the myths and limits of passive hedge fund replication

EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE

393-400 promenade des Anglais06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 78 24Fax: +33 (0)4 93 18 78 41E-mail: [email protected]: www.edhec-risk.com

The Myths and Limits of Passive Hedge Fund ReplicationAn Attractive Concept… Still a Work-in-Progress

June 2007

Noël AmencPhD, Professor of Finance and Director of the EDHEC Riskand Asset Management Research Centre

Walter GéhinResearch Associate, EDHEC Risk and Asset Management Research Centre and Business Analyst, Atos Euronext Market Solutions

Lionel MartelliniPhD, Professor of Finance and Scientific Director of the EDHEC Risk and Asset Management Research Centre

Jean-Christophe MeyfrediPhD, Professor of Finance and Research Associate with the EDHEC Risk and Asset Management Research Centre

In this paper we provide a detailed critical analysis of various methodologies involved in the so-called passive replication of hedge fund returns, a subject that has sparked renewed interest following recent initiatives by major investment banks such as Merrill Lynch and Goldman Sachs. In particular, we examine from both a theoretical and an empirical standpoint the respective benefits and limits of the two different and somewhat competing approaches to hedge fund replication, which are respectively known as "factor-based replication," and "payoff distribution replication." On the one hand, we argue that standard implementation efforts of the factor-based approach, arguably the most natural and straightforward way to tackle the hedge fund replication problem, have mostly failed in thorough empirical tests to produce satisfactory results on an out-of-sample basis. We also argue that the payoff distribution approach, on the other hand, while insightful and found to generate (relatively) satisfying results on an out-of-sample basis, unfortunately cannot be regarded as a method suitable for performing hedge fund replication, at least not in a sense likely to meet investors' expectations, due to its documented failure to match a number of relevant time-series properties of hedge fund returns. In conclusion, hedge fund replication, while obviously a powerful and attractive concept, is still, at least in terms of successful implementation, very much a work-in-progress. Our analysis suggests that it is only through the introduction of novel adapted econometric techniques allowing for a parsimonious statistical estimation of the dynamic and/or non-linear functions relating underlying factors to hedge fund returns that hedge fund replication could be turned from an attractive concept into a workable investment solution, and we discuss several possible directions for future research.

Abstract

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We would like to thank René Garcia for very useful comments.

The work presented herein is a detailed summary of academic research conducted by EDHEC. The opinions expressed are those of the authors and EDHEC Business School declines all reponsibility for any errors or omissions.

Noël Amenc, PhD is Professor of Finance and Director of Research and Development at EDHEC Business School, where he heads the Risk and Asset Management Research Centre. He has a Masters in Economics and a PhD in Finance and has conducted active research in the fields of quantitative equity management, portfolio performance analysis and active asset allocation, resulting in numerous academic and practitioner articles and books. He is an Associate Editor of the Journal of Alternative Investments and a member of the scientific advisory council of the AMF (French financial regulatory authority).

Walter Géhin is a Research Associate with the EDHEC Risk and Asset Management Research Centre. He is currently working at Atos Euronext Market Solutions as a Business Analyst. He holds a Master’s degree in Banking and Finance and an advanced graduate diploma in Financial Engineering. Walter has published articles on performance persistence, hedge fund performance measurement, hedge fund market capacity, and hedge fund indexes. He is in charge of the hedge fund performance research section of the www.edhec-risk.com website.

Lionel Martellini, PhD is Professor of Finance at EDHEC Business School and the Scientific Director of the EDHEC Risk and Asset Management Research Centre. He holds graduate degrees in Economics, Statistics and Mathematics, as well as a PhD in Finance from the University of California at Berkeley. Lionel is a member of the editorial board of the Journal of Portfolio Management and the Journal of Alternative Investments. An expert in quantitative asset management and derivatives valuation, Lionel has published widely in academic and practitioner journals, and has co-authored reference textbooks on alternative investment strategies and fixed-income securities.

Jean-Christophe Meyfredi, PhD is Head of the Accounting, Law and Finance Department at EDHEC Business School. His main teaching interests lie with portfolio management and quantitative trading. Jean-Christophe has published research on the pricing of bonds and derivatives, bond market microstructure and fixed-income portfolio risk management. He currently focuses on extreme risk modelling and hedge fund replication.

About the Authors

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4

1. Introduction ....................................................................................................................................................5

2. Factor Replication Approach .................................................................................................................8 2.1. Presentation of Basic Principles ...........................................................................................................................................8

2.2. Various Implementation Models ..........................................................................................................................................9

2.3. Further Empirical Analysis of the Factor-Based Approach .....................................................................................14

3. Payoff Distribution Approach ............................................................................................................ 22 3.1. Presentation of Basic Principles ........................................................................................................................................22

3.2. Various Implementation Models .......................................................................................................................................23

3.3. Empirical Results......................................................................................................................................................................27

4. Conclusion ................................................................................................................................................... 34 4.1. Conditional factor models ...................................................................................................................................................34

4.2. Non-linear factor models ....................................................................................................................................................35

5. References ..................................................................................................................................................... 37

Table of Contents

Following recent initiatives by major investment banks, there has been renewed interest in the financial industry concerning the subject of “passive hedge fund replication”, a question that had already been much discussed in academic circles over the last decade or so. In a nutshell, these initiatives are meant to enable investors to achieve returns similar to those of hedge funds with significantly lower fees through investment in a set of rules-based strategies based on liquid underlying assets aiming to replicate hedge fund performance, or at least the systematic factor exposure in hedge fund returns, i.e., their (traditional and alternative) beta components, as opposed to their alpha. Merrill Lynch and Goldman Sachs were the first to announce the launch of hedge fund replication tools, the “Merrill Lynch Factor Index” and the “Goldman Sachs Absolute Return Tracker Index.” Other attempts to introduce heuristic trading rules intended to replicate hedge fund manager decisions have recently been proposed, among others, by Partners Group with its “Alternative Beta Program”, and Merrill Lynch with its “Equity Volatility Arbitrage Index,” with JP Morgan apparently following suit with its announced “Alternative Beta Index.”

The purpose of this position paper is to provide an in-depth analysis of the subject, with an emphasis on the findings based on the last ten years of academic research on hedge fund performance analysis and replication, and a discussion of the implementation challenges related to a commercial offering based on these concepts. Overall, the emergence of these new forms of investment vehicles raises the questions of what exactly is to be understood behind the concept of a passive strategy. In the context of equity investment, where the concept was born, it appears that passive strategies are exclusively based on observable attributes, typically market cap weightings. Pure replication strategies are therefore subject neither to sample risk nor to model risk since the actual composition of the target portfolio (equity index) is continuously observable. Even in the context of statistical

replication strategies, where it is not necessary to replicate the target index composition, the tracking error between the index and the replicating portfolio can always be assessed explicitly since the composition of the index is known at all times. Hence, the replicating strategy can truthfully be regarded as purely passive, in the sense that no active views or modelling choices are involved in the process.

Now, this approach stands in sharp contrast to the aforementioned initiatives intended to provide “passive” replication of hedge fund returns. As we shall explain below, because the holdings of hedge fund managers are not observable, statistical models have to be introduced to capture the factor exposure or distributional properties of hedge fund returns. The success of these methods relies implicitly on a number of assumptions regarding the factors (absence of over- or under-specification and persistence) as well as the factor exposures (most notably functional form – linear or otherwise - and stationarity). These assumptions may in fact be regarded as implicit active decisions. This is even more obvious in the context of heuristic rule-based strategies that aim to replicate hedge fund managers’ trading patterns as opposed to managers’ returns. In the end, we will argue that so-called replication strategies appear to be nothing more than new systematic quantitative trading strategies, which can be useful additions to the investment space, but are hardly worthy of consideration as “passive replication strategies,” a serious misnomer in this context.

More specifically, we argue in this paper that there are two different, somewhat competing, approaches to hedge fund replication, known as factor-based replication (the technique involved in all of the aforementioned industry initiatives) and as payoff distribution replication (which has culminated in an independent industry initiative led by Professor Harry Kat). The two replication approaches are based on the same premise: namely, that assets with the same risk profile ought to earn the same returns.

1. Introduction

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In this position paper, we provide a critical assessment of both methods. Our conclusions can be broadly summarised as follows:

1/ The factor-based approach, while the most natural and straightforward way to tackle the hedge fund replication problem, has mostly failed in thorough empirical tests to produce satisfactory results on an out-of-sample basis. Intuitively, the reason the method fails is rather simple: because of the non-linear and dynamic exposure of hedge fund returns with respect to underlying risk factors, and in the absence of true modelling of the time-variations in these factor exposures, plain stepwise linear regression techniques, which simply match the average past exposures of hedge fund managers to underlying risk factors, are bound to perform poorly on an out-of-sample basis. In other words, capturing the conditional distribution of hedge fund returns, the focus of these strategies, would require truly conditional factor models that allow for time-varying factor exposures. For most hedge fund strategies, such satisfactory dynamic models for hedge fund returns have yet to be developed.

2/ The payoff distribution approach, while insightful and found to generate relatively satisfying results on an out-of-sample basis, unfortunately cannot be regarded as a method suitable for performing hedge fund replication, at least not in a sense likely to meet investors’ expectations. Intuitively, the reason the payoff distribution approach is not in essence a hedge fund replication method is that it focuses on matching the unconditional distributional properties of hedge fund returns, as opposed to their time-series properties. As such, the method is much more about fund design than about hedge fund replication, as actually acknowledged by its proponents. It should be emphasised, moreover, that the method aims to match the moments and co-moments of hedge fund returns … with the notable exception of the first moment. As a result, portfolio

performance, perhaps the most important aspect of portfolio return distribution for investors, is left unaddressed, and the average return on the replicating portfolio may be significantly lower (as illustrated by our empirical tests – see section 3) or higher than that of the hedge fund index it is meant to replicate, depending both on the choice of the risky portfolio involved in the replicating strategy and on the time period under consideration.

In addition to providing a detailed review of the literature, we also provide an empirical test of the performance of a benchmark implementation example for each of the two approaches. In what follows, we use the model introduced by Hasanhodzic and Lo (2006) as a generic template for the factor-based approach, and the model introduced by Amin and Kat (2003), as well as subsequent extensions by Kat and Palaro, as a generic template for the payoff distribution approach. We have chosen to focus on these two examples because both present rather elegant and parsimonious models that have appeared or are forthcoming in refereed academic journals and have therefore undergone the scrutiny that validates the methodological quality of the work presented. While we understand that these models could be, and have been, improved from a technical standpoint in the context of specific implementations, and while we understand that business secrets do not lend themselves to publication, we believe that these two models can be regarded as fair basic representations of the industry initiatives such as the factor-based approach led by Merrill Lynch or the payoff distribution approach led by Professor Harry Kat himself. In other words, our goal is not to focus on the benefits and limits of any particular model, but instead to present a series of tests of robustness that any model should be able to pass. It is therefore to be expected that the respective benefits and limits of the two models we examine can, regardless of their technical differences, shed light on other uses of these two approaches.

1. Introduction

The rest of the paper is organised as follows. In section 2 we present the factor replication approach. In section 3 we present the payoff distribution approach. In both cases we introduce the basic principles and describe the various implementation models and the numerical results that have been obtained. In section 4, we argue, as a conclusion, that hedge fund replication is in principle a powerful and attractive concept, but that, in terms of succesful implementation, it is still very much a work-in-progress. We also discuss possible directions for further research on the subject.

1. Introduction

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2. Factor Replication Approach

2.1. Presentation of Basic Principles

The factor replication approach involves looking for a portfolio composed of long and/or short positions in a set of suitably selected risk factors that minimises the tracking error with respect to the individual hedge fund or hedge fund index to be replicated. The best mimicking portfolio estimated from the in-sample analysis is then passively held in the out-of-sample period and out-of-sample performance is recorded and compared to the performance of the hedge fund target.

In a nutshell, factor-based replication is based on the following two-step process:

Step 1: Calibration of a satisfactory factor model for hedge fund returns

itkt

K

kik

HFt FR i εβ ˆˆ

1

+= ∑=

(1)where iHF

tR is the return at date t on hedge fund (or hedge fund style) i, ikβ̂ the (potentially time-varying) estimated exposure of the return on hedge fund i to factor k, ktF is the return at date t on factor k, and itε̂ is the estimated specific risk in the return of hedge fund i.

Step 2: Identification of the replicating portfolio strategy, or the clone, as:

kt

K

kik

Clonet FR i ∑

=

=1

β̂

The possible limits in the analysis are the following:• If the explanatory power (e.g., measured by a traditional R2 measure) of the in-sample regression analysis (1) is weak, a significant part of hedge fund return will not be captured in the returns of the clone.

• Even if the in-sample explanatory power is relatively high, the out-of-sample quality of replication could be low, because of a robustness problem, induced either by the presence of noise in the calibration sample (the model is fitted to reflect a sample-specific pattern and will not perform

well out-of-sample) and/or non-stationary series. In particular, we have strong evidence that hedge fund managers dynamically manage their exposure to various risk factors either in the context of active factor timing strategies or in the context of risk management strategies. While the true factor exposures ikβ are therefore likely to be time-varying, most commercial attempts to replicate hedge fund returns are based on models with constant parameters, undoubtedly because of the challenges involved in the estimation of a parsimonious conditional factor model.

The identification of a proper factor model in step 1 is therefore the main challenge. Given the wide range of risks to which hedge funds are exposed (see for example Géhin and Vaissié (2006)), one is naturally inclined to consider multi-factor approaches in the search for a good replicating factor model. This raises the key question of model specification. Over the in-sample period, hedge fund returns are regressed onto Asset-Based Style factors (henceforth ABS factors), through the following formula:

In-sample return ∑+= )*( ii ABSfactorβα + error term

ABS factors are selected through 1) arbitrary decisions, 2) optimisation, or 3) a suitably designed statistical analysis. Selecting the factors through arbitrary decisions typically involves the risk of under-specifying the model (omitting true factors) or over-specifying the model (including spurious factors). For this reason, several authors have preferred an optimisation-based approach, where the optimisation is typically implemented via stepwise regression. There are two types of stepwise regressions. Forward stepwise regression starts with no factor, and at each step the most statistically significant term is added to the model. Backward stepwise regression starts with all the factors, and at each step the least significant factor is removed.

In an attempt to alleviate the concern over specification risk, ABS factors can also be derived from principal component analysis, where Return-Based Style factors (henceforth RBS

factors) are estimated in the context of an implicit factor model. While this approach allows for lower misspecification risk, the drawback is in the economic significance of the implicit variables obtained, which are of little help in building mimicking portfolios. Consequently, an additional step is necessary to link RBS factors to ABS factors. For example, Sharpe’s (1992) return-based style analysis model can be used to link RBS factors to ABS factors. RBS factors are regressed onto several factors, and the most significant factors are kept.

To identify RBS/ABS factors, the in-sample regression is as follows:

RBS returns = ∑ )*ˆ( ii ABSβ + error term

During the out-of-sample period, the replication can be analysed in two ways. The first method consists of repeating the regression with the factors found during the in-sample period, and analysing the explanatory power of the replicating factors through out-of-sample R². The second method involves calculating the out-of-sample returns obtained by an exposure to the in-sample factors, and comparing the differences between replicating and replicated returns:

Out-of-sample return = ∑ )*ˆ( ii factorβ

2.2. Various Implementation Models

2.2.1. Literature Review – The ModelsFung and Hsieh (2002) study five fixed-income sub-strategies: Convertible Bond, High-Yield, Mortgage-Backed, Arbitrage and Diversified. To identify RBS factors, a principal component analysis is conducted on funds in the HFR database divided among peer groups. Next, RBS factors are turned into ABS factors. Considering the out-of-sample period, the authors report only the relationship between RBS and ABS factors.

Karavas, Kazemi and Schneeweis (2003) examine the replication of equally-weighted portfolios of

European-based hedge funds for five strategies, namely Funds of Funds, Convertible Arbitrage, Fixed Income Arbitrage, Event Driven and Long/Short Equity. The in-sample period is a rolling 24 month window, and the out-of-sample period is the month immediately following the sample window. The model is specified on the basis of two different approaches, a multi-factor model and a style-based analysis. The factors selected in the multi-factor analysis are European market factors that include equity market risk, interest rate risk, credit risk and volatility risk. The factors selected in the style-based analysis are a set of various futures and option contracts.

Agarwal, Fung, Loon and Naik (200�) create three ABS factors by simulating the returns generated by three Convertible Arbitrage sub-strategies: Positive Carry, Volatility Arbitrage and Credit Arbitrage. Since the ABS factor returns are obtained by using a sample of Japanese and US convertible bonds, the Japanese and US ABS factors are distinguished and used. The TASS, HFR, MSCI and CISDM Convertible Arbitrage indices are successively used as the dependent variable.

Studying eight strategies, Agarwal and Naik (2004) use a multi-factor model where the risk factors are “buy-and-hold” and “option-based.” The buy-and-hold risk factors are equities (four indices), bonds (three indices), currencies (one index), and commodities (one index). The authors add the Fama-French “size” and “book-to-market” factors, Carhart’s “momentum” factor, and a credit risk factor. The option-based risk factors are at-the-money (ATM) and out-of-the-money (OTM) European call and put options on the S&P �00. First, a stepwise regression is conducted to identify the significant factors, from 1990 to 2000, for eight HFR indices. Second, the authors examine whether “the replicating portfolios based on these factor loadings should do a good job of mimicking the out-of-sample performance of hedge funds.” A replicating portfolio is constructed, and the accuracy of the return replication is tested through a standard


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t-test and a Wilcoxon sign-test. Finally, for four strategies, the authors provide a comparison of the returns displayed by each replicating portfolio and its respective HFR index during the out-of-sample period.

The heterogeneity of the strategies used in the hedge fund industry raises the question of modelling the returns obtained by a diversified portfolio of hedge funds. Fung and Hsieh (2004) analyse whether a combination of the factors drawn from models specific to their respective strategy can lead to satisfactory modelling of the returns on a diversified hedge fund portfolio. The HFR Funds of Funds Index returns are regressed onto seven ABS factors obtained from three distinct strategies, from 1994 to 2002.

Jaeger and Wagner (200�) implement a multi-linear asset class factor model, from 1994 to 2004 (in-sample period). In a further step, the authors calculate the “Replicating Factor Strategy” returns (henceforth RFS): “the RFS returns in a given month were calculated using factors obtained by a regression over data for the previous five years ending with the previous month”, in order to “avoid the problem of data-mining and in-sample over-fitting.”

Hasanhodzic and Lo (2006) try to replicate the return distributions of eleven strategies extracted from the TASS Hedge Fund database, namely Convertible Arbitrage, Dedicated Short Bias, Emerging Markets, Equity Market Neutral, Event Driven, Fixed Income Arbitrage, Global Macro, Long/Short Equity, Managed Futures, Multi-Strategy and Fund of Funds. This constitutes a sample of 1,610 funds. Factor exposures of each strategy are measured by regressing returns onto six factors. These factors are the US Dollar Index, the Lehman Corporate AA Intermediate Bond Index, the spread between the Lehman Corporate BAA Bond Index and the Lehman Treasury Index, the S&P �00, the Goldman Sachs Commodity Index and the first difference of the end-of-month value of the CBOE Volatility Index. The authors select these factors because they can be

proxied by liquid instruments such as forwards or futures contracts. The factor exposures are calculated from February 1986 to September 200�. Hasanhodzic and Lo consider two types of replicating portfolios or clones: fixed-weight portfolios and linear clones based on rolling-window regressions. For the first type, weights for the instruments replicating the factors are estimated for each fund over the entire period, and they remain the same over this period. However, the “fixed-weight” approach suffers from a look-ahead bias and the authors also implement a 24-month rolling-window approach. This method involves dynamic rebalancing of the portfolio.

2.2.2. Literature Review – The ResultsFung and Hsieh (2002) obtain in-sample R² ranging from �9% to 78%. For three sub-strategies, lookback straddles are included to take into account the impact of dynamic trading, because lookback straddles disclose option-like payoffs. This generates an increase in R² of 1 to 6 points. No replication results are provided for the out-of-sample period.

With a multi-factor analysis, Karavas, Kazemi and Schneeweis (2003) obtain in-sample R² of 29.2% for Fixed Income Arbitrage, 31.9% for Convertible Arbitrage, �4.3% for Funds of Funds, 67.7% for Long/Short Equity, and 8�.8% for Event Driven. With the style-based analysis, they obtain in-sample R² of 30.2% for Fixed Income Arbitrage, 33.4% for Convertible Arbitrage, 46.7% for Long/Short Equity and 81.8% for Event Driven. The explanatory power of the different factors by strategy in the in-sample period is low, generally below 70%. Out-of-sample results are in line with the disappointing in-sample results. The correlation in performance between strategies and clones ranges from 12% to 91%, but is mainly below �0%. Only one strategy, Event Driven, reveals a satisfying correlation of 90% with the multi-factor analysis, and 91% with the style-based analysis. However, this result hides dramatic differences in performance and volatility. While the Event


Driven strategy discloses average performance of -2.67%, and volatility of 4.79%, the “multi-factor analysis” clone exhibits performance of -6.34% and volatility of 6.97% and the “style-based analysis” clone exhibits performance of -6.83% and volatility of 6.10%. In the same vein, while the Convertible Arbitrage strategy displays average performance of +8.28%, the “multi-

factor analysis” clone exhibits performance of +1.88% and the “style-based analysis” clone exhibits performance of +0.61%. Overall, for both methods and for all the strategies studied, clones considerably underperform hedge funds. Moreover, they are generally more volatile (see Exhibit 1 for an overview of the results in that paper).


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Exhibit 1. Results of a factor replication approach borrowed from Karavas, Kazemi and Schneeweis (2003), where both a multi-factor model and a style analysis are used

Strategies ABS/RBS In-sample results R²

Out-of-sample results

Mean strategy/clone // std dev

strategy/clone // correlation (in %)

Multi-factor analysis

Funds of Funds

ABS

�4.3% 4.19/-2.0� // 2.01/3.90 // 20%

Long/Short Equity 67.7% -0.98/-9.99 // 3.83/7.13 // 46%

Event Driven 8�.8% -2.67/-6.34 // 4.79/6.97 // 90%

Convertible Arbitrage 31.9% 8.28/1.88 // 1.82/1.�4 // 17%

Fixed Income Arbitrage 29.2% 7.87/2.89 // 2.96/2.�8 // 16%

Style-based analysis

Long/Short Equity

ABS

46.7% -0.98/-6.82 // 3.83/8.26 // 69%

Event Driven 81.8% -2.67/-6.83 // 4.79/6.10 // 91%

Convertible Arbitrage 33.4% 8.28/0.61 // 1.82/1.86 // 12%

Fixed Income Arbitrage 30.2% 7.87/3.�3 // 2.96/2.00 // 47%

Using Convertible Arbitrage indices as dependent variables, Agarwal, Fung, Loon and Naik (200�) find that the in-sample adjusted R² ranges from 14.67% to ��.9�% depending on the index provider. No replication results are provided for the out-of-sample period.

Agarwal and Naik (2004) obtain in-sample adjusted R² ranging from 40.�% to 91.63%. The standard t-test and Wilcoxon sign-test indicate that the difference in return (mean and median) between HFR indices and their respective replicating portfolio is statistically insignificant during the out-of-sample period from 2000 to 2001. According to the authors, the figure plotting the returns displayed by replicating portfolios and their respective HFR indices during the out-of-sample period “shows that the portfolios based on significant risk exposures estimated through our model closely track the hedge fund returns

during out-of-sample period.” However, as can be seen in Karavas, Kazemi and Schneeweis (2003) (see results above), this kind of graphical consideration should be handled with care in light of actual differences in average returns.

In Fung and Hsieh (2004), the regression of the HFR Fund-of-Funds Index returns onto seven ABS factors gives an in-sample adjusted R² of only ��%. No replication results are provided for the out-of-sample period.

Jaeger and Wagner (200�) show in-sample adjusted R² that ranges from 34.3% to 88.�%. The in-sample R² is lower than 70% for seven out of the ten strategies studied. On the out-of-sample period, the comparison between RFS cumulated returns and those of the corresponding HFR non-investable indices is presented for nine strategies. The differences range from

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-24.7 to +�.2 points. Clones underperform the non-investable indices in eight of the nine strategies and outperform them in one. The comparison between RFS cumulated returns and their corresponding HFR investable index is reported for seven strategies. The differences range from -3.2 to +12.9 points. Clones outperform the investable indices in six of the seven strategies and underperform them in one.

Hasanhodzic and Lo (2006) report highly contrasting results. Equal-weighted portfolios of fixed-weight clones underperform equal-weighted portfolios of their respective funds in four strategies (see details in Exhibit 2), outperform them in four others, and display comparable returns in three. Equal-weighted portfolios of rolling-window clones underperform equal-weighted portfolios of their respective funds in six strategies, outperform them in four, and have comparable returns in one.


Strategies ABS/RBS

Fixed-weight clones

Return1/ volatility2/

skewness3/ kurtosis4

Rolling-window clones

Return1/ volatility2/

skewness3/ kurtosis4

Convertible Arbitrage

ABS

= / = / = / - - / = / +p / -

Dedicated Short Bias - / + / = / - + / + / +p / =

Emerging Markets - / + / = / - - / + / +n / +

Equity Market Neutral + / + / = / - + / + / +p / -

Event Driven - / + / = / - - / + / -n / -

Fixed Income Arbitrage - / + / -n / - - / = / -n / -

Global Macro + / + / -p / - + / + / = / -

Long/Short Equity = / + / -n / - - / + / +n / +

Managed Futures + / = / -p / - = / + / -p / -

Multi-Strategy = / + / -p / - - / + / = / =

Funds of Funds + / = / -p / - + / + / -p / -

Exhibit 2. Results of a factor replication approach borrowed from Hasanhodzic and Lo (2006), where both fixed-weight clones and rolling-window clones are considered.

Focusing on volatility, equal-weighted portfolios of fixed-weight clones generate higher volatility than equal-weighted portfolios of their respective funds in eight strategies, and similar volatility in three. Equal-weighted portfolios of rolling-window clones display higher volatility than equal-weighted portfolios of their respective funds in nine strategies, and similar volatility in two. It also happens that at the third moment of the return distribution five fixed-weight clones show skewness similar to that of their respective funds, two fixed-weight clones show lower negative skewness compared to their respective funds, and four fixed-weight clones show lower positive skewness. Two rolling-window clones show skewness similar to that of their respective funds, two rolling-window clones show lower

negative skewness than their respective funds, two rolling-window clones show lower positive skewness, three rolling-window clones show higher positive skewness, and two rolling-window clones show higher negative skewness. Finally, concerning kurtosis, have funds from the 11 strategies exhibit fatter tails than fixed-weight clones. Using a rolling-window approach, this remains the case for seven strategies, while the opposite result holds for two other strategies, and similar kurtosis is found for the remaining two.

The replication of the correlation with market indices is of particular interest because it can be seen as an indication of the diversification power of hedge funds. The authors analyse the replication

1 + when clones outperform funds; - when they underperform; = when performance is similar.2 + when clones display higher volatility; - for lower volatility; = when volatility is similar.3 +p when clones display higher positive skewness; -p when they display lower positive skewness; +n when they display higher negative skewness; -n when they display lower negative skewness.4 + when clones display higher kurtosis; - for lower kurtosis; = when kurtosis is similar.

of the correlations with 28 market indices, from February 1986 to September 200�. Focusing on fixed-weight linear clones, the agreement in sign ranges from 79% to 100% across strategies, while the mean of the absolute difference in fund- and clone-correlation ranges from 11% to 23%. The results using rolling-window linear clones are similar, with an agreement in sign ranging from 71% to 96%, and absolute difference in correlation ranging from 9% to 19%. Again, the replication is not perfect, and it introduces significant deformation in the diversification properties. In section 2.3 below, we reproduce the results in Hasanhodzic and Lo (2006), so as to better understand the benefits and limits of the approach developed therein, with specific emphasis on analysing the robustness and quality of replicating strategies.

Overall, a review of the studies that attempt to replicate hedge fund return distributions through a factor replication approach leads to the conclusion that replication accuracy is not satisfying. In-sample R² is not sufficiently high to indicate satisfactory in-sample fit, while out-of-sample results suggest that hedge fund return replication is approximate at best (see Exhibit 3 for a summary of the results). Comparison of the first four moments of the returns displayed by the clones and their respective funds confirms that factor replication fails to provide a satisfactory substitute for hedge funds. Overall, no proper inference can be made when selecting the factors by stepwise regressions. The non-linear exposure to risk factors is not well captured by a linear projection even when some of the factors are option returns, and in the end using rolling


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Exhibit 3. In- and out-of-sample results of hedge fund replication based on ABS and RBS/ABS models

Authors Strategies ABS/RBS(In–sample period )In–sample results

(Out-of-sample period )Out-of-sample results

Agarwal, Fung, Loon, Naik (200�)

Convertible Arbitrage ABS R² 14.67 to ��.9�% n/a

Agarwal, Naik (2004)

Convertible ArbitrageEquity Long/ShortEquity Non Hedge

Event DrivenRelative ValueRestructuringRisk ArbitrageShort Selling

ABS

(1990-2000)R² 40.�0%R² 72.�0%R² 91.63%R² 73.40%R² �2.20%R² 6�.60%

R² 44%R² 82%

(2000-2001)approximate replication

of index returns(standard t-test &

Wilcoxon sign-test)

Fung, Hsieh (2002)

Convertible BondHigh-Yield

Mortgage-BackedArbitrageDiversified

RBS/ABS

(1990-2000)R² 70% to 7�%R² 78% to 79%R² �9% to 66%

R² 66%R² 64%

n/a

Fung, Hsieh (2004)Multi-Strategy

portfolioABS

(1994-2002)R² ��%

n/a

Jaeger, Wagner (200�)

Convertible ArbitrageDistressed SecuritiesEquity Long/Short

Equity Market NeutralEvent DrivenFixed IncomeGlobal Macro

Merger ArbitrageShort Selling

ABS

(1994-2004)

R² �4%R² 68.40%R² 88.�0%R² 3�.30%R² 79.30%R² 40.�%R² 49.70%R² �2.90%R² 81.20%

(2003-200�)RFS in % / HFRI / HFRX

+7.6 / +2.4 / -�.3+20.1 / +44.8 / +23.3+27.8 / +32.8 / +16+6.2 / +10.9 / -3.9+29.8 / +40 / +24.1+7.8 / +16.3 / n/a

+16.7 / +24.6 / +10.1+13 / +1�.3 / +10.9

-28.2 / -23 / n/a

14

samples is much too rudimentary to capture the conditional aspect of the relationship between hedge fund returns and the factors.

Since academic attempts to design factor models for hedge fund return replication have so far been unable to generate fully satisfactory results, the recent launch of a number of industry initiatives is perhaps surprising.

In an attempt to better assess the benefits and limits of factor-based hedge fund replication, we shall now attempt to calibrate a model and analyse the out-of-sample properties of hedge fund clones.

2.3. Further Empirical Analysis of the Factor-Based Approach

In this section, we have chosen to reproduce the results reported in Hasanhodzic and Lo (2006). The reason we have elected to focus on this particular model is essentially related to the fact that it is a simple parsimonious approach that is not suspected of data mining. As explained above, the authors have introduced a factor model based on the following six factors for all strategies: the US Dollar Index, the Lehman Corporate AA Intermediate Bond Index, the spread between the Lehman Corporate BAA Bond Index and the Lehman Treasury Index, the S&P �00, the Goldman Sachs Commodity Index and the first difference of the end-of-month value of the CBOE Volatility Index. Return series for these factors are taken from the Datastream database.

In what follows, we apply the model to the replication of EDHEC hedge fund indices using monthly data ranging from January 1997 to December 2006. As in Hasanhodzic and Lo (2006), we consider two types of replicating portfolios or clones, fixed-weight portfolios and linear clones based on rolling-window regressions. For the first type, weights for the instruments replicating the factors are estimated for each hedge fund index over the entire period, and they remain the

same over the whole period. To avoid the implied look-ahead bias, we also implement a 24-month rolling-window approach, which is the only experiment that generates truly out-of-sample results. Finally, we provide various robustness tests consisting of adjusting the size of the calibration and/or out-of-sample period.

2.3.1. In-Sample ResultsWe first consider in Exhibit 4 the result of a standard multiple regression analysis over the whole period.

Several comments are in order. First, the analysis of the results of the Fisher test (F-test) suggests that it is only in the case of Convertible Arbitrage that all factor exposures are simultaneously non-significantly different from zero (the test statistic for a �% p-value is 2.1798). In-sample (adjusted) R2 is relatively low, with a maximum of �9.43% for the short-selling index. Also, the analysis of the results from a standard Durbin-Watson test (not reported here) leads to acceptance of the null hypothesis of no significant auto-correlation in the residuals only for the CTA factor model. We also test for the presence of heteroscedasticity, using the Koenker (1981) version of the Breusch and Pagan (1979) test, and find that heteroscedasticity is actually present in the following strategies: Event Driven, Fixed Income Arbitrage, Global Macro and Merger Arbitrage.

In an effort to account for the presence of heteroscedasticity and auto-correlation, we then implement the Newey-West correction (Newey and West (1987)) for all models. The results are displayed in Exhibit �.

As can be seen in Exhibit �, after correcting for the presence of heteroscedasticity and auto-correlation, the exposure to the bond index factor is no longer significant for the Fixed Income Arbitrage and Relative Value indices. Similarly, the credit spread exposure is no longer significant for the Distressed Securities and Fixed Income Arbitrage indices, while the VIX exposure is no longer significant for the Fixed Income



1�

Exhibit 4. In-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 6-factor model. Sample period is January 1997 to December 2006. Negative values are indicated in parentheses.

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16

Arbitrage and Global Macro indices. Conversely, the commodity index exposure becomes significant for the Emerging Markets index. These results should, however, be regarded with caution, since a standard Jarque-Bera test signals (in unreported results) that none of the residual series is normally distributed. Overall, this analysis suggests that even simple in-sample results are rather sensitive to the methodology used and the assumptions underpinning the regression model. In section 2.3.3 below, we report the results of further empirical checks that confirm the relative lack of robustness of the model.

We now turn to replication results. We decide to follow Hasanhodzic and Lo (2006) who eventually chose to omit the VIX index as a factor out of concern for the insufficient liquidity of its

investable counterpart, and also to remove the constant term in the regression analysis. As in Hasanhodzic and Lo (2006), we also implement a further renormalisation in an attempt to impose the same volatility as the target hedge fund index on the replicated return series (for the case of the fixed-weight in-sample analysis).

Exhibit 6 presents the mean, volatility and minimum and maximum (non-annualised) monthly returns for both the hedge fund index and the linear clone, when weights are estimated over the whole sample period (and therefore subject to look-ahead bias). Our results show that the mean performance is always lower (and sometimes dramatically so) for the linear clone than for its corresponding hedge fund index. The VaR estimate is often higher as well.


Exhibit 6. In-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5-factor model. Sample period is January 1997 to December 2006.

Hedge Fund Index Mean SD Min Max VaR

Convertible Arbitrage 0.0076 0.0114 -0.0319 0.0344 -0.0133

CTA Global 0.0064 0.026 -0.0�43 0.0691 -0.0363

Distressed Securities 0.0101 0.01�3 -0.0836 0.0421 -0.0102

Emerging Markets 0.0102 0.0367 -0.1922 0.123 -0.0429

Equity Market Neutral 0.0074 0.0061 -0.0107 0.02�3 -0.0007

Event Driven 0.0092 0.016 -0.0886 0.0429 -0.0128

Fixed Income Arbitrage 0.00�2 0.0104 -0.0801 0.0208 -0.0061

Global Macro 0.0084 0.0173 -0.0304 0.0738 -0.0149

Long/Short Equity 0.009� 0.020� -0.0��2 0.074� -0.0203

Merger Arbitrage 0.007� 0.0107 -0.0�44 0.0272 -0.0093

Relative Value 0.0078 0.009� -0.0341 0.0333 -0.0107

Short Selling 0.003� 0.0�83 -0.134 0.2463 -0.099�

Funds of Funds 0.0079 0.016� -0.0616 0.0666 -0.014

Linear Clones


CTA Global 0.0016 0.026 -0.0786 0.0789 -0.039

Distressed Securities 0.0019 0.01�3 -0.0489 0.0482 -0.024

Emerging Markets 0.004� 0.0367 -0.13�3 0.0848 -0.0604

Equity Market Neutral 0.0008 0.0061 -0.0126 0.0212 -0.0099

Event Driven 0.0021 0.016 -0.0��2 0.046� -0.0247

Fixed Income Arbitrage 0.0004 0.0104 -0.0287 0.0372 -0.0164

Global Macro 0.0022 0.0173 -0.0�23 0.0�44 -0.0263

Long/Short Equity 0.0031 0.020� -0.072 0.0499 -0.0294

Merger Arbitrage 0.001� 0.0107 -0.032 0.03�4 -0.01�6

Relative Value 0.0012 0.009� -0.0288 0.032� -0.0144

Short Selling -0.00�8 0.0�83 -0.1267 0.198� -0.088�

Funds of Funds 0.0024 0.016� -0.0��2 0.0489 -0.0238

2.3.2. Out-of-Sample ResultsWe now turn to out-of-sample results. As in Hasanhodzic and Lo (2006), we use a 24-month rolling-window calibration period to estimate the model coefficients, and replicate the subsequent month’s return. The results are reported in Exhibit 7.

Analysis again confirms (Exhibit 7) that the mean return on the clone portfolio is always (often dramatically) lower than that of the hedge fund index, except for the short selling index. This is consistent with the results in Hasanhodzic and Lo (2006), who use a time period and hedge fund


17

Exhibit 7. Out-of-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5 factor model. Sample period is January 1997 to December 2006.



CTA Global 0.00�3 0.02�9 -0.0�43 0.0682 -0.036�

Distressed Securities 0.0112 0.012� -0.0209 0.0421 -0.008

Emerging Markets 0.0134 0.029 -0.0�41 0.123 -0.0386

Equity Market Neutral 0.0066 0.00�4 -0.0082 0.02�3 -0.0008

Event Driven 0.0094 0.0132 -0.03 0.0429 -0.013

Fixed Income Arbitrage 0.0061 0.00�1 -0.0092 0.0208 -0.002�

Global Macro 0.0074 0.0148 -0.0304 0.0612 -0.0114

Long/Short Equity 0.0084 0.0199 -0.0389 0.074� -0.0213

Merger Arbitrage 0.0069 0.009 -0.0267 0.0272 -0.009�

Relative Value 0.0076 0.0089 -0.0221 0.0333 -0.006

Short Selling 0.0009 0.0�37 -0.134 0.16�7 -0.102

Funds of Funds 0.0077 0.01� -0.0269 0.0666 -0.014

Linear Clone


CTA Global 0.0027 0.0293 -0.0724 0.0792 -0.0437

Distressed Securities 0.003 0.0164 -0.042� 0.0��9 -0.02�2

Emerging Markets 0.0039 0.0391 -0.1449 0.104 -0.0636

Equity Market Neutral 0.0011 0.006� -0.01� 0.026� -0.0103

Event Driven 0.0032 0.01�8 -0.0346 0.0�79 -0.019


Global Macro 0.0017 0.0166 -0.0�96 0.0�21 -0.0262

Long/Short Equity 0.0019 0.0202 -0.0816 0.0661 -0.0286

Merger Arbitrage 0.002� 0.0107 -0.023� 0.04� -0.0134

Relative Value 0.0019 0.0102 -0.0223 0.036 -0.0148

Short Selling 0.0031 0.070� -0.1332 0.336 -0.0987

Funds of Funds 0.0016 0.0181 -0.0867 0.06�8 -0.026

benchmarks different from ours, and overall report inferior performance of the clones. In addition, the VaR estimates are always higher for the clone than for the index to be replicated. Overall, these results do not support the belief that the risk-return trade-off of hedge fund returns can be satisfactorily replicated.

In Exhibit 8, we also report the out-of-sample correlation between the hedge fund index and the corresponding clone, as well as the associated R2.

Overall the replication power seems to be very low; there is, for example, a 2.6�% out-of-sample R2 in the case of the Convertible Arbitrage index! Even the Long/Short Equity index, arguably one

18


Exhibit 8. Out-of-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5 factor model. Sample period is January 1997 to December 2006

Correlation R2

Convertible Arbitrage 16.28% 2.6�%

CTA Global 40.99% 16.80%

Distressed Securities 31.33% 9.82%

Emerging Markets 60.22% 36.26%

Equity Market Neutral 27.41% 7.�1%

Event Driven 49.76% 24.76%

Fixed Income Arbitrage 18.28% 3.34%

Global Macro 36.11% 13.04%

Long/Short Equity 61.42% 37.72%

Merger Arbitrage 31.01% 9.62%

Relative Value 43.23% 18.69%

Short Selling 71.�8% �1.24%

Funds of Funds 4�.3�% 20.�7%

of the easiest to replicate, merely allows for 37.72% R2. With these results in mind, one can only be skeptical of the usefulness of simple factor models in the context of hedge fund return replication strategies.

We now report the results of a number of robustness checks that suggest that not only are standard factor models very unlikely to lead to

successful hedge fund replication, but also that they lead to results that are very sensitive to the sample period used in the analysis.

2.3.3. Robustness ChecksWe first conduct an in-sample analysis while splitting the overall sample period into two �-year sub-periods. We report the results we obtain in Exhibits 9 and 10.

As this analysis shows, the mean returns are still mostly lower for the clones than for the hedge fund index for the first sub-sample period. On the other hand, for the second sub-sample period, clones perform better than the corresponding hedge fund index for some strategies (CTA Global,

Global Macro, Long/Short Equity). Overall, an analysis of these results suggests that they are very sensitive to the choice of the sample period, and therefore any published results about the performance of factor-based replication models should be viewed with caution.


19


First � Year Sub-Period



CTA Global 0.006� 0.02�2 -0.0�43 0.0691 -0.03�4

Distressed Securities 0.008� 0.0186 -0.0836 0.0421 -0.0136

Emerging Markets 0.0069 0.0477 -0.1922 0.123 -0.0�43

Equity Market Neutral 0.0099 0.0067 -0.0107 0.02�3 0.0017

Event Driven 0.0099 0.019 -0.0886 0.0429 -0.0111


Global Macro 0.0098 0.0214 -0.0304 0.0738 -0.0182

Long/Short Equity 0.0121 0.0236 -0.0��2 0.074� -0.0204

Merger Arbitrage 0.0101 0.0124 -0.0�44 0.0239 -0.0062

Relative Value 0.0096 0.010� -0.0341 0.0333 -0.0036

Short Selling 0.008 0.07�3 -0.134 0.2463 -0.113�

Funds of Funds 0.0096 0.0211 -0.0616 0.0666 -0.014�

Fixed-Weight Linear Clones

Convertible Arbitrage 0.0036 0.0114 -0.0214 0.037� -0.014

CTA Global 0.004 0.02�2 -0.0�3 0.0723 -0.03�9

Distressed Securities 0.0037 0.01886 -0.0476 0.0�43 -0.0274

Emerging Markets 0.00�8 0.0477 -0.1�3 0.101� -0.0713

Equity Market Neutral 0.002� 0.0067 -0.0113 0.020� -0.0077

Event Driven 0.003� 0.019 -0.0�32 0.0�06 -0.0274


Global Macro 0.0037 0.0214 -0.0��6 0.0622 -0.0312

Long/Short Equity 0.0039 0.0236 -0.067� 0.0�09 -0.0303

Merger Arbitrage 0.003� 0.0124 -0.0268 0.0362 -0.01�8

Relative Value 0.0026 0.010� -0.023 0.0334 -0.01��

Short Selling 0.0013 0.07�3 -0.1229 0.2�39 -0.0894

Funds of Funds 0.0036 0.0211 -0.060� 0.0��6 -0.0286

20

As a last word of caution, we have investigated the robustness of adjusted R2 with respect to changes in the sample period. The results are reported in Exhibits 11, 12 and 13. Exhibit 11 shows the evolution of R2 when the initial date is fixed (January 1997) and the final date varies

from September 1998 on. Exhibit 12 shows the evolution of R2 when the final date is fixed (December 2006) and the initial date varies from January 1997 to April 200�. Finally, Exhibit 13 shows the evolution of R2 when the initial date is fixed, and a �-year rolling window is used.



Second � Year Sub-Period



CTA Global 0.0062 0.027 -0.0�32 0.06�� -0.0379

Distressed Securities 0.0117 0.0109 -0.0133 0.034� -0.0044

Emerging Markets 0.013� 0.020� -0.0389 0.0�26 -0.02�4


Event Driven 0.0086 0.0126 -0.03 0.0341 -0.013

Fixed Income Arbitrage 0.00�� 0.0046 -0.0092 0.0207 -0.0011

Global Macro 0.007 0.012 -0.0178 0.0397 -0.0088

Long/Short Equity 0.007 0.0166 -0.0389 0.0381 -0.0187

Merger Arbitrage 0.00� 0.008 -0.0174 0.0272 -0.0107

Relative Value 0.0061 0.0082 -0.018� 0.0238 -0.0107

Short Selling -0.001 0.0339 -0.06�6 0.0731 -0.0�48

Funds of Funds 0.0061 0.0098 -0.0149 0.0286 -0.0133

Fixed-Weight Linear Clones


CTA Global 0.0063 0.027 -0.0�96 0.0709 -0.04�3

Distressed Securities 0.0019 0.0109 -0.0218 0.0313 -0.018�

Emerging Markets 0.0046 0.020� -0.0492 0.0��1 -0.0271


Event Driven 0.0021 0.0126 -0.0302 0.0321 -0.0197


Global Macro 0.0026 0.012 -0.02�7 0.036� -0.0231

Long/Short Equity 0.0029 0.0166 -0.044� 0.038� -0.0226

Merger Arbitrage 0.0009 0.008 -0.01� 0.0199 -0.0137

Relative Value 0.0011 0.0082 -0.0172 0.021 -0.0136

Short Selling -0.00� 0.0339 -0.0727 0.1083 -0.0603

Funds of Funds 0.0017 0.0098 -0.018 0.0274 -0.016


21

Exhibit 11. In-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5 factor model. The graph shows the time-series evolution of adjusted R2 when the initial date is fixed (January 1997) and the final date varies from September 1998 on.

Exhibit 12. In-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5 factor model. The graph shows the time-series evolution of adjusted R2 when the final date is fixed (December 2006) and the initial date varies from January 1997 to April 2005.

Exhibit 13. In-sample results of EDHEC hedge fund index replication based on Hasanhodzic and Lo’s 5 factor model. The graph shows the time-series evolution of adjusted R2 when the initial date is fixed, and a 5-year rolling window is used

Overall, these results strongly suggest that R2 obtained on small samples can be somewhat misleading since the R2 obtained varies quite significantly depending on the sample period used in the analysis. They also suggest that it

is always possible to “maximise” reported R2 through data mining, when the sample period is chosen so as to display the most spectacular results.

We now turn to the competing approach to hedge fund replication, based on payoff distribution

replication, to see whether it can provide a more satisfying answer to this question.

22

The payoff distribution replication approach has been developed by Amin and Kat (2003) but finds its roots in the seminal work of Dybvig (1988).

The objective of this methodology is far less ambitious than the one pursued in the factor-based approach to hedge fund replication. While both approaches aim to generate a replicating portfolio for hedge fund returns, there is a dramatic difference in what is understood by the word “replication,” which is related to the several different senses in which random variables can be considered to be equivalent. In particular, two random variables can be equal almost surely or equal in distribution.

Two random variables (here the return on the hedge fund HF

tR and the return on the clone portfolio Clone

tR ) are said to be equal (almost surely) if their values agree with probability one; in other words, when we have for all t that:

( ) 1Pr == HFt

Clonet RR

This is precisely the ambitious objective that factor-based approaches attempt to achieve, with limited success, as was made apparent in the discussion from the previous section.

On the other hand, two random variables X and Y (here the return on the hedge fund and the return on the clone portfolio) are said to be equal in distribution if their probability distributions are identical; in other words, when we have for all x that:

( ) ( )xRxR HFt

Clonet <=< PrPr

As is obvious from the definitions above, while equality almost surely implies equality in distribution, the converse is not true. Hence, equality almost surely is a stronger, more ambitious, definition of “replication.”�

In what follows, we present a methodology attempting to replicate the distribution of hedge fund returns, with the notable exception of the

first moment (mean return). We argue that this weaker definition of replication is too limited to match investors’ expectations when it comes to replicating hedge fund returns.

3.1. Presentation of Basic Principles

This methodology was initially developed by Amin and Kat (2003) in the context of hedge fund performance measurement, and in an attempt to design a risk-adjusted measure of performance that would be better adapted to the hedge fund universe than a simple Sharpe ratio, which has been well documented as being ill-suited for assets exhibiting non-Gaussian distributions.

The keystone of this new approach is that “when buying fund participation, an investor acquires a claim to a certain payoff distribution.”6 If one is able to replicate the payoff distribution under consideration through a particular trading strategy, the cost of the replicating strategy could be compared to that of a direct investment in the hedge fund, and this could be used in assessing whether or not the manager is really adding value.

The principle of payoff replication is very simple and inspired from derivative pricing theory. It is based on the following two-step process. The first step consists of estimating the payoff function g that maps an index return onto a hedge fund return, while the second step consists of pricing the payoff and deriving the replicating strategy, in accordance with the Merton (1973) replicating portfolio interpretation of the Black and Scholes (1973) formula.

Regarding the first step (deriving the payoff distribution function), one key difference with a standard option pricing exercise is that the payoff function is created synthetically so as to match the hedge fund return distribution rather than being exogenously given as in the case of a standard option pricing problem.

3. Payoff Distribution Approach

5 - It should be noted that an investor is in general not indifferent when given a choice between two different portfolios even if these portfolios have the same return distribution. In particular, asset pricing theory (see for example Duffie (2006)) suggests that the price of a given claim is not only a function of the distribution of the claim, but also of its co-variation with a given pricing kernel or stochastic discount factor.6 - See Amin and Kat (2003) on page 255.

The payoff function can simply be written as:

( ) ( )( )xFFxg indexhf

1−= (2)

with x , the index return values, 1−hfF , the

quantile function of the hedge fund to replicate and indexF the cumulative density function of the index.

For example, assume there is a 10% probability that an index return over a horizon T=1 year takes on a value higher than 20%. Hence, we havethat ( ) ( ) 9.0%20%20Pr ==≤ indexT FIndex .

We then define g(20%) as the value that is such that the hedge fund return to be replicated has only a 10% chance of being higher, or: ( ) ( ) ( ) 9.0Prsuch that %20 ==≤= yFyHFyg hfT

Again, it can easily be seen that the condition

( )TT IndexgHF = (equality almost surely) implies condition (2), which defines an equality in distribution, while the converse is not true. Indeed, we have (for a strictly increasing function g): ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )[ ]ygFFyygFyF

ygIndexyIndexgyHF

IndexhfIndexhf

TTT

111

1PrPrPr−−−

−

=⇒=⇒

≤=≤=≤

Defining ( )xgy = , or ( )ygx 1−= , we finally obtain that ( ) ( )( )xFFxg indexhf

1−= .

Regarding the second step, once the payoff function is obtained, it can be priced (and dynamically replicated) with a suitable standard option pricing model, which allows one to obtain the “initial investment required by the dynamic trading strategy, trading the reference index and cash.”

Several alternatives could be explored both in the methodology for estimating the payoff function (empirical distribution, parametric Gaussian, parametric non-Gaussian), and in the option-pricing model used (standard Black-Scholes model with constant volatility, stochastic volatility model, etc.).

In what follows, we first present the original approach in Amin and Kat (2003), as well as its extensions to the bivariate case introduced by Kat and Palaro (200�, 2006a, 2006b, 2006c).

We will also briefly discuss various possible improvements.

3.2. Various Implementation Models

3.2.1. Presentation of the Original Approach

3.2.1.1. Step 1: Estimating the Payoff Distribution FunctionAmin and Kat (2003) chose to fit the observed cumulative distribution function of the underlying asset (the return of the nearby S&P �00 futures contract in their study) with a Gaussian distribution. In other words, they assume that

( ) ( )dt

txF

x

index ∫ ∞−

−=

2

2

2exp

2

1σ

μ

πσfor some parameters μ and σ to be empirically estimated as the sample mean and sample volatility of index return, respectively.

Exhibit 14 displays the different steps of payoff function construction. The top two graphs represent the empirical cumulative distribution functions for the hedge fund performance to be replicated hfF and a normal approximation for the underlying asset indexF (here the S&P �00 index), while the lower graph corresponds to the payoff distribution function g constructed as indicated in equation (2) above. In other words, to find the payoff function we simply match the quantile of the hedge fund density function with the value corresponding to the same quantile but for the S&P density function7.

It should be noted at this stage that imposing an assumption of normally distributed returns tends to induce a bias in the estimation of the payoff function and consequently the accuracy of the performance measure coming from this approach. Let us thus suppose that the S&P �00 returns actually follow a skew Student-t distribution with a mean of 100, a standard deviation of �, � degrees-of-freedom (dof) and a parameter of asymmetry of 0.8.


237 - As in Amin and Kat (2003), we assume that we invest $100 at the beginning of the period considered.

24


Exhibit 14. Construction of the Payoff Function. Construction of the payoff distribution function in the Amin and Kat (2003) context. The three graphs show the different steps involved in constructing the payoff distribution function. The graph in the lower part of this exhibit corresponds to the payoff distribution function constructed with an empirical cumulative distribution function for the hedge fund being replicated (upper left) and a normal approximation for the reference asset distribution (upper right), both assumptions corresponding to those of Amin and Kat (2003).

In Exhibit 1�, we show a simulation of this distribution, the probability density of which is given by:

with Γ the Euler function, δ a non-centrality parameter and ν the number of degrees of freedom.

Empirical cumulative distribution function to replicate (the fund distribution)

Normal cumulative distribution function of the underlying asset (S&P 500)

Payoff distribution function

f x( ) =ν( )ν / 2 exp −δ 2 / 2⎡⎣ ⎤⎦

Γ ν / 2( )π 1 / 2 ν + x 2( ) ν +1( ) / 2 . Γν + i +1

2⎛⎝⎜

⎞⎠⎟

xδi!

⎛⎝⎜

⎞⎠⎟

i 2ν + x 2

⎛⎝⎜

⎞⎠⎟

i / 2

i = 0

∞

∑


2�8 - In subsequent papers, Kat and Palaro (2005, 2006a, 2006b) explore other alternatives such as the Student-t or the Johnson SU distributions. Alternatively, one could naturally be inclined to use the empirical distribution itself.9 - All hedge fund data come from the MAR/Hedge database.

Of course, it is to be expected that the payoff function constructed on the basis of non-Gaussian return series will generate a discrepancy between the empirical payoff and the one obtained from a parametric fit based on a Gaussian distribution. For example, the existence of fat tails and a negative skewness will result in a payoff function that underestimates values in the tails, which introduces a negative bias in estimates of hedge fund performance reported in Amin and Kat (2003), as can be seen in Exhibit 168.

3.2.1.2. Step 2: Pricing the Payoff and Deriving the Replicating StrategyIn an attempt to find the fair price of the payoff, Amin and Kat (2003) use a Monte Carlo simulation whereby they generate 20,000 end-of-month index values using the assumption that index returns follow a geometric Brownian motion

with a drift equal to the risk-free rate under the risk-neutral probability measure. Hence, index prices are given by:

where ( )tS is the index value, normalised at 100 at the initial date, r is the risk-free rate (the one-month T-Bill rate), q is the dividend yield on the S&P �00, σ is the S&P �00 volatility, tδis the time step (one month) and φ is a random term assumed to follow a standard normal distribution. The price of the payoff function is obtained by discounting the average of the 20,000 different payoffs corresponding to the simulated end-of-month index values. Then the authors compare this price to $100 (amount initially invested) to analyse the value added (or destroyed) by the fund manager. Based on this set of assumptions, they found that over the period May 1990-April 2000, 72 of the 77 hedge fund profiles tested and 12 of the 13 hedge fund indices analysed were inefficient in the sense that they were outperformed by their replicating portfolio strategy9.

3.2.2. Presentation of Subsequent ImprovementsIn a more recent set of papers, Kat and Palaro (200�, 2006a, 2006b, 2006c, 2006d) transpose the methodology to hedge fund return replication, as opposed to merely performing hedge fund risk-adjusted performance evaluation. For this, it is necessary not only to consider the price of the replicated payoff but also to estimate the related replicating strategy, in accordance with the Merton (1973) replicating portfolio interpretation of the Black and Scholes (1973) formula.

Interestingly, the authors also extend the previous methodology to the bivariate context by considering the relationship between a specific fund and the investor’s existing portfolio in addition to the marginal distribution of hedge fund returns. The approach is very similar to the previous one, the improvement being that the

S t + δt( ) = S t( )exp r − q−1

2σ 2⎛

⎝⎜⎞⎠⎟δt + σ δtφ

⎡

⎣⎢

⎤

⎦⎥

Exhibit 15. Simulation of a Student-t distribution.

Exhibit 16: Existence of a bias when using a normal approximation (red curve) instead of the empirical distribution (black curve)

26

aim is not only to replicate the first moments of hedge fund return distribution but also its co-moments with an investor’s existing portfolio. The goal is to find the cheapest payoff function that transforms the joint distribution of the investor’s portfolio return and the return on the reserve asset into the joint distribution of the fund return and the investor’s portfolio return.

Two steps are involved again, the first one consisting of pricing this payoff function and the second one in estimating the dynamic portfolio strategy in the investor’s portfolio and the reserve asset that replicates it.

Using Kat and Palaro’s notation, the payoff function in the bivariate case is now given by:

g x, y( ) = κ X−1( )P , I κ x

P , R y( )( )

with:

FI / P =κ xP , I y( ) = P XI ≤ y X p = x( ) = ∂CP , I u,v( )

∂u u = F p x( ) ,v= F I y( )

FR / P =κ xP , R y( ) = P XR ≤ y X p = x( ) = ∂CP , R u,v( )

∂u u = F p x( ) ,v= FR y( )

with IPC , the copula between pX and IX ,

RPC , the copula between PX and RX and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

100log j

j

SX

while Sj, RPIj ,,= ,

represents the value of the end-of-month payoff of the fund, the investor’s portfolio and the reserve asset, respectively.

Kat and Palaro (200�, 2006a, 2006b, 2006c) test the Normal, Student-t and Johnson SU distribution as well as six bivariate copulas for integrating the dependence between hedge fund returns and the investor’s existing portfolio (Normal, Student-t, Gumbel, Clayton, Frank, Symmetrised Joe-Clayton). The objective is to find the best combination between marginal distributions and the dependence function to match the observed data considering symmetric/asymmetric, thin and fat-tailed joint distributions.

The estimation process is performed by following the Inference for Margins method, i.e., a two-step maximum likelihood method which consists of estimating the parameters of the margins, while the copula parameter(s) is (are) obtained in a second step. The best combination is selected on the basis of an Akaike Information Criterion. (It should be emphasised at this point that the AIC criterion, while usefully involving a penalty on the number of degrees-of-freedom, nevertheless tends to favour models - here distribution and copulas - which exhibit the best global fitting. In this context, one could benefit from other tests such as the Anderson Darling tests, the Integrated Anderson Darling test or the use of an Pl norm or tail concentration functions.10)

The end-of-month replicated value from a monthly initial investment of $100 becomes:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

100log,

100logexp100 Rp

g

SSgS

Kat and Palaro underline three practical limits to the approach. The first one is that the parameters of the true distribution are unknown and therefore have to be estimated. The second drawback is that parameter estimation could be biased by the fact that the distribution may not be stationary. Lastly, in practice, the desired payoff function cannot be generated exactly when the presence of market imperfections has to be accounted for.

They test their model on three fund return distributions estimated from the TASS database on a window whose first value corresponds to the starting date of their sample (January 1, 198�) and the end of the window corresponds to the (24+i)th month, with i ={0,204} corresponding to the date where the dynamic trading strategy is rebalanced. They assume the investor’s portfolio is invested half in the S&P �00 and half in US Treasury bonds. In a second paper, Kat and Palaro (2006c) introduce the Boyle and Lin (1997) multivariate option pricing model to price the payoff function. This particular model allows them to integrate transaction costs.


10 - See for example Meyfredi (2005).

3.3. Empirical Results

To get a better feel for its performance, we test in what follows a straightforward application of the payoff distribution approach to the 13 hedge fund indices in the EDHEC database between January 1997 and December 2006. As in Amin and Kat (2003) or Kat and Palaro (2006), we also used the nearby futures contract on the S&P �00 index and on the Eurodollar over the same period, taken from the Datastream database. The time series of these two asset returns are shown in Exhibit 17.

The out-of-sample period starts in January 1999, i.e., we use the first two years of data to calibrate the model. For each monthly date t starting in January 1999, we first estimate the payoff distribution function by using empirical asset return distributions so as to avoid imposing any overly restrictive assumption, as described in section 3.2.1, and on the basis of a sample period ranging from January 1997 to the current date. We then simulate 100,000 possible monthly sample paths for the risk-neutral S&P distribution, with a drift equal to the risk-free rate and volatility estimated over the period that ranges from the beginning of the sample to date t. The fair price of the empirical payoff is obtained by taking the risk-neutral discounted average value of the payoffs, while the delta of the option, which generates the replicating strategy, is generated on a daily basis by a numerical approximation of the first derivative of the value function with respect to the underlying asset price.

3.3.1. What Works (Relatively) Well: Matching Distributional PropertiesTo check the quality of our replication process, we have used various tests.

The first one consists of analysing the difference between the first four order moments observed and replicated. The upper part of Exhibit 18 displays those results, which show contrasting success for the replicating strategy. On the one hand, the average return on the clone appears in most cases very different from that of the index to be replicated. (In unreported results, we have obtained that the null hypothesis of an identical mean was rejected by a standard Student’s test in 11 out of 13 cases.) This was to be expected since the approach does not aim to replicate the first moment of the hedge fund return distribution. Interestingly, however, the average return obtained for the clone is always (significantly) lower than that of the index on our out-of-sample period, which can be explained by the bear equity market (March 2000 – March 2003), which spans a sizable fraction of the out-of-sample period. This result suggests that extreme caution should be used in choosing the reserve asset, and that the performance results of the replicating strategy are not robust with respect to the choice of the risky asset involved and the sample period considered11.

On the other hand, the volatility values obtained for the clones are relatively close to those obtained for the EDHEC indices. This is confirmed by the results of the F-test with only three rejections of the null hypothesis of equal


27

Exhibit 17: S&P 500 returns and Eurodollar time series.

11 - We have also repeated the analysis using as a reserve asset an equally-weighted portfolio of 3-month Eurodollar, 2-year note, 10-year note, S&P 500,Russell 2000 and GSCI futures as in Kat and Palaro (2006d). We have found a slight improvement in average performance for the clones, but they still underperform their respective index. This contrasts with the results reported in Kat and Palaro (2006d) who seem to use an artificially leveraged version of the underlying asset classes involved (see footnote 6 p. 9).

28

variances. Considering the other moments, and assuming that the sampling distribution of the skewness and the kurtosis follow a centred Gaussian distribution with a variance equal to 6/N and 24/N respectively, we have rejected the null hypothesis of equal higher moments in five cases for the skewness and in only one case for the kurtosis. Since the skewness and kurtosis parameters involve an estimation of the mean parameters, they are also highly impacted by the existence of outliers. Kat and Palaro have recommended a more robust estimator of the skewness and kurtosis to alleviate the dependency on mean return. They chose to use the quantile alternative proposed by Hinkley (197�) for the skewness and by Crow and Siddiqui (1967) for the kurtosis. Our results confirm that estimated skewness and kurtosis values for the clones are extremely close to those of the observed returns, except for the convertible bond arbitrage index. Going beyond individual moments, we also tested the quality of replication by using both Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests for the whole distribution12. Considering the two tests conducted, the best replication process was obtained for the Short Selling index and the worst was obtained with the Equity Market Neutral index13. The proposed approach works in 6 out of 13 cases with the KS test, but only works in two cases if we consider the AD tests. To prevent the differences in mean (replicating the mean is – unfortunately - not a target of the methodology) from having an excessive impact, we have also reported results of the tests with centred variables. In this case, the quality of replication cannot be statistically rejected for 12 of the 13 indices with both tests. Replication of the Fixed Income Arbitrage index was statistically rejected by our tests.

Those results imply that, even if one is willing to ignore the differences in mean returns, not all hedge fund distributions can be matched with relative satisfaction. This fact is also apparent from Exhibit 19, which shows the best and worst distribution fits based on the KS and AD tests, and signals poor quality replication for the Fixed

Income Arbitrage index. Finally, we have also reported the value of the Sharpe ratio and the historical Value-at-Risk at 9�%. In all cases, the Sharpe ratio was higher for the observed indices, which was to be expected given their higher mean returns. The VaR measure, on the other hand, seems to be fairly similar for the clones and the indices.

Exhibit 19 confirms that while a rather spectacular fit is obtained for some hedge fund strategy return distributions, the quality of fit is mediocre at best for others.

Another important point that should be emphasised at this stage is that the good results obtained for at least some hedge fund strategies are displayed only when a long out-of-sample period is used. Note in particular that the aforementioned results are based on an out-of-sample period made up of 96 monthly


12 - The main difference between the two tests is that the KS test is constructed around the maximum spread between the two distributions whereas the AD test gives more weight to the tails. These tests are based on the assumption of normally distributed returns, and should therefore be taken with caution. 13 - Contrary to Kat and Palaro (2006b) we have chosen to display only out-of-sample results (we omit the first 24 data that are considered in-sample) since the inclusion of the first 24 data points would have artificially increased the reported quality of the replication process.

Exhibit 19: Best and worst distribution fits based on KS (and AD)

tests


29

Exhibit 18: Monthly risk & performance indicators for the clones and the indices over the whole out-of-sample period 01/1999-12/2006

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observations, which corresponds to eight years! Considering the perspective of an investor whose patience may not be as extensive, we have repeated the analysis on a smaller out-of-sample period from January 1999 to December 2000, and report the results obtained in Exhibit 20. We observe that the null hypothesis of good replication quality is now rejected for the Merger Arbitrage and Relative Value indices, in addition to the Fixed Income Arbitrage index.

For an investor with more limited patience, the payoff distribution approach to hedge fund replication can lead to rather severe disappointment, as can be seen from Exhibit 20, which shows that a reduction in the out-of-sample test period has a dramatic effect on the results. To avoid introducing selection bias into our results, we have focused on the Convertible Arbitrage index, for which the quality of replication is the best. To obtain this graph we split our out-of-sample period into four sub-samples. The first one covers the period from January 1999 to December 2000, the second ends in December 2002, the third in December 2004 and the last integrates the whole out-of-sample period. As we reduce the length of the out-of-sample testing period, the quality of fit deteriorates dramatically, as can be seen in Exhibit 21. Hence, an investor with a four-year horizon (number of monthly out-of-sample observations N=48), who is arguably a very patient investor already, would not be very pleasantly surprised by the kind of replication quality obtained.

These results suggest that the method leads to satisfying results only for an investor who is willing to wait for more than six years before assessing the quality of the replication! So while the method may be suitable for replicating the distributional properties of hedge fund returns (with the notable exception of the mean return) over long-period samples, it can lead to disappointing results from the perspective of replicating the time-series properties of actual hedge fund returns.


Exhibit 21: Evolution of the quality of fit as a function of the

length of the out-of-sample period

From January 1999 to December 2000





31

Exhibit 20: Monthly risk & performance indicators for the clones and the indices over the (out-of-sample) period 01/1999-12/2000

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32

3.3.2. What Does Not Work Well: Matching Time-Series PropertiesIn practice, investors interested in hedge fund replication do not only expect to hold an investment that matches the horizon distribution of a specific target; they also would like to ensure that the time-series of returns of the replicating portfolio are reasonably close to those of the fund to be replicated. In an attempt to assess

the performance of the hedge fund replication process from a time-series standpoint, we have conducted a number of empirical tests.

We first report in Exhibit 22 the correlation between the clone and the corresponding index, as well as the tracking error of the clone with respect to the corresponding index, over the whole out-of-sample period.


Exhibit 22: Correlation and tracking error of the clones with respect to the whole out-of-sample period 01/1999-12/2006

Correlation between the clone

and the index

R2 of a regression of the clone return on the index

return

Tracking error of the clone with respect to the

corresponding index return

Convertible Bond Arbitrage 0.1402 1.97% 4.82%

CTA Global -0.1782 3.18% 12.49%

Distressed Securities 0.3701 13.70% �.30%

Emerging Markets 0.6233 38.8�% 10.86%

Equity Market Neutral 0.3106 9.6�% 2.18%

Event Driven 0.�861 34.3�% 4.37%

Fixed Income Arbitrage 0.0886 0.78% 4.28%

Global Macro 0.32�3 10.�8% 6.61%

Long/Short Equity 0.6740 4�.43% �.�0%

Merger Arbitrage 0.�369 28.83% 3.03%

Relative Value 0.6�� 42.97% 2.�9%

Short Selling - 0.749� �6.18% 38.79%

Funds of Funds 0.472� 22.33% �.74%

As can be seen in Exhibit 22, the results are rather disappointing, and show out-of-sample correlation and R2 values that are typically extremely low. One interesting comment relates to the Short Selling strategy, arguably one of the easiest to replicate by a short position in the underlying equity index. Since the methodology focuses only on matching the distributional properties, we obtain a large negative correlation (-0.749�) between the clone and the corresponding index. This can be explained by the fact that the distributions of X and –X are identical for a symmetric distribution. Hence, the methodology does not actually replicate the Short Selling index returns, but the return on a hypothetical index with a long bias of the same magnitude as the short bias displayed by the short-selling index.

This example perhaps epitomises the severe limitations involved in matching the distributional properties, as opposed to matching the actual returns.

As another illustration of the failure of the methodology to replicate the time-series properties of hedge fund returns, we analyse the difference between the replicated values and the actual prices over a monthly horizon. Exhibit 23 shows the different individual replication errors in percentage, and more specifically the minimum, average and maximum differences between monthly values. In our sample, 7 out of our 13 indices show a monthly difference within a -�0/+�0 basis-points interval, 11 show a difference within a range of -100/+100 basis points, while

the differences are beyond that range for the remaining two strategies. Obviously, such monthly differences in returns are simply too large to make this approach suitable for an investor who aims to replicate hedge fund returns.

In Exhibit 24, we again show the performance of $100 invested in the clone strategy as opposed to the actual index for the best replicated index, i.e., the Convertible Arbitrage index. In this specific case, the investor will underperform the hedge fund index by more than $76 at the end of our sample test, for an initial investment of $100, which corresponds to a monthly compound loss of 0.2%.

It should be noted, finally, that this analysis abstracts away from a transaction cost analysis, which is bound to further impact the performance of the replicating strategy. Our empirical analysis actually suggests that the replicating strategy involves a healthy turnover rate, which might result in costly implementation.


33

Exhibit 23: 3D plot of replication errors for the 13 indices

Exhibit 24: Evolution of the difference in time series between a

synthetic and real investment in the Convertible Arbitrage index

over the out-of-sample period.

The black curve corresponds to the observed index while the red curve corresponds to the evolution of $100 invested in the replication process.

34

After carefully analysing the two competing approches currently tested for hedge fund return replication, we have concluded that neither is yet ready to generate fully satisfactory results.

The factor replicating approach, while actually addressing the essence of the problem, mostly fails in out-of-sample tests. This failure can be traced to the difficulty both of identifying the right factors and of using simple regression methodologies (which can pick up only the past average exposures of the managers) to replicate hedge fund managers' dynamic exposure to these factors.

On the other hand, the distribution replication approach has been found to generate better results, but the problem is that its success is related merely to a possible replication of long-horizon returns, with no success in replicating their time-series properties. A serious concern also remains over the robustness of the results, in particular those related to the difference in average returns with respect to the choice of reserve asset and sample period. In other words, the most important aspect of the hedge fund return distribution (the average performance) is not an output of the process but a mere function of the choice of reserve asset and its performance over the period. As a result, the investor is left with the question of selecting/designing a well-diversified strategic and/or tactical benchmark that could be used as a reserve asset so as to generate the highest risk-adjusted return. Unfortunately, this is precisely the kind of challenge that an investor interested in hedge funds as absolute return vehicles typically tries to avoid addressing.

In conclusion, while the replication of hedge fund factor exposures may be a very attractive concept from a conceptual standpoint, one is compelled to conclude that it still is very much a work-in-progress. We believe that it is only through the introduction of novel econometric techniques allowing for either the statistical estimation of non-linear functions relating underlying factors to hedge fund returns,

or equivalently the statistical estimation of parsimonious conditional factor models, that hedge fund replication could be turned from an attractive concept into a workable investment solution.

Overall, it seems that it is only by combining the best of the two competing approaches that one can possibly hope to achieve truly satisfying results: the factor approach could facilitate time-series replication while the payoff distribution approach could help generate the replicating portfolio for the non-linear factors involved in the process.

We argue below that there may be hope in designing successful hedge fund replication models in the following two categories of factor models: conditional factor models and non-linear factor models. These approaches are somewhat consistent given that the Merton (1973) reinterpretation of the Black and Scholes (1973) option pricing model in terms of the replicating portfolio approach has emphasised the correspondence between non-linear payoff and dynamic trading strategies.

4.1. Conditional factor models

The idea is to introduce a model with time-varying factor exposure for hedge fund returns. These models can be built on rule-based and econometric approaches:

4.1.1. Rule-based approachesSome authors have argued that various heuristic dynamic strategies can generate an improvement in the in-sample performance of hedge fund factor models. Such strategies have been branded “rule-based strategies” by Jaeger and Wagner (200�), and also labelled “primitive trading strategies” by Fung and Hsieh (2007)). Overall, in the absence of suitably-designed econometric techniques designed to provide statistical estimates of such rule-based factor strategies, these heuristic approaches, even when economically motivated by an analysis of hedge

4. Conclusion

fund managers’ trading strategies, are likely to generate low out-of-sample performance because of the magnitude of misspecification risk involved in the model specification phase. Overall, we recognise that it is possible that such trading strategies may eventually emerge as attractive low-cost systematic alternatives to hedge funds, but in no circumstances should such strategies, for which the use of the word “passive” would obviously be a serious misnomer, be assumed to enable the replication of hedge fund returns.

4.1.2. Econometric approachesGiven that a rolling-window analysis implem-entation of an unconditional factor model can provide only a backward-looking view of past hedge fund exposures, and given that changes in hedge fund managers' risk exposures cannot be expected to depend solely on their past values, several authors have recommended using instrumental variables to explore the time- and state-dependencies of hedge fund return exposures with respect to risk factors. For example, Kazemi and Li (2007) document evidence of predictability in changes in hedge fund factor exposure based on daily hedge fund return data. In a related effort, Billio, Getmansky and Pelizzon (2006) use switching regime beta models to analyse the changes in hedge fund exposure conditional on different states of various risk factors. Also related to an attempt to forecast future changes in factor exposure are a paper by Avramov et al. (2007), who evaluate hedge fund performance through portfolio strategies that incorporate predictability in fund risk loadings and benchmark returns, and a paper by Basu, Hung and Stremme (2007), who use an augmented CAPM that incorporates skewness and kurtosis factors with time-varying risk-premia that are functions of term-structure variables to evaluate the performance of hedge fund indices.

4.2. Non-linear factor models

Two approaches to dealing with non-linearity in the context of factor models should be distinguished: 1) heuristic attempts to introduce ad-hoc option portfolios to improve the performance of a hedge fund factor model and 2) statistical models designed to extract implied option payoffs from hedge fund return observations.

Academic literature on the first approach has suggested trying to capture the non-linear dependency of hedge fund returns with respect to risk factors by including new regressors with non-linear exposure to standard asset classes to proxy dynamic trading strategies in a linear regression. Natural candidates for new regressors are buy-and-hold or dynamic positions in derivatives. This fruitful line of research has been pursued systematically by Schneeweis and Spurgin (2000) and Agarwal and Naik (2004) and specifically applied to strategies such as Pair Trading (Gatev, Goetzmann and Rouwenhorst (2006)), Event Arbitrage (Mitchell and Pulvino (2001)), Trend-Following Strategies (Fung and Hsieh (2004)) or Fixed Income Arbitrage (Fung and Hsieh (2002)). A detailed summary of the research in this area can be found in Fung and Hsieh (2004)).

The problem here again is the lack of robustness of these otherwise insightful heuristic approaches, leading to a concern over out-of-sample replicating performance. In all of these studies, hedge fund returns are characterised by a complex set of risk factors, and returns on a set of well chosen traded option indices are added to the regression to capture non-linear risk exposure. For example, Agarwal and Naik (2004) choose at-the-money and out-of-the-money puts and calls on the S&P �00 index. Although the introduction of arbitrarily-chosen option portfolios can improve the in-sample explanatory power, the out-of-sample quality of replication is likely to be affected by a robustness problem induced by specification

4. Conclusion

3�

36

risk. In particular, nothing guarantees that the chosen underlying assets and levels of moneyness accurately represent the true state-dependent factor exposure of hedge fund managers.

While the second approach represents a real challenge from a statistical standpoint, recent research by Díez de los Ríos and García (2006) suggests that suitably designed statistical techniques can be used to estimate implicit option positions in hedge fund returns. In particular, they argue that suitably designed statistical techniques can be used to (a) determine the portfolio of options that best approximates the returns of a given hedge fund, (b) use options on any benchmark portfolio deemed to best characterise the strategies of the fund (and not simply traded options on an equity index), (c) estimate the corresponding moneyness of the options that best characterise the returns of a particular fund, and (d) assess whether the presence of the estimated non-linearities is statistically significant. The application of these techniques for hedge fund replication is left for further research.

4. Conclusion

• Agarwal, V., W. Fung, Y. C. Loon, and N. Y. Naik, (200�), “Risks in Hedge Fund Strategies: The Case of

Convertible Arbitrage”, Working Paper, Georgia State University and London Business School.

• Agarwal, V., and N. Naik, (2004), “Risk and Portfolio Decisions Involving Hedge Funds”, Review of Financial Studies.

• Amin, G. and H. Kat, “Hedge Fund Performance (1990-2000): Do the "Money Machines" Really Add

Value?”, Journal of Financial and Quantitative Analysis, 38, 2003, 2�1-274.

• Avramov, D., R. Kosowski, N. Naik, and M. Teo, (2007), “Investing in Hedge Funds when Returns are

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Business School, School of Business, Singapore Management University.

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