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The Brave New World of Hedge Fund Indexes Desperately Seeking Pure Style Indexes Lionel Martellini Marshall School of Business University of Southern California Chercheur Associé EDHEC Joint work with Noël Amenc © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Outline Motivation –Performance Mesurement and Asset Allocation –Problems with Hedge Fund Indexes The World of Hedge Fund Indexes –Overview of Popular Hedge Fund Strategies –Survey of Existing Hedge Fund Indexes Hedge Fund Indexes are not Created Equal –Heterogeity in Competing Hedge Fund Index Returns –Implications for Asset Allocation Desperately Seeking Pure Style Indexes –Statistical Approach –Portfolio Approach How Pure is Pure? –The Two Basic Theorems of Pure Indexing –Testing Representativeness of Pure Indexes © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
HF managers often use risk-free rate as a benchmark This absolute return approach is theoretically valid if and only if –CAPM is the true model –Hedge fund beta is zero Hedge fund indexes and sub-indexes are a natural choice for benchmarking hedge fund returns Right benchmarking is a fundamental problem in the presence of incentive fees Reliable HF indexes are also needed for –Strategic Asset Allocation –Tactical Asset Allocation Motivation From an Absolute to a Relative Return Perspective © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Universe of Equity (20000+) Wilshire Sub-Universe (5000 approx) Russell Sub Universe MSCI Sub-Universe S&P Indices (value, growth) Russell Indices (growth,value) Motivation Equity Universe, Sub-Universes, and Specialized Indexes Composite of World Unknown Sub-universe (e.g., MSCI) may or may not represent World Index Competing indexes for the same universe © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Universe of Hedge Funds (6000+) Zurich Sub-Universe (1300 approx) HFR Sub Universe TASS Sub-Universe Zurich Hedge Fund Indices CSFB/Tremont Index EACM 100 Composite of World Unknown Sub-universe (e.g., HFR, TASS) may or may not represent World Index Competing indexes for the same universe Motivation HF Universe, Sub-Universes, and Specialized Indexes © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Motivation Problems are Amplified for Hedge Fund Indexes Existing indexes are not fully representative –Because of the lack of regulation on hedge fund performance disclosure, existing data bases only cover a relatively small fraction of the hedge fund population –Probably only a little more than half of existing hedge funds choose to self- report their performance to one of the major hedge fund databases –One of the most popular hedge fund indexes, the EACM 100, does not account for more than a tiny percentage of all existing hedge funds –Most HF indexes are equally-weighted (all but CSFB/Tremont) Existing indexes are biased –Most hedge fund indexes are based upon managers' self-proclaimed styles –Given that hedge fund managers jealously protect the secret of their investment strategies (the so-called black-box problem), relying on managers' self-proclaimed style is actually almost a necessity –This procedure only makes sense under the following two conditions: (1) a manager follows a unique investment and (2) a manager's self-proclaimed style matches the manager's actual trading strategies –Style drift problem (see for example Lhabitant (2001)) © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Motivation Pure Style Indexes are not Observable Sample of hedge funds in the database used by a given commercial index Population of hedge funds following a given strategy Lack of representativeness Presence of a style bias © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
World of HF Indexes – Popular Strategies Hedge Funds Classification © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Directional strategies: aimed at benefiting from market movements and trends –Global macro –Hedge (long bias) –Long (e.g. growth or value stocks) –Short (e.g. overvalued, glamour stocks) Non directional strategies: –Event driven (corporate events such as takeovers, spin-offs, mergers, etc.) –Restructuring (buying or shorting securities of companies under Chapter 11 and/or undergoing some form of reorganization) –Fixed-income arbitrage (long and short via treasuries, corporate and/or asset-backed securities) –Capital structure arbitrage (buying and selling different securities of the same issuer, e.g., convertibles/common stock) –Equity market neutral (long-short zero beta strategies) World of HF Indexes – Popular Strategies Hedge Funds Styles © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
World of HF Indexes – Competing Providers There exist at least a Dozen HF Index Providers © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
World of HF Indexes – Competing Providers Strategies Covered in this Paper © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
World of HF Indexes – Competing Providers Strategies Not Covered in this Paper © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Hedge Fund Indexes are not Created Equal Heterogenity in Hedge Fund Indexes – Max Difference © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Hedge Fund Indexes are not Created Equal Heterogeity in Hedge Fund Indexes – Correlation © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Hedge Fund Indexes are not Created Equal Implications for Asset Allocation Efficient frontiers based on monthly data for the period extending from January 1996 to October 2001 © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Investment in Hedge Funds Problems –Competing indexes disagree –All are potentially flawed Can not tell which is best –All existing indexes have both advantages and drawbacks –For example, Zurich indexes may be less biased than some of their competitors, but they are less representative Statistical approach Portfolio approach –Maximization of Representativeness –Minimization of Bias © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Statistical Approach – Kalman Filter Simple model – R kt is the return on competing index k – I t is the (unobservable) return on pure index – kt is the noise, measurement error resulting from presence of biases and absence of representativeness Assume normally distributed pure indexes and measurement errors Kalman filter is used both to evaluate the likelihood function and to forecast and smooth the unobserved state variables Here, our motivation is to estimate (smooth) the unobserved pure index, based on the observed returns on competing indexes: © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Statistical Approach – The Results Performance of Kalman filter pure index (in parenthesis, the average of competing index) © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Statistical Approach – Convertible Arbitrage © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – The Method Black-Box problem –A pure index generated Kalman filtering techniques has an appealing built-in element of optimality (minimized mean squared error) –It can not, however, be regarded as index in its own right, since it can not be expressed as a portfolio of individual hedge funds Portfolio approach –Taking a portfolio of existing indexes should intuitively be better than selecting any of them –Is equally-weighting a good scheme? Maximization of representativeness –Principal component analysis Minimization of bias –Minimum-variance analysis © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – PCA Use factor analysis techniques to generate a set of pure indexes –They can be thought of as the best possible one-dimensional summaries of information conveyed by competing indexes for a given style, in the sense of the larger fraction of the variance explained. –Here, we are looking for the portfolio weights that make the combination of competing indexes capture the largest possible fraction of the information contained in the data from the various competing indexes The method –From a mathematical standpoint, it involves transforming a set of K correlated competing indexes into a set of orthogonal variables, or implicit factors, which reproduces the original information present in the correlation structure –Each implicit factor is defined as a linear combination of original variables –Use the first component of a PCA of competing indexes as a candidate for a pure style index (typically captures a large proportion of cross-sectional variations because competing styles tend to be at least somewhat positively correlated) © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes PCA – The Results © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – Min Var Analysis Pure hedge fund indexes generated as the first component in a factor analysis should be as representative as possible since there is no other linear combination of competing indexes that implies a lower information loss Another approach consists in focusing on minimization of the bias Use same model as before Add assumptions –Noise term is zero on average –Noise term is uncorrelated with return on pure index –Homoscedastic model © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – Min Var Analysis Define the return on a portfolio of indexes Under previous assumptions, we have that The problem of minimizing the bias is Solution is © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – Additional Assumption If one is willing to make the additional assumption of no correlation between noise terms for various competing indexes, then the variance-covariance matrix of residuals is diagonal, and one obtains the following simple solution May also impose positivity constraints to ensure that resulting index is a long-only portfolio of individual hedge funds © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Desperately Seeking Pure Style Indexes Portfolio Approach – Additional Assumption © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
How Pure is Pure? The Two (Obvious) Theorems of Pure Indexing PCA and min-var analysis have appealing element of optimality However, any portfolio (e.g., equally-weighted portfolio) should perform better than competing indexes Theorem 1: An index of the indexes is always less biased than the average of the set of indexes it is extracted from Theorem 2: An index of the indexes is always more representative than any competing index Index 1 Index 2 Index 3 © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
How Pure is Pure? Testing Representativeness – The Method Reduction of bias is very hard to test (chicken-and-egg problem) We have tested enhancement of representativeness using the following test experiment –We have merged the 3 major databases providing information on individual fund return (MAR, TASS and HFR) –We have also added data on funds which do not report to any data base, that had been directly obtained from administrators –We have gathered monthly returns on a total of 7,422 hedge funds, including 2,317 funds that do not report their returns to the major data bases –We have formed equally-weighted portfolios for each style based on managers self-proclaimed styles and compute correlation with pure indexes These portfolios are –Arguably biased –Undoubtly representative © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
How Pure is Pure? Testing Representativeness – The Results © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
Contribution –Document heterogeneity in competing hedge fund index providers –Attempt to provide remedies to the problem Extensions –We have suggested that a database of pure style indexes be maintained at the EDHEC-MISYS Risk and Asset Management Research Center, and posted on a dedicated web site –The index construction methodology Step 1: Use the first 3 years of monthly returns to calibrate the model. Step 2: Perform a PCA analysis of competing indexes for each strategy on the data used for calibration purposes Step 3: These portfolios are held for 3 months, their monthly returns are recorded, and the same process is repeated –Our results can easily be extended to traditional investment styles such as growth/value, small-cap/large-cap Conclusion © Edhec 2002Mesure de la Performance et des Risques de la Gestion Alternative18/06/02
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