Impact of Fund Size on Hedge Fund Performance

Transcription

1 Impact of Fund Sze on Hedge Fund Performance Manuel Ammann Patrck Moerth Workng Paper Seres n Fnance Paper No January 2005

2 IMPACT OF FUND SIZE ON HEDGE FUND PERFORMANCE Manuel Ammann, Patrck Moerth Swss Insttute of Bankng and Fnance Unversty of St. Gallen Rosenbergstrasse 52 CH-9000 St. Gallen Swtzerland May 2004, revsed January 2005 Keywords: Hedge funds, performance measurement, sze Manuel Ammann s professor of fnance; Patrck Moerth s a hedge fund analyst at Credt Susse and a Ph.D. student at the Unversty of St. Gallen. Contact detals are Tel: , Fax: , emal of correspondng authors: manuel.ammann@unsg.ch, patrckmoerth@yahoo.com

3 ABSTRACT In ths artcle we nvestgate whether the ncrease n assets flowng nto the hedge fund ndustry dmnshes returns and, n partcular, whether larger hedge funds underperform smaller hedge funds, as s often conjectured due to lmted capacty n certan hedge fund strateges. The mpact of fund szes s analyzed wth respect to fund returns, standard devatons, Sharpe Ratos and alphas derved from a mult asset class factor model. In the last years hedge funds have ganed a wdespread acceptance due to ther nterestng rsk-return characterstcs and ther low correlatons to tradtonal asset classes. Many studes have nvestgated the factors mpactng hedge fund returns. The factors vary between the dfferent hedge fund strateges and, addtonally, the varety of strateges s ncreasng n the hedge fund ndustry. At the same tme, the hedge fund ndustry has experenced mpressve growth such that capacty s becomng a serous ssue not only for large hedge fund nvestors that are lookng for nvestment opportuntes to employ the captal, but also for hedge fund nvestors that are lookng for good hedge funds that are open for nvestment. Increasng effcency of fnancal markets results n decreasng arbtrage opportuntes that are the prmary source for returns of some hedge fund strateges. In ths paper, we nvestgate whether an ncreasng asset base of hedge funds s dlutng performance. On one sde we can conjecture that small hedge funds are underperformng larger hedge funds due to a hgher expense rato. On the other sde many nvestment professonals argue that smaller funds are outperformng larger funds due to a hgher rsk appette or ther enhanced flexblty to concentrate the captal under management on ther best nvestment deas. The best ten nvestment deas of a hedge fund manager are generally better than the best 100 deas. Smaller hedge funds are also more nmble and therefore more lqud due to smaller poston szes. Large funds may face dffcultes n lqudatng ther postons n dffcult market envronments. We fnd that some studes are touchng the subject wth varyng results. Whle the effect of fund sze on performance s one of the largest concerns n the hedge fund ndustry t has receved lttle research attenton of studes exclusvely focusng on ths subject. Ths artcle attempts to fll ths gap by evaluatng the relatonshp between fund szes and performance

4 from dfferent angles. The study s supported by emprcal evdence based on a large data sample of hedge fund returns and fund szes. A number of dfferent mult asset class factor models have been used to derve alphas of hedge fund returns. In contrast to the standard asset-class factor model of Sharpe [1992] we use excess returns to derve alphas from our factor models n order to nvestgate the mpact of fund szes. Our emprcal study provdes some evdence for a postve relatonshp between fund szes and hedge fund performance. The structure of the paper s the followng: We start wth a lterature overvew, followed by a descrpton of the data set used for our analyses. Next, we ntroduce the methodology and an emprcal analyss concernng the mpact of fund szes on hedge fund returns, standard devatons, Sharpe Ratos and alphas. LITERATURE OVERVIEW Several authors propose factor models to explan hedge fund returns. Fung and Hseh [1997] employ Sharpe s [1992] model and use a prncpal component analyss to explan hedge fund returns. Agarwal and Nak [2000] develop a factor model and pont out the nonlnear opton-lke exposures of varous hedge fund strateges. An nterestng approach to overcome the short hstory of hedge fund returns has been proposed by Agarwal and Nak [2002]. The authors use the underlyng rsk factors estmated wth a mult-factor model to smulate the effects of the major stock market crses of 1929 and 1987 on hedge fund returns. Schneewes, Kazem and Martn [2003] nvestgate the dfferences between a sngle factor and a mult factor model to explan hedge fund strategy returns. Brealey and Kaplans [2001] nvestgate changes n factor exposures to explan hedge fund returns over tme. Clark [2003] provdes a comprehensve study about the relatonshp of fund assets and performance for mutual funds and concludes that no sgnfcant return dfferences can be found between small and large funds on a varety of holdng perods from 1991 through Herzberg and Mozes [2003] nvestgate the mpact of several factors on hedge funds performance and fnd that smaller hedge funds dsplay a performance that s slghtly better and barely sgnfcant compared to larger funds, whle the dfference s postvely sgnfcant regardng Sharpe Ratos. Hedges [2003] shows that smaller funds outperform larger funds, but fnds that md-szed funds perform the worst. Ths phenomenon s explaned wth the concept of md-lfe crses for hedge funds managers.

5 Gregorou and Rouah [2003] fnd no correlaton between the sze of hedge funds and ther performance. The relatonshp s tested wth Pearson s correlaton coeffcent and Spearman s rank correlaton from January 1994 to December Usng the geometrc mean, the Sharpe Rato and the Treynor Rato, the correlatons are not statstcally sgnfcant. The sample s composed of 204 hedge funds and 72 funds of hedge funds and s therefore sgnfcantly smaller than n our study and not necessarly representatve for the hedge fund ndustry. Edwards and Caglayan [2003] argue that hedge funds performance ncreases at a declnng rate as fund szes ncrease. The authors derve sx-factor alphas from a smlar framework than that of Fama and French [1993, 1996]. The sx-factor alphas are then regressed on fve varables: sze, the recprocal of sze to capture nonlnearty n the szeperformance relatonshp, age, and both management and ncentve fees. Both sze varables are statstcally sgnfcant for all hedge funds and for all nvestment styles except global macro and global. A postve coeffcent on the sze varable together wth a negatve coeffcent on the sze recprocal varable ndcates that hedge funds performance ncreases at a declnng rate as fund szes ncrease. Lang [1999] nvestgates the mpact of fund characterstcs wth a cross-sectonal regresson and fnds a sgnfcant postve relatonshp between fund assets and performance. The assets of the funds are taken only from one pont n tme at the end of the perod. Therefore the result may smply suggest that successful funds attract more money over tme and therefore have a postve correlaton to past performance. The study does therefore not necessarly measure the mpact of fund assets on performance, but the mpact of performance on fund assets. The data set used contans only 385 funds nvestgated over a three years tme horzon from January 1994 to December Amenc and Martelln [2003] support the vew by nvestgatng two equally szed groups wth large and small funds. For each group, the average alpha s computed based on a number of dfferent models, such as the standard CAPM, an adjusted CAPM for the presence of stale prces and an mplct factor model extracted from a prncpal component analyss. For all models the mean alpha for large funds exceeds the mean alpha for small funds. The separaton of the data n small and large funds s smplstc and not suffcent to measure the relatonshp between fund szes and performance. A smlar approach has been chosen by Kazem and Schneewes [2003]. At the begnnng of each year funds wthn each style are ether assgned a large or a small subgroup dependng on the sze of assets under management. The authors fnd that large or small funds do not

6 unformly outperform the other group. The study contans only 15 to 30 hedge funds n each subgroup. Getmansky [2004] analyses the effects of fund and strategy specfc factors on lfe cycles of hedge funds and confrms that better performng funds are more lkely to attract assets than poorly performng funds. The performance-asset sze relatonshp takes on dfferent functonal forms for dfferent strateges. For nstance, for llqud strateges such as emergng markets and convertble arbtrage the relatonshp s concave so that the top performng funds do not grow proportonally as much as the average fund n the market. The use of quadratc regressons rases the queston of data fttng. The use of the relatonshp between fund szes and the performance of ndvdual strateges s lmted to the relatve low number of funds per strategy. DATA SET We combne three databases from TASS, one of today s largest commercal hedge fund data provders. Lang [2000, 2003] nvestgated the data qualty of varous hedge fund data provders and concluded that the TASS database has a hgh data qualty n comparson to other commercal database provders. Many emprcal studes are based on the TASS database. TASS mantans three separate databases, a hedge fund database wth lvng hedge funds, one graveyard database wth funds that stopped reportng and one CTA database. The combnaton of the three databases contanng lvng and dead funds allows us to derve the survvorshp bas. 1 TASS started to buld ther databases n 1993/1994. The data pror to 1994 has been backflled by hedge fund managers startng to report n 1994 or later. Therefore data pror to 1994 contans a number of bases and has not been used for our analyss. All three TASS databases combned contan a total of 4327 funds: 2367 funds n the hedge fund database, 1542 funds n the graveyard database and 418 funds n the CTA database. From our data sample we elmnate 265 funds due to poor data qualty or double countng. 707 of the remanng funds are classfed as fund of hedge funds and have been excluded from the hedge fund data sample. For the remanng 3355 funds we have found the nconsstences shown n Exhbt 1. Our fndngs confrm the concerns of other performance 1 Survvorshp bas occurs f the database only contans nformaton on survvng funds. Followng Malkel s method [1995], the bas s evaluated va the dfference n the performance of the observable portfolo (nvestment n each fund n the database from the begnnng of the data sample) and the portfolo of survvng funds.

7 studes that asset data of hedge funds s very often ncomplete. The nconsstences n the asset development can partly be explaned by the fact that some hedge funds have been reportng ther assets only on a quarterly, sem-annual or annual bass, partcularly n ther early years of reportng. On one hand we want to elmnate funds wth ncomplete data n order to avod any addtonal bases n our data sample and on the other hand we do not want to lose too many funds for our analyss, snce our sample would then be less representatve for the Hedge Fund unverse. We therefore need to fnd a compromse between data qualty and data quantty. We decde to elmnate all funds wth more than 10% mssng data ponts n return data or more than 20% mssng data ponts n asset data. The ntal sample based on 3355 hedge funds s capturng assets of 241 bllon USD n June Wth the adjustment we exclude 1027 funds wth assets of 54 bllon USD n June Mssng data ponts n assets under management s the man restrcton. Only 17 funds are not provdng enough return data, whle 1033 funds are provdng less than 90% of the development of the assets under management. In addton to that the data s screened for extreme outlers exceedng 60% per month. If the extreme return data was not n lne wth the fund profle we conjectured a data error and elmnated the fund. We therefore excluded another eleven funds out of the sample due to a hgh lkelhood of data errors n the return seres. Fnally we use a sample of 2317 funds wth assets of 186 bllon USD n June The data qualty of the sample used n the performance study s llustrated n Exhbt 2. METHODOLOGY In order to analyze the mpact of the fund sze of hedge funds on the returns we use the followng approaches. In a frst step we analyze the dfference between asset-weghted and equally-weghted returns. We also nvestgate the dfference n the survvorshp bas usng equally and asset weghted returns. Logarthmc data s used for the return analyss. In a second step we use an asset class factor model to explan excess returns. 2 We therefore defne eleven asset class factors. The eleven asset class factors are the MSCI World, 2 Schneewes and Kazem [2003] nvestgate three dfferent methods to explan excess returns: (a) sngle-factor approach usng a small captalzaton equty ndex (b) a mult-factor lnear uncondtonal model and (c) a SDF/GMM approach. The authors fnd that n most cases the alphas are rather smlar regardless of the emprcal methodology appled.

8 the NASDAQ Composte Index, the Russell 2000 Index, the Wlshre Mcro Cap Index, the Lehman Aggregate Bond Index, the Lehman Hgh Yeld Credt Bond Index, the JP Morgan Government Bond Index, the Goldman Sachs Commodty Index, the IPE Brent Crude Ol Index, the London Gold Bullon USD Index, the 90-day T-Bll rate and the Chcago Board Optons Exchange SPX Volatlty Index. Our factor model s smlar to the Sharpe s [1992] style regresson wth the dfference that we dstract the rsk-free rate and therefore use the excess return as the dependent varable. The equaton r t r f = Alpha+ n k=1 β x + ε k kt t (1) wth k factors and the factor loadngs β k s used. We use the least square method n our regressons. We assume a factor structure for returns accordng to the arbtrage prcng theory. In order to facltate the nterpretatons of the results we reduce the number of factors by usng two dfferent approaches. In our frst approach we dvde the eleven factors n four asset classes and test for the optmal factor of each asset class. In our second approach we use a stepwse regresson. 3 In both factor reducton approaches we derve to the same factor model used for further analyss. In a thrd step we break the sample accordng to the fund szes nto 100 percentles. The average fund sze and the average returns are calculated for each percentle. Monthly data s used to conduct the analyss. The natural logarthm of the average asset szes and the natural logarthm of the quadratc average asset szes are then regressed on the average annualzed returns for the 100 percentles. We therefore specfy lnear regressons of the form r = α + β ) + 1 log( Assets ε (2) and quadratc regresson of the form r = α + β log( + ε (3) 2 1 Assets ) + β 2 log( Assets ) Smlarly we regress the fund szes on annualzed standard devatons and on annualzed rsk-adjusted returns usng the followng lnear and quadratc specfcatons. The standard devatons and Sharpe Ratos are referrng to percentles and not to ndvdual funds. Each percentle can be consdered as a portfolo of funds wth smlar fund szes. 3 Stepwse regresson has been used by other researchers ncludng Lang [1999] and Agarwal [2001]. Please note that stepwse regresson procedure can be used wth dfferent selecton crtera. We conscously do not use maxmzng the n-sample R-Squared as our selecton crtera.

9 = α + β1 log( Assets ) ε (4) σ + σ = α + β log( + ε (5) 2 1 Assets ) + β 2 log( Assets ) SR SR = α + β ) + 1 log( Assets ε (6) = α + β log( + ε (7) 2 1 Assets ) + β 2 log( Assets ) For the calculaton of the Sharpe ratos we use 90-days T-Bll rates as rsk free rate. We also calculate the alphas for each ndvdual percentle and therefore apply the model descrbed n equaton (1) 100 tmes to derve equaton (8). r t r ft = Alpha + n k=1 β x + ε k kt t (8) We then test the relatonshp between the alphas derved from the 100 factor models and the average fund szes for the 100 percentles wth the followng lnear and quadratc regressons. Alpha = α + β ) + 1 log( Assets ε (9) Alpha = α + β log( + ε (10) 2 1 Assets ) + β 2 log( Assets ) HEDGE FUND RETURNS In ths secton we conduct our emprcal analyss wth the data sample descrbed before. Equally versus asset weghted returns Dfferent from prevous studes, we focus on the average returns acheved by hedge fund nvestors n contrast to the returns acheved by the average hedge fund. The dfference les n the measurement method of hedge fund returns. To our knowledge most exstng studes about the performance of hedge funds are usng equally weghted returns n order to estmate returns of the unobservable hedge fund unverse. We use asset weghted returns of a large sample of hedge funds for performance measurement purposes. One reason why most studes have focused on equally weghted hedge fund returns s the poor data qualty of hedge funds assets under management. Snce the qualty of avalable data mproved sgnfcantly n the last years, t s now feasble to calculate asset weghted returns. A number of ndex provders developed dfferent methodologes to benchmark hedge fund returns. Most hedge fund ndces

10 are equally weghted wth the excepton of the CSFB-Tremont Hedge Fund Indces 4 and some of the recently launched MSCI Hedge Fund Indces. We calculate both, equally weghted and asset weghted returns for our data sample based on the TASS databases for varous tme perods from January 1994 to June The results can be found n Exhbt 3. Our results are not subject to the survvorshp bas, snce both dead and lvng funds are ncluded n the analyss. The nstant hstory bas and the selecton bas are hard to avod and are therefore affectng the results. We calculate the survvorshp bas of the TASS hedge fund database n the followng subsecton. The annualzed return dfferences between equally and asset weghted returns s 0.95% over the 9.5 years tme horzon. Interestngly the dfference s postve n the tme perod from January 1994 to June 1998, but negatve from July 1998 to June The dfferences are not statstcally sgnfcant at the 5% sgnfcance level. Exhbt 4 shows a comparson between the rollng 12-months equally weghted and the rollng 12-months asset weghted returns. We can observe temporary dfferences between equally and asset weghted returns n the late 90s. Ths phenomenon can be explaned by the fact that very few large hedge funds were domnatng the returns of the hedge fund ndustry at that tme drawng the asset weghted returns up n 1997 and 1998, but causng an underperformance n Dfferences n the survvorshp bas We also nvestgate whether smaller funds have a larger survvorshp bas than larger funds. The results are llustrated n Exhbt 5. The annualzed dfferences between the survvorshp bas calculated wth equally weghted returns and the survvorshp bas calculated wth asset weghted returns s sgnfcant at the 5% sgnfcance level for the 9.5- year perod as well as for the frst sub perod from January 1994 to June We can therefore state that small frms face an ncreased rsk of extng the database. If we are wllng to beleve that asset weghted returns are more representatve for the hedge fund ndustry then we need to be careful n nterpretng the results of many exstng studes based on equally weghted returns. Our fndngs based on asset weghted returns are useful from a macro perspectve and relevant to measure the survvorshp bas of the hedge 4 The CSFB-Tremont Hedge Fund Indces contan approxmately 400 Hedge Funds and represents 160 bllon USD n assets. The ndex clearly focuses on large Hedge Funds. Hedge Funds requre a mnmum of 10 mllon USD assets under management, a mnmum track record of one year and audted fnancal statements n order to become part of the ndex.

11 fund ndustry n general. If we want to nterpret the results from the perspectve of the ndvdual hedge fund nvestor who s equally dstrbutng hs nvestments to a number of hedge funds regardless of the hedge fund sze, then the survvorshp bas derved from equally weghted returns s the more accurate measure. Exhbt 6 llustrates the development of the survvorshp bas n equally and asset weghted returns over tme. Usng 12-months rollng return data we can see that the survvorshp bas wth asset weghted returns s less stable. Partcularly n the perod around 1998 the rollng survvorshp bas wth asset weghted returns s devatng sgnfcantly from the long term average. We are aware that funds may voluntarly stop reportng because they do not want to publsh bad performance and harm ther reputaton. We cannot assume that funds droppng from the database are really ceasng ther operaton. TASS classfes to some extent extng funds n seven categores. A summary can be found n Exhbt 7. 51% of the extng funds have been lqudated and 3.5% of the funds merged wth other funds. Other reasons for extng are more dffcult to nterpret. Only sx funds ndcated that they closed for new nvestments, but the number mght be much hgher, snce more than 45% of the funds refused to state any reason for extng or were unreachable. Factor Model We want to explan excess returns of hedge funds wth an asset class factor model n order to derve hedge fund alphas. Exhbt 8 represents the results of a multple regresson of the eleven asset class factors on the returns of our sample wth 2317 funds. Almost 54% of the excess returns can be explaned wth the eleven-factor asset-class model. The Goldman Sachs Commodty Index, the Lehman Aggregate Bond Index and the Wlshre Mcro Cap Index are the only three factors that are statstcally sgnfcant at the 5% sgnfcance level. It s nterestng to see that other equty ndces ncludng large cap and md cap stocks are less useful to explan excess returns of hedge funds. Ths s an ndcaton that hedge funds tend to focus on small cap stocks that have typcally less research coverage from nvestment banks. We therefore reduce the number of factor n order to better nterpret the result. In our frst factor reducton approach we start wth a stepwse regresson. The hghest adjusted R-squared can be found n a factor model wth seven factors represented n Exhbt 9. The Akake nfo crteron s mnmzed n a model wth four factors ncludng the same factors as n Exhbt 9 wth the excepton of the MSCI World, the JPM Government Bond Index and crude ol. We

12 contnue elmnatng factors untl all remanng factors are sgnfcant at the 5% sgnfcance level. We then fnd three factors that are sgnfcant. The Wlshre Mcro Cap ndex and the Lehman Aggregate Bond ndex are sgnfcant at the 1% sgnfcance level. The Goldman Sachs Commodty ndex s sgnfcant at the 5% sgnfcance level. The results are presented n Exhbt 10. The constant, ndcatng the alpha of hedge funds, s postve, but not statstcally sgnfcant. The annualzed alpha of the model s 0.81%. The asset class factor model wth three factors s explanng more than 50% of the returns. The adjusted R-squared of the three factor model s above the adjusted R-squared of the factor model wth eleven factors. As ndcated n the eleven factor model, commodtes, bonds and small-cap stocks are the most common rsk-exposures of hedge funds. In our second approach we start wth sngle factor models for each asset class factor llustrated n Exhbt 11. We then take for each of the four asset classes the factor wth the hghest explanatory power and combne them n a factor model wth four asset class factors. We then test for each asset class other potental factors wth the objectve to fnd a model wth a hgher adjusted R-Squared. An overvew s gven n Exhbt 12. We fnally conclude that volatlty as an asset class represented by the Chcago Board Optons Exchange SPX Volatlty Index s not sgnfcantly ncreasng the adjusted R-Squared of the model and s therefore excluded. We are therefore left wth the same three asset class factors as n the frst factor reducton approach representng commodtes, equtes and currences. We further explore the relevance of the three factors chosen n our three factor asset class model. We break the sample data nto 100 percentles accordng to ther fund szes and conduct a prncpal component analyss on the 100 tme seres representng the average returns of each percentle. The result of the prncpal component analyss s presented n Exhbt 13. The frst component explans 40.84% of the varance and the frst ten components together explan 61.53% of the varance. The cumulatve varance explaned of the frst three components s 47.60% and therefore lower than the varance explaned by the asset class factor model wth three factors shown n Exhbt 10. The scree plot of the prncpal component analyss presented n Exhbt 14 llustrates the domnance of the frst component versus the other components based on the Egenvalues. Cross-sectonal regressons

13 In ths sub-secton we use cross-sectonal regressons to dentfy the mpact of fund szes on excess, returns, standard devatons, Sharpe Ratos and alphas. We use the breakdown of the sample data nto 100 percentles based on ther fund szes. Each percentle can be regarded as a sub-sample. The consttuton of the sub-samples changes n each month as the assets under management are changng and funds wth ncreasng assets relatve to ts peer group fall nto a hgher percentle. In our frst regresson we regress the logarthm of the average fund szes of the 100 sub-samples on the average excess returns of the sub-samples. Exhbt 15 shows the decles of fund szes and the annualzed average returns of the hedge funds n each decle. We can see that very small funds of the bottom decle are underperformng. Ths phenomenon can be explaned by an economy of scale effect. The operatonal expenses play a sgnfcant role for smaller funds and make t uneconomc to run a fund wth a very small asset base. Funds from the 21 st to the 50 th percentle have the best performance. At ths stage hedge funds are stll relatvely small wth an asset base of less than 20 mllon USD. Many nsttutonal nvestors are focussng on hedge funds wth a larger asset base and are therefore elmnatng funds wth the hghest return potental based on the ndcaton of asset szes. An nsttutonal nvestor who s lookng for funds wth a mnmum of 50 mllon USD under management wll only focus on funds n the top three decles. We know that many of the largest funds are closed for nvestment and many nvestors are therefore left wth a relatvely small unverse of a few hundred hedge funds. Funds from the 51 st to the 100 th percentle tend to have smlar average returns. Ths result suggests that t would not make a large dfference of pckng a hedge fund wth 30 mllon USD assets or a mult-bllon USD hedge fund. We further nvestgate the relatonshp between fund szes and returns. Therefore we apply a smple regresson analyss of the logarthm of the fund szes on the average returns of the asset percentles. The results of the lnear regresson analyss are presented n Exhbt 16 and Exhbt 17. Each data pont n Exhbt 16 represents an average annualzed return for one asset percentle n our data sample. The lnear relatonshp between returns and fund szes s statstcally sgnfcant at the 1% sgnfcance level. Investgatng the scatter plot n Exhbt 16 we conjecture a non-lnear relatonshp between fund-szes and returns. Applyng a quadratc regresson analyss we can fnd a concave relatonshp that bascally confrms the fndng of Getmansky [2004]. The curve n Exhbt 16 represents the results of the quadratc regresson. The quadratc term n the regresson analyss s sgnfcant at the 1% sgnfcance level.

14 We can see that partcularly very small funds wth less than USD n assets under management are very often underperformng. The underperformance can be explaned by hgher relatve operatonal costs. Mnmum fxed costs for the management of the hedge fund vehcle, fund admnstraton and custody are the major operatonal costs that are dmnshng net returns for the nvestor. It s dffcult to derve the mpact of fund szes on gross returns, snce the total expense rato of ndvdual hedge funds s generally not reported. We also explore the mpact of fund szes on volatltes n a lnear and quadratc regresson analyss. The volatltes refer to percentles and therefore portfolos of hedge funds, rather than ndvdual hedge funds. The volatlty of portfolos of hedge funds s generally lower than the volatlty of ndvdual hedge funds due to dversfcaton effects. The lnear regresson ndcates a sgnfcant relatonshp at the 5% sgnfcance level. We also test the relatonshp for convexty and fnd that the quadratc term n the quadratc regresson s not statstcally sgnfcant at the 5% sgnfcance level. The results can be found n Exhbt 18 and Exhbt 19. The relatonshp between fund szes and standard devatons s ntutve snce large funds generally beneft from a broader dversfcaton and therefore a reducton of volatltes. Larger funds very often attracted assets based on a proven track record and mght therefore shft ther focus on captal preservaton. Hghly aggressve strateges wth concentrated bets are sometmes more dffcult to mplement wth a large asset base due to capacty constrants. Larger funds are very often n a poston to better control the asset flows and therefore beneft from a more stable ncome. To control the asset flows larger funds can more easly afford less favourable lqudty condtons for nvestors and keep nvestors n the fund by applyng lockup perods, maxmum redempton gates or redempton fees for early wthdrawal of nvestments. A stable asset base allows for better plannng of nvestments. The manager can therefore more consstently apply hs strategy or also nvest n llqud securtes that are not prced on a daly bass and therefore dmnsh the volatlty of the fund. Next, we explore the relatonshp between asset szes and Sharpe Ratos wth a lnear and a quadratc regresson approach. Smlar to the standard devatons, the Sharpe Ratos are referrng to asset percentles that are representng portfolos of hedge funds wth smlar sze. Due to the dversfcaton effect of portfolos the Sharpe Ratos of ndvdual hedge funds are typcally lower than the Sharpe Ratos of portfolos. The fndngs of our regresson analyss are llustrated n Exhbt 20 and Exhbt 21. In the lnear regresson the relatonshp between fund szes and Sharpe Ratos s sgnfcant at the 1% sgnfcance level. In the quadratc

15 regresson the quadratc term s sgnfcant at the 5% sgnfcance level. Usng the results of the quadratc regresson t s therefore generally possble to fnd an optmal asset sze wth the hghest Sharpe Rato, although the quadratc relatonshp s not necessarly obvous from the scatter plot. Addtonally, as can be seen from Exhbt 20, nonlnearty tends to be drven by very small funds. Thus, t cannot be concluded that very large hedge funds have lower Sharpe Ratos than medum-szed funds. In the next step we are explorng the relatonshp between fund szes and the alphas derved from our asset class factor model wth three factors. We are usng the three factors that have the hghest explanatory power gven the eleven ntal factors. We remember that the alphas n our 3-factor model account for rsk exposures to commodtes, small-cap stocks and bonds. Exhbt 22 and Exhbt 23 llustrate the results of our regresson analyss. The coeffcent of our lnear regresson s sgnfcant at the 1% sgnfcance level and ndcates hgher alphas for larger hedge funds. The coeffcents of the quadratc regresson are sgnfcant and therefore ndcate a concave relatonshp between fund szes and alphas. We nvestgate f the Wlshre Mcro Cap Index as a factor n the 3-factor model s beneftng small hedge funds. Based on the conjecture that smaller funds may have a dfferent nvestment focus and more exposure to a small-cap ndex we replace the Wlshre Mcro Cap Index n our 3-factor model wth the MSCI World and then apply the same cross-sectonal regresson between asset percentles and the new alphas. We fnd that the lnear relatonshp between fund szes and the alphas from the adjusted factor model s even stronger. The results are dsplayed n Exhbt 24. We therefore have to reject ths conjecture and confrm the robustness of the results based on our orgnal 3-factor model. We also test whether large hedge funds have more nterest rate exposure snce fxed ncome related hedge funds typcally have a larger asset base due to the nature of ther strategy. We therefore drop the Lehman Aggregate Bond Index as an explanatory factor n our 3-factor model and derve the alphas from an adjusted 2-factor model based on the Goldman Sachs Commodtes Index and the Wlshre Mcro Cap Index. Our results llustrated n Exhbt 25 ndcate a weaker relatonshp between fund szes and alphas that s stll sgnfcant at the 5% sgnfcance level. It remans dffcult to estmate whether large funds are takng more exposure to rsk factors that are not captured by the smple asset class factor model compared to small funds. In any case the hgher relatve expense rato of smaller funds s decreasng the alphas based on net returns. For very small funds the alpha s even negatve.

16 Based on the parameters found n our quadratc regresson analyss we can derve optmal asset szes. However, we generally conclude that the convex and concave relatonshps are weak and the usefulness of the quadratc regressons s therefore lmted. CONCLUSION Ths artcle contrbutes to exstng lterature about hedge funds performance wth a detaled analyss of the mpact of fund szes on returns, Sharpe Ratos and alphas derved from a mult asset class factor model. Based on a large sample of hedge fund returns we reveal emprcal evdence for a postve relatonshp between fund szes and returns usng cross sectonal regresson technques. We fnd that partcularly very small funds are underperformng on average. We conjecture that the underperformance s based on hgher total expense ratos of small funds. On the other sde the observed relatonshp between standard devatons and fund assets s negatve. Generally larger funds tend to have lower volatltes, but hgher Sharpe Ratos. Very small funds have a clear dsadvantage to compete wth medum- and larger-szed funds. We show that the average alphas generated by hedge funds derved from a smple asset class factor model are not statstcally sgnfcant n the long term. It s nterestng to see that the postve relatonshp between fund szes and returns also holds after adjustng the returns for the rsk free rate and factor exposures to commodtes, small-cap stocks and bonds. Smlar than our analyss based on annualzed returns and Sharpe Ratos, our analyss based on alphas reveals that very small funds are underperformng on average. Hedge fund managers are prmarly remunerated wth a performance fee. In absolute terms the performance fee can be ncreased by hgher returns, but also by a larger asset base. A hedge fund manager who s maxmzng hs ncome may therefore be wllng to grow a fund above ts optmal sze from a pure performance perspectve. In the long-term a good performance s nstrumental n attractng assets and also enhances the reputaton of the manager. Therefore the manager faces a trade-off between optmzng the performance of the fund and optmzng hs revenues. Nevertheless, n ths study, the emprcal evdence for managers ncreasng ther fund sze beyond the optmal pont s very weak. The number of large funds exceedng 100 mllon USD assets under management s very small compared to the total number of hedge funds. Dfferent hedge fund strateges have dfferent capacty lmts. The strategy-specfc characterstcs of the asset-return relatonshp open opportuntes for further research projects.

17 For our percentles based approach, however, the number of hedge funds avalable for each hedge fund strategy s not suffcent to break the strategy-specfc samples further down nto 100 sub-samples over a 9.5 year perod. The results of strategy-specfc analyss are therefore lmted by the data avalable.

18 REFERENCES Agarwal, V., and N. Nak, Performance Evaluaton of Hedge Funds wth opton-based and Buy-and-Hold Strateges, Workng Paper, London Busness School, Agarwal, V., and N. Nak, Characterzng Hedge Fund Strateges wth Buy-and-Hold and Opton-Based Strateges, Workng Paper, London Busness School, 2001 Agarwal, N., and N. Nak, Rsks and Portfolo Decsons nvolvng Hedge Funds, Workng Paper, London Busness School, Amenc, M., Curts, S., and L. Martelln, The Alpha and Omega of Hedge Fund Performance, Workng Paper, Edhec/USC, Brealey, R. A., and E. Kaplans, Changes n Factor Exposures of Hedge Funds, Workng Paper, Clark, A., Does Fund Sze Affect Performance?, Lpper Research Study, Fama, E. F., and K. R. French, Common Rsk Factors n the Returns on Stocks and Bonds, Journal of Fnancal Economcs (1993), pp3-56. Fama, E. F., and K. R. French, Mult-factor Explanatons of Asset Prcng Anomales, Journal of Fnance (1996), pp Fung, W., and D. Hseh, Emprcal Characterstcs of Dynamc Tradng Strateges: The Case of Hedge Funds, The Revew of Fnancal Studes (1997), pp Getmansky, M., The Lfe Cycle of Hedge Funds: Fund Flows, Sze and Performance, Workng Paper MIT, Goetzmann, W., Ingersoll, J., and S. Ross, Hgh Water Marks and Hedge Fund Management Contracts, Journal of Fnance (2003), pp Gregorou, G. N., and F. Rouah, Large versus Small Hedge Funds: Does Sze Affect Performance?, Journal of Alternatve Investments (2003), pp Hedges, R. J., Sze vs. performance n the hedge fund ndustry, Journal of fnancal transformaton, The capco nsttute, Herzberg M. M., and H. A. Mozes, The Persstence of Hedge Fund Rsk: Evdence and Implcatons for Investors, Journal of Alternatve Investments (2003), pp Edwards, F. R., and M. Caglayan, Hedge Fund Performance and Manager Skll, Journal of Futures Markets (2001), pp Kazem, H., and Schneewes, Condtonal Performance of Hedge Funds, Isenberg School of Management, Unversty of Massachusetts, Lang, B., On the Performance of Hedge Funds, Fnancal Analysts Journal (1999), pp72-85.

19 Lang, B., Hedge Funds: The Lvng and the Dead, Journal of Fnancal and Quanttatve Analyss (2000), pp Lang, B., The Accuracy of Hedge Fund Returns, The Journal of Portfolo Management (2003), pp Malkel, B., Returns from Investng n Equty Mutual Funds 1971 to 1991, Journal of Fnance (1995), pp Sharpe, W. F., Asset Allocaton: Management Style and Performance Measurement, The Journal of Portfolo Management (1992), pp7-19. Schneewes, T., Kazem H., and G. Martn, Understandng Hedge Fund Performance: Research Issues Revsted Part II, The Journal of Alternatve Investments (2003), pp8-30.

20 EXHIBIT 1 Data qualty n the TASS database Mssng Returns n Funds n % More than 1 datapont % More than 5% % More than 10% % More than 20% % More than 30% % More than 40% % Mssng Assets n Funds n% More than 1 mssng datapont % More than 10% % More than 30% % More than 50% % A large number of funds have mssng data ponts n ther development of assets under management. The data qualty s hgher for return data. The data source s the TASS database.

21 EXHIBIT 2 Data qualty of our data sample wth 2317 funds Mssng Returns n Funds n % Funds wth at least 1 mssng return % More than 1 mssng datapont % More than 1% % More than 3% % More than 5% % More than 10% % Mssng Assets n Funds n % At least 1 datapont % More than 1 mssng datapont % More than 5% % More than 10% % More than 20% % Ths exhbt descrbes the qualty of the data set used n the emprcal part of our study. The data source s the TASS database hedge funds are used for the emprcal analyss.

22 EXHIBIT 3 Return comparson based on a sample wth 2317 hedge funds Jan 94 - Jun 03 Jan 94 - Jun 98 Jul 98 - Jun 03 Equally weghted returns p.a. 9.88% 11.32% 8.59% Standard Devaton p.a. 5.78% 5.81% 5.78% Sharpe Rato Asset weghted returns p.a % 14.08% 7.92% Standard Devaton p.a. 5.73% 6.24% 5.13% Sharpe Rato Annualzed dfferences n returns 0.95% 2.76% -0.67% Standard Devaton p.a. 3.24% 3.44% 3.00% t-statstc Equally weghted and asset weghted returns are calculated for a 9.5 years tme horzon and for two sub-perods from January 1994 to June 1998 and from July 1998 to June The sgnfcance of the return dfferences s tested wth a t-statstc.

23 EXHIBIT 4 Equally versus asset weghted logarthmc returns 30% 25% 20% 15% 10% 5% 0% -5% Dec-94 Jun-95 Dec-95 Jun-96 Dec-96 Jun-97 Dec-97 Jun-98 Dec-98 Jun-99 Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02 Dec-02 Jun-03 Rollng 12 months equally weghted returns Rollng 12 months asset weghted returns

24 EXHIBIT 5 Survvorshp Bas based on a sample wth 2317 hedge funds Jan 94 - Jun 03 Jan 94 - Jun 98 Jul 98 - Jun 03 Surv. Bas wth equally weghted returns 2.44% 2.72% 2.20% Surv. Bas wth asset weghted returns 0.85% -1.33% 2.82% Dfferences n Survvorshp Bas p.a. 1.59% 4.05% -0.62% Standard Devaton p.a. 2.35% 2.58% 1.94% t-statstc The survvorshp bases of equally weghted and asset weghted returns are calculated for a 9.5 years tme horzon and for two sub-perods from January 1994 to June 1998 and from July 1998 to June The sgnfcance of the dfferences n the survvorshp bases are tested wth a t-statstc.

25 EXHIBIT 6 12-months rollng survvorshp bases 15% 10% 5% 0% -5% Dez 94 Dez 95 Dez 96 Dez 97 Dez 98 Dez 99 Dez 00 Dez 01 Dez 02-10% -15% Rollng equally weghted survvorshp bas Rollng asset weghted survvorshp bas

26 EXHIBIT 7 TASS Graveyard database - reasons for extng Funds lqudated 800 Funds no longer reportng to TASS 438 TASS has been unable to contact the manger for updated nformaton 146 Funds closed to new nvestment 6 Funds merged nto another entty 56 Funds dormant 2 Unknown 134 All Funds 1582 The hedge funds n the TASS Graveyard database are classfed accordng to ther reasons for extng. The exhbt llustrates the classfcaton takng data untl June 2003 nto account.

27 EXHIBIT 8 Asset class factor model wth eleven factors Varable Coeffcent Std. Error t-statstc Prob. ALPHA VIX GOLD GSCI JPM GOV BOND LEHM BOND LEHMAN HY MSCI WORLD NASDAQ CRUDE OIL RUSSEL WILSHIREMICRO R-squared Adjusted R-squared A mult asset class factor model wth eleven factors s used to explan excess returns of hedge funds. Standard errors and t-statstcs as well as p-probabltes are calculated for each factor.

28 EXHIBIT 9 Asset class factor model wth seven factors Varable Coeffcent Std. Error t-statstc Prob. ALPHA GSCI JPM GOV BOND LEHM BOND MSCI WORLD NASDAQ CRUDE OIL WILSHIRE MICRO R-squared Adjusted R-squared A mult asset class factor model wth seven factors s used to explan excess returns of hedge funds. Standard errors and t-statstcs as well as p-probabltes are calculated for each factor.

29 EXHIBIT 10 Asset class factor model wth three factors Varable Coeffcent Std. Error t-statstc Prob. ALPHA GSCI LEHM BOND WILSHIRE MICRO R-squared Adjusted R-squared A mult asset class factor model wth three factors s used to explan excess hedge fund returns. Standard errors and t-statstcs as well as p-probabltes are calculated for each factor.

30 EXHIBIT 11 Sngle Factor Models for eleven asset class factors ASSET CLASSES FACTORS MONTHLY ALPHA ADJUSTED R- SQUARED EQUITIES WILSHIRE MICRO 0.26% MSCI WORLD 0.41% NASDAQ 0.35% RUSSELL % BONDS LEHMAN HY 0.29% LEHM BOND 0.30% JPM GOV BOND 0.44% COMMODITIES GOLD 0.47% GSCI 0.41% OIL 0.44% VOLATILITY VIX 0.54% A sngle asset class factor model for all eleven asset class factors s used to explan excess returns of hedge funds. Monthly alphas and adjusted R-Squares are calculated for each model.

31 EXHIBIT 12 R-Squares of factor models based on four asset classes EQUITIES BONDS COMMODITIES VOLATILITY ADJUSTED R-SQARED WILSHIRE MICRO LEHMAN HY GOLD VIX MSCI WORLD LEHMAN HY GOLD VIX NASDAQ LEHMAN HY GOLD VIX RUSSELL 2000 LEHMAN HY GOLD VIX WILSHIRE MICRO LEHM BOND GOLD VIX WILSHIRE MICRO JPM GOV BOND GOLD VIX WILSHIRE MICRO LEHM BOND GSCI VIX WILSHIRE MICRO LEHM BOND OIL VIX WILSHIRE MICRO LEHM BOND GSCI A mult asset class factor model wth four factors, one of each asset class s used to explan excess hedge fund returns. The ntal asset class factors are derved from the hghest adjusted R-Squares wthn each asset class from the sngle factor models. For each asset class all asset class factors are tested gven the asset class factors of the other asset classes. Adjusted R-Squared are calculated for each model.

32 EXHIBIT 13 Prncpal Component Analyss Component % of Varance Explaned Cumulatve % of Varance Explaned The funds are ranked accordng to ther fund szes and 100 asset percentles are bult n each month. The prncpal component analyss s based on the returns of the 100 percentles. The frst component explans 40.84% of the varance and the frst ten components explan 61.53% of the varance.

33 EXHIBIT 14 Scree Plot of Prncpal Component Analyss Egenvalue Component Number The scree plot of the prncpal component analyss llustrates the varance explaned by each of the frst ten components.

34 EXHIBIT 15 Hedge fund assets and returns Percentle Average Assets Average Returns 91st - 100th 535'395' % 81st - 90th 124'524' % 71st - 80th 65'117' % 61st - 70th 37'614' % 51st - 60th 22'934' % 41st - 50th 14'230' % 31st - 40th 8'567' % 21st - 30th 4'832' % 11th - 20th 2'348' % 1st - 10th 697' % Our sample of 2317 hedge funds s classfed n percentles accordng to ther fund szes. The second column llustrates the average fund szes of each decle. The thrd column shows the average returns for each decle.

35 Exhbt 16 Log of asset szes versus annualzed returns Annualzed Returns , ,000 1,000,000 10,000, ,000,000 1,000,000,000 10,000,000,000 Assets

36 EXHIBIT 17 Regresson results of fund szes versus annualzed returns Dependent varable Annualzed Returns Lnear regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) R-squared Adjusted R-squared Quadratc regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) LOG(ASSETS)^ R-squared Adjusted R-squared The funds are ranked accordng to ther fund szes and 100 asset percentles are bult n each month. In the lnear regresson the logarthm of the average assets of each of the 100 percentles are regressed on the average annualzed returns. In the quadratc regresson the logarthm of the assets and squared logarthm of the assets are regressed on the average annualzed returns.

37 EXHIBIT 18 Log of asset szes versus annualzed standard devatons 0.14 Annualzed Standard Devatons , ,000 1,000,000 10,000, ,000,000 1,000,000,000 10,000,000,000 Assets

38 EXHIBIT 19 Regresson results of fund szes versus standard devatons Dependent varable Annualzed Standard Devatons Lnear regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) R-squared Adjusted R-squared Quadratc regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) LOG(ASSETS)^ R-squared Adjusted R-squared The funds are ranked accordng to ther fund szes and 100 asset percentles are bult n each month. In the lnear regresson the logarthm of the average assets of each of the 100 percentles are regressed on the standard devatons of the funds. In the quadratc regresson the logarthm of the assets and squared logarthm of the assets are regressed on the standard devatons.

39 EXHIBIT 20 Log of asset szes versus annualzed Sharpe Ratos Annualzed Sharpe Ratos , ,000 1,000,000 10,000, ,000,000 1,000,000,000 10,000,000, Assets

40 EXHIBIT 21 Regresson results of fund szes versus Sharpe Ratos Dependent varable Annualzed Sharpe Ratos Lnear regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) R-squared Adjusted R-squared Quadratc regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) LOG(ASSETS)^ R-squared Adjusted R-squared The funds are ranked accordng to ther fund szes and 100 asset percentles are bult n each month. In the lnear regresson the logarthm of the average assets of each of the 100 percentles are regressed on the Sharpe Ratos. In the quadratc regresson the logarthm of the assets and squared logarthm of the assets are regressed on the Sharpe Ratos.

41 EXHIBIT 22 Log of asset szes versus annualzed alphas Annualzed Alphas , ,000 1,000,000 10,000, ,000,000 1,000,000,000 10,000,000, Assets The alphas are derved from excess returns and a mult asset class factor model wth three factors. The factors are the Goldman Sachs Commodty Index, the Wlshre Mcrocap Index and the Lehman Aggregate Bond Index.

42 EXHIBIT 23 Regresson results of fund szes versus annualzed alphas Dependent varable Annualzed Alphas (3-factor model) Lnear regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) R-squared Adjusted R-squared Quadratc regresson Independent varables Coeffcent Std. Error t-statstc Prob. C LOG(ASSETS) LOG(ASSETS)^ R-squared Adjusted R-squared The alphas are derved from excess returns and a mult asset class factor model wth three factors. The factors are the Goldman Sachs Commodty Index, the Wlshre Mcrocap Index and the Lehman Aggregate Bond Index. For the regresson analyss the funds are ranked accordng to ther fund szes and 100 asset percentles are bult n each month. In the lnear regresson the logarthm of the average assets of each of the 100 percentles are regressed on the alphas derved from the 3-factor model. In the quadratc regresson the logarthm of the assets and squared logarthm of the assets are regressed on the alphas derved from the 3-factor model.