MUTUAL FUND PERFORMANCE: SKILL OR LUCK?

Transcription

1 Frst draft 11/11/04. Ths verson : 4/7/05 MUTUAL FUND PERFORMANCE: SKILL OR LUCK? Abstract: Keth Cuthbertson*, Drk Ntzsche* and Nall O Sullvan** Usng a comprehensve data set on (survvng and non-survvng) UK equty mutual funds (Aprl 1975 December 2002), we use a bootstrap methodology to dstngush between skll and luck for ndvdual funds. Ths methodology allows for non-normalty n the dosyncratc rsks of the funds a major ssue when consderng those funds whch appear to be ether very good or very bad performers, snce these are the funds whch nvestors are prmarly nterested n dentfyng. Our study ponts to the exstence of genune stock pckng ablty among a relatvely small number of top performng UK equty mutual funds (.e. performance whch s not solely due to good luck). At the negatve end of the performance scale, our analyss strongly rejects the hypothess that most poor performng funds are merely unlucky. Most of these funds demonstrate bad skll. Recursve estmaton and Kalman smoothed coeffcents ndcate temporal stablty n the performance alpha s of wnner and loser portfolos. Keywords : Mutual fund performance, Bootstrappng, Fama-French model JEL Classfcaton: C15, G11 * Cass Busness School, Cty Unversty, London ** Department of Economcs, Unversty College Cork, Ireland Correspondng Author : Professor Keth Cuthbertson Cass Busness School, Cty Unversty London 106 Bunhll Row, London, EC1Y 8TZ. Tel. : +44-(0) Fax : +44-(0) E-mal : k.cuthbertson@cty.ac.uk 1. Nall O Sullvan s grateful for fnancal assstance from the Arts Faculty Research Fund at Unversty College Cork. We gratefully acknowledge the provson of mutual fund return data by Fenchurch Corporate Servces usng Standard & Poor's Analytcal Software and Data. Man programmes use GAUSS. 2. We thank, Don Bredn, Stuart Hyde, Davd Mles, Turalay Kenc, Ales Cerny, Aneel Keswan, Ian Marsh, Gulnar Muradoglu, Andrew Sykes (FSA), Dylan Thomas, Govann Urga and partcpants at Smrft Busness School, Dubln, Unversty of Bordeaux IV, Unversty of Bar, MMF Conference London 2004, Internatonal Fnance Conference- Unversty College Cork 2004, Global Fnance Conference-Trnty College, Dubln

2 MUTUAL FUND PERFORMANCE: SKILL OR LUCK? Two key ssues on fund performance have been central to recent academc and polcy debates. The frst s whether average rsk adjusted abnormal fund performance (after expenses are taken nto account) s postve, negatve or zero. On balance, US studes of mutual (and penson) funds suggest lttle or no superor performance but somewhat stronger evdence of underperformance (e.g. Lakonshok et al 1992, Grnblatt, Ttman and Wermers 1995, Danel et al 1997, Carhart 1997, Chevaler and Ellson 1999, Wermers 2000, Baks et al 2001, Pastor and Stambaugh 2002). Results usng UK data on mutual and penson funds gve smlar results (e.g. Blake and Tmmermann 1998, Blake, Lehmann and Tmmerman 1999, Thomas and Tonks 2001). A second major ssue s whether abnormal performance can be dentfed ex-ante and for how long t perssts. Persstence s examned usng ether a contngency table approach or performance ranked portfolo strateges or by observng actual trades of mutual funds. Usng the frst two technques the evdence s rather mxed. For US funds t seems that selectng funds wth superor future performance s rather dffcult and probably mpossble, unless portfolo rebalancng s frequent (e.g. at least once per year) and the performance horzon s not longer than about one-year (e.g. Grnblatt and Ttman 1992, Hendrcks, Patel and Zechauser 1993, Brown and Goetzmann 1995, Carhart 1997, Wermers 2003, Blake and Morey 2000, Bollen and Busse 2005, Mamaysky, Spegel and Zhang 2004). A recent excepton s Teo and Woo (2001) who fnd persstence n style adjusted returns for up to sx years. Studes usng actual trades of mutual funds fnd that one-year persstence amongst wnner funds s due to stocks passvely carred over, rather than newly purchased stocks of wnner funds performng better than newly purchased stocks of loser funds (Chen et al 2000). Followng on from ths Wermers (2003) fnds that persstent large cash nflows to wnner funds are nvested wth a lag and the average dollar nvested n past wnner funds does not earn more than that nvested n past loser funds. Ths s consstent wth the hypothess of Berk and Green (2004) where excess fund returns are quckly bd away n a compettve market. For UK data on mutual and penson funds there s lttle evdence of persstence n superor performance but much stronger evdence that poor performers contnue to underperform (e.g. Blake and Tmmermann 1998, Allen and Tan 1999, Fletcher and Forbes 2002, Blake, Lehmann and Tmmermann 1999, Tonks 2004). Ths study examnes the performance of open-end mutual funds nvestng n UK equty (Unt Trusts and Open Ended Investment Companes OEICs) durng the perod Aprl 1975 to 1

3 December A data set of over 900 equty funds s examned. Ths represents almost the entre UK equty mutual fund ndustry at the end of the sample perod. In comparson wth the US mutual fund ndustry, there have been comparatvely few studes of the performance of UK mutual funds (unt trusts). Unlke many prevous studes the focus of ths paper s on ndvdual fund performance (partcularly n the tals of the performance dstrbuton) and n determnng the role of luck versus skll. In contrast to earler studes whch use conventonal statstcal measures, often appled to portfolos of funds, we use a cross-secton bootstrap procedure across all ndvdual funds. Ths enables our luck dstrbuton for any chosen fund (e.g. the best fund), to encapsulate possble outcomes of luck not just for our chosen fund but across all the funds n our data set. We are then able to separate skll from luck n the performance of ndvdual funds, even when the dstrbuton of dosyncratc rsk across many funds s hghly non-normal. Ths methodology has not been appled to UK data and was frst appled to US mutual funds by Kosowsk, Tmmermann, Whte and Wermers (2004). As noted above, the absolute performance of mutual (and penson) funds and the relatve performance of actve versus passve (ndex) funds are central to recent polcy debates, partcularly n Europe. Wth ncreasng longevty and gven projected state pensons, a savngs gap s predcted for many European countres n 20 years tme (Turner 2004, OECD 2003). Wll voluntary savng n mutual and penson funds over the next 20 years be suffcent to fll ths gap, so that those reachng retrement age have suffcent savngs to provde an adequate standard of lvng? A key element here s the attractveness of savngs products n general and also the choce between actvely managed and passve (or ndex/tracker) funds. In recent theoretcal and emprcal work, the allocaton across dfferent asset classes (manly bonds versus stocks, but n prncpal across all asset classes) has been examned n an ntertemporal framework. The rule of thumb that the percentage nvestment n rsky assets (stocks) should equal 100 mnus your age s not robust ether n the face of uncertan ncome (whch gves rse to hedgng demands Bode, Merton and Samuelson 1992, Campbell and Vcera 1999, Vcera 2001) or, when return predctablty s present (Brennan et al 1997, Campbell et al 2003) or, when there s uncertanty about parameters n the predcton equaton for returns (Barbers 2000, Xa 2001). In practce, the lack of a consensus model of asset allocaton at both the strategc and tactcal level s starkly llustrated by Boots (the UK chemst) swtchng all ts penson fund assets nto bonds n 2001 (for strategc not market tmng reasons), whle most UK penson funds contnue to hold around 70% of ther assets n stocks. In the US, partcpants n 401(K) retrement plans (Benartz and Thaler 2001), when faced wth the choce between several funds each of whch has alternatve proportons of stocks and bonds, tend to use a smple 2

4 1/n allocaton rule - so the actual allocaton to each asset class s not determned by any sophstcated optmzaton problem and s changed nfrequently. Such naïve asset allocaton decsons may carry over to nvestment n mutual funds (and even trustees decsons for penson fund asset allocatons), so that poor funds survve and exacerbate the savngs gap. The behavoral fnance lterature (see Barbers and Thaler 2003 for a survey) has provded theoretcal models and emprcal evdence whch suggests that actve stock pckng styles such as value-growth (LaPorta et al 1997, Chan and Lakonshok 2004) and momentum (Jegadeesh and Ttman 1993, 2001, Chan et al 1996, 2000, Hon and Tonks 2003), as well as market tmng strateges (Pesaran and Tmmermann 1994, 1995, 2000, Ang and Bekaert 2004) can earn abnormal returns after correctng for rsk and transactons costs. Large sectons of the mutual fund sector follow these actve strateges and more recently there s an ongong debate on whether mutual (and penson) funds should be allowed to nvest n hedge funds and prvate equty, whch also follow a wde varety of actve strateges. The queston s therefore whether one can fnd actvely managed funds whch outperform ndex funds (after correctng for rsk and transactons costs). The Presdental Commsson on Socal Securty Reform (2001) and the State of the Unon Address (2005) envsage the part-prvatzaton of US Socal Securty. Ths wll ncrease debate on all aspects of the fund management ndustry, partcularly n the lght of the market tmng abuses uncovered n the US by New York Attorney General Ellot Sptzer (Goetzmann, Ivkovc and Rouwenhorst 2001) - whch has reduced confdence n the fnancal servce sector s ablty to provde adequate and far treatment of retal nvestors. In the UK, the contnung swtch from defned beneft to defned contrbuton penson schemes wll strengthen the argument for a closer analyss of actve versus passve strateges (as well as the competence and ndependence of trustee governance arrangements-myners 2001). The Fnancal Servces Authorty (FSA) n the UK s concerned that (retal) nvestors may be msled by mutual fund advertsng. In ts comparatve tables t currently does not enter a fund s rankng vs-a-vs compettor funds, n terms of (raw) returns. The FSA beleves ths could encourage more nvestment n funds whch may smply have hgh returns because they are more rsky (Blake and Tmmermann 1998 and 2003 and Charles Rver Assocates 2002). To the extent that any savngs gap s to be flled by nvestment n mutual funds, the need to evaluate rsk adjusted performance n a tractable and ntutve way, whle takng account of the nherent uncertanty n performance measures, wll be of ncreasng mportance. Ths paper drectly addresses the ssue of skll versus luck. We use alpha α and the t-statstc of alpha 3

5 t α, as our measures of rsk adjusted performance of mutual funds. However, we do not assume, as many earler studes do, that a fund s dosyncratc rsk has a known parametrc dstrbuton. Instead we bootstrap the emprcal dstrbuton of dosyncratc rsk not just fund-by-fund, but across the whole cross-secton of funds. Ths allows us to obtan a performance dstrbuton for funds whch are n the tals of the cross-secton dstrbuton precsely the funds that nvestors are lkely to be most nterested n (.e. extreme wnners or losers ). In fact, we manly use t α rather than alpha α as our performance statstc snce t has superor statstcal propertes and helps mtgate survval bas problems (Brown, Goetzmann, Ibbotson and Ross 1992). We also perform a number of bootstrap technques to account for any seral correlaton or heteroscedastcty n the dosyncratc rsk of each fund and possble contemporaneous cross-secton correlaton. The bootstrap procedure s robust to possble msspecfcaton but reported results are of course dependent on the chosen performance model. We therefore examne a wde range of alternatve models whch we dvde nto three broad classes () uncondtonal models (Jensen 1968, Fama and French 1993, Carhart 1997) () condtonal-beta models, n whch factor loadngs are allowed to change wth condtonng publc nformaton (Ferson and Schadt 1996) and () condtonal alpha-beta models where condtonng nformaton also allows for tme varyng alphas (Chrstopherson, Ferson and Glassman 1998). We control for survvor bas by ncludng 236 nonsurvvng funds n the analyss. We now antcpate some of our key fndngs. Frst the good news. The bootstrap procedure ndcates there s strong evdence n support of genune stock pckng ablty on the part of a relatvely small number of top ranked UK equty mutual funds. For example (usng the Fama-French 3 factor uncondtonal model), of the top 20 ranked funds n the postve tal of the performance dstrbuton, 7 funds exhbt levels of performance whch cannot be attrbutable to luck at 5% sgnfcance level, whle 12 funds exhbt such performance at 10% sgnfcance level. As we move further towards the centre of the performance dstrbuton (e.g. below the 97% percentle) many funds have postve alphas but ths can be attrbuted to luck rather than skll. In the left tal of the performance dstrbuton, from the worst (ex-post) fund manager to the fund manager at the 40 th percentle, we fnd that an economcally sgnfcant negatve abnormal performance cannot be attrbuted to bad luck but s due to bad skll. Therefore there are a large number of poorly performng actve funds n the unverse of UK equty mutual funds. Ths s consstent wth fndngs from the behavoral fnance lterature where retal nvestors often use smple rules of thumb n asset allocaton and who face nerta, learnng and search costs when tryng to evaluate alternatves. 4

6 When examnng dfferent fund styles, we fnd genune outperformance among the top equty ncome funds but there s lttle evdence of skll for the top performers amongst the all company and small stock funds. For all companes and small stock funds the extreme left tal of the performance dstrbuton ndcates bad skll rather than bad luck but for ncome funds the converse apples the poor performance of ncome funds s due to bad luck rather than bad skll. We also fnd that the top ranked onshore funds have genune skll, whereas the postve alphas for the best offshore funds are due to luck. In the left tals of these dstrbutons, we fnd that extreme poor performers (negatve alphas), whether they are onshore or offshore, demonstrate bad skll rather than bad luck. Broadly speakng, the above results are robust across all three classes of model we nvestgate, across several varants of the bootstrap and do not appear to be subject to survvorshp bas. The strong message from these results s that there are a few top funds who have genune skll but the majorty have ether no skll and do well because of luck or, perform worse than bad luck and essentally waste nvestors tme and money. If you choose your actve funds by throwng darts at the Fnancal Tmes mutual fund pages, then you are hghly lkely to choose a fund whch has no skll - you would be better off choosng an ndex fund (especally after transactons costs). On the other hand, a careful analyss of rsk adjusted performance takng full account of luck across all funds, can dentfy wth reasonable probablty, those few funds wth genune skll. In the rest of the paper we proceed as follows. Secton 1 descrbes the data used n the study. In secton 2 we dscuss performance measurement models appled to mutual fund returns. Secton 3 detals the bootstrap methodology. In secton 4 we evaluate the performance measurement models and select a subset of best models to whch we apply the bootstrap procedure. Secton 5 examnes the results of the bootstrap analyss and secton 6 concludes. 1. Data Our mutual fund data set comprses 935 equty Unt Trusts and Open Ended Investment Companes (OEICs). These funds nvest prmarly n UK equty (.e. mnmum 80% must be n UK equtes) and represent almost the entre set of equty funds whch have exsted at any pont durng the sample perod under consderaton, Aprl 1975 December Unt trusts are open ended mutual funds, they can only be traded between the nvestor and the trust manager and there s no secondary market. They dffer from nvestment trusts whch are closed end funds. Mutual fund monthly returns data have been obtaned from Fenchurch Corporate Servces usng Standard & Poor's Analytcal Software and Data. By restrctng funds to those nvestng n UK 5

7 equty, more accurate benchmark factor portfolos may be used n estmatng rsk adjusted abnormal performance. In our database of 935 funds, we remove second unts. These arse because of mergers or splts and n the vast majorty of cases the mergers occur early and the splts occur late n the fund s lfe, and therefore these second unts report relatvely few ndependent returns. Furthermore, 93 of the funds n the database are market (FTSE 250) ndex/tracker funds and as we are nterested n stock selecton ablty, these are also excluded. Ths leaves 842 non-tracker ndependent (.e. non-second unt) funds for our analyss. The equty funds are categorzed by the nvestment objectves of the funds whch nclude: equty ncome (162 funds), all companes (.e. formerly general equty and equty growth, 553 funds) and smaller companes (127 funds). The data set ncludes both survvng funds (699) and nonsurvvng funds (236). Nonsurvvng funds may cease to exst because they were merged wth other funds or they may have been forced to close due to bad performance. Because of the latter scenaro, t s crtcal to nclude nonsurvvng funds n any performance analyss of the mutual fund ndustry, as falure to do so may bas performance fndngs upwards (Carhart et al 2002). In addton, funds are also categorzed by the locaton of operaton. Onshore funds (731) are managed n the UK whle offshore funds (204) are operated from Dubln, Luxembourg, Denmark, the Channel Islands or some other European locatons. All fund returns are measured gross of taxes on dvdends and captal gans and net of management fees. Because our focus s on the performance of the fund s managers rather than on returns to nvestors/customers, our returns data s calculated bd-prce to bd-prce (wth ncome renvested). The market factor used s the FT All Share Index of total returns (.e. ncludng renvested dvdends). Excess returns are calculated usng the one-month UK T-bll rate. The factor mmckng portfolo for the sze effect, SMB, s the dfference between the monthly returns on the Hoare Govett Small Companes (HGSC) Index and the returns on the FT 100 ndex 1. The value premum, HML, s the dfference between the monthly returns of the Morgan Stanley Captal Internatonal (MSCI) UK value ndex and the returns on the MSCI UK growth ndex 2. The factor 1 The HGSC ndex measures the performance of the lowest 10% of stocks by market captalzaton, of the man UK equty market. Both ndces are total return measures. 2 These ndces are constructed by Morgan Stanley who ranks all the stocks n ther UK natonal ndex by ther book-to-market rato. Startng wth the hghest book-to-market rato stocks, these are attrbuted to the value ndex untl 50% of the market captalzaton of the natonal ndex s reached. The remanng stocks are attrbuted to the growth ndex. The MSCI natonal ndces have a market coverage of 6

8 mmckng portfolo s momentum behavor, MOM, has been constructed usng the consttuents of the London Share Prce Data Base, (total return) ndex 3. Other varables used n condtonal and market tmng models nclude the one-month UK T-bll rate, the dvdend yeld on the FT-All Share ndex and the slope of the term structure (.e. the yeld on the UK 20 year glt mnus the yeld on the UK three-month T-bll). 2. Performance Models The alternatve models of performance we consder are well known factor models and therefore we only descrbe these brefly. Each model can be represented n ts uncondtonal, condtonal-beta and condtonal alpha-beta form. For all models the ntercept ( alpha ) α and n partcular the t-statstc of alpha t α, are our measures of rsk adjusted abnormal performance. Uncondtonal Models These have factor loadngs that are tme nvarant. The alpha model (Jensen 1968) s gven by the regresson: α of the CAPM or market (1) r, t = α + β rm, t + ε, t where r, t ( R, t R f, t ) ( R R ) r m, t m, t f, t =, R, t = return on fund- n perod t, = s the excess return on the market portfolo. R, = rsk free rate, f t Carhart s (1997) performance measure s the alpha estmate from a four-factor model: r = α + β r + β SMB + β HML + β MOM + ε (2) t, 1 mt, 2 t 3 t 4 t t, where SMB t, HML t and MOM t are factor mmckng portfolos for sze, book-to-market value and momentum effects, respectvely. On US data, Fama and French (1993) fnd that a threefactor model ncludng r m, t, SMB t and HML t factors, provdes sgnfcantly greater power than at least 60% (more recently ths has been ncreased to 85%). Total return ndces are used for the constructon of the HML varable. 3 For each month, the equally weghted average returns of stocks wth the hghest and lowest 30% returns, over the prevous sx months are calculated. The MOM varable s constructed by takng the dfference between these two varables. The unverse of stocks s the London Share Prce Data Base. 7

9 the CAPM. In addton, Carhart(1997) fnds that momentum s statstcally sgnfcant n explanng (decle) returns on US mutual funds although the latter varable s less prevalent n studes on UK data (e.g. Blake and Tmmermann 1998, Qugley and Snquefeld 2000, Tonks 2004). Condtonal-Beta Models Condtonal models (Ferson and Schadt 1996) allow for the possblty that a fund s factor betas depend on lagged publc nformaton varables. Ths may arse because of under and overprcng (Chan 1988 and Ball and Kothar 1989), or changng fnancal characterstcs of companes such as gearng, earnngs varablty and dvdend polcy (Foster 1986, Mandelker and Rhee 1984, Hochman 1983, Bldersee 1975). Also, an actve fund manager may alter portfolo weghts and consequently portfolo betas dependng on publc nformaton. Thus there may well be tme ' varaton n the portfolo betas dependng on the nformaton set Z t so that β ( ) t, = b0 + B2 zt, where z t s the vector of devatons of Z t from ts uncondtonal mean. For the CAPM ths gves: (3) r = α + b ( r ) + B( z * r ) + ε ' t, bt, + 1 t bt, + 1 t, + 1 where r bt, + 1 = the excess return on a benchmark portfolo (.e. market portfolo n ths case). The null hypothess of zero abnormal performance s H 0 : α = 0. Condtonal Alpha-Beta Models Chrstopherson, Ferson and Glassman (1998) assume that alpha (as well as the beta s) may depend lnearly on t z so that = α A '( z ) α and the performance model s:, t 0 + t (4) r = α + A( z ) + b ( r ) + B( z * r ) + ε ' ' t, t 0 bt, + 1 t bt, + 1 t, + 1 Here, α0 measures abnormal performance after controllng for () publcly avalable nformaton, z t and () adjustment of the factor loadngs based on publcly avalable nformaton. Followng earler studes (Ferson and Schadt 1996, Chrstopherson, Ferson and Glassman 1998) our Z t varables nclude permutatons of : the one-month T-Bll yeld, the dvdend yeld of the market factor and the term spread. 8

10 Market Tmng In addton to stock selecton sklls, models of portfolo performance also attempt to dentfy whether fund managers have the ablty to market-tme. Can fund managers successfully assess the future drecton of the market n aggregate and alter the market beta accordngly? (see Admat et al 1986). In the model of Treynor and Mazuy (1966) a successful market tmer adjusts the market factor loadng βt = θ + γm[ rm, t ] so that (1) may now be wrtten: (5) r = α + θ ( r ) + γ [ r ] + ε 2 t, mt, m mt, t, where γ m > 0 s the uncondtonal measure of market tmng ablty. Alternatvely, the Merton and Henrksson (1981) model of market tmng s: r = α + θ ( r ) + γ [ r ] + ε + (6) t, mt, m mt, t, max 0,r m, t and m where [ r ] + m, t = { } γ s the uncondtonal measure of market tmng ablty. These two models can be easly generalzed to a condtonal-beta model, where β also depends on the publc nformaton set, z t (Ferson and Schadt 1996). As a test of robustness, each of the above models s estmated for each mutual fund. Results are then averaged across funds n order to select a sngle best ft model from each of the three classes: uncondtonal, condtonal-beta and condtonal alpha-beta models. These three best models are used n the subsequent (computatonally ntensve) bootstrap analyss. 3. Bootstrap Methodology Prevous studes of UK unt trust performance all use conventonal statstcal measures, and generally fnd (usng a three or four factor model) that there s lttle or no postve abnormal performance by (portfolos of) best funds, whereas the worst funds have statstcally sgnfcant negatve rsk adjusted performance (see nter ala, Blake and Tmmermann 1998, Qugley and Snquefeld 2000, Fletcher and Forbes 2002). Among US mutual funds there s lttle evdence of postve abnormal performance but stronger evdence of poor performng funds - Carhart (1997), Chrstopherson et al (1998), Hendrcks et al (1993). It has been argued that abnormal performance may be due to a momentum effect n exstng stock holdngs rather than genune 9

11 stock pckng skll (Carhart 1997, Chen et al 2000), although the evdence s not entrely defntve (Chen et al 2000 and Wermers 2000). In ths paper we use a cross-secton bootstrap procedure and are able to separate skll from luck for ndvdual funds, even when dosyncratc rsks are hghly non-normal as s the case for funds n the extreme tals, n whch nvestors are partcularly nterested. We begn wth a largely ntutve exposton of our bootstrap analyss, usng alpha as our measure of rsk adjusted abnormal performance and the market model (CAPM) as the true model of expected fund returns. In a large unverse of funds (say n = 1,000) there wll always be some funds that perform well (badly), smply due to good (bad) luck. Assume that when all funds have no stock pckng ablty (.e. H 0 : α = 0 for = 1, 2,, n) each fund s true alpha s normally dstrbuted and each fund has a dfferent but known standard devaton σ. Suppose we are nterested n the performance of the best fund. If we replay hstory just for the best fund, where we mpose α = 0 (here = best fund) but luck s represented by the normal dstrbuton wth known standard devaton σ, we would sample a dfferent estmate of alpha. Of course there s a hgh probablty that we sample a value of alpha close to zero, but luck mples that we may sample a value for alpha whch s n the extreme tals of the dstrbuton. Smlarly, when we resample the alpha for all the other n-1 funds, wth all α = 0 (but wth dfferent σ ), t s qute concevable that the second or thrd etc. ranked fund n the ex-post data, now has the hghest alpha. Ths would hold a fortor f the dstrbutons of the second, or thrd, etc. ranked funds have relatvely large values of σ. From ths sngle replay of hstory, wth α = 0 across all funds, we have ( α, α,... α ) from whch we choose the largest value α (1). So, takng the luck (1) (1) (1) 1 2 n dstrbuton across all funds nto consderaton (wth dfferent σ s), we now have one value α (1) max for the best fund whch arses purely due to samplng varablty or luck. However, by ( k ) repeatng the above (B-tmes) and each tme choosng α (for k = 1, 2,, B trals) we can obtan the complete dstrbuton of α max under the null of no outperformance, whch we denote f ( α max ). max max 10

12 Note that the dstrbuton f ( α max ) uses the nformaton about luck represented by all the funds and not just the luck encountered by the best fund n the ex-post rankng. Ths s a key dfference between our study and many earler studes that have used ths type of methodology. It s mportant to measure the performance dstrbuton of the best fund not just by re-samplng from the dstrbuton of the best fund ex-post, snce ths s a sngle realzaton of luck for one partcular fund. Clearly, re-runnng hstory for just the ex-post best fund gnores the other possble dstrbutons of luck (here just the dfferent standard devatons) encountered by all other funds these other luck dstrbutons provde hghly valuable and relevant nformaton. Havng obtaned our luck dstrbuton, we now compare the best fund s actual ex-post performance gven by ts estmated exceeds the 5% rght tal cut off pont n performance of the best fund s attrbutable to luck. ˆα max aganst the luck dstrbuton for the best fund. If ˆα max f ( α ), we can reject the null hypothess that the max Above, we could have chosen any fund (e.g. the 2 nd best fund) on whch to base the luck dstrbuton. So, we can compare the actual ex-post rankng for any chosen fund aganst ts luck dstrbuton and separate luck from skll, for all ndvdual funds n our sample. The above demonstrates the man features and ntuton behnd our analyss of fund performance - whch s based on the theory of order statstcs. But a key dfference n our study (whch we hghlght below) s that under the null of no out-performance, we do not assume the dstrbuton of alpha for each fund s normal and each fund s alpha can n prncpal take on any dstrbuton. The dstrbuton for each fund s luck s represented by the emprcal dstrbuton observed n the hstorc data and ths dstrbuton can be dfferent for each fund. Hence the dstrbuton under the null f ( α ), encapsulates all of the dfferent ndvdual fund s luck max dstrbutons (and n a multvarate context ths cannot be derved analytcally from the theory of order statstcs). Investors are partcularly nterested n funds n the tals of the performance dstrbuton, such as the best fund, the second best fund, and so on. We fnd that the emprcal luck dstrbuton of alpha for these funds are hghly non-normal, thus nvaldatng the usual test statstcs. Ths motvates the use of the cross-secton bootstrap to ascertan whether the outstandng or abysmal performance of tal funds s due to ether, good or bad skll or good or bad luck, respectvely. 11

13 There are a number of possble explanatons as to why non-normal securty returns can reman at the portfolo (mutual fund) level. As noted by Kosowsk et al (2004), co-skewness of ndvdual consttuent non-normal securty returns may not be dversfed away n a fund 4. Also, funds may hold dervatves to hedge return outcomes and ths may result n a non-normal return dstrbuton. Basc Bootstrap Kosowsk et al (2004) provde a thorough analyss of the bootstrap methodology appled to mutual fund performance so we provde only a bref exposton of the basc procedure (see Polts and Romano 1994). Consder an estmated model of equlbrum returns of the form: (7) t = ˆ α + ˆ β X t + e t r, ', for = {1, 2,, n) funds, where T = number of observatons on fund-, matrx of rsk factors and r, t = ( Rt, R f, t), X t = e, t = resduals of fund-. For our basc bootstrap we use resdual-only resamplng, under the null of no outperformance. Ths nvolves the followng steps (Efron and Tbshran 1993). Frst, estmate the chosen model for each fund (separately) and save the vectors { ˆβ, } e, t from the resduals. Next, for each fund-, draw a random sample (wth replacement) of length T e, t. Whle retanng the orgnal chronologcal orderng of X t, use these resampled bootstrap resduals ~ to generate a smulated excess return seres ~ t for fund-, e, t r, under the null hypothess of no abnormal performance (.e. settng α = 0): (8) ~ t = 0 + ˆ β ' X + t e~ t r,, By constructon, the true abnormal performance, for fund- s zero. Ths s then repeated for all funds. Next, usng the smulated returns ~, the performance model s estmated and the r, t resultng estmate of alpha ~ (1) α for each fund s obtaned. The ~ (1) α estmates for each of the n- funds represent samplng varaton around a true value of zero (by constructon) and are entrely due to luck. The ~ (1) α { = 1, 2,, n} are then ordered from hghest to lowest. The above process s repeated B tmes for each of the n funds, where B (= 1,000) denotes the number of 4 The central lmt theorem mples that a large, well dversfed and equal weghted portfolo of non-normally dstrbuted securtes wll approxmate normalty. However, many funds do not have these characterstcs. 12

14 bootstrap smulatons. The bootstrap estmates of (n x B) as follows. ~ α may be gathered n a matrx of dmenson ~ α1 ~ α 2 M ~ α n (1) (1) (1) ~ α ~ α (2) 1 (2) 2 M ~ α (2) n L L O L ~ α ~ α ( B) 1 ( B) 2 M ~ α ( B) n The frst row of ths sorted bootstrap matrx now contans the hghest values of ~ α from the B bootstrap smulatons, under the null hypothess α = 0. Ths s the luck dstrbuton for the extreme top performer, f ( α% max ). The second row contans the second hghest values of α% etc. Therefore each row of the bootstrap matrx provdes a separate dstrbuton of performance f ( α% ), for each pont n the performance dstrbuton, from the extreme best performer to the extreme worst performer, all of whch are solely due to luck. We can now compare any ex-post ˆ α wth ts approprate luck dstrbuton Suppose we are nterested n whether the performance of the ex- post best fund ˆα max s due to skll or luck. If ˆα max s greater than the 5% upper tal cut off pont from f ( α% max ) then we reject the null that ts performance s due to luck (at 95% confdence). We nfer that the fund has genune skll. Ths can be repeated for any other pont n the performance dstrbuton, rght down to the ex-post worst performng fund n the data. However, notwthstandng the above exposton n terms of the luck dstrbuton for alpha, our bootstrap analyss manly focuses on the luck dstrbuton for the t-statstc of alpha because t has better statstcal propertes (.e. t s a pvotal statstc, see Kosowsk et al 2004 and Hall 1992, for further dscusson). The ntutve reason for ths s straghtforward. The dosyncratc rsk of funds wth few observatons may have hgh varance and wll n consequence tend to generate outler alphas. These funds may dsproportonately occupy the extreme tals of the bootstrapped alpha dstrbutons leadng to a very hgh varance n ther luck dstrbuton. However, t ~ α, scales alpha by ts estmated standard error and therefore s ndependent of the 2 nusance parameter σ ε and has superor statstcal propertes. The argument apples a fortor at the extremes of the performance dstrbuton whch are of partcular nterest. t ~ α 13

15 Throughout ths study both the ex-post actual and bootstrap t-statstcs are based on Newey-West heteroscedastcty and autocorrelaton adjusted standard errors. In our baselne bootstrap we set the mnmum number of observatons for the ncluson of any fund n the analyss at T, mn = 36 months to mnmze survvorshp bas. 4. Model Selecton In ths secton, the equlbrum models descrbed n secton 2 are examned. All tests are conducted at a 5% sgnfcance level unless stated otherwse and results presented relate to all UK equty mutual funds over the perod Aprl 1975 December 2002 and are based on funds wth T,mn = 36, to mnmze survvorshp bas. For each model, cross-sectonal (across funds) average statstcs are calculated. A sngle best model s chosen from each of the 3 model classes; () uncondtonal, () condtonal-beta and () condtonal alpha-beta, and these results are reported n table 1. (In all, we examned over 50 models wthn the three classes of model and these results are avalable on request). In none of our models are the Treynor-Mazuy (1966) and Merton-Henrksson (1981) market tmng varables sgnfcant. The key model selecton metrcs are the Schwartz Informaton Crteron (SIC) and the statstcal sgnfcance of the ndvdual parameters. [Table 1 here] In the best three models, the cross-sectonal average alpha takes on a small and statstcally nsgnfcant negatve value. The fndng of negatve abnormal performance (on average) s consstent wth Blake and Tmmermann (1998). They report results for equally weghted portfolos of UK mutual funds, whch are n lne wth our results n the bottom half of table 1, where we fnd that the alpha of an equally weghted portfolo of all funds have statstcally sgnfcant negatve alphas (for all three models). However, of key mportance for ths study (and for nvestors) s the relatvely large crosssectonal standard devatons of the alpha estmates whch s around 0.26% p.m. (3.1% p.a.), for the uncondtonal and condtonal-beta models and somewhat larger at 0.75%p.m. for the condtonal alpha-beta model. Ths mples that the extreme tals of the dstrbuton of abnormal performance may contan a substantal number of funds. Ths s mportant snce nvestors are more nterested n holdng funds n the rght tal of the performance dstrbuton and avodng those n the extreme left tal, than they are n the average fund s performance. 14

16 The market excess return, r m, t, and the SMB factor betas are consstently found to be statstcally sgnfcant across all three classes of model, whereas the HML factor beta s often not statstcally sgnfcant, even at a 10% sgnfcance level (as dscussed further at the end of the next secton). We fnd that the momentum factor ( MOM ) s generally not statstcally sgnfcant at the ndvdual UK fund level (e.g. Blake and Tmmermann 1998, Tonks 2004), n contrast to US studes (Carhart 1997). For the condtonal-beta model (2 nd column, table 1) only the dvdend yeld varable produces near statstcally sgnfcant results. In the condtonal alpha-beta model we fnd that none of the condtonal alphas has a t-statstc greater than 1.1 but some of the condtonal betas are borderng on statstcal sgnfcance and our best model s shown n column 3. The above results suggest that the uncondtonal Fama-French 3 factor model explans UK equty mutual fund returns data reasonably well. There s lttle addtonal explanatory power from the condtonal and market tmng varables (not reported). The latter fndng s consstent wth exstng studes of UK market tmng (Fletcher 1995, Leger 1997) whle Jang (2003) also fnds aganst superor market tmng usng nonparametrc tests on US equty mutual funds. Turnng now to dagnostcs (bottom half of table 1), the adjusted R 2 across all three models s around 0.8, whle around 64% of funds have non-normal errors (Bera-Jarque statstc), and around 40% of funds have seral correlaton (whch s of order one LM statstc). The Schwartz Informaton Crteron (SIC) s lowest for the uncondtonal model. The Fama-French 3 factor model was selected as the best model for all three categores: uncondtonal, condtonal beta and condtonal alpha-beta model. Fgures 1, 2 and 3 respectvely, show hstograms of the cross-secton dstrbuton of the actual alphas estmated from these three models, appled to all funds. There s a wde spread of alpha estmates across all three models wth a reasonable number of funds n each of the tals of the dstrbuton [Fgures 1, 2 and 3 here] 5. Emprcal Results: Bootstrap Analyss In ths secton we present the man fndngs from the applcaton of the baselne bootstrap procedure. As dscussed prevously, we mpose a mnmum requrement of 36 observatons for a fund to be ncluded n the analyss. Ths leaves a sample of 675 funds, of whch 189 are nonsurvvor funds (.e. have ceased to exst at some pont before the end of the sample perod), whle 486 are survvor funds. 15

17 In Table 1, we reported that around 64% of mutual funds reject normalty n ther regresson resduals. As ths fndng partly motvates the use of the nonparametrc bootstrap, we provde further nformaton on ths aspect. Fgure 4 shows the dstrbuton of the resduals for selected funds n the upper (.e. best and 90th percentle fund) and lower tal (.e. worst and 10 th percentle fund) of the (ex-post) performance dstrbuton. Resduals from funds n the extreme tals (e.g. best and worst funds) tend to exhbt hgher varance and a greater degree of nonnormalty than resduals from funds closer to the centre of the performance dstrbuton (e.g. 90 th and 10 th percentles). For funds n the upper tal, t s ths hgh varance coupled wth large postve resduals, that causes these best funds to populate the very top end of the bootstrap dstrbutons. In turn ths generates wde dsperson and non-normalty n the performance of the very top performng funds. Ths s evdent n fgure 5a whch shows the bootstrap hstograms of t ~ α at selected ponts of the performance dstrbuton. Fgure 5b presents an almost mrror mage for the lower end of the performance dstrbuton. Ths vvdly llustrates that although funds n the center of the performance dstrbuton may exhbt near normal dosyncratc rsks, those n each of the tals do not, and t s the latter n whch nvestors are partcularly nterested. [Fgure 4 here] [Fgures 5a and 5b here] Table 2 shows bootstrap results for the full set of mutual funds (.e. ncludng all nvestment objectves) for the uncondtonal (Panel A), condtonal-beta (Panel B) and condtonal alpha-beta (Panel C) models, all of whch use the Fama-French (FF) three-factor model. The frst row n each panel shows each fund s actual (ex-post) t-alpha, ranked from lowest to hghest (left to rght) and the second row shows ts assocated value of alpha. Row 3 ( p-tstat ) reports the bootstrap p-values of the ranked t-statstcs n row 1, based on the luck dstrbuton for t α% under the null of no outperformance. [Table 2 here] For example, usng the uncondtonal model the max fund (Table 2, Panel A) has an actual ex-post t ˆ α = and acheved an abnormal performance of ˆα max = 0.412% p.m. However, the bootstrap p-value of t-alpha for the max fund s (row 4). The latter ndcates that from among the 1,000 bootstrap smulatons across all funds, under the null hypothess of 16

18 zero abnormal performance, 43.7% of the bootstrap t-statstcs for the hghest ranked fund were greater than t ˆ α = Ths can be seen n the hstogram n top left of fgure 4b, where the vertcal lne shows the actual t ˆ α = 3.389, relatve to the luck dstrbuton. Thus usng a 5% upper tal cut off pont, we cannot reject the hypothess that the best fund s actual t ˆ α = may be explaned by luck alone. Thus whle the conventonal t ˆ α = of the best fund ndcates genune skll, the non-parametrc bootstrap ndcates good luck. Ths apparent contradcton s due to the hghly non-normal dstrbuton of dosyncratc rsk across our top performng funds n the rght tal of the performance dstrbuton. It demonstrates that standard test statstcs may gve msleadng nferences when we look at funds n the extreme tals as can be seen for example, for funds up to 7 max n table 2, panel A. Our complete set of bootstrap results show that of the top 20 ranked funds, 12 acheve genune outperformance at a 10% sgnfcance level whle 7 funds outperform at a 5% sgnfcance level. However, as one moves nto the centre of the performance dstrbuton (.e. at or below the top 3% of funds) there s no evdence of stock pckng ablty the bootstrap ndcates that any postve t ˆ α s are due to luck rather than skll (see table 2, panel A and fgure 4b). In the left tal of the dstrbuton, (.e. the left sde of Panel A, table 2), the lowest ranked fund has t ˆ α = wth a bootstrap p-value of Hence for the ex-post worst fund there s a near zero probablty that ths s due to bad luck rather than bad skll. Ths fund has produced truly nferor performance. Ths can be seen n the upper left panel of fgure 5b, where the vertcal lne ndcates an actual t ˆ α = 5.358, whch s to the left of the luck dstrbuton. It s clear from the left hand sde of Panel A, table 2 (and fgure 5b), that all funds n the left tal (up to the mn 40% pont) have genunely poor skll. An alternatve nterpretaton of the bootstrap results s to see how many funds one mght expect to acheve a gven level of alpha performance by random chance alone and compare ths wth the number of funds whch actually dd acheve ths level of alpha n the real world. For example, based on the uncondtonal (FF) model we would expect 10 funds to acheve αˆ 0.5% p.m. (6% p.a.) based on random chance alone, whereas 19 funds exhbt ths level of performance (or hgher). However, αˆ 0.1% p.m. (1.2% p.a.) s expected to be acheved by 173 funds solely based on chance, whle n fact only 142 funds are observed to have reached ths level of performance. Of course, ths nterpretaton s consstent wth the dscusson of p-values 17

19 above. There s greater evdence of genune outperformance just wthn the extreme rght tal, than nearer the centre of the performance dstrbuton. [Fgure 6 here] Fgure 6 renforces the above ponts by showng Kernel densty estmates of the dstrbutons of t αˆ n the real data and the bootstrap dstrbuton for t α% - under the null of zero outperformance (.e. the luck dstrbuton ). It shows that the left tal of the dstrbuton of actual t s usng the real data (dashed lne), les largely to the left of the bootstrap dstrbuton αˆ (contnuous lne). Such poor performng funds cannot attrbute ther performance to bad luck but have bad skll. In contrast, the extreme rght tal of the dstrbuton of t ˆ α for the real data les outsde the luck dstrbuton. Ths means there are some, but not many, genune outperformers. Panels B and C of Table 2 reports fndngs from the condtonal-beta and condtonal alpha-beta FF models. Inferences from the bootstrap (rows t-alpha, p-tstat ), for the left tal of the performance dstrbuton are very smlar to those for the uncondtonal FF model n Panel A - bad luck s agan not a defense for bad performance.. The results for the rght tal of the dstrbuton usng the two condtonal FF models (Panels B and C, Table 2) are broadly consstent wth those for the uncondtonal model (Panel A). Genune stock pckng ablty s found for around 7% of funds usng the condton-beta model and for about 4% of funds usng the condtonal alpha-beta model, but t s luck rather than skll whch accounts for the postve performance of many funds further towards the centre of the performance dstrbuton. In other results (avalable on request) we fnd that the removal of the HML t varable produces vrtually no changes from those reported n table 2, whle addton of the momentum varable produces slghtly more wnners (around 5%) than the uncondtonal 3 factor model (of table 1, Panel A). These models also support the vew that many poorly performng funds have bad skll rather than bad luck. Our results are qualtatvely consstent wth Kosowsk et al (2004) who fnd strong evdence of stock pckng ablty among top performng 5-10% of US funds (dependng on the model chosen) and genune poor performance for the funds n most of the left tal of the performance dstrbuton. 18

20 Above we appled the bootstrap across all funds usng each of our 3 best models. However, recall from Table 1 that the set of condtonng nformaton varables were shown to be only weakly statstcally determned (on average across funds) and these varables are also statstcally nsgnfcant for more than 90% of the funds. Therefore, there s lttle evdence that condtonal models offer addtonal explanatory power or are lkely canddates for the true equlbrum model of returns. We are nclned to place greater weght on results from the uncondtonal FF 3 factor model of panel A and our varants (descrbed below) use ths baselne model. Performance and Investment Styles Havng found some good skll and lots of bad skll when analyzng all UK mutual funds together, the queston now arses whether these skllful and not so skllful funds are equally dstrbuted across dfferent fund classfcatons or, whether they are concentrated n partcular nvestment styles. From the US mutual fund performance lterature, there s some evdence that funds wth a growth nvestment style tend to be among the top performng funds (see Chen, Jegadeesh and Wermers 2000). In our data set 675 funds have a mnmum of 36 monthly observatons of whch 143 (21%) are ncome funds, 423 (63%) are all companes funds and 109 (16%) are small stock funds. To further address the style queston we apply the bootstrap procedure separately for each style, snce the dstrbuton of dosyncratc rsk may dffer across dfferent styles. [Table 3 here] Table 3, Panels A, B and C re-estmate the performance statstcs of table 2, for the three nvestment styles. Lookng at the left sde of all three Panels n Table 3 (t-alpha, p-tstat ) t s clear that genune bad skll n the left tal s common across all companes and small stock funds, whereas poorly performng ncome funds experence bad luck rather than bad skll. Lookng at the rght sde of all three panels of Table 3, t s manly hgh rankng equty ncome funds (Panel A) whch acheve postve levels of performance, whch cannot be accounted for by luck. In partcular, we fnd that most equty ncome funds ranked from the 3rd hghest to the max 10% generally beat the bootstrap estmate of luck (at a 5% sgnfcance level), whle the performance of the two hghest ranked ncome funds could have been acheved by luck alone. In contrast to the above, for UK All Companes and small stocks (Table 3, Panels B and C, talphat, p-tstat ), there are hardly any funds whch have genune stock pckng sklls n the rght tal of the performance dstrbuton. Note that amongst these top performng funds, standard t- 19

21 tests would often gve dfferent nferences to the bootstrap (e.g. see the max, 2 max and 3 max funds for equty ncome and UK all companes ). The above results are consstent wth those n table 2, where of the 7 funds wth genune skll (at a 5% sgnfcance level), 6 can be dentfed as ncome funds and one as a small company fund, whereas at a 10% sgnfcance level we have 12 sklled funds of whch 6 are ncome funds, 5 are all companes and 1 s a small company fund. Hence, n table 2 ncome funds are proportonately more representatve of skll, than the other fund styles. Our fndngs for the UK of skll manly among some top performng UK ncome funds are n contrast to results n Kosowsk et al (2004) for US mutual funds, who fnd that t s the top performng growth funds that have genune skll. (But note that Kosowsk et al 2004 do not have a small companes style classfcaton). [Fgure 7 here] Fgure 7 shows Kernel densty estmates for the three nvestment styles. For equty ncome funds the extreme rght tal of the dstrbuton of actual t-statstcs les outsde that of the luck dstrbuton, ndcatng the presence of some funds wth good skll rather than good luck. But for the other two style categores, actual ex-post performance does not exceed random samplng varaton 5. We also see that the left tals of the actual ex-post t-statstcs for all companes and small companes, le largely to the left of the luck dstrbutons, ndcatng that poor performance s unlkely to be due to bad luck. Performance and Fund Locaton All mutual funds n ths study nvest only n UK equty but funds are operated from both onshore UK and offshore locatons such as Dubln, Luxembourg, Channel Islands and some other European locatons. Dfferental performance may arse due to possble nformaton asymmetres between offshore versus onshore operatons or smply dfferental skll gven dentcal nformaton. 5 It should be noted that Kernel densty plots need not necessarly lead to the same concluson as the bootstrap analyss. Ths s because the Kernels compare the frequency of a gven level of performance from among the actual funds, aganst the frequency of ths same level of performance n the entre bootstrap matrx. The bootstrap p-value s a more sophstcated measure and compares the actual performance measure tαˆ aganst the bootstrap dstrbuton of performance t ~ α, at the same pont n the cross-sectonal performance dstrbuton. 20