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Lionel Martellini Risk and Asset Management Research Center, EDHEC Graduate School of Business

Conférence Gestion Alternative 2 Avril 2004. The Alpha and Omega of Hedge Fund Performance Evaluation Joint work with Noël Amenc (EDHEC) and Susan Curtis (USC). Lionel Martellini Risk and Asset Management Research Center, EDHEC Graduate School of Business lionel.martellini@edhec.edu.

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Lionel Martellini Risk and Asset Management Research Center, EDHEC Graduate School of Business

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  1. Conférence Gestion Alternative 2 Avril 2004 The Alpha and Omega of Hedge Fund Performance Evaluation Joint work with Noël Amenc (EDHEC) and Susan Curtis (USC) Lionel Martellini Risk and Asset Management Research Center, EDHEC Graduate School of Business lionel.martellini@edhec.edu

  2. Outline • Introduction • Standard CAPM Model • Adjusting CAPM for the Presence of Stale Prices • Payoff Distribution Model • Multi-Factor Models • Implicit Factor Model • Explicit Factor Model • Explicit Index Model • Peer Benchmarking • Comparative Performance Analysis • Impact on Attributes on Funds’ Performance • Conclusion

  3. IntroductionPapers on MF and HF Performance • There is ample evidence that portfolio managers following traditional active strategies on average under-perform passive investment strategies • Examples are: Jensen (1968), Sharpe (1966), Treynor (1966), Grinblatt and Titman (1992), Hendricks, Patel and Zeckhauser (1993), Elton, Gruber, Das and Hlavka (1993), Brown and Goeztman (1995), Malkiel (1995), Elton, Gruber and Blake (1996), or Carhart (1997), among many others • Recently, many papers have focused on hedge fund performance evaluation • Examples are: Ackermann, McEnally, and Ravenscraft (1999), Amin and Kat (2001), Agarwal and Naik (2000a, 2000b), Brown, Goetzmann and Ibbotson (1999), Edwards and Caglayan (2001), Fung and Hsieh (1997, 2001a, 2001b), Gatev, Goetzmann and Rouwenhorst (1999), Liang (2000), Lhabitant (2001), Lo (2001), Mitchell and Pulvino (2001), Schneeweis and Spurgin (1999, 2000)

  4. IntroductionModels • Because these studies are based on a variety of models for risk-adjustment, and also differ in terms of data used and time period under consideration, they yield very contrasted results • The present paper can be viewed as an attempt to provide an unified picture of hedge fund managers to generate superior performance • To alleviate the concern of model risk on the results of performance measurement, we consider an almost exhaustive set of pricing models that can be used for assessing the risk-adjusted performance of hedge fund managers.

  5. IntroductionPreview of the Results • While we find significantly positive alphas for a sub-set of hedge funds across all possible models, our main finding is perhaps that the dispersion of alphas across models is very large • Hedge funds appear to have significantly positive alphas on average when normal returns are measured by an explicit factor model, even when multiple factors serving as proxies for credit or liquidity risks are accounted for • However, hedge funds on average do not have significantly positive alphas once the entire distribution is considered or implicit factors are included • On the other hand, all pairs of models have probabilities of agreement greater than .50 • In other words, while different models strongly disagree on the absolute risk-adjusted performance of hedge funds, they largely agree on their relative performance in the sense that they tend to rank order the funds in the same way

  6. IntroductionData • Our analysis is conducted on a proprietary data base of 1,500 individual hedge fund managers (MAR-CISDM data base) • We use the 581 hedge funds in the MAR database that have performance data as early as January 1996 • The data base contains monthly returns and also • Fund size (asset under management) • Fund type (MAR classification system) • Fund age (defined as the length of time in operation prior to the beginning of our study) • Location (US versus non US) • Incentive fees • Management fees • Minimum purchase amount

  7. Standard CAPM ModelNormal and Abnormal Returns • Factor models allow us to decompose managers’ (excess) returns into • Normal returns (risk premium) • Abnormal returns (investment opportunity) • Statistical noise (illusion) • Normal returns are generated as a fair reward for the risk(s) taken by fund managers • Abnormal returns are generated managers’ unique ability to “beat the market” in a risk-adjusted sense, generated through superior access to information or better ability to process commonly available information • Need some model to understand what a “normal” return is; benchmark model is the CAPM (Sharpe (1964))

  8. Standard CAPM ModelResults • The average alpha across all funds is significantly positive • The majority of hedge funds have positive alphas, and about a third are statistically significant • Very few funds have significantly negative alphas

  9. Standard CAPM ModelDistribution of CAPM Alphas

  10. Standard CAPM ModelDistribution of CAPM Betas

  11. Adjusting CAPM for Stale Prices The Presence of Stale Prices • It has been documented (Asness, Krail and Liew (2001)) that a fair number of hedge funds hold illiquid securities • For monthly reporting purposes, they typically price these securities using either the last available traded price or estimates of current market prices • Such non-synchronous return data can lead to understated estimates of actual market exposure, and therefore to mismeasurement of hedge fund risk-adjusted performance • In that context, some adjustment must be performed to account for the presence of stale prices

  12. Adjusting CAPM for Stale Prices Model and Results • Model: run regressions of returns on both contemporaneous and lagged market returns • Results: the number of funds with alpha values significantly greater than zero has been cut in half

  13. Adjusting CAPM for Stale Prices Model and Results • CAPM has more funds with alphas near 10%, and lagged CAPM model has more funds with alphas between -10% and 0

  14. Payoff Distribution Function ApproachNon-Linear Exposure to Standard Asset Classes • Hedge fund returns exhibit non-linear option-like exposures to standard asset classes because • They can use derivatives • They follow dynamic trading strategies • Furthermore, the explicit sharing of the upside profits under the form of incentive fees implies that post-fee returns have option-like element even if pre-fee returns do not • In this context, mean-variance CAPM based performance measures will fail to account for non trivial preferences about skewness and kurtosis • There exists a method allowing an investor to account for the whole distribution of returns

  15. Payoff Distribution Function ApproachMethodology • Methodology introduced by Dybvig (see Dybvig (1988a, 1988b)), applied to hedge fund performance evaluation by Amin and Kat (2001) • First step: recover the cumulative probability distribution of the monthly hedge fund payoffs as well as the S&P 500 from the available data set assuming $100 are invested at the beginning of the period • A normal distribution is assumed for the S&P 500 (i.e., we only need to estimate the mean and standard deviation of the monthly return on the S&P 500 over the period), but not for the hedge funds • Second step: generate payoff functions for each hedge fund • A payoff function is a function f that maps the return distribution of the S&P 500 into a relevant return distribution for the hedge fund

  16. Payoff Distribution Function ApproachMethodology (con’t) • Third step: we use a discrete version of a geometric Brownian motion as a model for the underlying S&P price process generate 20,000 end-of-month value • From these 20,000 values, we generate 20,000 corresponding payoffs for each hedge fund, average them, and discount them back to the present to obtain a fair price for the payoff • This “price” thus obtained can be thought of the minimum initial amount that needs to be invested in a dynamic strategy involving the S&P and cash to generate the hedge fund payoff function • If the price thus obtained is higher than 100, this means that more than $100 needs to be invested in S\&P to generate a random terminal payoff comparable to the one obtained from investing a mere $100 in the hedge fund. We therefore take this as evidence of superior performance. • On the other hand, if the price obtained is lower than $100, we conclude that one may achieve a payoff comparable to that of the hedge fund for a lower initial amount. • The percentage difference is computed as a relative measure of efficiency loss

  17. Payoff Distribution Function ApproachCumulative Probability Distributions • The slope for the average hedge fund is much steeper than for the S&P 500, indicating a much narrow distribution of returns

  18. Payoff Distribution Function ApproachPerformance of High- and Low-Rated Funds • The low-rated fund has a wide distribution of returns • Top rated funds were found to be of two types: high volatility funds with exceptionally high returns, and low volatility funds

  19. Payoff Distribution Function ApproachDistribution of Efficiency Gain or Loss

  20. Payoff Distribution Function ApproachPerformance of Hedge Funds • On average, hedge funds do not outperform the market • However, the statistics are influenced by a few funds with large negative efficiencies. Over half of the funds have positive efficiency measures • Note that in the implementation of PDPM, we must make an assumption about the volatility of the S&P 500 • Here, we used a16% volatility as measured during the time period of the data • When we repeated the analysis with a higher volatility of 20%, the average hedge fund has a slightly higher efficiency and is no longer significantly different from zero

  21. Multi-Factor ModelsOther Sources of Risk • Hedge funds are typically exposed to a variety of risk sources including volatility risks, credit or default risks, liquidity risks, etc., on top of standard market risks • If one uses CAPM while the “true” model is a multi-factor model, then estimated alpha will be higher than true alpha • Modern portfolio theory and practice is based upon multi-factor models (Merton (1973), Ross (1976)) • The return on asset or find i is Rit = mi+bi1F1t+ ... +biKFKt+ eit • Fktis factork at date t (k = 1,…,K) • eitis the asset specific return • bikmeasures the sensitivity ofRito factork, (k = 1,…,K)

  22. Multi-Factor Models Four Types of Factor Models • Implicit factor models • Factors: principal components, i.e., uncorrelated linear combinations of asset returns • Explicit factor models – macro factors • Factors (Chen, Roll, Ross (1986)): inflation rate, growth in industrial production, spread long-short treasuries, spread high-low grade corporate interest rate • Explicit factor models – micro “factors” • Factors (actually attributes): size, country, industry, etc. • Explicit factor model – index “factors” • Factors are stock and bond market indices

  23. Implicit Factor ModelThe Model • Use principal component analysis to extract statistical factors (linear combinations of returns) • Challenge is determine the optimal # of factors (K’) • Select K’ by applying results from the theory of random matrices • Compare the properties of an empirical covariance matrix to a null hypothesis purely random matrix • Distribution function for eigenvalues under the null hypothesis • Here, we regard as statistical noise all factors associated with an eigenvalue lower than lambda min

  24. Implicit Factor ModelThe Results • The mean alpha is less than zero under this model • This suggests that there are factors influencing hedge fund performance that are captured in the Implicit Factor Model but not captured in CAPM

  25. Explicit Factor ModelThe Model • We test an explicit macro factor model where we use financial variables to proxy deeper economic effects • The following factors are used • US equity risk is proxied by the return on the S\&P 500 index, and world equity risk is proxied by the return on the MSCI World Index ex US • Equity volatility risk, proxied by using the changes in the average of intra-month values of the VIX • Fixed-income level risk is proxied by the 3 months T-Bill rate • Slope risk or term premium risk is proxied by monthly differences between the yield on 3 months Treasuries and 10-year Treasuries • Currency risk is proxied by changes in the level of an exchange volume-weighted index of currencies versus US dollar • Commodity risk is proxied by changes in the level of a volume-weighted index of commodity prices • Credit risk is proxied by changes in the monthly observations of the difference between the yield on long term Baa bonds and the yield on long term AAA bonds • Liquidity risk is proxied by changes in the monthly market volume on then NYSE

  26. Explicit Factor ModelThe Results • Some of the above variables do not appear to command a premium • The following variables seems to be rewarded risk factors: S&P 500, MSCI ex-US, oil price return, change in credit spread, change in VIX • This is the 5-factor model that we use

  27. Explicit Factor ModelThe Analysis • The average alpha is higher than under the CAPM • This is primarily due to the inclusion of the MSCI ex-US index • Many of the funds in our database are global funds, with a higher beta on the MSCI index than on the S&P 500. • Since the MSCI index underperformed the S&P 500 during this time period, the inclusion of the MSCI factor makes the alpha values higher for these funds • Thus, in fact, these funds are outperforming the world index but are underrated in the CAPM\ model since their performance is compared to the S&P 500 • The inclusion of other factors (oil prices, change in credit spread, change in VIX) tends to lower the alphas (compared to a 2-variable model using S&P500 and MSCI only), but does not erase the gains made by including the MSCI • Thus, we conclude that when all of these significant factors are included, the average hedge fund still has a positive alpha

  28. Explicit Multi-Index ModelReturn-Based Style Analysis • Style analysis • Sharpe (1992): equity styles are as important as asset classes • Examples: growth/value, small cap/large cap, etc. • Model: Rit= wi1F1t+ wi2F2t+ ... wikFkt+ eit • Rit= (net of fees) excess return on a given portfolio or fund • Fkt= excess return on index j for the period t • wik= style weight (add up to one) • eit= error term • Divide the fund return into two parts • Style:wi1F1t+ wi2F2t+ ... wikFkt (part attributable to market movements) • “Skill”: eit(part unique to the manager), emanates from 3 sources • manager’s exposure to other asset classes not included in the analysis • manager’s active bets: active picking within classes and/or class timing • statistical error: if zero, Var(eit) can be regarded as selection return risk

  29. Explicit Multi-Index ModelStyle Analysis and Performance Evaluation • Step 1: select a set of indices to perform return-based style analysis • CSFB/Tremont indices or (preferably!) EDHEC indices • Step 2: perform style analysis of fund returns • Constrained regression • Step 3: form peer groups • Use cluster analysis on style exposure • Step 4: perform a risk-adjusted analysis of each fund’s performance • Unconstrained regression

  30. Explicit Multi-Index ModelPerformance Results • We measure the excess return of hedge funds using the primary indexes appropriate to each cluster as factors in the model (average weight > 10%) • The mean hedge fund has alpha not significantly different from zero • These results suggest that the CSFB indexes effectively capture risk factors that are not captured by the standard CAPM, and that fund managers with positive CAPM alphas are often not outperforming hedge fund indexes

  31. Peer BenchmarkingCluster-Based Index • Next, we regress hedge fund excess returns on the excess return of the equally-weighted portfolio of all hedge funds within a cluster • This is formally similar to Sharpe's (1963) single-index model except that perform a relevant peer benchmarking • Cluster-index model has average alpha very close to zero • This should not be surprising since the same funds are used in the computation of the index as are used for computation of alpha • Useful to spot the best performing funds in a peer group

  32. Comparative Performance AnalysisSynthesis • The standard deviations are across funds (not across time periods)

  33. Comparative Performance AnalysisSynthesis (con’t) • Hedge funds appear to have significantly positive alphas for CAPM-like models, even with multiple factors • However, hedge funds on average do not have significantly positive alphas once the entire distribution is considered (PDPM) or implicit factors are included (PCA) • Nevertheless, many individual funds do have significantly positive alphas

  34. Comparative Performance AnalysisCross-Sectional Distribution of Average Alphas • The mean of that distribution is 4.07%, the standard deviation is 9.56% • 276 (out of 581 hedge funds) have an average alpha across methods larger than 4.5%

  35. Comparative Performance AnalysisCross-Sectional Distribution of Standard Deviation • The mean of that distribution is 7.66%, the standard deviation is 4.60% • One fund has a dispersion of alpha across methods larger than 40%!

  36. Comparative Performance AnalysisCorrelation of Alphas • CAPM-related methods are highly correlated with each other, indicating that the adjustments have small effects • The implicit factor model and clustering based methods have a smaller correlation with the other methods, indicating that they pick up different factors

  37. Comparative Performance AnalysisProbability of Agreement • For any two models, we compute the probability that the two models will agree on the rank order of a randomly-chosen pair of hedge funds • All pairs of models have probabilities of agreement > 0.50, even the model that only computes the average return

  38. Impact on Attributes on Fund PerformanceImpact of Fund Size on Performance • Note that for all methods, the mean alpha for large funds exceeds the mean alpha for small funds • This fact, combined with the observation that most of the results are statistically significant, suggests that large funds do indeed outperform small funds on average

  39. Impact on Attributes on Fund PerformanceImpact of Fund Type on Performance • Most models rate market neutral funds as outperforming the average of other funds at a statistically significant level. However, two of the factor models do not. Presumably a typical market neutral fund has a favorable probability distribution of returns but is subject to some implicit or macroeconomic risks not well captured by the other models. • The CAPM models rate short-selling funds the highest, although other models did not. Short-selling funds tend to have negative betas, so even absolute performance near the risk-free rate will result in positive CAPM alphas.

  40. Impact on Attributes on Fund PerformanceImpact of Fund Age on Performance • Note that for all methods, the mean alpha for newer funds exceeds the mean alpha for older funds • The differences vary in significance across the methods

  41. Impact on Attributes on Fund PerformanceImpact of Incentive Fee on Performance • High incentive fees (>=20%; most were exactly 20%) versus low incentive fees (<20%) • Note that for all methods, the mean alpha for high incentive funds exceeds the mean alpha for low incentive funds • A strong significant effect is obtained with almost all of the methods

  42. Impact on Attributes on Fund PerformanceImpact of Management Fee on Performance • High management fees (>=2%) versus low management fees (<2%) • None of the reported differences is significant at the 0.05 level • This suggests that there is no significant difference between funds with higher or lower administrative fees

  43. Impact on Attributes on Fund PerformanceImpact of Minimum Purchase Amount on Performance • Minimum purchase amounts for the hedge funds in our study ranged from 0 to \$25 million. • High MPA (>=$300,000) versus low MPA (<$300,000) • For all methods, the mean alpha for funds with the larger minimum purchase amounts exceeds the mean alpha for the other funds (statistically significant difference)

  44. ConclusionAt the HF world's lies the Impersonal • Alphas on active strategies, if they exist, are not easy to measure with any degree of certainty • In a companion paper, we test the impact of uncertainty in alpha estimates on optimal allocation decision in a CT Bayesian setting • Hedge fund are exposed to a variety of risk factors, and, as a result, generate normal, as opposed to abnormal, returns • The hedge fund industry should perhaps focus on promoting the beta-benefits of hedge fund investing, which are significant and less arguable, as opposed to promoting the alpha-benefits of hedge fund investing, which are very hard to measure with any degree of accuracy • This also suggests that the future of alternative investments may lie in “the impersonal”, i.e., in passive indexing strategies

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