Return Persistence, Risk Dynamics and Momentum Exposures of Equity and Bond Mutual Funds

Transcription

1 Reurn Perssence, Rsk Dynamcs and Momenum Exposures of Equy and Bond Muual Funds Joop Hu, Marn Marens, and Therry Pos Ths Verson: Absrac To analyze perssence n muual fund performance, s common pracce o consruc porfolos of funds based on pas fund reurns. Usng a large sample of equy and bond funds, we show ha hs approach nroduces dynamc exposures o common sock and bond rsk facors. Correcng for rsk dynamcs subsanally reduces he level of perssence n rskadused performance and drves ou he explanaory power of sock and bond momenum facors. Keywords: momenum, performance, rsk dynamcs, muual funds Joop Hu s a RSM Erasmus Unversy, Marn Marens and Therry Pos are a he Erasmus School of Economcs. E-mal addresses are: hu@rsm.nl, mmarens@few.eur.nl and gpos@few.eur.nl. We hank Sandra Szer for excellen edoral asssance. Also, we graefully acknowledge research suppor by Tnbergen Insue and Erasmus Research Insue of Managemen. We welcome commens, ncludng references o papers we have nadverenly overlooked. The usual dsclamer apples. Elecronc copy avalable a: hp://ssrn.com/absrac=

2 Reurn Perssence, Rsk Dynamcs and Momenum Exposures of Equy and Bond Muual Funds Ths Verson: Absrac To analyze perssence n muual fund performance, s common pracce o consruc porfolos of funds based on pas fund reurns. Usng a large sample of equy and bond funds, we show ha hs approach nroduces dynamc exposures o common sock and bond rsk facors. Correcng for rsk dynamcs subsanally reduces he level of perssence n rskadused performance and drves ou he explanaory power of sock and bond momenum facors. Keywords: momenum, performance, rsk dynamcs, muual funds 1 Elecronc copy avalable a: hp://ssrn.com/absrac=

3 The perssence of nvesmen reurns s a key heme n asse managemen and emprcal asse prcng research. Many nvesors are guded by pas performance, as wnessed by he emphass hey place on pas reurns for selecng muual funds. A he same me, he ably o sysemacally prof from perssence challenges heores based on marke effcency and equlbrum, as he nflow and ouflow of smar money can be expeced o reduce he profably of smple nvesmen rules (see, e.g., Berk and Green, 2004). When analyzng nvesmen performance, nvesors need o correc raw reurns for he exposure o sysemac rsk facors. The mos popular rsk facor model for evaluang he performance of sock porfolos s he hree-facor model of Fama and French (1993, 1995, 1996), henceforh 3FM, whch ncludes marke, sze, and value facors. Compared o he classcal, sngle-facor marke model, he 3FM gves a superor descrpon of he reurns of ndvdual socks and sock porfolos. However, he 3FM does no explan why sock momenum sraeges appear o be hghly profable. Nor why equy muual funds appear o exhb srong performance perssence. Carhar (1997) argues ha sock momenum s an mporan facor n explanng perssence n muual fund performance. Hs four-facor model (henceforh, 4FM) adds a sock momenum facor o he 3FM, and has become he sandard for evaluang equy muual fund performance. For bond muual funds, several rsk facor models have been developed, dependng among oher hngs on he relevan marke segmen. Governmen and nvesmen-grade corporae bond funds appear o be affeced by erm and cred spread facors (see, e.g., Fama and French (1993), and Gebhard, Hvdkaer, and Swamnahan (2005)). For speculavegrade bond funds, oher rsk facors are also relevan (see, e.g., Blake, Elon, and Gruber (1993)). Correcng for rsk s no always smple, because he exposures o rsk facors can vary over me. Ths s especally rue for momenum sraeges, whch ypcally nvolve a hgh urnover. Several sudes fnd ha sock momenum sraeges exhb me-varyng rsk exposures (see, e.g., Chorda and Shvakumar, 2002, Grffn, J, and Marn, 2003, Wang, 2003, Chen and De Bond, 2004, and Cooper, Guerrez, and Hammeed, 2004). Mos noably, Grundy and Marn (2001) show ha sock momenum sraeges nroduce me-varyng exposure o he marke, sze and value facors. For example, n up (down) markes, hgh-bea (low-bea) socks wll be overrepresened among he wnner socks and underrepresened among loser socks and a momenum sraegy wll have a posve (negave) marke bea. The same reasonng apples for he sze and value facors, and any oher rsk facor. Momenum 2 Elecronc copy avalable a: hp://ssrn.com/absrac=

4 sraeges are posvely (negavely) exposed o rsk facors ha have performed well (poorly) n he recen pas. In a smlar way, he emprcal researcher or fund-of-funds manager nroduces rsk dynamcs by consrucng porfolos of funds based on he funds pas reurns. Wnner funds,.e., funds wh hgh pas reurns, wll have a hgh posve exposure o rsk facors ha have recenly performed well, and loser funds wll have a low or negave exposure o hose facors. In hs way, fund porfolos based on par reurns wll have dynamc rsk exposures even when he ndvdual funds have sable rsk profles. Dynamc rsk exposures can affec performance measuremen f he exposures are correlaed wh he rsk facor premums. In he conex of he sngle-facor marke model, Jaganahan and Wang (1996) show ha correlaon beween he marke bea and he marke rsk premum affecs he esmaed alphas. Our sudy examnes f perssence n muual fund performance s affeced by he abovemenoned rsk dynamcs. We use large daa ses of equy and bond muual funds from he 2006 CRSP Muual Fund Survvor Bas Free Daabase. To nvesgae he relaon beween perssence and rsk exposures, we develop condonal versons of popular sock and bond rsk facor models. In he spr of Grundy and Marn (2001), our facor exposures depend on he lagged values of he rsk facors. Ths approach dffers from he radonal approaches o modellng me-varyng rsk, whch ypcally use lagged macroeconomc varables as condonng nformaon. Nowhsandng he mers of hese approaches, condonng on lagged facor reurns seems a more approprae approach for capurng rsk dynamcs caused by sorng funds on pas reurns. We fnd ha he equy fund sample and bond fund sample boh show a comparable paern. Accounng for rsk dynamcs srongly ncreases he explanaory power for he meseres of reurns of loser and wnner fund porfolos. For example, he adused R-squared for he reurn dfferenal beween he wnner decle and he loser decle ncreases from 9% o 63% for equy funds, and from 45% o 77% for bond funds. The ndvdual funds generally have relavely sable rsk profles. Our resuls ndcae ha he rsk dynamcs of he fund porfolos are nroduced by sorng funds based on pas reurns. Furhermore, he exposures of wnners and losers are correlaed wh he facor premums, so rsk dynamcs affecs he esmaed alphas. The common sock and bond rsk facors exhb momenum, nroducng a posve (negave) correlaon beween he premums and he loadngs of wnner (loser) funds. Once rsk dynamcs are aken no accoun, he wnner-loser alpha spread falls from 11.91% o 6.35% per annum for equy 3

5 funds, and from 1.46% o 0.70% per annum for bond funds. Ths ndcaes ha a subsanal par of fund perssence s arbuable o me-varyng rsk exposures. Condonng also drves ou he explanaory power of sock and bond wnner-mnusloser (WML) momenum facors. The dynamc rsk exposures of wnner and loser funds are smlar o he dynamc rsk exposures of wnner and loser socks documened n Grundy and Marn (2001). Consequenly, WML facors can serve as proxes for dynamc rsk exposures. Ths approach however requres ha he rsk facors explan he bulk of he reurns of he WML facors. If he alphas and resdual varance of he momenum facors devae sgnfcanly from zero, as s rue n our samples, bas arses for he esmaon of he perssence levels. An alernave approach s o adus he sorng mehod by sorng funds based on pas alphas raher han pas raw reurns. Snce he alphas correc for facor loadngs and are reesmaed every me he porfolos are rebalanced, hs approach correcs for me varaon n he porfolos rsk exposures. Alpha-sorng s somemes used n sudes on muual fund performance evaluaon (see, e.g., Bollen and Busse, 2005). We use here as a conrol for rsk dynamcs. An advanage of hs approach s ha ha s does no requre he specfcaon of he dynamc paern and apples under general condons. Also, hs approach maxmzes he alpha spread and hence reduces he probably ha perssence s no deeced. A lmaon s ha he alphas are dffcul o esmae based on a shor me-seres, and ha esmaon error agan nroduces dynamc exposures and he need o condon on pas rsk facors. The paper s srucured as follows. Secon 1 descrbes our daa ses of equy and bond muual funds, common sock and bond rsk facors, and sock and bond WML facors. Secon 2 derves our condonal model and presens he emprcal resuls. Secon 3 shows ha momenum facors are poenal proxes for rsk dynamcs. Secon 4 shows he resuls of sorng funds based on pas alphas raher han pas reurns. Secon 5 concludes. 4

6 1. Daa Our sudy uses wo large daa ses of muual fund reurns, one of whch s a sample of equy funds and he second of whch s a sample of bond funds. We also use daa on several common sock and bond rsk facors and a sock WML facor, and consruc our own bond WML facor Sock muual funds and rsk facors We exrac monhly reurn daa on U.S. equy funds from he 2006 CRSP Muual Fund Survvorshp Bas Free daabase. CRSP covers all daa on all muual funds, boh exan and delsed, n he Uned Saes for any gven dae snce 1962, ncludng he daa for he monhly oal reurns of more han 21,400 open-ended muual funds and daa on approxmaely 7,000 delsed funds. The daabase also provdes mporan supplemenary daa, such as fund classfcaons by Wesenberger, Mcropal/Invesmen Company Daa, Inc., Sraegc Insgh, and S&P. We selec equy funds ha are classfed as small cap/growh, growh, growh and ncome, or secor fund, followng Carhar (1997). We drop funds wh less han 12 consecuve reurn observaons over he enre sample perod from our sample. The resulng sample covers 12,717 equy funds over January 1962 o December Our daa are free from survvorshp bas as documened by Brown (1992) and Brown and Goezmann (1995). To consruc decle porfolos, we follow Hendrcks, Pael, and Zechauser (1993) and assgn funds o equally weghed decle porfolos based on her cumulave reurn durng one-year rankng perods. We readus he porfolo weghs f a fund dsappears afer formaon. We evaluae porfolos based on her excess reurn durng he frs monh afer porfolo formaon. The frs rankng perod n our sample s from January 1962 o December 1962, and he frs nvesmen perod s January The second rankng perod moves one monh ahead. The las rankng perod s from December 2004 o November 2005, and he las nvesmen perod s December We evaluae equy funds relave o he Fama and French hree-facor model (3FM), whch ncludes he marke facor (RMRF), he small-mnus-bg (SMB) sze facor, and he hgh-mnus-low (HML) value facor of he 3FM. We oban reurn daa on he Fama and French facors from Kenneh French's (DATE) daa lbrary. As a proxy for he rsk-free rae, 5

7 we use he one-monh Treasury bll rae from Ibboson and Assocaes. We use he Carhar (1997) sock WML facor Bond muual funds and rsk facors We also oban monhly reurn daa on bond funds from CRSP. We selec all bond funds ha CRSP classfes as nvesmen-grade funds. We drop from our sample any funds wh less han 12 consecuve reurn observaons over he enre sample perod. The resulng sample covers 1,090 bond funds for he January 1987 o December 2003 perod. Agan, we consruc decle porfolos by usng he Hendrcks, Pael, and Zechauser (1993) approach. The frs rankng perod n our sample of bond funds s from January 1987 o December 1987, and he frs nvesmen perod s January We evaluae bond funds by usng he erm facor (TERM) and defaul facor (DEF) of he Gebhard, Hvdkaer, and Swamnahan (2005) wo-facor model (henceforh, 2FM). We fnd ha hs model s relevan for evaluang nvesmen-grade bonds; oher rsk facors apply mosly o hgh-yeld bond funds. Followng Carhar s (1997) mehod, we develop a bond WML facor for sudyng bond funds. We ake he equal-weghed average of bonds wh he hghes 30% 11-monh reurns lagged one monh mnus he equal-weghed average of bonds wh he lowes 30% 11-monh reurns lagged one monh. The porfolos nclude all bonds from he 2003 Lehman Brohers Fxed Income daabase (LBFI) and are re-formed monhly. LBFI ncludes all U.S. nvesmen-grade corporae bonds ha are ncluded n he Lehman US Cred Index, formerly known as he U.S. Corporae Invesmen Grade Index. To be ncluded n LBFI, corporae bonds need o mee lqudy, maury and qualy requremens. Each bond mus be raed nvesmen grade by a leas wo rang agences (Moody's, S&P or Fch), publcly ssued, dollar-denomnaed, coupon-bearng, nonconverble, and mus have a mnmum remanng maury of one year. The bond WML facor has an expeced reurn of -0.33% per annum and herefore, n conras o he sock WML facor, whch has an expeced reurn of 10.92% per annum, hs facor s unlkely o represen acual acve bond nvesmen sraeges. Sll, as wll be shown below, he bond WML facor does pck up dynamc rsk facor exposures. Table 1 lss some sascs for he sock and bond rsk facors of he 3FM and he 2FM. The able shows he uncondonal average reurns and he condonal average reurns 1 The auhors would lke o hank Mark Carhar for provdng he daa on he momenum facor used n Carhar (1997). 6

8 afer posve and negave one-year lagged reurns. All values are annualzed. The resuls ndcae ha he rsk facor premums end o ncrease (decrease) afer posve (negave) facor reurns. Togeher wh he dynamc rsk facor exposures documened below, hs dynamc paern nroduces correlaon beween exposures and premums. [Inser Table 1 abou here] 2. Condonal rsk facor models Ths secon descrbes our use of condonal rsk facor models and compares he esmaon resuls of uncondonal versus condonal models. 2.1 Model formulaon For smplcy, we here focus on ndvdual funds, alhough our emprcal analyss focuses mosly on porfolos of funds. For fund, = 1,, n, we represen he excess reurn r, a me as a funcon of rsk facors f,, = 1,, m, by he followng facor model: (1) r, = α + β,, f, + ε,. m = 1 In hs expresson, α s he fund s alpha, whch measures fund manager skll, β, = 1,,m, are dynamc loadngs, and ε, s a zero-mean resdual. Agan, our sudy uses he hree sock rsk facors, RMRF, SMB and HML, and wo bond rsk facors, TERM and DEF. We also consder models o whch we add a sock or bond WML facor. Uncondonal models assume consan loadngs, or,, β,,, β =, = 1,, m. If he loadngs are dynamc and correlaed wh he rsk premums, hen hese models wll lead o based esmaes of he alphas. More formally, akng he uncondonal expecaon of boh sdes of Equaon (1), and usng C β, f ) = E[ β f ] E[ β ] E[ f ], we oban: (,,, (2) E[ r, ] = { α + C( β,, f )} + E[ β, ] E[ f ]. m = 1 m = 1 7

9 Ths expresson bascally generalzes he resul for he sngle-facor marke model of Jaganahan and Wang (1999; Equaon (4)). The nercep { α + C( β, f )} ncludes he m bas erm C(β,, f ). Thus, f a fund s loadngs are posvely (negavely) correlaed o = 1 he rsk premums, s alpha wll be overesmaed (underesmaed). Ths resul shows he mporance of properly accounng for dynamc loadngs. + Arguably, he smples possble dynamc specfcaon s β β + β d ), m = 1,,, = (,,, + where d 1 f > 0 s a dummy varable ha akes a value of one when he facor reurn s, =, posve and a value of zero when he facor s negave. We nclude wo exposure coeffcens for every rsk facor: he basc loadng β, and he ncremenal loadng β,. When he reurn o facor s negave, he exposure s measured by β, ; when he facor reurn s posve, s measured by β + β ). Usng hs specfcaon, our facor model reduces o (,, + (3) r, = α + β, f, + β, d, f, + ε,. m = 1 m = 1 The model s n fac a smplfcaon of he Grundy and Marn (2001) model for equy momenum reurns. Ths model uses hree exposure coeffcens for every rsk facor for large negave, and medum and large posve values. Our model uses wo exposure coeffcens for every rsk facor, whch reduces he number of free parameers and smplfes he exposon. In addon, unrepored ess show ha he wo approaches gve smlar resuls and ha a hrd exposure coeffcen does no add sgnfcan explanaory power for wnner and loser funds. A smulaon sudy also shows ha our smple condonal model s surprsngly effecve n capurng he dynamcs. Frs, we esmae all equy funds alphas, facor exposures, and resduals usng he 3FM. Nex, we use hs nformaon o smulae fund reurns assumng zero alphas. Thus, we creae a suaon n whch here should be no perssence. We subec he smulaed reurns o he same porfolo formaon procedure as he emprcal reurns. Each monh, we assgn funds o equally weghed decle porfolos based on her cumulave reurns durng one-year rankng perods. The uncondonal 3FM yelds he wrong concluson of perssence, fndng a sgnfcan alpha of nearly 5% per annum for he 8

10 reurn dfferenal beween wnner and loser funds. By conras, our condonal model n equaon (3) yelds an nonsgnfcan alpha close o zero. Hence, he condonal model resuls correcly show ha here s no perssence n muual fund performance. The approach of condonng on lagged rsk facors dffers from he radonal approaches o modellng me-varaon, whch ypcally use lagged macroeconomc varables as condonng nformaon (see, e.g., Ferson and Schad, 1996, and Ferson and Harvey, 1999). Usng S condonng varables, z s,, ( β,, = ( β, + β, S s= 1 s) z s, s = 1,, S, hs seup amouns o assumng ). Ths approach s well sued for capurng he changng rsk profle of ndvdual secures and funds as he economy evolves hrough dfferen sages of he busness cycle. However, we beleve ha condonng on lagged facor reurns s a more approprae approach for me varaon caused by sorng funds on pas reurns. To demonsrae hs pon, we compare he resuls of our condonal model wh several wellesablshed condonal models Emprcal resuls Table II shows our esmaon resuls of he uncondonal rsk facor models. The resuls are comparable o he resuls n oher muual fund performance evaluaon sudes, for example, Hendrcks, Pael, and Zechauser (1993), and Carhar (1997), and serve prmarly o summarze hese known resuls and as a benchmark for he esmaon resuls for he condonal models. [Inser Table II abou here] Panel A shows he resuls for our sample of equy funds. The op decle D1 has an average reurn of 11.48% per annum and he boom decle D10 has a mean of -0.76%, a oal mean spread of 12.24%. The 3FM model acheves an adused R-squared of 77% for he opdecle D1, 64% for he boom-decle porfolo D10, and only 9% for he reurn dfferenal D1-D10. All decle porfolos have a smlar marke facor exposure, wh a RMRF loadng around 0.8. The dfferences n value facor exposure are also lmed: D1 has an HML loadng of and D10 has an HML loadng of zero. Only he sze facor exposure shows subsanal varaon: D1 has a SMB loadng of 0.55, and D10 has an SMB loadng of

11 No surprsngly, he uncondonal model s no very successful n explanng he large dfferences n average reurn beween he decles: D1 and D10 have alphas of 6.06% and -5.85%, respecvely, a oal alpha spread of 11.91%. These resuls clearly sugges ha equy funds dsplay srong performance perssence. Panel B shows he esmaon resuls for bond funds. Snce nvesmen-grade bond reurns are generally less volale han sock reurns, he level and dsperson of he reurns s more modes for he bond fund decles han for he equy fund decles, as wnessed by he average reurn spread of 2.95%. The 2FM acheves an adused R-squared of 90% for D1, 71% for D10, and 45% for D1-D10. The facor exposures show subsanal varaon: D1 has a TERM loadng of 0.91 and DEF loadng of 0.42, whle D10 has a TERM loadng of 0.56 and DEF loadng of The loadngs are relavely successful n explanng he crosssecon of average reurns, wh alphas of 0.39% for D1 and -1.07% for D10, a oal alpha spread of 1.46%. Sll, hs sample oo shows ha buyng wnner funds and sellng loser funds appears o be profable. [Inser Table III abou here] Table III presens he esmaon resuls of he condonal rsk facor models. Panel A shows he equy muual fund resuls. Condonng sgnfcanly ncreases he explanaory power for he me-seres of reurns. The adused R-squared ncreases from 77% o 89% for D1, from 64% o 80% for D10, and, mos sgnfcanly, from 9% o 63% for D1-D10. Clearly, he rsk facor loadngs of wnner funds ncrease afer posve facor reurns. For example, he RMRF loadng of D1 s 0.56 afer down markes and ncreases wh 0.48 o 1.04 afer up markes. We observe a smlar paern for he SMB and HML facors. Loser funds show exacly he oppose paern of decreased loadngs afer posve facor reurns. In conras, he exposures of he mddle decles D4-7 show lle me varaon. Ths paern helps o explan he bulk of he cross-seconal varaon n average reurns beween he decles. The op-oboom alpha spread falls from 11.91% for he uncondonal model o 6.35%, a 47% reducon. These resuls clearly sugges ha dynamc exposures play an mporan role n undersandng he reurns o wnner and loser equy funds. Panel B shows smlar resuls for bond funds. The adused R-squared ncreases from 90% o 94% for D1, and from 71% o 85% for D10. Agan, we observe a large ncrease n adused R-squared for D1-D10 from 45% o 77%. The TERM and DEF loadngs of wnner (loser) funds ncrease (decrease) subsanally afer posve facor reurns. Ths paern helps 10

12 o explan he bulk of he cross-seconal varaon n average reurns beween he decles. The op-boom alpha spread falls from 1.46% o 0.70%, a reducon of 52%. Snce boh samples show smlar paerns, we conclude ha condonng srongly ncreases he adused R-squared and subsanally reduces he cross-seconal alpha spread for loser and wnner funds. I appears ha we can arbue a subsanal par of fund perssence o me-varyng facor loadngs. So we ask wha causes he rsk dynamcs. One compellng answer s ha he rsk dynamcs are nroduced by usng he mehod of formng porfolos of funds based on her pas reurns, n he spr of he dynamc exposures of sock momenum porfolo shown by Grundy and Marn (2001). Wnner funds ha have recenly performed well wll have a hgh posve exposure o facors smply because a hgh exposure o hose facors places hem wh he wnners. Smlarly, loser funds wll have a low or negave exposure o successful facors. In hs way, wnner and loser porfolos wll have dynamc exposures even when he ndvdual funds have sable rsk profles. Ths nerpreaon s suppored by he abovemenoned observaon ha ha he mddle decle D4-7 do no dsplay srong rsk dynamcs. Addonal resuls provde furher suppor for our sorng-based explanaon. Frs, we esmae several well-known condonal models ha use macroeconomc varables. Followng Ferson and Schad (1996) and Ferson and Harvey (1999), we use he followng condonng varables for equy muual funds: he dfference beween he hreemonh and one-monh Treasury bll reurn; he dvdend yeld of all NYSE, Amex, and Nasdaq socks; he spread beween a en-year and a hree-monh Treasury bond yeld; he spread beween Moody s Baa and Aaa corporae bond yelds; and he one-monh Treasury bll yeld. For bond muual funds, we follow Ferson, Henry, and Ksgen (2006), and use he followng varables: he one-monh Treasury bll rae, he corporae defaul spread n he nvesmen-grade bond marke, and a measure of ndusral producon and capacy ulzaon from he Federal Reserve Board webse. The resulng condonal models are unsuccessful n capurng a meanngful par of he reurn dynamcs of he wnner and loser funds n our samples. Alhough he adused R-squareds for D1-D10 of 25% for equy muual funds and 59% for bond muual funds are hgher han for he uncondonal 3FM, he values are subsanally lower han for our condonal model. More o he pon, he op-boom alpha spread ncreases o 14.63% for equy muual funds and o 2.11% for bond muual funds. These resuls sugges ha he me-varyng exposures of wnner and loser funds s no relaed o known sources of me varaon. 11

13 Second, we esmae he condonal models for ndvdual funds raher han porfolos of funds based on pas reurns. Our resuls show ha our condonal models based on lagged rsk facors have lle explanaory power, bu he models based on lagged macro-economc varables sll have some explanaory power for ndvdual funds. These fndngs sugges ha he me varaon for decle porfolos s no caused by he rsk dynamcs of ndvdual funds, bu nsead reflecs he sorng mehod. 3. WML momenum facors The dynamc rsk facor exposures of wnner and loser funds resemble he dynamc exposures of wnner and loser socks ha Grundy and Marn repor (2001). The queson arses f wnner-mnus-loser (WML) momenum facors can serve as proxes for dynamc exposures. To answer hs queson, we frs nvesgae he condons under whch WML facors can serve as proxes for dynamc exposures and hen compare he emprcal resuls of WML-based models wh hose of our condonal models. Throughou he analyss, we assume ha our dynamc model (3) s he correc specfcaon. The model assumes a smple funconal form and does no nclude an explc momenum facor. The prevous secon shows srong emprcal suppor for hs model for wnner and loser funds. Neverheless, a smlar analyss would apply when anoher dynamc specfcaon were used or f an explc momenum facor were ncluded. 3.1 WML facors as proxes for me varaon We consder an uncondonal facor model wh an addonal WML facor: (5) m * * *, = α + β, f, + δ WML + ε, = 1 r, * where α, β, = 1,, m, δ, and *, and resdual, respecvely. * ε, denoe he alpha, facor loadngs, WML loadng, We llusrae he relaon beween hs WML-based model and our dynamc model (3) by frs applyng he dynamc model o he WML facor: 12

14 m + (6) WML = α WML + βwml, f, + βwml, d, f, + ε WML,, = 1 m = 1 where he WML alpha α WML measures he average WML reurn ha s no arbuable o facor exposure and ε WML, s a resdual WML facor. The WML facor wll have negave basc loadngs, or β 0, = 1,, m, and posve ncremenal loadngs, or β 0, WML, < WML, > = 1,,m, because wnner (loser) secures end o have hgh facor loadngs afer posve (negave) facor reurns. Grundy and Marn (2001) fnd hs resul for a sock momenum facor n he conex of he 3FM. The condonal model has (2m+1) free parameers. In conras, he WML-based model (5) has only (m+2) free parameers. I follows drecly ha he WML facor can capure he ncremenal loadngs only f he followng condon s sasfed: (7) β = δ WML, = 1,, n, = 1,, m., β, In oher words, we mus assume ha all facor loadngs change n he same proporon o he facor loadngs of he WML porfolo. If hs proporonaly condon does no hold, hen he WML facor generally canno capure he compose dynamc effec of he varous facor loadngs. For example, he WML-based model canno cope wh he suaon n whch a fund s ncremenal RMRF loadng s posve, bu s ncremenal HML loadng s negave. In general, he proporonaly condon s hghly resrcve. However, for fund porfolos formed on pas reurns, s less resrcve, because wnner funds end o have posve ncremenal loadngs for all rsk facors, us as does he WML facor, and loser funds end o have negave ncremenal loadngs for all rsk facors. Assumng ha he proporonaly condon (7) holds, we can rewre (6) as follows: m 1 + (8) WML = α WML + βwml, f, + δ β, d, f, + ε WML, m = 1 = 1 m = 1 m +, d, f, = δ ( WML αwml βwml, f, ε WML, = 1 β ) Subsung hs equaly n (3) and rearrangng erms yelds: 13

15 m (9) r, = ( α δ αwml ) + ( β, δ βwml, ) f, + δ WML + ( ε, δ ε WML, ) = 1 We can draw several conclusons from hs formulaon. Frs, ha he nercep * α = ( α δ αwml ) s generally a based esmaor of he fund s alpha α. If he WML facor has a non-zero alpha, hen he WML premum devaes from he rue rsk premum for condonal facor loadngs and he nercep s a based esmaor of he rue alphas. If momenum s profable ( α > 0 ), hen we wll have underesmaed he alpha of a wnner WML * * fund ( δ > 0 ) ( α < α ) and overesmaed he alpha of a loser fund ( δ < 0 ) ( α > α ). The oppose s rue when momenum s loss-makng ( α < 0 ). Consequenly, he WML-based model yelds unbased esmaes for he condonal alphas only f he WML alpha s small or he WML alpha s removed. * Second, ha he basc rsk facor loadngs β = β δ β ), = 1,, m, are WML, (, WML, generally based esmaors for he fund s basc loadngs β, = 1,, m. The bas occurs, because he WML facor no only pcks up ncremenal facor loadngs, bu also basc facor loadngs. The WML facor wll have negave basc loadngs, or β 0. Therefore, we WML, < * wll have overesmaed he loadngs of a wnner fund ( δ > 0 ), or β >, and, β, * underesmaed he loadngs of a loser fund ( δ < 0 ), or β <. For measurng, β, perssence, he precse values of he facor loadngs are no of drec neres, so we gnore hs ssue here. * Thrd, we conclude ha he compose error erm ε = ε δ ε ) ncludes he, (, WML, negave of he resdual WML facor ε WML, and herefore s negavely correlaed wh he WML facor. We know ha hs errors-n-varables problem nroduces bas for OLS esmaors. In our case, he OLS esmaor for he WML loadng δ s based o zero, wh he magnude of he bas dependng on he relave mporance of he resdual WML facor. Ths bas reduces he qualy of WML exposure as a proxy for ncremenal rsk facor exposures and moves he WML-based model owards he uncondonal facor model. Our reasonng assumes ha he condonal facor model s correc and ha he WML facor only serves as a poenal proxy for rsk dynamcs. If many funds sysemacally follow 14

16 momenum sraeges, hen may be desrable o nclude a momenum facor n he model o capure hese sraeges, and we mus hen ask f he classcal WML hedge porfolos are he mos approprae choce. As we noed above, he WML porfolos are heavly nfluenced by rsk dynamcs and herefore he WML loadngs are srongly confounded wh condonal rsk facor loadngs. We can crcumven hs problem by replacng he WML facor wh he resdual WML facor, or ε WML,. An alernave approach o correcng he WML facor for rsk facor exposures s o sor funds based on pas alphas raher han pas reurns. However, we fnd no sgnfcan mprovemen n he me-seres f from ncludng such resdual momenum facors n our samples Emprcal resuls Table IV shows he resuls of applyng he condonal and uncondonal facor models o he WML porfolos. Panel A shows he resuls for he sock WML facor and he 3FM. Usng he uncondonal 3FM, we see ha he sock WML facor s only weakly relaed o he RMRF, SMB and HML facors, as wnessed by he modes values for he facor loadngs and an adused R-squared of only 4%. However, he condonal model resuls show ha he sock WML facor exhbs a srong dynamc paern. Conssen wh he fndngs of Grundy and Marn (2001), we fnd ha he facor loadngs are sgnfcanly negave afer negave facor reurns and sgnfcanly posve afer posve facor reurns. For example, he RMRF loadng s afer down markes and ncreases by 0.65 o 0.21 afer up markes. Ths paern s smlar o he paern for he decle porfolos. The proporonaly condon (7) s approxmaely sasfed wh δ > 0 for he wnner decles D1-3, δ 0 for he mddle decles D4-7 and δ < 0 for he loser decles D8-10. Condonng ncreases he adused R- squared from 4% o 38% and reduces he WML alpha from 13.68% o 8.68%. Panel B shows he resuls for he bond WML facor and he 2FM. In comparson wh he sock WML facor, he me-seres f of he uncondonal model s beer and he momenum sraegy has a negave uncondonal alpha, ha s, he bond momenum sraegy nduces losses. Condonng has smlar effecs as for he sock WML facor. The condonal TERM loadng ncreases from afer negave TERM reurns and ncreases by 1.41 o 0.76 afer posve TERM reurns, and he DEF loadng ncreases from afer negave DEF reurns and ncreases by 1.36 o 0.71 afer posve DEF reurns. Ths dynamc paern ncreases he adused R-squared from 37% o 66%. Condonng n hs case lowers he WML alpha from -2.00% o -2.72%. 15

17 [Inser Table IV abou here] Due o he common paern of me-varaon, WML loadngs can serve as proxes for he dynamc rsk facor exposures of wnner and loser funds. However, because he alphas and he resdual varance devae sgnfcanly from zero, he proxes are based and nosy, whch may bas he esmaon of he alphas of he funds. To llusrae hese pons, Table V shows he esmaon resuls of he WML-based models. Table VI dsplays mpled rsk facor loadngs. We oban hese rsk facor loadngs by combnng he WML facor loadngs from Table V wh he rsk facor loadngs of he WML facors from Table IV. For example, n he equy muual fund sample, D1-D10 has a basc RMRF loadng of = and an ncremenal RMRF loadng of = Panel A of Table IV shows he resuls for he equy funds and he sock WML facor. The decle porfolos have srong sock WML exposures, rangng from 0.34 for D1 o for he D10, leadng o a loadng of 0.64 for D1-D10. The model yelds an adused R-squared of 86% for D1, 74% for D10, and 47% for D1-D10. Ths me-seres f s worse han ha of he condonal 3FM (see Table III), bu sll subsanally beer han he uncondonal 3FM (see Table II). Thus, seems ha WML exposure can ndeed serve as a proxy for dynamc rsk facor exposures. However, as we show above, he sock WML facor has a sgnfcan posve alpha, so ncludng wll narrow he op-o-boom alpha spread compared o he condonal model. Indeed, he alpha spread falls o 3.17%, whch s below he value of 6.35% for he condonal model. In addon, he WML facor s a nosy proxy for rsk dynamcs, as wnessed by he adused R-squared of 38% n Table IV; 62% of he varaon of he WML facor s no explaned by he rsk dynamcs. The nose bases he esmaes owards he uncondonal model and underesmaon of he (mpled) rsk facor loadngs. For example, he mpled RMRF loadng of and ncremenal RMRF loadng of 0.42 for D1-D10 are closer o zero han he values of and 1.07 n Table III. We see a smlar paern for he SMB and HML rsk facors. Underesmang he loadngs wdens he alpha spread, bu hs effec does no cancel he effec of he large posve sock WML alpha n hs sample. Agan, n Panel B we observe a smlar paern for he bond WML loadngs and meseres f of bond funds. However, n hs case we see a wdenng of he op-o-boom alpha spread from 0.70% for he condonal model o 2.52% for he WML-based model. Ths wdenng happens because he bond WML facor n conras o he sock WML facor has a 16

18 sgnfcan negave alpha and hus he alpha of wnner (loser) funds ncreases (decreases). Furhermore, he WML-based model agan underesmaes he (mpled) rsk facor loadngs, whch furher wdens he alpha spread. [Inser Table V abou here] [Inser Table VI abou here] These resuls confrm our rsk-based explanaon for he role of he WML facors. Several unrepored follow-up analyses furher suppor hs nerpreaon. Frs, Carhar (1997) shows ha ndvdual equy muual funds n conras o fund porfolos generally do no have conssen WML loadngs. We fnd he same resul n our samples of equy and bond muual funds. Furhermore, f we om he ndvdual funds wh sgnfcan WML loadngs from our samples and nsead form decle porfolos whou hese funds, hen we sll fnd smlar alpha spreads, dynamc rsk facor exposures, and WML exposures as for he full sample. Second, n he smulaon expermen we descrbe n Secon 2.1, each equy fund has a zero alpha and a zero loadng on he equy WML facor. Sll, formng decle porfolos based on pas fund reurns, he reurn dfferenal D1-D10 does load sgnfcanly on he WML facor. In hs case, we can arbue he explanaory power of he WML facor enrely o he facor s me-varyng loadngs on he 3FM facors. 4. Alpha-sored porfolos An alernave approach o correc for me-varaon s o adus he sorng mehod by sorng funds based on pas alphas raher han pas raw reurns. Snce he alphas correc for facor loadngs and are re-esmaed every me he porfolo s rebalanced, hs approach can correc for me varaon. Alpha sorng s somemes used n he fund performance sudes (see, for example, Bollen and Busse, 2005). Here, we use as a conrol for rsk dynamcs. An advanage of hs approach s ha does no requre ha we specfy he dynamc paern. Also, hs approach maxmzes he alpha spread and hence reduces he probably ha we wll no deec perssence. In conras, sorng on pas reurns leads o a lower alpha spread f pas reurns and pas alphas are mperfecly correlaed, as s rue n our sample of bond muual funds (for example, n Panel B of Table III, he low-reurn decle D9 has he hghes alpha). 17

19 Alpha sorng s unlkely o compleely sablze he rsk profles. I s noorously dffcul o oban accurae esmaes of alphas of ndvdual funds for a shor me perod, especally when here are mulple rsk facors a work. If he alphas are esmaed wh nose, hen sorng on esmaed alphas mgh sll nroduce some rsk dynamcs. Apar from funds wh a hgh (low) rue alpha, he op (boom) decle wll also nclude some funds wh a hgh exposure o rsk facors wh a posve (negave) pas reurn. Usng hgh-frequency daa or Bayesan esmaon and shrnkage mehods may help o mprove he esmaon. However, gven he lengh of he formaon perod, a hgh level of accuracy may no be feasble. When alpha sorng does no compleely sablze he rsk profles, a condonal rsk facor model may sll be requred. To mplemen he alpha-based approach, we form sock and bond decle porfolos based on pas alphas ha we esmae n he 12-monh formaon perod. Table VII shows he resuls of he uncondonal facor models appled o he alpha decles. Clearly, sorng on pas alphas raher han pas reurns reduces he reurn spread and he sensvy o he lagged rsk facors. The reurn spread drops from 12.24% for reurn decles o 7.48% for alpha decles for equy funds, and from 2.95% o 1.13% for bond funds. Ths s he frs ndcaon ha alpha sorng leads o a more sable rsk profle. [Inser Table VII abou here] Table VIII shows he resuls when we apply he condonal facor models o he alpha decles. The facor models are agan successful n capurng he me-seres varaon of he decles; for example, wness he adused R-squared of 26% for he equy D1-D10 and 33% for he bond D1-D10. The conrbuon of condonng s smaller han ha for he reurn decles, bu remans sgnfcan. For example, he equy D1-D10 has an ncremenal RMRF loadng of 0.07, ncremenal SMB loadng of 0.16 and ncremenal HML loadng of 0.57; he bond D1-D10 has an ncremenal TERM loadng of and ncremenal DEF loadng of As noed above, he remanng rsk dynamcs presumably reflecs error n he esmaon of fund alphas based on 12 monhly observaons. For he equy funds, he op-boom alpha spread s reduced o 6.51%, whch s very close o he 6.35% for he reurn decles. In conras, he alpha spread for bond muual funds s 2.24%, whch s subsanally hgher han he 0.7% for he reurn decles. The dfference arses because he correlaon beween reurns and alphas s lower for bond funds han for equy 18

20 funds. Therefore, alpha sorng nroduces a larger alpha spread for bond funds han does reurn sorng. [Inser Table VIII abou here] For he sake of brevy, resuls for he WML-based model are no shown here. As for he decles formed on pas reurns, he WML-based model has lower me-seres explanaory power han he condonal model and yelds based esmaes for he alpha spread. Sll, we ask f WML facors could add somehng o our condonal model. To answer hs queson, we esmae he condonal models augmened wh he equy or bond WML facor. The adused R-squared ncreases from 26% o 29% for sock D1-D10 and from 33% o 39% for bond D1-D10. So condonng seems o compleely drve ou he WML facors. 5. Concluson In hs sudy we analyze he relaon beween reurn perssence, dynamc rsk facor exposures, and WML momenum facor exposures for large samples of equy and bond muual funds. We conclude ha: 1. The rsk profles of wnner and loser funds are hghly dynamc. The facor loadngs of wnner (loser) funds end o ncrease (decrease) afer posve facor reurns and decrease (ncrease) afer negave reurns. Accounng for hs dynamc paern subsanally mproves our undersandng of he me-seres and cross-secon of fund reurns. The adused R-squared for he op-boom reurn dfferenal ncreases from 9% o 63% for equy funds, and from 45% o 77% for bond funds. In addon, he alpha spread beween wnner and loser funds falls from 11.91% o 6.35%, and from 1.46% o 0.70% per annum, respecvely. These resuls sugges ha a subsanal par of performance perssence s arbuable o me-varyng rsk exposures. 2. The rsk dynamcs arses by sorng funds on pas reurns and consrucng wnner and loser porfolos and does no reflec he rsk dynamcs of ndvdual funds. In conras o wnner and loser porfolos, he exposures of ndvdual funds generally are no relaed o lagged facor values. 19

21 3. We use he smples possble funconal form for our condonal model, based on a sngle dummy varable for every rsk facor. We nvesgaed several alernave funconal forms, bu found no sgnfcan mprovemens beyond hs smple model. We also nvesgaed models ha use macroeconomc varables as condonng nformaon. These models were unsuccessful n capurng he reurn dynamcs n our daa ses. Nowhsandng he mers of hese models, we fnd ha condonng on pas rsk facors s more approprae for capurng me varaon ha s caused by sorng on pas reurns. 4. Alhough he explanaory power of wnner-mnus-loser (WML) momenum facors s well known, he nerpreaon of hese facors s he subec of an ongong debae. We provde a rsk-based explanaon based on dynamc exposures. WML facors have an ncreased exposure o facors ha have performed well recenly, a resul ha was shown earler by Grundy and Marn (2001) n he conex of sock momenum and he 3FM. Smlarly, wnner (loser) funds have an ncreased (decreased) exposure o facors ha have performed well recenly. Due o hs common paern of me varaon, WML loadngs can serve as proxes for he reurns dynamcs of wnner and loser funds. Indeed, ncludng sock and/or bond WML facors n he uncondonal facor models subsanally mproves he me-seres f n our samples. Accordng o our rsk-based explanaon, he WML exposures of wnner and loser porfolos do no reflec ndvdual funds followng momenum sraeges, bu raher he rsk dynamcs nroduced by sorng funds based on pas reurns. 5. Despe he mproved me-seres f, WML facors can nroduce bas for esmang he level of fund perssence, when her alpha or resdual varance devaes sgnfcanly from zero, as s rue n our samples. If a WML facor has a posve alpha, hen he WML premum exceeds he rue rsk premum for condonal facor exposures, leadng o underesmaon (overesmaon) of he rue alphas for wnner (loser) funds. Ths paern appears for he sock WML facor n our sample of equy funds. In conras, f he WML alpha s negave, hen he rue alphas for wnner (loser) funds are overesmaed (underesmaed), as s he case for he bond WML facor n our sample of bond funds. If he rsk facors do no explan he bulk of he varaon of a WML facor and he resdual varance devaes sgnfcanly from zero, hen a classcal errors-n-varables problem arses. The WML facors hen are nosy 20

22 proxes for he rsk dynamcs and he regresson esmaes wll be based owards he esmaes of he uncondonal model. 6. By sorng socks or bonds on pas reurns, WML porfolos are heavly nfluenced by rsk dynamcs. Ths problem may speak o he dea of replacng he orgnal WML facors wh resdual WML facors ha are uncorrelaed wh he rsk dynamcs. Such resdual WML facors can be consruced eher as he resduals of condonal rsk facor models appled o he orgnal WML facor, or, alernavely, by sorng socks or bonds based on pas alphas raher han pas raw reurns. However, resdual momenum facors do no have sgnfcan explanaory power for our me seres. 7. To accoun for changng exposures, we have used condonal models. An alernave mehod o correc for rsk dynamcs s o adus he sorng mehod by sorng socks based on pas alphas raher han on pas raw reurns. The man advanage of hs approach s ha does no requre he researcher o specfy he funconal form of he rsk dynamcs, and apples under general condons. Furhermore, hs approach maxmzes he alpha spread and hus reduces he probably ha perssence s no deeced. However, a lmaon of hs approach s ha he alphas can be dffcul o esmae f hey are based on he shor me-seres of he formaon perod, especally when mulple rsk facors are a work. Due o esmaon error, he alpha-based approach sll nroduces dynamc exposures and he need o condon on pas rsk facors. 21

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26 Table I Rsk Facor Reurns The able lss he uncondonal average reurns of he sock and bond rsk facors of Fama and French (1993, 1995, 1996) and Gebhard, Hvdkaer, and Swamnahan (2005). The able also lss he average reurns of he rsk facors posve and negave one-year lagged reurns. All values are annualzed. Afer posve reurns Afer negave reurns Uncondonal average RMRF 5.49% 6.50% 3.12% SMB 2.70% 6.63% -2.49% HML 5.71% 7.19% 2.26% TERM 3.25% 3.78% 1.72% DEF 0.71% 0.91% 0.30% 25

27 Table II Uncondonal Rsk Facor Models We oban our daa on reurns of equy and bond muual funds from he 2006 CRSP Muual Fund Survvor Bas Free Daabase. Our sample comprses 12,717 equy muual funds for he perod and 1,090 bond muual funds for he perod We sor funds no en quanle porfolos based on lagged one-year reurns. For he resulng porfolos of equy muual funds we run uncondonal Fama and French (1993, 1995, 1996) hree-facor model regressons; for he porfolos of bond muual funds we run uncondonal Gebhard, Hvdkaer, and Swamnahan (2005) wo-facor model regressons. The able lss he porfolos parameer esmaes, and adused R-squared values. All values are annualzed. We esmae all models by means of OLS. Reurn Alpha Alpha- RMRF SMB HML TERM DEF Ad.Rsq Panel A. Equy muual funds D % 6.06% D2 8.11% 2.63% D3 6.37% 0.82% D4 5.08% -0.54% D5 4.13% -1.19% D6 3.65% -1.34% D7 3.14% -1.71% D8 2.68% -2.40% D9 1.00% -3.99% D % -5.85% D1-D % 11.91%

28 Table II-Connued Reurn Alpha Alpha- RMRF SMB HML TERM DEF Ad.Rsq Panel B. Bond muual funds D1 4.06% 0.39% D2 3.50% 0.29% D3 3.05% -0.14% D4 2.96% -0.14% D5 2.83% -0.21% D6 2.68% -0.25% D7 2.48% -0.28% D8 2.23% -0.25% D9 2.15% -0.08% D % -1.07% D1-D % 1.46%

29 Table III Condonal Rsk Facor Models We oban our daa on reurns of equy and bond muual funds from he 2006 CRSP Muual Fund Survvor Bas Free Daabase. Our sample comprses 12,717 equy muual funds for he perod , and 1,090 bond muual funds for he perod We sor funds no en quanle porfolos based on lagged one-year reurns. For he resulng porfolos of equy muual funds we run condonal Fama and French (1993, 1995, 1996) hree-facor model regressons; for he porfolos of bond muual funds we run condonal Gebhard, Hvdkaer, and Swamnahan (2005) wo-facor model regressons. The able lss he porfolos parameer esmaes, and adused R-squared values. All values are annualzed. We esmae all models by means of OLS. Alpha Alpha- RMRF SMB HML TERM DEF RMRF SMB HML TERM DEF Ad.Rsq Panel A. Equy muual funds D1 3.21% D2 0.85% D3-0.14% D4-1.04% D5-1.12% D6-0.95% D7-0.83% D8-0.82% D9-1.93% D % D1-D %

30 Table III-Connued Alpha Alpha- RMRF SMB HML TERM DEF RMRF SMB HML TERM DEF Ad.Rsq Panel B. Bond muual funds D1 0.02% D2 0.08% D3-0.26% D4-0.18% D5-0.21% D6-0.20% D7-0.17% D8-0.07% D9 0.18% D % D1-D %