when we include the crisis (notably Short Bias and Event Driven) whereas for many othersthe APM decreases(decreases almost 45bps per month for Convertible Arbitrage). The re-sults are however quite robust to the crisis, because although the magnitude of the excessreturns changes, the same 6 indices exhibit negative excess returns when 2008 is removedfrom the study.

7 Conclusion

In this paper, we present a new performance measure (APM) which encapsulates the manydimensions that need to be addressed when deciding whether or not to invest in an alter-native investment. The methodology is adapted from American option theory and basedon earlier work of Dybvig (1988), Kat and Palaro (2005) and Papageorgiou et al. (2008).This bivariate performance measure values both the stand alone properties of a specific fundas well as the diversification effect with respect to the investor’s portfolio. This approachdissociates itself from the factor model approach, which aims to identify a relationship be-tween the time-series of the fund and some systematic risk factor(s). We no longer concernourselves with the actual time-series of returns, instead, we evaluate the distributional prop-erties of the fund.

We provide empirical evidence of the uniqueness of the APM, by comparing boththe absolute and relative performance of 22 HFRI hedge fund indices using four differentperformance measures, specifically the Sharpe ratio, the Omega measure, the alpha of amulti-factor regression model and the APM. The results demonstrate the importance of notonly accounting for the higher moments but also the dependence structure (correlation) withthe investor’s portfolio.

25

References

Ackermann, C., McEnally, R., and Ravenscraft, D. (1999). The performance of hedge fund:Risk, return and incentives. Journal of Finance, 54(3):833–874.

Agarwal, V. and Naik, N. (2004). Risks and portfolio decisions involving hedge funds. Reviewof Financial Studies, 17:63–98.

Amin, G. and Kat, H. (2003). Hedge fund performanc 1990-2000: Do the “money machines”really add value. Journal of Financial and Quantitative Analysis, 38(2):251–275.

Bernardo, A. and Ledoit, O. (2000). Gain, loss, and asset pricing. Journal of PoliticalEconomy, 108:144–172.

Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incom-plete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B, 39:1–38.

Diez de los Rios, A. and Garcia, R. (2005). Assessing and valuing the nonlinear structure ofhedge funds. Technical report, CIRANO.

Durbin, J. (1973). Weak convergence of the sample distribution function when parametersare estimated. Ann. Statist., 1(2):279–290.

Dybvig, P. (1988). Distributional analysis of portfolio choice. The Journal of Business,61(3):369–393.

Fung, W. and Hsieh, D. (1997). Empirical characteristics of dynamic trading stategies: Thecase of hedge funds. Review of Financial Studies, 10:275–302.

Fung, W. and Hsieh, D. (2001). The risk in hedge fund stategies: Theory and evidence fromtrend followers. Review of Financial Studies, 14:313–341.

Genest, C., Ghoudi, K., and Rivest, L.-P. (1995). A semiparametric estimation procedure ofdependence parameters in multivariate families of distributions. Biometrika, 82:543–552.

Genest, C., Quessy, J.-F., and Remillard, B. (2006). Goodness-of-fit procedures for copulamodels based on the integral probability transformation. Scand. J. Statist., 33:337–366.

Genest, C. and Remillard, B. (2005). Validity of the parametric bootsrap for goodness-of-fittesting in semiparametric models. Technical Report G-2005-51, GERAD.

Genest, C., Remillard, B., and Beaudoin, D. (2007). Omnibus goodness-of-fit tests forcopulas: A review and a power study. Insurance Math. Econom., 40:in press.

26

Goetzmann, W., Ingersoll, J., Spiegel, M., and Welch, I. (2006). Portfolio performancemanipulation and manipulation-proof performance measures.

Grinblatt, M. and Titman, S. (1989). Mutual fund performance: An analysis of quarterlyportfolio holdings. Journal of Business, 63:393–416.

Hafner, C. and Rombouts, J. (2007). Estimation of temporally aggregated multivariateGARCH models. Journal of Statistical Computation and Simulation, 77:629–650.

Hasanhodzic, J. and Lo, A. W. (2007). Can hedge-fund returns be replicated?: The linearcase. Journal of Investment Management, 5(2):5–45.

Kat, H. and Palaro, H. (2005). Who needs hedge funds? A copula-based approach to hedgefund return replication. Technical report, Cass Business School, City University.

Keating, C. and Shadwick, W. F. (2002). A universal performance measure. Technicalreport, The Finance Development Centre, London.

Lambert, M., Hubner, G., and Papageorgiou, N. (2009). Directional and non-directionalcomponents in hedge fund returns. Technical report, DGAM-HEC Alternative InvestmentResearch.

Liang, B. (1999). On the performance of hedge funds. Financial Analysts Journal, 55:72–85.

Lo, A. (2002). The statistics of sharpe ratio,. Financial Analysts Journal, 1.

Mahdavi, M. (2004). Risk-adjusted return when returns are not normally distributed: Ad-justed sharpe ratio,. Journalof Alternative Investments, 2.

Mitchell, M. and Pulvino, T. (2001). Characteristics of risk and return in risk arbitrage.Journal of Finance, LVI:2135–2175.

Nelsen, R. B. (1999). An introduction to copulas, volume 139 of Lecture Notes in Statistics.Springer-Verlag, New York.

Papageorgiou, N., Remillard, B., and Hocquard, A. (2008). Replicating the properties ofhedge fund returns. Journal of Alternative Invesments, (in press).

Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Stat., 23:470–472.

Schweizer, M. (1995). Variance-optimal hedging in discrete time. Math. Oper. Res., 20:1–32.

Sharpe, W. (1966). Mutual fund performance. Journal of Business, 39:119–138.

27

Sharpe, W. (1992). Asset allocation: Management style and performance measurement.Journal of Portfolio Management, 18:7–19.

Stute, W., Gonzales Manteiga, W., and Presedo Quindimil, M. (1993). Bootstrap basedgoodness-of-fit tests. Metrika, 40:243–256.

28

A Numerical Results

A.1 Composition of Benchmark Portfolio

Table 3: Factors Decomposition for HFRI Fund Weighted Composite Index

Factor 1990 − 2008 1990 − 1999 1999 − 2008

S&P 500 7.96% 14.25% -2.69%Nasdaq 100 5.45% 3.49% 6.46%MSCI EAFE 6.31% 5.83% 11.33%MSCI Emerging Markets 11.44% 10.55% 11.69%S&P GSCI 3.66% 2.59% 3.75%Dollar Spot Index 11.88% 11.35% 8.73%Lehman US Aggregate 19.63% 15.98% 15.34%Spread Russell 2000 - S&P 500 15.64% 24.44% 10.40%Libor US 1Month 17.33% 10.50% 34.58%Alpha 0.57% 0.85% 0.27%R2 84.00% 86.56% 87.61%

This table presents the weights of the different factors in the benchmark portfolio for the entire sample aswell as for the two sub-samples.

29

A.2 Analysis of Performance Measures

Table 4: Sharpe Ratio rankings

Mean/Std Skewness ρ1,3 = −0.5 ρ1,3 = 0 ρ1,3 = 0.5−1 1 1 1

μ = 10% 0 1 1 1σ = 6% 1 1 1 1

−1 10 10 10μ = 10% 0 10 10 10σ = 14% 1 10 10 10

−1 19 19 19μ = 5% 0 19 19 19σ = 6% 1 19 19 19

−1 19 19 19μ = 5% 0 19 19 19σ = 14% 1 19 19 19

This table presents the ranking using the Sharpe ratio of funds with different levels of returns, volatility,skewness and correlation with the investor’s portfolio.

30

Table 5: Omega Ratio rankings


μ = 10% 0 4 4 4σ = 6% 1 1 1 1

−1 16 16 16μ = 10% 0 13 13 13σ = 14% 1 10 10 10

−1 31 31 31μ = 5% 0 25 25 25σ = 6% 1 19 19 19

−1 34 34 34μ = 5% 0 28 28 28σ = 14% 1 22 22 22

This table presents the ranking using the Omega measure for funds with different levels of returns,volatility, skewness and correlation with the investor’s portfolio.

Table 6: APM rankings


μ = 10% 0 2 8 11σ = 6% 1 1 7 10

−1 6 19 27μ = 10% 0 5 14 26σ = 14% 1 4 13 25

−1 18 24 30μ = 5% 0 17 23 29σ = 6% 1 15 22 28

−1 21 33 36μ = 5% 0 20 32 35σ = 14% 1 16 31 34

This table presents the ranking using the APM measure for funds with different levels of returns, volatility,skewness and correlation with the investor’s portfolio.

31

A.3 Hedge Fund Index Performance Analysis

A.3.1 Results for entire sample period - 1990 − 2008

Table 7: Factors Descriptive Statistics for 1990− 2008

Factor Mean Std.Dev. Skew. Kurt. Min. Max.S&P 500 0.65% 4.18% -0.96 5.38 -18.39% 10.84%Nasdaq 100 0.85% 8.02% -0.61 4.48 -30.66% 22.30%MSCI EAFE 0.16% 4.55% -0.96 5.56 -22.61% 9.79%MSCI Emerging Markets 0.45% 6.86% -1.34 7.38 -34.65% 14.47%S&P GSCI 0.20% 6.22% -0.88 6.35 -33.13% 15.61%Dollar Spot Index -0.03% 2.39% 0.40 4.09 -6.21% 9.02%Lehman US Aggregate 0.57% 1.11% -0.35 3.85 -3.42% 3.80%Spread Russell 2000 - S&P 500 0.09% 3.45% 0.10 6.94 -16.16% 17.19%

This table presents the descriptive statistics for the market factors that comprise the benchmark portfolioover the entire sample period.

32

Table 8: Statistics for 1990 − 2008

Fund Mean Std.Dev. Skew. Kurt. Min. Max. Corr.HFRI Distressed/Restructuring 0.99% 1.89% -1.14 8.94 -8.50% 7.06% 49.12%HFRI Merger Arbitrage 0.79% 1.10% -1.68 8.82 -5.69% 3.12% 52.24%HFRI Equity Market Neutral 0.62% 0.96% -0.20 4.28 -2.87% 3.59% 24.01%HFRI Quantitative Directional 1.23% 3.88% -0.45 3.61 -13.34% 10.74% 76.59%HFRI Short Bias 0.25% 5.61% 0.18 5.30 -21.21% 22.84% -66.65%HFRI Emerging Markets 1.18% 4.18% -0.89 6.94 -21.02% 14.80% 59.70%HFRI Emerging Markets: Asia ex-Japan 0.90% 3.88% 0.07 3.70 -10.84% 13.30% 51.37%HFRI Equity Hedge 1.11% 2.65% -0.26 5.20 -9.23% 10.88% 70.52%HFRI Event-Driven 1.02% 1.93% -1.43 8.00 -8.90% 5.13% 68.20%HFRI FOF: Conservative 0.53% 1.19% -1.76 10.73 -5.91% 3.96% 57.45%HFRI FOF: Diversified 0.58% 1.85% -0.45 6.60 -7.75% 7.73% 53.53%HFRI FOF: Market Defensive 0.70% 1.58% -0.04 3.56 -5.42% 4.93% 12.55%HFRI FOF: Strategic 0.80% 2.62% -0.56 6.32 -12.11% 9.47% 58.10%HFRI Fund of Funds Composite 0.62% 1.77% -0.69 6.80 -7.47% 6.85% 56.71%HFRI Fund Weighted Composite 0.98% 2.05% -0.79 5.90 -8.70% 7.65% 71.77%HFRI Macro 1.17% 2.27% 0.45 3.83 -6.40% 7.88% 35.53%HFRI Macro: Systematic Diversified 1.07% 2.12% 0.14 2.81 -4.41% 6.49% 48.77%HFRI Relative Value 0.79% 1.31% -2.33 17.29 -7.93% 5.72% 51.02%HFRI Fixed Income-Convertible Arbitrage 0.59% 1.81% -5.56 47.24 -16.37% 3.33% 48.66%HFRI Fixed Income-Corporate 0.68% 1.77% -1.16 13.21 -10.20% 9.54% 51.36%HFRI Multi-Strategy 0.66% 1.29% -2.58 19.40 -8.56% 5.34% 53.60%Benchmark Portfolio 0.35% 1.88% -1.22 6.63 -9.07% 4.65% 78.65%Investor’s portfolio 0.62% 2.60% -0.92 5.55 -11.99% 7.67% 100.00%

This table presents the descriptive statistics for the 22 HFR Indices, the incvestor’s portfolio and thebenchmark portfolio over the entire sample period

33

Table 9: Copula Parameters for 1990 − 2008

Fund Copula P-value Kendall’s TauHFRI Distressed/Restructuring Student 74% 29.65%HFRI Merger Arbitrage Student 55% 27.55%HFRI Equity Market Neutral Frank 58% 14.17%HFRI Quantitative Directional Frank 55% 56.40%HFRI Short Bias Student 47% -54.03%HFRI Emerging Markets Student 68% 35.41%HFRI Emerging Markets: Asia ex-Japan Clayton 86% 32.12%HFRI Equity Hedge Student 63% 49.83%HFRI Event-Driven Student 71% 43.98%HFRI FOF: Conservative Student 71% 32.60%HFRI FOF: Diversified Student 70% 32.23%HFRI FOF: Market Defensive Frank 55% 6.50%HFRI FOF: Strategic Student 72% 37.37%HFRI Fund of Funds Composite Student 74% 33.60%HFRI Fund Weighted Composite Student 69% 49.63%HFRI Macro Student 67% 25.88%HFRI Macro: Systematic Diversified Frank 53% 45.67%HFRI Relative Value Clayton 88% 28.58%HFRI Fixed Income-Convertible Arbitrage Gaussian 62% 22.59%HFRI Fixed Income-Corporate Student 60% 30.06%HFRI Multi-Strategy Student 65% 34.69%

This table presents the dependence between the HFR indices and the investor’s portfolio as measured byKendall’s tau for the entire sample period.

34

Table 10: Perf. Measures Rankings for 1990− 2008

Fund Sharpe Omega Alpha APMM. R. M. R. M. R. M. R.

HFRI Distressed/Restructuring 0.327 5 2.534 4 0.643% 6 0.497% 4HFRI Merger Arbitrage 0.380 1 2.630 1 0.376% 12 0.468% 7HFRI Equity Market Neutral 0.254 9 1.991 9 0.230% 18 0.162% 19HFRI Quantitative Directional 0.219 11 1.724 11 0.676% 4 0.397% 10HFRI Short Bias -0.023 21 0.939 21 0.116% 21 0.552% 3HFRI Emerging Markets 0.192 13 1.654 14 0.850% 1 0.620% 1HFRI Emerging Markets: Asia ex-Japan 0.134 17 1.410 19 0.618% 7 0.492% 5HFRI Equity Hedge 0.276 8 2.068 8 0.657% 5 0.386% 11HFRI Event-Driven 0.330 3 2.387 5 0.606% 8 0.556% 2HFRI FOF: Conservative 0.132 18 1.461 18 0.173% 19 0.177% 18HFRI FOF: Diversified 0.108 20 1.358 20 0.248% 17 0.013% 21HFRI FOF: Market Defensive 0.207 12 1.698 12 0.332% 13 0.224% 16HFRI FOF: Strategic 0.161 15 1.540 15 0.441% 10 0.205% 17HFRI Fund of Funds Composite 0.138 16 1.467 17 0.273% 15 0.105% 20HFRI Fund Weighted Composite 0.295 7 2.140 7 0.565% 9 0.450% 8HFRI Macro 0.347 2 2.627 2 0.788% 2 0.310% 12HFRI Macro: Systematic Diversified 0.327 4 2.252 6 0.689% 3 0.293% 14HFRI Relative Value 0.316 6 2.589 3 0.414% 11 0.470% 6HFRI Fixed Income-Convertible Arbitrage 0.118 19 1.511 16 0.163% 20 0.406% 9HFRI Fixed Income-Corporate 0.172 14 1.683 13 0.308% 14 0.253% 15HFRI Multi-Strategy 0.222 10 1.983 10 0.268% 16 0.295% 13

This table presents the rankings for each HFR index over the entire sample period using the four differentperformance measures

Table 11: Spearman Rank’s correlation between ranks for 1990 − 2008

Spearman Rho Sharpe Omega Alpha APMSharpe 1 0.978 0.588 0.281Omega 1 0.509 0.279Alpha 1 0.387APM 1

Table 12: Benchmarks Funds Performance Measures 1990 − 2008

Fund Sharpe OmegaInvestor’s Portfolio 0.093 1.276HFRI FW. Composite Replica -0.012 0.968

35

Table 13: Adjusted R2 of factor model for 1990 − 2008

Fund R2

HFRI Distressed/Restructuring 49.96%HFRI Merger Arbitrage 43.77%HFRI Equity Market Neutral 13.00%HFRI Quantitative Directional 88.38%HFRI Short Bias 76.32%HFRI Emerging Markets 81.92%HFRI Emerging Markets: Asia ex-Japan 68.36%HFRI Equity Hedge 78.62%HFRI Event-Driven 73.65%HFRI FOF: Conservative 53.31%HFRI FOF: Diversified 62.02%HFRI FOF: Market Defensive 21.30%HFRI FOF: Strategic 65.06%HFRI Fund of Funds Composite 63.56%HFRI Fund Weighted Composite 84.00%HFRI Macro 37.27%HFRI Macro: Systematic Diversified 36.76%HFRI Relative Value 46.20%HFRI Fixed Income-Convertible Arbitrage 39.31%HFRI Fixed Income-Corporate 45.90%HFRI Multi-Strategy 50.39%

This table presents the adjusted R-square obtained from regressing the eight market factors against each ofthe HFR indices over the entire sample period

Table 14: Bivariate Monthly Gaussian Mixture Parameters for 1990− 2008

4 Regimes / P-value = 35.4%Regime Probability Mean P Mean R Var P Var R Corr. (P,R)1 0.46 0.0154 0.0126 0.00025 0.00009 0.702 0.44 -0.0002 -0.0048 0.00068 0.00023 0.783 0.05 0.0201 0.0205 0.00046 0.00022 -0.824 0.05 -0.0312 -0.0199 0.00183 0.00151 0.98

36

A.3.2 Results for first sub-sample period - 1990 − 1999

Table 15: Factors Descriptive Statistics for 1990 − 1999

Factor Mean Std.Dev. Skew. Kurt. Min. Max.S&P 500 1.58% 3.66% -0.94 6.23 -15.62% 10.84%Nasdaq 100 2.71% 6.51% 0.05 3.71 -18.88% 22.30%MSCI EAFE 0.63% 4.31% -0.43 3.25 -13.36% 9.79%MSCI Emerging Markets 0.76% 6.88% -1.47 8.61 -34.65% 14.47%S&P GSCI 0.10% 4.43% -0.01 4.14 -12.97% 15.61%Dollar Spot Index 0.17% 2.43% 0.63 4.33 -5.43% 9.02%Lehman US Aggregate 0.63% 1.09% -0.23 3.29 -2.50% 3.80%Spread Russell 2000 - S&P 500 -0.20% 3.10% 0.19 3.15 -7.37% 9.66%


Fund Mean Std.Dev. Skew. Kurt. Min. Max. Corr.HFRI Distressed/Restructuring 1.39% 1.83% -0.89 11.07 -8.50% 7.06% 34.00%HFRI Merger Arbitrage 1.11% 1.03% -3.08 19.77 -5.69% 2.83% 40.31%HFRI Equity Market Neutral 0.87% 0.93% -0.04 3.29 -1.67% 3.59% 34.94%HFRI Quantitative Directional 2.01% 3.67% -0.55 4.91 -13.34% 10.74% 69.55%HFRI Short Bias -0.29% 5.59% 0.22 4.07 -16.24% 19.40% -65.23%HFRI Emerging Markets 1.64% 4.65% -0.87 7.42 -21.02% 14.80% 48.17%HFRI Emerging Markets: Asia ex-Japan 1.26% 3.99% 0.60 3.94 -6.96% 13.30% 35.65%HFRI Equity Hedge 1.89% 2.45% -0.02 5.23 -7.65% 10.88% 62.93%HFRI Event-Driven 1.51% 1.72% -1.87 13.66 -8.90% 5.13% 57.43%HFRI FOF: Conservative 0.81% 1.04% -0.73 6.94 -3.88% 3.96% 47.32%HFRI FOF: Diversified 0.88% 1.95% -0.22 6.65 -7.75% 7.73% 47.89%HFRI FOF: Market Defensive 0.73% 1.68% -0.12 3.69 -5.42% 4.70% 26.50%HFRI FOF: Strategic 1.35% 2.74% -0.80 7.64 -12.11% 9.47% 48.68%HFRI Fund of Funds Composite 0.96% 1.80% -0.49 7.22 -7.47% 6.85% 49.87%HFRI Fund Weighted Composite 1.53% 1.97% -1.09 8.80 -8.70% 7.65% 64.96%HFRI Macro 1.67% 2.69% 0.16 3.08 -6.40% 7.88% 46.94%HFRI Macro: Systematic Diversified 1.30% 1.94% 0.11 2.37 -2.99% 5.49% 66.18%HFRI Relative Value 1.12% 1.23% -1.17 12.03 -5.80% 5.72% 33.17%HFRI Fixed Income-Convertible Arbitrage 0.98% 0.97% -1.65 7.75 -3.19% 3.33% 39.61%HFRI Fixed Income-Corporate 1.10% 1.76% 0.04 10.87 -7.16% 9.54% 36.71%HFRI Multi-Strategy 1.02% 1.05% -0.49 9.09 -3.27% 5.34% 34.06%Benchmark Portfolio 0.56% 1.83% -1.28 7.84 -8.73% 4.53% 69.88%Investor’s portfolio 1.20% 2.38% -0.72 4.81 -8.72% 7.67% 100.00%

37

Table 17: Copula Parameters for 1990− 1999

Fund Copula P-value Kendall’s TauHFRI Distressed/Restructuring Student 59% 18.96%HFRI Merger Arbitrage Clayton 83% 18.62%HFRI Equity Market Neutral Student 53% 19.54%HFRI Quantitative Directional Clayton 76% 47.13%HFRI Short Bias Gaussian 69% -43.84%HFRI Emerging Markets Student 63% 24.33%HFRI Emerging Markets: Asia ex-Japan Gaussian 82% 23.27%HFRI Equity Hedge Gaussian 65% 40.82%HFRI Event-Driven Student 70% 34.01%HFRI FOF: Conservative Student 71% 23.41%HFRI FOF: Diversified Student 71% 25.28%HFRI FOF: Market Defensive Frank 76% 8.90%HFRI FOF: Strategic Student 65% 24.91%HFRI Fund of Funds Composite Student 67% 24.84%HFRI Fund Weighted Composite Student 62% 40.13%HFRI Macro Clayton 73% 28.24%HFRI Macro: Systematic Diversified Student 62% 51.10%HFRI Relative Value Student 76% 15.80%HFRI Fixed Income-Convertible Arbitrage Student 68% 21.71%HFRI Fixed Income-Corporate Frank 54% 20.66%HFRI Multi-Strategy Student 60% 25.04%

This table presents the dependence between the HFR index and the investor’s portfolio as measured byKendall’s tau for the first sub-sample period (1990-1999).

38

Table 18: Perf. Measures Rankings for 1990− 1999


HFRI Distressed/Restructuring 0.503 8 4.467 4 1.016% 4 0.934% 6HFRI Merger Arbitrage 0.623 1 5.002 2 0.589% 12 0.671% 12HFRI Equity Market Neutral 0.432 10 3.098 10 0.275% 21 0.411% 18HFRI Quantitative Directional 0.421 12 2.874 12 0.867% 6 1.237% 1HFRI Short Bias -0.136 21 0.699 21 0.484% 15 1.038% 5HFRI Emerging Markets 0.251 17 1.977 17 1.146% 2 1.156% 3HFRI Emerging Markets: Asia ex-Japan 0.198 19 1.712 19 0.866% 7 0.800% 10HFRI Equity Hedge 0.578 3 4.707 3 1.023% 3 1.144% 4HFRI Event-Driven 0.604 2 5.166 1 0.969% 5 0.913% 7HFRI FOF: Conservative 0.329 14 2.479 14 0.304% 20 0.325% 21HFRI FOF: Diversified 0.212 18 1.837 18 0.376% 18 0.367% 20HFRI FOF: Market Defensive 0.156 20 1.504 20 0.321% 19 0.395% 19HFRI FOF: Strategic 0.321 15 2.369 15 0.755% 9 0.838% 8HFRI Fund of Funds Composite 0.273 16 2.152 16 0.441% 17 0.452% 17HFRI Fund Weighted Composite 0.539 4 4.073 7 0.854% 8 0.824% 9HFRI Macro 0.445 9 3.347 9 1.176% 1 1.226% 2HFRI Macro: Systematic Diversified 0.429 11 2.859 13 0.492% 14 0.507% 15HFRI Relative Value 0.531 5 4.361 6 0.658% 11 0.674% 11HFRI Fixed Income-Convertible Arbitrage 0.527 6 3.772 8 0.472% 16 0.506% 16HFRI Fixed Income-Corporate 0.359 13 2.961 11 0.670% 10 0.618% 13HFRI Multi-Strategy 0.524 7 4.394 5 0.566% 13 0.523% 14




Fund Sharpe OmegaBalanced Fund Proxy 0.307 2.142HFRI Composite Replica 0.050 1.141

39

Table 21: OLS R2 for 1990 − 1999

Fund R2



2 Regimes / P-value = 24.2%Regime Probability Mean P Mean R Var P Var R Corr. (P,R)1 0.95 0.0143 0.0072 0.0004 0.0002 0.602 0.05 -0.0352 -0.0274 0.0008 0.0011 0.96

40

A.3.3 Results for second sub-sample period - 1999 − 2008


Factor Mean Std.Dev. Skew. Kurt. Min. Max.S&P 500 -0.28% 4.48% -0.85 4.78 -18.39% 9.34%Nasdaq 100 -1.04% 8.94% -0.61 3.88 -30.66% 17.83%MSCI EAFE -0.31% 4.75% -1.34 6.74 -22.61% 8.93%MSCI Emerging Markets 0.14% 6.85% -1.23 6.19 -32.16% 12.39%S&P GSCI 0.30% 7.63% -0.98 5.25 -33.13% 14.33%Dollar Spot Index -0.24% 2.35% 0.13 3.64 -6.21% 7.50%Lehman US Aggregate 0.51% 1.12% -0.45 4.31 -3.42% 3.66%Spread Russell 2000 - S&P 500 0.38% 3.76% -0.03 8.47 -16.16% 17.19%


Fund Mean Std.Dev. Skew. Kurt. Min. Max. Corr.HFRI Distressed/Restructuring 0.60% 1.87% -1.51 7.47 -7.90% 4.00% 57.72%HFRI Merger Arbitrage 0.48% 1.08% -0.82 3.93 -2.90% 3.12% 56.51%HFRI Equity Market Neutral 0.37% 0.93% -0.41 5.31 -2.87% 3.06% 5.33%HFRI Quantitative Directional 0.43% 3.94% -0.34 2.77 -8.97% 9.41% 80.48%HFRI Short Bias 0.80% 5.60% 0.14 6.66 -21.21% 22.84% -67.80%HFRI Emerging Markets 0.71% 3.59% -1.18 4.71 -13.81% 6.61% 73.96%HFRI Emerging Markets: Asia ex-Japan 0.53% 3.74% -0.62 2.85 -10.84% 6.25% 65.58%HFRI Equity Hedge 0.32% 2.62% -0.40 5.49 -9.23% 10.00% 73.40%HFRI Event-Driven 0.52% 2.01% -1.18 5.76 -8.01% 4.59% 73.06%HFRI FOF: Conservative 0.25% 1.26% -2.35 11.62 -5.91% 2.18% 60.85%HFRI FOF: Diversified 0.27% 1.69% -1.04 6.21 -6.53% 5.40% 56.81%HFRI FOF: Market Defensive 0.68% 1.49% 0.07 3.24 -2.85% 4.93% -1.45%HFRI FOF: Strategic 0.25% 2.37% -0.52 5.12 -7.67% 8.75% 64.59%HFRI Fund of Funds Composite 0.28% 1.68% -1.11 6.47 -6.54% 5.21% 59.95%HFRI Fund Weighted Composite 0.43% 1.99% -0.67 4.48 -6.68% 6.16% 74.72%HFRI Macro 0.66% 1.62% 0.12 3.50 -3.68% 5.67% 13.99%HFRI Macro: Systematic Diversified 0.84% 2.28% 0.24 3.00 -4.41% 6.49% 33.84%HFRI Relative Value 0.45% 1.30% -3.72 22.54 -7.93% 2.20% 60.71%HFRI Fixed Income-Convertible Arbitrage 0.20% 2.31% -4.85 32.30 -16.37% 2.73% 51.95%HFRI Fixed Income-Corporate 0.26% 1.69% -2.80 15.83 -10.20% 2.71% 59.99%HFRI Multi-Strategy 0.30% 1.40% -3.51 21.10 -8.56% 3.28% 61.93%Benchmark Portfolio 0.13% 1.85% -1.14 6.02 -8.38% 4.35% 80.23%Investor’s portfolio 0.03% 2.69% -1.03 5.86 -11.99% 6.13% 100.00%

41


Fund Copula P-value Kendall’s TauHFRI Distressed/Restructuring Clayton 73% 34.20%HFRI Merger Arbitrage Student 54% 33.44%HFRI Equity Market Neutral Frank 44% 3.95%HFRI Quantitative Directional Frank 53% 64.24%HFRI Short Bias Student 55% -66.18%HFRI Emerging Markets Clayton 89% 47.18%HFRI Emerging Markets: Asia ex-Japan Student 59% 44.03%HFRI Equity Hedge Student 60% 55.45%HFRI Event-Driven Student 67% 51.37%HFRI FOF: Conservative Student 61% 38.35%HFRI FOF: Diversified Student 60% 38.28%HFRI FOF: Market Defensive Frank 50% 4.85%HFRI FOF: Strategic Student 59% 47.87%HFRI Fund of Funds Composite Student 59% 40.36%HFRI Fund Weighted Composite Student 59% 55.87%HFRI Macro Frank 52% 15.96%HFRI Macro: Systematic Diversified Frank 50% 40.20%HFRI Relative Value Clayton 82% 33.61%HFRI Fixed Income-Convertible Arbitrage Frank 52% 17.27%HFRI Fixed Income-Corporate Student 66% 34.96%HFRI Multi-Strategy Student 60% 38.42%

This table presents the dependence between the HFR index and the investor’s portfolio as measured byKendall’s tau for the second sub-sample period (1999-2008).

42

Table 26: HFR Index Rankings for 1999 − 2008


HFRI Distressed/Restructuring 0.167 5 1.587 4 0.362% 5 0.427% 3HFRI Merger Arbitrage 0.178 4 1.575 5 0.219% 11 0.130% 12HFRI Equity Market Neutral 0.092 9 1.299 10 0.123% 14 0.012% 15HFRI Quantitative Directional 0.038 13 1.099 13 0.437% 3 0.282% 8HFRI Short Bias 0.092 10 1.306 9 -0.117% 20 0.323% 7HFRI Emerging Markets 0.119 7 1.342 8 0.583% 2 0.539% 2HFRI Emerging Markets: Asia ex-Japan 0.066 12 1.173 12 0.375% 4 0.348% 6HFRI Equity Hedge 0.014 14 1.039 15 0.242% 9 0.116% 13HFRI Event-Driven 0.116 8 1.356 7 0.297% 6 0.224% 10HFRI FOF: Conservative -0.025 20 0.929 20 0.035% 17 -0.147% 19HFRI FOF: Diversified -0.011 17 0.971 17 0.060% 16 0.108% 14HFRI FOF: Market Defensive 0.263 1 1.937 1 0.228% 10 0.390% 4HFRI FOF: Strategic -0.016 19 0.957 19 0.131% 13 -0.006% 16HFRI Fund of Funds Composite -0.002 16 0.994 16 0.073% 15 -0.067% 17HFRI Fund Weighted Composite 0.074 11 1.207 11 0.268% 7 0.182% 11HFRI Macro 0.232 3 1.822 3 0.260% 8 0.369% 5HFRI Macro: Systematic Diversified 0.243 2 1.847 2 0.694% 1 0.576% 1HFRI Relative Value 0.130 6 1.531 6 0.175% 12 0.259% 9HFRI Fixed Income-Convertible Arbitrage -0.039 21 0.859 21 -0.121% 21 -0.719% 21HFRI Fixed Income-Corporate -0.014 18 0.958 18 -0.010% 19 -0.226% 20HFRI Multi-Strategy 0.012 15 1.042 14 0.001% 18 -0.121% 18




Fund Sharpe OmegaBalanced Fund Proxy -0.093 0.771HFRI Composite Replica -0.084 0.800

43

Table 29: OLS R2 for 1999 − 2008

Fund R2



3 Regimes / P-value = 35.6%Regime Probability Mean P Mean R Var P Var R Corr. (P,R)1 0.21 -0.00966 -0.00322 0.00088 0.00030 0.482 0.49 0.01003 0.01124 0.00019 0.00008 0.723 0.30 -0.00857 -0.01177 0.00116 0.00045 0.96

44

A.3.4 Results for second sub-sample period - excluding 2008


Factor Mean Std.Dev. Skew. Kurt. Min. Max.S&P 500 0.16% 4.00% -0.41 3.43 -11.51% 9.34%Nasdaq 100 -0.60% 8.90% -0.68 4.27 -30.66% 17.83%MSCI EAFE 0.23% 3.92% -0.49 3.06 -11.52% 8.93%MSCI Emerging Markets 0.91% 5.67% -0.43 2.46 -11.91% 12.39%S&P GSCI 0.99% 6.38% -0.29 2.82 -15.83% 14.33%Dollar Spot Index -0.32% 2.13% -0.17 2.84 -5.47% 4.67%Lehman US Aggregate 0.52% 1.03% -0.87 4.59 -3.42% 2.61%Spread Russell 2000 - S&P 500 0.38% 3.85% 0.00 8.62 -16.16% 17.19%


Fund Mean Std.Dev. Skew. Kurt. Min. Max. Corr.HFRI Distressed/Restructuring 0.96% 1.34% -0.08 2.92 -2.49% 4.00% 44.31%HFRI Merger Arbitrage 0.59% 0.97% -0.65 4.05 -2.72% 3.12% 43.09%HFRI Equity Market Neutral 0.48% 0.78% 0.51 4.60 -1.57% 3.06% -5.46%HFRI Quantitative Directional 0.74% 3.87% -0.35 2.81 -8.65% 9.41% 80.20%HFRI Short Bias 0.63% 5.77% 0.19 6.63 -21.21% 22.84% -68.73%HFRI Emerging Markets 1.25% 3.00% -0.79 2.89 -6.59% 6.61% 68.07%HFRI Emerging Markets: Asia ex-Japan 1.00% 3.34% -0.45 2.47 -6.84% 6.25% 57.58%HFRI Equity Hedge 0.67% 2.25% 0.42 5.01 -4.30% 10.00% 66.56%HFRI Event-Driven 0.83% 1.67% -0.57 3.65 -4.35% 4.59% 64.90%HFRI FOF: Conservative 0.50% 0.78% -0.31 2.77 -1.79% 2.18% 51.85%HFRI FOF: Diversified 0.53% 1.34% -0.13 5.06 -4.22% 5.40% 44.71%HFRI FOF: Market Defensive 0.69% 1.41% 0.01 3.11 -2.85% 4.50% 0.82%HFRI FOF: Strategic 0.57% 2.06% 0.06 5.12 -5.51% 8.75% 56.35%HFRI Fund of Funds Composite 0.55% 1.30% 0.03 4.18 -3.37% 5.21% 48.57%HFRI Fund Weighted Composite 0.69% 1.71% -0.09 3.27 -3.49% 6.16% 69.78%HFRI Macro 0.69% 1.60% 0.11 3.60 -3.68% 5.67% 22.04%HFRI Macro: Systematic Diversified 0.77% 2.20% 0.20 2.96 -4.41% 6.43% 63.45%HFRI Relative Value 0.70% 0.65% -0.07 2.92 -0.84% 2.20% 38.79%HFRI Fixed Income-Convertible Arbitrage 0.63% 0.97% -0.59 3.51 -2.64% 2.73% 12.70%HFRI Fixed Income-Corporate 0.55% 1.12% -1.27 5.95 -3.92% 2.71% 39.96%HFRI Multi-Strategy 0.57% 0.75% -0.25 4.74 -1.72% 3.28% 45.07%Benchmark Portfolio 0.35% 1.60% -0.37 2.79 -3.78% 4.40% 74.41%Investor’s portfolio 0.30% 2.34% -0.33 3.34 -6.26% 6.13% 100.00%

45


Fund Copula P-value Kendall’s TauHFRI Distressed/Restructuring Student 61% 31.71%HFRI Merger Arbitrage Student 52% 27.72%HFRI Equity Market Neutral Student 64% 3.25%HFRI Quantitative Directional Student 59% 64.25%HFRI Short Bias Student 53% -64.74%HFRI Emerging Markets Clayton 87% 46.58%HFRI Emerging Markets: Asia ex-Japan Student 66% 40.66%HFRI Equity Hedge Student 61% 55.75%HFRI Event-Driven Student 61% 51.23%HFRI FOF: Conservative Student 55% 38.77%HFRI FOF: Diversified Student 66% 38.29%HFRI FOF: Market Defensive Frank 67% 7.37%HFRI FOF: Strategic Student 64% 48.38%HFRI Fund of Funds Composite Student 60% 40.83%HFRI Fund Weighted Composite Student 57% 57.11%HFRI Macro Student 61% 20.39%HFRI Macro: Systematic Diversified Frank 54% 53.11%HFRI Relative Value Gumbel 72% 28.68%HFRI Fixed Income-Convertible Arbitrage Frank 72% 8.33%HFRI Fixed Income-Corporate Student 54% 30.61%HFRI Multi-Strategy Student 62% 35.92%

This table presents the dependence between the HFR index and the investor’s portfolio as measured byKendall’s tau for the second sub-sample excluding 2008.

46

Table 34: Ranking of HFR indices for 1999 − 2007


HFRI Distressed/Restructuring 0.498 2 3.555 2 0.571% 2 0.580% 2HFRI Merger Arbitrage 0.301 7 2.147 6 0.191% 12 0.219% 13HFRI Equity Market Neutral 0.237 11 1.953 9 0.133% 17 0.128% 20HFRI Quantitative Directional 0.114 20 1.325 20 0.281% 9 0.273% 10HFRI Short Bias 0.058 21 1.185 21 0.005% 21 0.165% 16HFRI Emerging Markets 0.319 5 2.113 7 0.679% 1 0.813% 1HFRI Emerging Markets: Asia ex-Japan 0.211 15 1.650 16 0.368% 4 0.537% 3HFRI Equity Hedge 0.165 18 1.530 18 0.302% 7 0.228% 12HFRI Event-Driven 0.319 6 2.205 5 0.377% 3 0.425% 4HFRI FOF: Conservative 0.263 9 1.903 10 0.139% 15 -0.150% 18HFRI FOF: Diversified 0.178 17 1.604 17 0.126% 18 0.134% 19HFRI FOF: Market Defensive 0.279 8 2.011 8 0.105% 19 0.298% 7HFRI FOF: Strategic 0.131 19 1.407 19 0.205% 11 -0.126% 21HFRI Fund of Funds Composite 0.198 16 1.670 15 0.151% 14 -0.161% 17HFRI Fund Weighted Composite 0.232 12 1.778 13 0.295% 8 0.291% 8HFRI Macro 0.245 10 1.899 11 0.095% 20 0.278% 9HFRI Macro: Systematic Diversified 0.214 14 1.711 14 0.347% 5 0.344% 6HFRI Relative Value 0.626 1 4.724 1 0.329% 6 0.363% 5HFRI Fixed Income-Convertible Arbitrage 0.350 4 2.355 4 0.259% 10 -0.268% 11HFRI Fixed Income-Corporate 0.224 13 1.787 12 0.136% 16 -0.170% 15HFRI Multi-Strategy 0.364 3 2.517 3 0.157% 13 -0.217% 14

Table 35: Spearman Rank correlations between ranks for 1999 − 2007



Fund Sharpe OmegaBenchmark Portfolio 0.004 1.010Investor’s portfolio 0.035 1.090

47

Table 37: OLS R2 for 1999 − 2007

Fund R2



1 Regime / P-value = 35%Regime Probability Mean P Mean R Var P Var R Corr. (P,R)1 1.00 0.00304 0.00352 0.00054 0.00025 0.74

48

B Estimation and goodness-of-fit

In this section, we describe the estimation procedure and the goodness-of-fit tests.

B.1 EM algorithm for bivariate Gaussian mixtures

Let y1, . . . , yn be a random sample from a bivariate Gaussian mixture with parameters π =(πk)

mk=1, μ = (μk)

mk=1 and A = (Ak)

mk=1. Start with an initial estimator θ(0). Given an

estimator θ(�) =(π(�), μ(�), A(�)

)of the parameters θ = (π, μ, A), set

πk

(yi, θ

(�))

=π

(�)k φ2

(yi; μ

(�)k , A

(�)k

)∑m

j=1 π(�)j φ2

(yi; μ

(�)j , A

(�)j

) , i = 1, . . . , n,

and define the new estimator θ(�+1) =(π(�+1), μ(�+1), A(�+1)

)viz.

π(�+1)k =

1

n

n∑i=1

πk

(yi, θ

(�)),

μ(�+1)k =

1

n

n∑i=1

yiπk

(yi, θ

(�))/

π(�+1)k ,

and

A(�+1)k =

1

n

n∑i=1

(yi − μ

(�+1)k

)(yi − μ

(�+1)k

)�πk

(yi, θ

(�)) /

π(�+1)k ,

for k = 1, . . . , m. As � increases, the numbers {πk

(yi, θ

(�)); k = 1, . . . , i = 1, . . . , n} stabilize

and the estimators converge.

B.2 Tests of goodness-of-fit

Testing goodness-of-fit is an essential step for modeling data. There are many tests availablebut to our knowledge, the best ones are based on empirical processes (Genest and Remillard,2005, Genest et al., 2007). Here, we only consider two tests based on the so-called Rosen-blatt‘s transform. The first one is due to Durbin (1973) but the calculation of P -values is re-cent (Stute et al., 1993). For the second test designed for testing goodness-of-fit for bivariatedata, the validity of the algorithm for calculating P -values follows from Genest and Remillard(2005).

49

B.3 Tests of goodness-of-fit for a bivariate parametric distribution

Let (X1, Y1) . . . , (Xn, Yn) be a sample of size n from a (continuous) distribution F on R2.

Suppose that the hypotheses to be tested are

H0 : F ∈ F = {Fθ; θ ∈ Θ} vs H1 : F �∈ F

For example, the parametric family F could be the family of bivariate Gaussian mixtureswith m regimes. Denote by Gθ the distribution function of Xi and let Hθ be the conditionaldistribution function of Yi given Xi, i.e., Hθ(x, y) = P (Yi ≤ y|Xi = x).

The proposed test statistic is based on Durbin (1973) and the Rosenblatt’s transform(Rosenblatt, 1952).

Suppose that θn = Tn(X1, Y1, . . . , Xn, Yn) is a regular estimator of θ, in the sense ofGenest and Remillard (2005) and set

Dn(u, v) =1

n

n∑i=1

I(Ui ≤ u, Vi ≤ v), u, v ∈ [0, 1],

where Ui = Gθn(Xi), Vi = Hθn(Xi, Yi), i = 1, . . . , n. To test H0 against H1, one may use theCramer-von Mises type statistic

Sn = n

∫ 1

0

∫ 1

0

{Dn(u, v) − uv}2dudv

=1

n

n∑i=1

n∑j=1

[1

9− 1

4(1 − U2

i )(1 − V 2i ) − 1

4(1 − U2

j )(1 − V 2j )

+{1 − max(Ui, Uj)}{1 − max(Vi, Vj)}].

Since the pairs (Ui, Vi)’s are “almost uniformly distributed on [0, 1]2” under the null hypothe-sis, large values of Sn should lead to rejection of the null hypothesis. However, in general thelimiting distribution of Sn depend on the unknown parameter θ. To calculate the P -valueof Sn, one can use a parametric bootstrap approach as described below.

a) Calculate θn and Sn.

b) For some large integer N (say 1000), repeat the following steps for every k ∈ {1, . . . , N}:(i) Generate a random sample (X1,k, Y1,k), . . . , (Xn,k, Yn,k) from distribution Fθn.

50

(ii) Calculate

θ∗n,k = Tn (X1,k, Y1,k, . . . , Xn,k, Yn,k) ,

Ui,k = Gθn,k(Xi,k), Vi,k = Hθn,k

(Xi,k, Yi,k), i = 1, . . . , n

Sn,k =1

n

n∑i=1

n∑j=1

[1

9− 1

4(1 − U2

i,k)(1 − V 2i,k) −

1

4(1 − U2

j,k)(1 − V 2j,k)

+{1 − max(Ui,k, Uj,k)}{1 − max(Vi,k, Vj,k)}].

An approximate P -value for the test based on the Cramer–von Mises statistic Sn is thengiven by

1

N

N∑k=1

I(Sn,k > Sn).

51

C Optimal hedging and Alternative Performance Mea-

sure

In this section we describe the methodology used to derive the APM measure and the optimalhedging strategy.

Suppose that (Ω, P,F) is a probability space with filtration F = {F0, . . . ,FT}, underwhich the stochastic processes are defined. Assume that the price process St is d-dimensional,

i.e. St =(S

(1)t , . . . , S

(d)t

).

A dynamic replicating strategy can be described by a (deterministic) initial value V0

and a sequence of random weight vectors ϕ = (ϕt)Tt=0, where for any j = 1, . . . , d, ϕ

(j)t denotes

the number of parts of assets S(j) invested during period (t − 1, t]. Because ϕt may dependonly on the values values S0, . . . , St−1, the stochastic process ϕt is assumed to be predictable.Initially, ϕ0 = ϕ1, and the portfolio initial value is V0. It follows that the amount initiallyinvested in the non risky asset is V0 −

∑dj=1 ϕ

(j)1 S

(j)0 = V0 − ϕ�

1 S0.

Since the hedging strategy must be self-financing, it follows that for all t = 1, . . . , T ,

βtVt(V0, ϕ) − βt−1Vt−1(V0, ϕ) = ϕ�t (βtSt − βt−1St−1). (11)

Using the self-financing condition (11), it follows that

βT VT = βT VT (V0, ϕ) = V0 +T∑

t=1

ϕ�t (βtSt − βt−1St−1). (12)

The replication strategy problem for a given payoff C is thus equivalent to findingthe strategy (V0, ϕ) so that the hedging error

GT (V0, ϕ) = βT VT (V0, ϕ) − βT C (13)

is as small as possible. Here, the RMSHE (root mean square hedging error) measures

the quality of replication. It is therefore natural to suppose that the prices S(j)t have finite

second moments. We further assume that the hedging strategy ϕ satisfies a similar property,namely that for any t = 1, . . . , T , ϕ�

t (βtSt − βt−1St−1) have finite second moments. Notethat these two technical conditions were also made by Schweizer (1995).

For simplicity, set Δt = St − E(St|Ft−1), t = 1, . . . , T . Under the above momentconditions, the conditional covariance matrix Σt of Δt exists and is given by

Σt = E{ΔtΔ

�t |Ft−1

}, 1 ≤ t ≤ T. (14)

52

V0 is equivalent to what we call the Alternative Performance Measure (APM) in thesense that it represents the fair optimal price of the strategy at time t = 0.

In Schweizer (1995), the author treats the case d = 1 and assumes a restrictiveboundedness condition. Here, in contrast, we treat the general d-dimensional case and weonly suppose that Σt is invertible for all t = 1, . . . , T . This was implicitly part of theboundedness condition of Schweizer (1995).

If Σt is not invertible for some t, there would exists a ϕt ∈ Ft−1 such that ϕ�t St =

ϕ�t E(St|Ft−1), that is, ϕ�

t St is predictable. Our assumption can be interpreted as sayingthat the genuine dimension of the assets is d.

53

Documents

An Alternative Performance Measure · the distributional properties of this portfolio present an excellent basis for evaluating the performance of these funds. The underlying notion