Do hedge funds deliver alpha? A Bayesian and bootstrap analysis *

Transcription

1 Journal of Fnancal Economcs 00 (0000) Do hedge funds delver alpha? A Bayesan and bootstrap analyss * Robert Kosowsk a, Narayan Y. Nak b,**, Melvyn Teo c a INSEAD b London Busness School c School of Busness, Sngapore Management Unversty Receved 17 October 2005; receved n revsed format 18 November 2005; accepted 19 December 2005 Abstract Usng a robust bootstrap procedure, we fnd that top hedge fund performance cannot be explaned by luck, and hedge fund performance perssts at annual horzons. Moreover, we show that Bayesan measures, whch help overcome the short-sample problem nherent n hedge fund returns, lead to superor performance predctablty. Sortng on Bayesan alphas, relatve to OLS alphas, yelds a 5.5% per year ncrease n the alpha of the spread between the top, and bottom hedge fund decles. Our results are robust and relevant to nvestors as they are nether confned to small funds, nor drven by ncubaton bas, backfll bas, or seral correlaton. JEL Classfcatons: G11, G12, G23 Keywords: hedge fund performance, persstence, alpha, factor models, Bayesan, bootstrap *We would lke to thank Vkas Agarwal, Wllam Fung, Robert Korajczyk, Tobas Moskowtz, Luboš Pástor, Allan Tmmermann, Russ Wermers, and an anonymous referee for many helpful comments and constructve suggestons. Nak s grateful for fundng from the BNP Parbas Hedge Fund Centre at the London Busness School. Teo s thankful for fundng from the Offce of Research at the Sngapore Management Unversty. We are ndebted to CISDM, HFR, MSCI, and TASS for provdng us wth the data. We alone are responsble for all errors. **Correspondng author: London Busness School, Sussex Place, Regent's Park, London NW1 4SA, Unted Kngdom. E-mal: nnak@london.edu, Tel: , extenson 3579; Fax: /00 see front matter 0000Publshed by Elsever Scence B.V. All rghts reserved.

2 1. Introducton Are stellar hedge funds such as George Soros' Quantum Fund just lucky? Is the exstence of such funds to be expected gven the large sample of hedge funds that exst? If not, does the abnormal performance of these funds persst and can t be exploted by means of tradng strateges? These questons are of great relevance to nsttutonal and retal nvestors who have recently rased ther portfolo allocatons to hedge funds, 1 and to the ssue of market effcency n general. Our paper answers these questons by employng nnovatve statstcal technques on a large database of hedge funds. Evaluatng the sgnfcance and persstence of hedge fund returns s fraught wth many dffcultes. Frst, top performers are drawn from a large cross-secton of hedge funds, whch ncreases the potental for some managers to do partcularly well by chance alone. Second, hedge fund performance measures typcally do not follow parametrc normal dstrbutons gven the funds' dynamc tradng strateges and ther holdngs of dervatves securtes lke optons. 2 Thrd, the complexty of hedge fund strateges makes benchmarkng ther returns partcularly challengng and rases the possblty of model msspecfcaton. Fourth, hedge fund return seres are often short; hence, tradtonal performance measures (e.g., multvarate generalzaton of Jensen s alpha) may be measured mprecsely. Ffth, hedge fund portfolo holdngs are hghly confdental and rarely, avalable, f at all, to researchers. Methods usng portfolo holdngs to assess fund performance are not applcable n hedge fund space. In ths paper, we address these dffcultes by employng robust bootstrap methodologes proposed by Kosowsk, Tmmermann, Wermers, and Whte (2006) (henceforth, KTWW) and the Bayesan approach of Pástor and Stambaugh (2002a) (henceforth, PS). The nonparametrc bootstrap approach of KTWW permts a comprehensve examnaton of hedge fund performance that controls explctly for luck, mnmzes the potental bas from msspecfcaton (Horowtz, Härdle, and Kress, 2003), and avods havng to parametrcally model the jont dstrbuton of hedge fund performance across thousands of funds, most of whch are sparsely overlappng. The seemngly unrelated assets (henceforth, SURA) Bayesan approach of PS s also robust to model 1 About one-thrd of nsttutonal nvestors, ncludng CalPERS n the Unted States, plan to ncrease ther allocatons to hedge funds. See For the fortunate few, The Economst, 3 rd July Also, the hedge fund ndustry has expanded consderably over the last decade. The 2005 Hedge Fund Research report ndcates that there were 530 hedge funds managng under US$39 bllons n 1990, whle there are over 7,000 hedge funds managng over US$970 bllons by the end of December See, e.g., Fung and Hseh (2001), Mtchell and Pulvno (2001), and Agarwal and Nak (2004). 1

3 msspecfcaton. More mportantly, t takes advantage of nformaton n seemngly unrelated assets to overcome short sample problems and mprove on the precson of performance estmates. Both of these approaches have been appled wth some success to mutual fund returns [see KTWW and Busse and Irvne (2006)] but not to hedge fund returns. Gven ther complex and dynamc tradng strateges, we beleve that hedge funds are more lkely to suffer from return nonnormalty, model msspecfcaton, and short sample problems, as compared to mutual funds. Therefore, these approaches should be even more relevant for hedge funds. Our emprcal results are strkng. Accordng to our bootstrap estmates, the performance of the top hedge funds ranked by the t-statstc of alpha (whch s smlar to sortng by the nformaton rato often used to rank fund managers) cannot be attrbuted to sample varablty or luck alone. Ths s true across all fund nvestment categores, and s robust to controllng for ncubaton and backfll bas (Ackermann, McEnally, and Ravenscraft, 1999; Fung and Hseh, 2000; Malkel and Saha, 2004; Ibbotson and Chen, 2005), short-term seral correlaton n returns (Getmansky, Lo, and Makarov, 2004), and structural breaks (Ba and Perron, 1998). However, the bootstrap estmates also ndcate that the performance of funds wth hgh alphas cannot be dstngushed from luck. Further nvestgaton reveals that the Ordnary Least Squares (OLS) alphas of the top funds are often mprecsely estmated overstatng hedge fund performance. Our results emprcally valdate the ndustry practce of rankng fund managers on ther nformaton ratos as opposed to ther Jensen s alpha. The bootstrap results also have mplcatons for hedge fund return persstence. We report that funds sorted on the alpha t-statstc formed over the past two years persst more than funds sorted on alpha over the same perod. More mportantly, relatve to sortng on OLS performance measures, sortng on Bayesan performance measures dramatcally mproves predctablty n hedge fund returns. The sort on past two-year Bayesan posteror alpha yelds a decle spread wth an economcally and statstcally sgnfcant alpha of 5.81 % per annum (t-statstc = 2.65) whch s 5.48 % greater than that from the sort on OLS alpha. Moreover, the alpha of the top decle portfolo from the Bayesan alpha sort s 8.21% per annum (t-statstc = 4.35), whch s 54% hgher than that from the OLS alpha sort. Indeed, we observe a smlar phenomenon when we sort funds on the t-statstc of Bayesan alpha versus the t-statstc of OLS alpha. The Bayesan approach ncreases the magntude of the alphas for both the top decle portfolo and the spread portfolo relatve to the Frequentst approach. In partcular, the sort on the t-statstc of Bayesan 2

4 alpha yelds a decle spread wth an alpha of 3.15 % per annum (t-statstc = 1.76) and a onepercentle spread wth an alpha of 6.53 % per annum (t-statstc = 2.30). Collectvely, the persstence results suggest that the top hedge fund managers (at least based on the more precse Bayesan performance measures) possess asset selecton skll, and that one can take advantage of ths fact va smple tradng strateges. We consder alternatve explanatons such as persstence n fund fees and short-term seral correlaton n fund returns. However, our results suggest these stores cannot explan the bulk of the performance persstence. For nstance, we obtan even stronger results wth pre-fee returns. We explctly show that our results are not drven by seral correlaton, whch we take nto account usng varous approaches. Senstvty tests ndcate that our results are of partcular relevance to nvestors as they are nether drven by ncubaton and backfll bas, nor confned solely to small funds. Our fndngs are also robust to varyng the formaton and evaluaton perods used to form portfolos of hedge funds. The results challenge the classcal theoretcal vew n fnance that the top hedge funds are just lucky, that s that hedge fund returns do not persst. In dong so, we buld on several themes. Fung and Hseh (1999, 2000, 2001), Mtchell and Pulvno (2001) and Agarwal and Nak (2004) show that hedge fund returns relate to conventonal asset class returns and opton-based strategy returns. 3 They fnd that a sgnfcant part of the varaton n hedge fund returns over tme can be explaned by these systematc rsk factors. We buld on ther poneerng work and show that after controllng for hedge fund exposures to these systematc rsk factors, the manageral - specfc component of fund returns persst, and for the top funds, cannot be attrbuted to luck. Brown, Goetzmann, and Ibbotson (1999), Agarwal and Nak (2000), and Lang (2000) conclude that hedge funds persst at quarterly horzons but not at longer horzons. Getmansky, Lo, and Makarov (2004) ascrbe the short-term persstence to llqudty n stock returns. Our Bayesan persstence results suggest that one reason why they do not fnd long-term persstence s that they rely on relatvely mprecse Frequentst performance measures. KTWW and Busse and Irvne (2006) apply the bootstrap and Bayesan methodologes, respectvely, to mutual fund returns. We adopt both these complementary approaches n analyzng hedge fund returns. The results n KTWW ndcate that performance of the top alpha mutual funds cannot be attrbuted to samplng varablty. We show that due to the extreme varablty of hedge fund returns, hedge fund alphas 3 Fung and Hseh (1999, 2001) show that Global Macro funds delver collar lke payoffs whle Trend Follower funds exhbt look-back straddle lke payoffs. Mtchell and Pulvno (2001) and Agarwal and Nak (2004) demonstrate that a number of equty-based hedge fund strategy payoffs resemble those obtaned from wrtng an uncovered put opton on the equty market. 3

5 do not convey as much nformaton about future returns as do mutual fund alphas. Busse and Irvne (2006) report that, wth mutual funds, Bayesan sorts mprove on the equal-weghted decle spread by at most 100% relatve to Frequentst sorts [see Table 1 n Busse and Irvne (2006)]. Our results ndcate that, wth hedge funds, the spread portfolo alpha ncreases by sxteen-fold wth Bayesan alpha - versus OLS alpha - based sorts, suggestng that Bayesan methods are even more powerful n hedge fund studes gven the severty of the small sample problem. Our study jons a nascent body of work that apples Bayesan methodology to the study of fund performance. 4 Baks, Metrck, and Wachter (2001) estmate mutual funds alphas based on nformatve pror belefs about ndvdual fund alphas. Pástor and Stambaugh (2002b) develop and mplement a Bayesan framework n whch pror vews and emprcal evdence about prcng models and manageral skll can be ncorporated nto the nvestment decson. Stambaugh (2003) shows how Baysan methods can help the nference about the returns on a fund that has survved whle others have faled due to lower returns. Jones and Shanken (2005) take advantage of pror belefs about aggregate fund performance to estmate ndvdual mutual fund performance. Smlar nformaton poolng across funds takes place n the approach developed by Cohen, Coval, and Pástor (2005) n whch the hstorcal returns and holdngs of other mutual funds are used to evaluate the performance of a sngle mutual fund. Avramov and Wermers (2006) present a Bayesan technque to select mutual funds n the presence of general forms of stock return predctablty. None of these papers apply Bayesan methods to the study of hedge funds. The rest of the paper s structured as follows. Secton 2 descrbes the data whle Secton 3 presents the bootstrap and Bayesan performance measures. Secton 4 reports the emprcal results ncludng the bootstrap analyss, a comparson of Bayesan and standard performance measures, and the Bayesan performance measure-based persstence tests. Secton 5 concludes. 2. Data We evaluate the performance of hedge funds usng monthly net-of-fee 5 returns of lve and dead hedge funds reported n the CSFB/Tremont TASS (TASS), Hedge Fund Report (HFR), Centre for Internatonal Securtes and Dervatves Markets (CISDM), and Morgan Stanley 4 In a smlar ven, Huj and Verbeek (2003) fnd that shrnkage estmators that explot nformaton n the cross-secton of fund returns are useful n analyzng mutual fund performance and persstence. 5 Our results are robust to usng pre-fee returns. 4

6 Captal Internatonal Inc. (MSCI) data sets over January 1990 to December a tme perod that covers both market upturns and downturns, as well as relatvely calm and turbulent perods. The unon of the TASS, HFR, CISDM, and MSCI databases represents the largest known data set of hedge funds to date. In our fund unverse, we have a total of 6,392 lve hedge funds and 2,946 dead hedge funds. However, due to concerns that funds wth assets under management (henceforth, AUM) below US$20 mllon may be too small for many nsttutonal nvestors, we exclude such funds from the analyss. 6 Ths leaves us wth a total of 4,300 lve hedge funds and 1,233 dead hedge funds. Fg. 1 llustrates the breakdown of funds by database. The Venn dagram n Fg. 1 reveals that the funds are roughly evenly splt among TASS, HFR, and CISDM/MSCI 7. Whle there are overlaps among the databases, many funds belong to only one specfc database. For example, there are 1,410 funds and 1,513 funds pecular to the TASS and HFR databases, respectvely. Ths hghlghts the advantage of obtanng our funds from a varety of data vendors. [Fgure 1 here] Although the term hedge fund orgnates from the Long/Short Equty strategy employed by managers lke Alfred Wnslow Jones, the current defnton of hedge funds covers a multtude of dfferent strateges. Because there does not exst a unversally accepted norm wth whch to classfy hedge funds nto dfferent strategy classes, we follow Agarwal, Danel, and Nak (2005) and group funds nto fve broad nvestment categores: Drectonal Traders, Relatve Value, Securty Selecton, Mult-process, and Fund of Funds. Drectonal Trader funds usually bet on the drecton of the prces of currences, commodtes, equtes, and bonds n the futures and cash markets. Relatve Value funds take postons on spread relatons between prces of fnancal assets and am to mnmze market exposure. Securty Selecton funds take long and short postons n undervalued and overvalued securtes, respectvely, and reduce systematc rsk n the process; usually, they take postons n equty markets. Multprocess funds employ multple strateges usually nvolvng nvestments n opportuntes created by sgnfcant transactonal events such as spn-offs, mergers and acqustons, bankruptcy reorganzatons, recaptalzatons, 6 The AUM cutoff s mplemented every month. Snce there may be concerns that ths may bas the sample n favor of fndng alpha, we also run the basc bootstrap and persstence tests wth the full sample of fund return observatons and obtan even stronger results. 7 The CISDM and MSCI databases are combned n Fg 1 to facltate llustraton. A further breakdown of funds nto each database s avalable upon request. 5

7 and share buybacks. Fund of Funds nvest n a pool of hedge funds and typcally have lower mnmum nvestment requrements. We also sngle out Long/Short Equty funds, whch are a subset of Securty Selecton funds, for further scrutny as ths strategy has grown consderably over tme (now representng the sngle largest strategy accordng to HFR) and has the hghest alpha n Agarwal and Nak (2004, Table 4). For rest of the paper, we focus on the funds for whch we have nvestment style nformaton. It s well known that hedge fund data are assocated wth many bases (Fung and Hseh, 2000). These bases are drven by the fact that due to the lack of regulaton, hedge fund data are self-reported and hence subject to self-selecton bas. For example, funds often undergo an ncubaton perod durng whch they buld up a track record usng manager s/sponsor s money before seekng captal from outsde nvestors. Only the funds wth good track records go on to approach outsde nvestors. Snce hedge funds are prohbted from advertsng, one way they can dssemnate nformaton about ther track record s by reportng ther return hstory to dfferent databases. Unfortunately, funds wth poor track records do not reach ths stage, whch nduces an ncubaton bas n the fund returns reported n the databases. Moreover, funds often report return data pror to ther lstng date n the database, thereby creatng a backfll bas. Snce wellperformng funds have strong ncentves to lst, the backflled returns are usually hgher than the nonbackflled returns. To ensure that our fndngs are robust to ncubaton and backfll bases, we repeat our analyss by excludng the frst 12 months of data. In addton, snce most database vendors started dstrbutng ther data n 1994, the data sets do not contan nformaton on funds that ded before December Ths gves rse to survvorshp bas. We mtgate ths bas by focusng on post-january 1994 data. 3. Methodology 3.1. Factor benchmarks and performance measure α In order to examne the abnormal performance of hedge funds, we regress the net-of-fee monthly excess return (n excess of the rsk-free rate) of a hedge fund on the excess returns earned by tradtonal buy-and-hold and prmtve trend followng strateges. That s, we use as 6

8 performance benchmarks the seven-factor model developed by Fung and Hseh (2004). 8 The Fung and Hseh (2004) factors are S&P 500 return mnus rsk-free rate (SNPMRF), Wlshre small cap mnus large cap return (SCMLC), change n the constant maturty yeld of the 10-year Treasury (BD10RET), change n the spread of Moody's Baa mnus the 10-year Treasury (BAAMTSY), bond PTFS (PTFSBD), currency PTFS (PTFSFX), and commodtes PTFS (PTFSCOM), where PTFS denotes prmtve trend followng strategy. Fung and Hseh (2004) show that ther seven factor model strongly explans varaton n ndvdual hedge fund returns. The ntercept αˆ n the regresson below represents the abnormal performance of the manager of hedge fund after controllng for her rsk exposures. In partcular, to evaluate the performance of hedge funds, we run the regresson K r t = ˆ α + ˆ β F, + ˆ ε, (1) k = 1 k k t t where rt s the net-of-fees excess return (n excess of the rsk-free rate) on an ndvdual hedge fund for month t, fund over the regresson tme perod, αˆ s the alpha performance measure or the abnormal performance of hedge βˆ k s the factor loadng of hedge fund on factor k durng the regresson perod, F, s the return for factor k for month t, and k t εˆ t s the error term. As a prelude to the bootstrap analyss, n Table 1 we report tests of normalty, heteroskedastcty, and seral correlaton on hedge fund resduals to examne the behavor of fund returns n our sample broken down by nvestment category. The frst column n Table 1 reveals that the average seven-factor model alpha s postve 9 and about 0.42 % per month (5.04 % per year) across all funds. The average alpha t-statstc (1.43) s low and ndcates that on average the alphas are not statstcally dfferent from zero at the 10 % level of sgnfcance. However, ths does not rule out the possblty of fndng performance persstence or top hedge funds wth statstcally sgnfcant performance. 8 Smlar results obtan when we measure performance relatve to the Agarwal and Nak (2004) optonbased factor model. 9 The average fund alpha may not be closer to zero because of ncubaton and backfll bas, whch artfcally nflates hedge fund returns. Hence, we nclude tests that control for the effects of ncubaton and backfll bas n both the bootstrap and the Bayesan analyses. More recently, Fung, Hseh, Nak and Ramadora (2006) use funds of hedge fund returns to reduce these bases. 7

9 [Table 1 here] Lookng across fund categores, Long/Short Equty funds n partcular appear to have resduals that are hghly negatvely skewed, whle Relatve Value funds exhbt hgh kurtoss or fat tals. In addton, many funds fal the test for normalty. For two nvestment categores, Long/Short Equty and Drectonal Traders, the null hypothess of normalty s rejected wth the Jarque Bera test for over 50 % of the funds. Clearly, nonparametrc methods lke the bootstrap, whch avod mposng assumptons of normalty on hedge fund returns, are useful n evaluatng hedge funds. Table 1 also reveals that fund returns are often serally correlated, and that fund resduals often heteroskedastc. Drectonal Traders fund returns appear to be most serally correlated whle Long/Short Equty fund resduals are most prone to heteroskedastcty. The former s not surprsng gven that Drectonal Traders often explot momentum-based strateges that bet on the contnuaton of prce trends The bootstrap approach The bootstrap s a nonparametrc approach to statstcal nference. 10 It s especally relevant to the study of top hedge fund performance for three reasons. Frst, the bootstrap allows the researcher to avod havng to make a pror assumptons about the shape of the dstrbuton from whch ndvdual fund alphas are drawn. As Table 1 shows, the emprcal dstrbuton of resduals from multfactor performance regressons s nonnormal for many hedge funds. Thus, the dstrbuton of αˆ may be poorly approxmated by the normal dstrbuton, wth ts statstcal sgnfcance better evaluated usng a nonparametrc approach such as the bootstrap. Second, the bootstrap frees the researcher from havng to estmate the entre covarance matrx characterzng the jont dstrbuton of ndvdual funds. Specfcally, the dstrbuton of the maxmum alpha depends on ths covarance matrx, whch s generally mpossble to estmate wth precson. Even f ndvdual fund resduals are adequately approxmated by a normal dstrbuton, the very large dmenson of ths matrx wth thousands of funds and the entry and ext of funds (whch mply that many funds do not even have overlappng return records) all conspre to make estmaton nfeasble. Thrd, refnements of the bootstrap (whch we mplement) provde a general approach for dealng wth unknown tme-seres dependences that are due, for example, to 10 Our approach s based on the bootstrap ntroduced by Efron (1979). For a detaled dscusson of the propertes of the bootstrap, see, for example, Efron and Tbshran (1993) or Hall (1992). 8

10 heteroskedastcty or seral correlaton n the resduals from performance regressons. The basc ntuton underlyng the bootstrap mplementaton s smple. We seek to compare the observed top fund performance to the performance of top funds n artfcally generated data samples n whch varaton n fund performance s entrely due to sample varablty or luck. To prepare for our bootstrap procedure, for each fund we measure performance relatve to the multfactor model n Eq (1). We then save the coeffcent estmates { ˆ α, ˆ } β { ˆ,, t = 1,..., T } t, the t-statstc of alpha ε. tˆ 11 αˆ, and the tme seres of estmated resduals For the baselne resdual-only resamplng bootstrap, we draw a sample wth replacement from the fund - resduals that are saved n the frst step, thereby creatng a tme seres of ˆ, t = 1 2 b b b b resampled resduals {, t s, s,..., s } ε, where b=1 (for bootstrap resample number one). T The sample s drawn such that t has the same number of resduals (e.g., the same number of tme perods T ) as the orgnal sample for each fund. Ths resamplng procedure s repeated for the remanng bootstrap teratons b = 2,...,B (n all of our bootstrap tests, we set B = 1,000). Next, for each bootstrap teraton b, we construct a tme seres of (bootstrapped) monthly net returns for ths fund, mposng the null hypothess of zero true performance ( α = 0, or equvalently, t α = 0 ): K b b b b r ˆ t k F ˆ, = k, t + ε, t, t = s1, s2,..., k = 1 β s, (2) b T where b b s, s2,..., b 1 st s the tme reorderng mposed by resamplng the resduals n bootstrap teraton b. As Eq (2) ndcates, ths sequence of artfcal returns has a true alpha (and t-statstc of alpha) of zero, snce the resduals are drawn from a sample that s mean zero by constructon. However, when we regress the returns for a gven bootstrap sample, b, on the multfactor model (below), a postve estmated alpha (and t-statstc) may result, snce that bootstrap may have drawn an abnormally hgh number of postve resduals, or, conversely, a negatve alpha (and t- 11 We estmate ˆt αˆ usng the Newey and West (1987) heteroskedastcty and autocorrelaton consstent estmate of the standard error. 9

11 statstc) may result f an abnormally hgh number of negatve resduals are drawn. Repeatng the above steps across funds = 1,...,N and bootstrap teratons b = 1,...,B, we buld the cross-sectonal dstrbuton of the alpha estmates b αˆ, or ther t-statstcs b tˆ αˆ, whch result purely from samplng varaton as we mpose the null of no abnormal performance. If we fnd that very few of the bootstrap teratons generate values of αˆ or ˆt αˆ, that are as large as those that obtan n the actual data, ths would suggest that samplng varaton (luck) s not the source of performance, and thus that genune asset selecton sklls may exst. Gven the superor statstcal propertes of the alpha t-statstc, n addton to evaluatng top funds ranked on alpha, we also evaluate top funds based on the alpha t-statstc. Although alpha measures the economc sze of abnormal performance, t has a relatvely hgh coverage error n the constructon of confdence ntervals. Also, funds wth a shorter hstory of monthly net returns wll have an alpha estmated wth less precson, and wll tend to generate alphas that are outlers. The alpha t-statstc provdes a correcton for these spurous outlers by normalzng the estmated alpha by the estmated precson of the alpha estmate. Moreover, the t-statstc s related to the well-known nformaton rato of Treynor and Black (1973), whch s commonly used by practtoners to rate fund managers. For the nterested reader, KTWW provdes further detals on the bootstrap procedure and the ratonale for sortng on the alpha t-statstc The Bayesan approach The average hedge fund has a short tme seres. Ths necessarly reduces the precson wth whch performance measures such as alpha can be estmated. However, as PS pont out, t s possble to substantally mprove on the alpha estmates by usng hstorcal returns on seemngly unrelated assets not used n the defnton of the alpha performance measure. These so-called nonbenchmark passve assets have longer tme seres than the benchmark seres and are correlated wth hedge fund returns. The correlaton between the hedge fund and non-benchmark passve returns can be exploted to mprove our alpha estmates ndependent of whether these passve returns are prced by the benchmarks. PS use seemngly unrelated asset returns to mprove the precson of alpha estmates of mutual funds They fnd a medan dfference between ther Bayesan posteror alphas and the OLS alphas of 2.3 % per annum and 8.1 % per annum for all funds and for small-company growth funds, respectvely. 10

12 Gven that the average hedge fund has a much shorter returns seres than the average mutual fund, ths methodology s even more relevant to hedge funds than to mutual funds. Indeed, by usng nformaton on passve non-benchmark returns, we can double the length of the tme seres used for estmatng alphas. Stambaugh (1997) shows how assets wth longer hstorcal tme seres provde nformaton about the moments of assets wth shorter hstores. We follow the PS methodology and regress non-benchmark passve returns on benchmark returns. Let N F m, t denote the m 1 vector of non-benchmark passve asset returns n month t regressed on the k benchmark returns F, : B k t F N t K N = ˆ α + k = 1 ˆ β F N k B k, t + ˆ ε. (3) N t Importantly, εˆ t n Eq. (1) and N εˆ t n Eq. (3) are allowed to be correlated. PS show that the mprovement n the estmaton of alpha performance measure does not depend on whether the benchmarks F, perfectly prce the non-benchmark passve assets B k t N F m, t mpled. We also defne the regresson of a fund s return on p ( = m + k ) benchmark and non-benchmark assets: M K N N B B Rt = ˆ δ + cˆ m Fm t cˆ k Fk t uˆ, +, + t, = 1,..., L. (4) m= 1 k = 1 Usng Eq. (1) and the fact that F, s uncorrelated wth both B k t N εˆ t and û t, PS show that α = δ + c α. (5) N N PS also show how to derve the posteror estmate ~ α of α n Eq. (5) from the posteror moments of δ, c N, and α N. PS provde analytcal expressons for the posteror moments ~ α, ~ δ, c ~ N, and ~ α N. As we shall show, the mean posteror alpha estmate OLS alpha αˆ from Eq. (1) for hedge funds. ~ α s below the mean 11

13 We follow PS and apply an emprcal Bayesan approach to estmate the pror dstrbuton of varous varables. The pror dstrbuton of the covarance matrx of denoted by Σ, s specfed as an nverted Wshart dstrbuton, N εˆ t n Eq. (3), 1 Σ ~ W ( H 1, v). We follow PS n specfyng the emprcal estmates of the prors. We set the degrees of freedom to v = m + 3, whch mples that the pror contans very lttle nformaton about Σ. Moreover, we specfy H ) s 2 2 = ( v m 1 I m and E( Σ ) = s I m. The value of 2 s s set equal to the average of the dagonal elements of the sample estmates of Σ obtaned usng OLS regressons 2 n Eq. (3). The parameters n Eq. (4) are specfed as follows; The pror for σ u, the varance of u, t, s an nverted gamma dstrbuton, or v s σ u ~ 2 χ v 0, (6) 2 where χ v 0 represents a ch-square varate wth v 0 degrees of freedom. Defne c = ( ) L cln clb. Condtonal on ndependent of each other: 2 σ u, the prors for δ L and c L are assumed to follow normal dstrbutons, and 2 2 σ 2 δ L σ u ~ N δ 0, u 2 σ δ, (7) E ( σ u ) 2 c L σ u ~ N c 2 σ u 0, E 2 ( σ u Φc. (8) ) Followng PS, all of our estmates are based on dffuse or completely nonnformatve prors. 13 In the applcaton of the Bayesan framework, decsons regardng the non-benchmark seres need to be made. As non-benchmark passve assets we use the HFR style ndex for each 13 As PS pont out, the Bayesan framework can also accommodate nformatve belefs. 12

14 respectve nvestment objectve. For example, Long/Short Equty funds are matched wth the HFR Long/Short Equty ndex. The lmted number of benchmark factors s motvated by the observaton of PS that f the number of non-benchmark assets ncreases wthout a suffcent ncrease n R 2, then the posteror alpha estmate may be less precse. Thus, we use the same set of non-benchmark assets for funds that share the same nvestment objectve. As PS pont out, the non-benchmark passve factor should be hghly correlated wth fund returns, a condton that the respectve HFR style ndces fulfl. To rase the explanatory power of our regressors wthout unduly ncreasng the number of non-benchmark passve assets, we also nclude the HFR Fund of Funds ndex as an addtonal non-benchmark passve asset for all ndvdual funds snce Fund of Funds hold a varety of hedge funds n ther portfolos. 4. Emprcal results 4.1. A bootstrap analyss of top hedge fund performance The frst order of busness s to evaluate the statstcal sgnfcance of top hedge fund performance wth the bootstrap. Panel A of Table 2 dsplays the results from our applcaton of the bootstrap algorthm. Gven we requre suffcent return data to estmate the factor loadngs, only funds wth at least 24 months of return data are ncluded n the bootstrap sample. 14 Ths yelds a sample of 2,734 funds for whch we have nvestment style nformaton. We rank all funds n two dfferent ways. The frst two rows of Panel A rank funds accordng to ther estmated alphas. The thrd and fourth rows rank funds based on the estmated alpha t-statstcs. The results are dsplayed for the extreme top fve and bottom fve funds as well as funds at the 1st, 3rd, 5th, and 10th percentles on both ends of the alpha and alpha t-statstc spectra. [Table 2 here] The results n Panel A of Table 2 ndcate that the performance of the top alpha hedge funds can be attrbuted to samplng varablty. Both bootstrapped p-values for the top two funds ranked on fund alphas are greater than 0.1, suggestng that we cannot reject the null hypothess that ther alphas are drven by samplng varablty at the 10 % level of sgnfcance. In contrast to 14 As a robustness check, we also perform the bootstrap on funds wth at least 30 and at least 48 months of return data and fnd qualtatvely smlar results. These results are avalable upon request. 13

15 the performance of top alpha hedge funds, the performance of the top alpha t-statstc hedge funds cannot be attrbuted to samplng varablty. Ther bootstrapped p-values are all below Indeed, ths s not surprsng gven the superor statstcal propertes of the alpha t-statstc. By penalzng the hgh alpha funds, whch have short nvestment hstores and hgh standard devatons, the alpha t-statstc better dscrmnates between funds that generate superor performance through skll and funds that are smply lucky. Accordng to our bootstrap estmates, of the 2,734 funds n our sample, by chance alone we would expect at most three funds to acheve alpha t-statstcs of at least In realty, 155 hedge funds exceed ths alpha t-statstc. It s mportant to note that most of the top funds sorted on alpha have bootstrapped p-values below 0.1. Also n contrast to the performance of the top hedge funds, the performance of the bottom funds can be explaned by samplng varablty, or rather, bad luck, to be precse. To examne potental dfferences n performance between nvestment objectves, we report the statstcal sgnfcance of performance measures by nvestment objectve n Panels B to G of Table 2. To group funds by broad nvestment category, we use the sx broad nvestment categores dscussed n Secton 2. The results dsplayed n Table 2 suggest that top alpha t- statstc funds consstently outperform ther benchmarks n every nvestment category. We also note that the best and worst funds n the sample are Drectonal Trader funds. These funds make aggressve bets on the drecton of the market and thus t s no concdence that the top alpha, top alpha t-statstc, bottom alpha, and bottom alpha t-statstc funds are all Drectonal Trader funds. Fund of Funds (reported n Panel G) appear to perform worse than other nvestment objectves. The top Fund of Funds generates an alpha of 1.6 % per month (19.2 % per year) compared to the top Long/Short Equty fund that acheves an alpha of 4.1 % per month (49.2 % per year). The performances of the rght-tal funds n the Fund of Funds category appear statstcally less sgnfcant than those n other nvestment objectves as ther hgher p-values show. The relatvely poor performance of Funds of Funds may be due to the fact that Funds of Funds charge an addtonal layer of fees for combnng ndvdual hedge funds nto a portfolo. Another nterpretaton s that due to selecton bas, the ndvdual hedge funds that are not n the database but are held by Fund of Funds have lower returns than the hedge funds n the database. 15 To further llustrate the baselne results of the bootstrap, we plot the kernel densty estmate of the bootstrapped and actual alpha t-statstc dstrbutons for all funds n Fg. 2. The 15 We thank an anonymous referee for ths mportant nsght. In results avalable upon request, we rerun the entre bootstrap analyss wthout Fund of Funds and obtan almost dentcal results. 14

16 two denstes are rather dfferent. The dstrbuton of the alpha t-statstcs has much more mass on the rght tal than the bootstrapped alpha t-statstc dstrbuton. Ths suggests that we are lkely to fnd funds n the rght tal wth superor alpha t-statstcs that cannot be explaned by random resamplng alone. [Fgure 2 here] To gauge the robustness of our bootstrap results, we control for varous ssues pecular to hedge fund returns, namely, ncubaton and backfll bas, short-term seral correlaton, and structural breaks. Hedge funds often ncubate ther funds wth the manager s own money before seekng outsde nvestors. Also, hedge funds often report returns for ther entre hstory n the fund databases, ncludng backdated returns for the perod pror to a funds lstng on the database. Snce hedge fund ncluson n databases s done on a voluntary bass, hedge funds wth poor track records wll have a dsncentve to report to databases whereas hedge funds wth good track records wll have an ncentve to report to databases. In response to these concerns, we control for ncubaton and backfll bas n our bootstrap analyss. That s, we exclude the frst 12 months of each fund s hstory, regardless of whether that hstory s backflled or not (snce ncubaton bas may occur n a fund s hstory even though t s not backflled). Then, we bootstrap the resduals of the funds from the ncubaton and backfll bas-adjusted sample. Other hedge fund studes document that hedge fund returns are often hghly serally correlated, n contrast to mutual fund returns, for example. Accordng to Getmansky, Lo, and Makarov (2004), the most lkely reason for ths seral correlaton s funds' llqudty exposure: hedge funds trade n securtes that are not actvely traded and whose market prces are not readly avalable. To remove the effects of artfcal seral correlaton nduced by llqudty exposure, we adopt the methodology n Getmansky, Lo, and Makarov (2004) to unsmooth hedge fund returns and reduce seral correlaton. In partcular, we map the fund categores n Table 8 of Getmansky, Lo, and Makarov (2004) to our fund categores and use the average θ 0, θ1, and θ 2 estmates for each fund category from ther Table 8 to unsmooth fund returns. Then, we re-run the bootstrap analyss on the unsmoothed sample of hedge fund returns. The Appendx detals how we map the Getmansky, Lo, and Makarov (2004) fund categores to our categores. If the regresson coeffcents n hedge fund performance regressons exhbt structural breaks over tme, then constant coeffcent regressons and bootstrap results are lkely to be 15

17 msspecfed. Recent papers document tme varaton n the return characterstcs of hedge funds (Fung and Hseh, 2004; Fung, Hseh, Nak, and Ramadora, 2006). There s evdence that hedge funds change ther loadngs on dfferent rsk factors over tme. Anecdotal evdence suggests that hedge funds suffer from sudden shocks such as the Asan and Russan crses, whch can lead to structural breaks n ther return seres. Ba and Perron (1998) provde a least-squares method for optmally determnng the unknown breakponts. We apply the Ba and Perron (1998) test to HFR hedge fund return ndces and fnd that most categores have a common structural break n December The structural break n December 2000 concdes wth the heght of the bull market n the late 1990s. Based on the structural break evdence n December 2000, we repeat our bootstrap procedure allowng for a break n the beta slope coeffcents usng a dummy regresson. [Table 3 here] Panels A, B, and C of Table 3 ndcate that smlar bootstrap results obtan when we control for ncubaton and backfll bas, short-term seral correlaton n returns, and the presence of a structural break n December 2000, respectvely. Thus, whether we confne the analyss to ncubaton and backfll bas-adjusted return data, to unsmoothed hedge fund returns 17, or to data that allow for varaton n beta coeffcents before and after December 2000, one cannot explan the performance of the top alpha t-statstc funds wth sample varablty or luck. The bootstrap results thus far have reled on the smple resdual-only resamplng bootstrap. Gven the complexty of hedge fund strateges, there may be concerns that ther return seres may not satsfy the assumptons underlyng the smple bootstrap approach. In partcular, the fund resduals may be cross-sectonally dependent. Ths could arse from funds holdng the same securtes or from funds followng the same tradng strateges. To take nto account the potental cross-sectonal dependence between fund returns we apply a cross-sectonal bootstrap. Ths procedure dffers from the algorthm descrbed n Secton 3.2 n that, for a gven bootstrap, we now employ the same bootstrap ndex across all funds. Rather than drawng sequences of tme perods, t, that are unque to each fund,, we draw T tme perods from the set (t = 1,...,T), then resample resduals for ths rendexed tme sequence across all funds, thus preservng any cross- 16 HFR hedge fund returns ndces are regressed on factor benchmarks. Factor benchmarks are determned for each hedge fund ndex category by means of the regresson approach descrbed n Secton 3.1 above. Both the ntercept and the slope coeffcents are allowed to vary. 17 Inferences also do not change when we use the methodology of Okunev and Whte (2003) to elmnate return seral correlaton of up to order two. 16

18 sectonal correlaton n the resduals. Snce as a result some funds may be allocated bootstrap ndex entres from perods when they dd not have a return, we drop a fund f t does not have at least 24 observatons after applyng the bootstrap ndex. The results n Panel D of Table 3 ndcate that our basc bootstrap results are not drven by cross-sectonal dependence n fund returns. The p-values show that the performance of the best alpha t-statstc funds cannot be explaned by samplng varaton. Other devatons from the assumptons underlyng the smple bootstrap approach can occur. Frst, there may be correlaton between the factor returns and regresson resduals. Ths correlaton could arse f the hedge fund managers trade n securtes that have a return coskewness wth the factor returns. Second, there may exst an omtted factor that s not accounted for by the Fung and Hseh (2004) multfactor model. If funds have hgh alpha t- statstcs because they load on ths omtted factor then we may be led erroneously to conclude that fund managers have securty selecton ablty f we do not nclude the omtted factor, n our performance measurement model. Thrd, condtonal on factor realzatons, the fund resduals may not be ndependently dstrbuted across tme. To address these concerns, we adopt the alternatve bootstrap extensons descrbed n KTWW. Specfcally, we perform the factor-resdual resamplng bootstrap that accommodates correlaton between factor returns and fund regresson resduals, run Monte Carlo smulatons for a persstent omtted factor, and adopt the statonary bootstrap procedure of Polts and Romano (1994) to allow for tme-seres dependence n fund resduals. The results from these bootstrap extensons, whch are avalable upon request, confrm that our concluson of sgnfcant securty selecton ablty amongst top hedge fund managers s robust to the choce of bootstrap methodology Bayesan hedge fund performance measures The bootstrap results n the prevous secton suggest that standard OLS fund alpha may not be as ndcatve of fund manager performance as OLS fund alpha t-statstcs. The bootstrapped p-values of top funds sorted on alpha are typcally hgher than those for top funds sorted on alpha t-statstcs. One reason may be that the OLS fund alphas are estmated mprecsely. In ths secton, we ntroduce the Bayesan posteror fund alpha and compare t to the standard OLS fund alpha. If the standard OLS alpha s measured mprecsely, the Bayesan alpha 17

19 (whch s more precsely measured) should dffer sgnfcantly from t, whch n turn would suggest that one may get greater mleage from sortng on Bayesan fund alpha than from sortng on standard OLS fund alpha n persstence tests. Table 4 reports the estmates of the Bayesan posteror alpha ~ α and the OLS alpha αˆ (from Eq. 1). In Panel A, the frst column reports the Bayesan alpha estmate. The second and thrd columns report the OLS alpha estmate and the mean dfference between the Bayesan and OLS alpha. As column three shows, the mean Bayesan alpha s very smlar to the mean OLS alpha. However, wth the top percentle and decle funds (ranked by OLS alpha), the OLS alpha tends to overestmate performance relatve to the Bayesan alpha. Conversely, wth bottom percentle and decle funds, the OLS alpha tends to underestmate performance relatve to the Bayesan alpha. Column fve reveals that ths s because the Bayesan alphas are measured more precsely than the OLS alphas. The mean standard error of the Bayesan alpha s lower than that of the OLS alpha for each portfolo of funds. Columns seven and eght show that n general, smlar results obtan for dfferent prors of σ α N. 18 Panel B of Table 4 shows the analogous statstcs for Long/Short Equty funds. We sngle out Long/Short Equty funds as ther strateges are partcularly easy to understand and ther rsks are easly captured by the multfactor model. Wth ths group of funds, the OLS alpha tends to mldly overestmate the true fund performance on average. The mean Bayesan alpha s 0.4 % per year lower than the mean OLS alpha. More mportantly, we fnd that the OLS alpha overestmates the performance of the top funds (ranked by OLS alpha) and underestmates the performance of the bottom funds. Moreover, the Bayesan alphas are also measured more precsely than the OLS alphas wth ths group of funds. To get a sense of the dfference between the Bayesan and OLS alphas for the top funds, Panel C of Table 4 reports the Bayesan alphas for the top 20 Long/Short Equty funds ranked by OLS alpha. For ths group of funds, the Bayesan alpha s on average 8.4 % per year below the OLS alpha. The reducton n partcularly pronounced for funds wth relatvely short sample perods. It s nterestng to note that many of the top funds have very short return hstores, 18 The baselne pror σ of 0.02 s based on Pastor and Stambaugh (2002b). In results avalable from the α N authors upon request we carry out a senstvty analyss to varous pror values between zero and nfnty. We fnd that the results are qualtatvely smlar to those wth the baselne value of 0.02 and thus our conclusons are not senstve to ths partcular choce of pror. 18

20 reflectng the severty of the short sample problem n hedge fund studes. Moreover, the dfference n standard errors reported n column fve ndcates that the Bayesan alpha s more precsely estmated than the OLS alpha for all funds. Overall, Table 4 shows that the longer tme seres of the HFR style ndces (our choce for the seemngly unrelated assets) provde mportant nformaton. By neglectng such nformaton, the OLS alphas often overestmate the performance of the top funds and underestmate the performance of the bottom funds. Moreover, the Bayesan estmates are almost always measured wth greater precson. Snce the Bayesan alpha measure provdes a more accurate estmate of performance, t should also be better at pckng the good and bad funds, and generatng performance persstence A Bayesan analyss of hedge fund performance persstence The results thus far have shown that the performance of the top hedge funds (at least when sorted on alpha t-statstc) cannot be explaned by luck. Also, relatve to the standard Frequentst approach, the Bayesan SUR approach yelds both more precse and more conservatve performance estmates. However, those results may not be relevant to nvestors f they cannot be parlayed nto tradng profts. In ths secton, we test whether hedge fund performance perssts usng the Bayesan SUR approach. Concretely, we rank funds on ther Bayesan posteror alphas and alpha t-statstcs, and evaluate the abnormal performance of the resultng spreads and top portfolos. The use of the Bayesan performance measures allevates the short sample problem and maxmzes the power of the persstence tests. By analyzng performance persstence, we are ostensbly measurng the profts that may be harvested by nvestors from smple performance-based tradng strateges. Snce t s well known that Fund of Funds have lower average returns than ndvdual hedge funds (Table 1 confrms ths), we do not nclude Fund of Funds n any of our baselne persstence tests so as not to nflate the spreads from those tests. Performance persstence has been extensvely studed n the mutual fund lterature. Hendrcks, Patel, and Zeckhauser (1993) and others show that mutual fund returns persst n the medum term (one to three years). Carhart (1997) argues that ths s ether due to managers adoptng momentum strateges or to the persstence of fund expenses. However, recent studes 19

21 such as Mamaysky, Spegel, and Zhang (2005), Cohen, Coval, and Pastor (2005), and KTWW show that more sophstcated econometrc methods allow one to pck out funds whose returns cannot be explaned by four-factor covaraton or expense ratos. Wth respect to hedge funds, prevous studes fnd lttle evdence of persstence n returns. Agarwal and Nak (2000) show that hedge fund returns only persst n the short term (one to three months). Getmansky, Lo, and Makarov (2004) credt ths fndng to the llqudty nduced by assets that hedge funds trade. Lke Agarwal and Nak (2000), Brown, Goetzmann, and Ibbotson (1999) and Lang (2000) fnd no evdence of persstence n hedge fund returns at annual horzons. What of hedge fund alphas? As a prelude to the Bayesan analyss of fund performance persstence, we frst nvestgate performance persstence wth the standard Frequentst performance measures. Specfcally, we follow Carhart (1997) and sort hedge funds on January 1 of each year (from 1996 to 2002) nto decle portfolos, based on ther Fung and Hseh (2004) seven-factor alphas estmated over the pror two years. 19 The portfolos are equally weghted monthly, so the weghts are readjusted whenever a fund dsappears. Funds wth the hghest past two-year alpha comprse decle 1, and funds wth the lowest past two-year alpha comprse decle 10. The resultng decle portfolos and annualzed performance measures are reported n Panel A of Table 5. We also sort funds based on ther alpha t-statstcs estmated over the past two years and report the performance of the resultant portfolos n Panel B of Table 5. Gven the superor statstcal qualtes of the alpha t-statstc (see Table 2), we expect performance persstence to be stronger wth these sorts than wth the sorts on fund alpha. [Table 5 here] Accordng to Panel A of Table 5, n the sort on alpha, the top decle portfolo generates an alpha of 5.32 % per year. As the parametrc p-value n column fve shows, ths alpha s statstcally sgnfcant. The alpha of the second decle s also statstcally sgnfcant. However, the top one percentle and top fve percentle portfolos do not generate statstcally sgnfcant alphas. Ths s consstent wth our fndng n Table 2 that the top two or three hedge funds do not generate statstcally sgnfcant performance. One explanaton of ths result s that these funds follow partcularly rsky strateges that fal to perform well n the future. Also, the alpha for the 19 We requre a mnmum of 24 monthly net return observatons for ths estmate. For funds wth mssng observatons, observatons from the 12 months precedng the two-year wndow are added to obtan 24 observatons. Ths ensures that funds wth mssng observatons are not automatcally excluded. 20