Centre for Investment Research Discussion Paper Series. Momentum Profits and Time-Varying Unsystematic Risk

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1 Cenre for Invesmen Research Discussion Paper Series Discussion Paper # 08-0* Momenum Profis and Time-Varying Unsysemaic Risk Cenre for Invesmen Research O'Rahilly Building, Room 3.0 Universiy College Cork College Road Cork Ireland T +353 (0) /765 F +353 (0) /390 E W *These Discussion Papers ofen represen preliminary or incomplee work, circulaed o encourage discussion and commens. Ciaion and use of such a paper should ake accoun of is provisional characer. A revised version may be available direcly from he auhor(s).

2 Momenum Profis and Time-Varying Unsysemaic Risk Xiafei Li*, Joëlle Miffre**, Chris Brooks***, and Niall O Sullivan**** Absrac This sudy assesses wheher he widely documened momenum profis can be aribued o ime-varying risk as described by a GJR-GARCH(1,1)-M model. We reveal ha momenum profis are a compensaion for ime-varying unsysemaic risks, which are common o he winner and loser socks bu affec he former more han he laer. In addiion, we find ha, perhaps because losers have a higher propensiy han winners o disclose bad news, negaive reurn shocks increase heir volailiy more han i increases hose of he winners. The volailiy of he losers is also found o respond o news more slowly, bu evenually o a greaer exen, han ha of he winners. Keywords: Momenum profis, Common unsysemaic risk, GJR-GARCH(1,1)-M JEL classificaions: G1, G14 This version: February 007 * Cass Business School, Ciy Universiy (UK), ** EDHEC (France), *** ICMA Cenre, Universiy of Reading (UK), **** Deparmen of Economics and Cenre for Invesmen Research, Universiy College Cork, Ireland. Auhor for correspondence: Chris Brooks, ICMA Cenre, Universiy of Reading, Whieknighs, PO Box 4, Reading, RG6 6BA, UK; el: (+44) ; fax: (+44) ;

3 1. Inroducion Momenum sraegies ha buy recen winners and sell recen losers are profiable over shor horizons of 3 o 1 monhs (Jegadeesh and Timan, 1993). Price coninuaion has prevailed over ime (Jegadeesh and Timan, 001), across counries (Griffin e al., 003; Liu e al., 1999; Ellis and Thomas, 003), across indusries (Moskowiz and Grinbla, 1999), across equiy syles (Chen and De Bond, 004) and across asse classes (Okunev and Whie, 003). While he profiabiliy of relaive-srengh porfolios is no dispued, here is sill a lo of conroversy as o why hese abnormal reurns occur. Two explanaions have been pu forward. The firs is based on psychology and marke inefficiency. Behavioral proponens relae price under- and over-reacion o cogniive errors ha invesors make when incorporaing informaion ino prices. For example, invesors may be oo quick o draw he conclusion ha a given sock follows a paricular ideal ype (he represenaiveness heurisic), and hey may be oo slow o updae heir beliefs when confroned wih new, especially conradicory, evidence (he conservaism bias). These behavioral aribues lead firs o momenum as sock prices reac wih delay o firm-specific informaion and, once deviaions from equilibrium are acknowledged, o subsequen mean reversion (Barberis e al., 1998; Daniel e al., 1998; Hong and Sein, 1999). 1 This suggess ha he irraionaliy from which agens suffer may push prices away from fundamenals and allow profiable mispricings o survive. The second explanaion relies on he noion of marke efficiency and argues ha he reurns of he relaive-srengh porfolios are a fair compensaion for he risk and/or rading coss of implemening he sraegies. On balance, however, he evidence suggess ha he profiabiliy of he relaive-srengh porfolios is no solely a compensaion for exposure o higher risks (Jegadeesh and Timan, 1993; Chan e al., 1996; Fama and French, 1996; Griffin e al., 003; Karolyi and Kho, 004; Sadka, 006). Sudies ha allow for ime-variaion in sysemaic risks reach conflicing conclusions. While Chordia and Shivakumar (00), Wu (00) and Wang (003) explain he profiabiliy of momenum sraegies hrough ime-variaion in expeced reurns, Grundy and Marin (001), Griffin e al. (003) and Nagel and Lewellen 1 Oher behavioral deficiencies ha invesors may suffer from include biased self-aribuion and overconfidence (Daniel e al., 1998), and bounded raionaliy (Hong and Sein, 1999). Jegadeesh and Timan (1993) esimae a marke model, o which Chan e al. (1996) and Fama and French (1996) add he reurn of porfolios sored on size and book-o-marke value. Griffin e al. (003) look a macroeconomic and financial facors ha are in he spiri of he model of Chen e al. (1986). Sadka (006) looks a he role of liquidiy risk. Karolyi and Kho (004) use boosrap experimens and a wide range of reurn-generaing processes. 1

4 (003) argue ha he momenum reurns are oo large o be accouned for in erms of imevarying risks. I is imporan o noe also ha a raionale relaed o ransacion coss has been pu forward as an explanaion for he momenum profis. Lesmond e al. (004) indeed argue ha momenum profis have lile o do wih risk as hey are simply an illusion induced by rading coss. 3 The conribuion of his aricle o he momenum lieraure is wih regards o he ime-varying unsysemaic risk of he winners and he losers and o he role i may have in explaining he abnormal reurns of momenum sraegies. While several sudies look a variaions in sysemaic risk (Grundy and Marin, 001; Chordia and Shivakumar, 00; Wu, 00; Griffin e al., 003; Wang, 003; Nagel and Lewellen, 006), his sudy is he firs o look a variaions in he unsysemaic risks of he winner and loser porfolios. We do his wihin a GJR-GARCH(1,1)-M framework. 4 The raionale for choosing a GARCH(1,1)-M model sems from he idea ha in raional markes, volailiy is ofen viewed as being commensurae wih news or informaion flow, and indeed, he auocorrelaion in informaion arrival ( news evens happen in bunches ) is one of he primary raionalizaions of he volailiy clusering ha is almos universally observed in asse reurns. The condiional sandard deviaion erm in he mean equaion capures he ime-varying relaionship beween oal risk and reurns, and hus our conribuion is o link momenum profis wih he impac of news on reurns. We use he GJR varian of he basic GARCH model in order o allow for a possible asymmery in he relaionship beween he reurns o he winner and loser porfolios and he volailiy. By using a condiional model, we are able o capure he possibiliy ha he risks of he winners and he losers may change in a predicable, bu differen, way over ime. This suggess ha a model ha explicily allows for risk o be ime-dependen migh explain he abnormal reurns of he momenum sraegies. Our approach is an alernaive o one where pre-specified condiional variables, such as macroeconomic or firm-specific influences, are used as he risk facors in a condiional pricing model. Mos such models have largely failed o explain he profiabiliy of relaive srengh porfolios, and hus an advanage of he mehod ha we employ is ha i does no require any a priori specificaion of he se of risk facors in order o allow for imevarying risk. 3 Lesmond e al. (004) show ha momenum sraegies are highly rading inensive and pick up socks ha are expensive and risky (small, high bea, illiquid, off-nyse exreme performers). Besides, he momenum profis are mainly driven by he losers (Hong e al., 000) and hus shor-sale coss also need o be aken ino accoun. 4 GJR-GARCH(1,1)-M sands for Glosen e al. (1993) Generalized Auoregressive Condiional Heeroscedasiciy of order 1,1 wih a Mean erm ha models he condiional risk premium. A number of sudies (Nelson, 1991; Glosen e al., 1993; Rabemananjara and Zakoian, 1993) show ha good news (measured by posiive reurn shocks) and bad news (measured by negaive reurn shocks) have an asymmeric impac on he condiional variance of sock reurns.

5 We draw he following wo conclusions from our analysis. Firs, we idenify some clear paerns in he volailiy of he winner and loser porfolios. The volailiy of he winners is found o be more sensiive o recen news han ha of he losers, whereas by conras, he volailiy of he losers is found o be more sensiive o disan news han ha of he winners. Besides, he volailiy of he losers (wih an average volailiy half-life of 4 monhs and 13 days) shows a higher level of persisence han ha of he winners (whose volailiy half-life only equals 3 monhs and 5 days on average). The second conclusion of his aricle is wih regards o he hypohesis ha he momenum reurns are a compensaion for ime-varying unsysemaic risk as modeled by he GJR- GARCH(1,1)-M model. We show ha he GJR-GARCH(1,1)-M erms, when added o he radiional marke and Fama and French models, explain he abnormal performance of he momenum sraegies wihou he need o resor o he ransacions cos and illiquidiy issues ha were he focus of Lesmond e al. (004) or Sadka (006). Ineresingly, neiher he GJR- GARCH(1,1) nor he GARCH(1,1)-M specificaions alone could accoun for he abnormal reurn of he relaive-srengh porfolios. I is herefore boh he asymmeric response of he losers o good and bad news and he condiional risk premium embedded in he GARCH(1,1)- M model ha explain he profiabiliy of he momenum sraegies. The remainder of he paper is organized as follows. Secion inroduces he daase, he mehodology employed o consruc he momenum porfolios and he models used o adjus for risk. Secion 3 analyzes how recen news, disan news and negaive reurn shocks impac he volailiy of he winners and losers. I also ess wheher he momenum profis are a compensaion for ime-varying unsysemaic risk common o he winners and losers. Finally, secion 4 concludes he paper wih a summary of our findings.. Daa and Mehodology Monhly UK sock prices adjused for dividends are obained from he London Share Price Daabase over he period 8 February 1975 o 31 December To address problems of survivorship bias, we also include socks ha were delised due o merger, acquisiion or 5 The reurns o he Fama-French facor porfolios ha we employ subsequenly are only available o December 001, which necessiaes his runcaion of our sample period. 3

6 bankrupcy. The sample includes all companies wih a leas 3 monhs of available reurns. A oal of 6,155 companies are considered. All socks are ranked and sored ino 10 equally-weighed porfolios based on heir pas J- monh cumulaive reurns (J = 3, 6, 1 monhs). The decile porfolio wih he highes cumulaive reurn is ermed he winner porfolio, while he decile porfolio wih he lowes cumulaive reurn is called he loser porfolio. The reurn on he momenum porfolio is hen measured as he reurn difference beween he winner and loser porfolios over he nex K monhs (K = 3, 6, 1 monhs). 6 The resuling porfolio is referred o as he J-K momenum porfolio. The procedure is rolled forward a he end of each holding period o produce new winner, loser and momenum porfolios. The formaion of he relaive-srengh porfolios is herefore non-overlapping, hus reducing he rading frequency and he ransacion coss incurred in porfolio consrucion and ensuring ha saisical ess are valid wihou requiring modificaion of he sandard errors. Our framework is also more realisic in erms of he behavior of invesors han one based on overlapping porfolios where hey would presumably have o vary he amoun of wealh devoed o he sraegies over ime. Tradiionally, performance has been measured by regressing a porfolio s reurns on a se of sysemaic risk facors emanaing from he CAPM of Sharpe (1964) or he hree-facor model of Fama and French (1993), which can be expressed respecively as R R ( RM R f ) ε ( RM R f ) + ssmb + hhml ε = α + β + (1) = α + β + () where R is eiher he reurn on he momenum porfolio or he reurn of he winner and loser porfolios in excess of he risk-free rae, R f is he hree-monh Treasury bill rae, R M is he value-weighed marke reurn on all socks quoed on he London Sock Exchange, SMB and HML are UK-based reurns of Fama and French (1993) size and book-o-marke value porfolios as provided by Nagel 7 and ε is a whie noise error erm. The performance of he porfolios is hen evaluaed by esing he saisical significance of he α coefficien in (1) and (). 8 6 We also employed holding periods of 15 monhs duraion, bu he resuls were qualiaively idenical o hose employing a 1-monh horizon and are herefore no repored. 7 These daa are available a hp://faculy-gsb.sanford.edu/nagel/daa/uk_fffac.csv 8 The Carhar (1997) four-facor version of he Fama-French model is ofen used in performance aribuion for muual funds. The fourh facor, known as UMD ( up-minus-down ) is a measure of he reurn o momenum porfolios. The key disincion beween his approach and wha we propose here is ha we are rying o explain he profiabiliy of momenum porfolios using a previously unexamined measure of risk, whereas he UMD erm 4

7 Embedded in equaions (1) and () is he assumpion ha ε ~ N( 0, σ ) and, hus, ha here is no condiional volailiy in he marke. Since Engle (198), numerous sudies have been wrien on he family of GARCH models (Poon and Granger, 003; Andersen e al., 006; Bauwens e al., 006). The araciveness of he GARCH models sems from he fac ha hey model he condiional variance of asse reurns by aking ino accoun persisence in volailiy (where volailiy shocks oday influence expeced volailiy many monhs from now) and leverage effecs (where negaive reurn shocks impac volailiy more han posiive reurn shocks of he same magniude). These wo feaures are cenral o our hypoheses ha he losers volailiies show more persisence and asymmery han hose of he winners. We invesigae wheher momenum profis in he UK are a compensaion for ime-varying risk wihin GJR-GARCH(1,1)-M versions of he marke and Fama and French models: R = α + β σ = ω + γε R = α + β σ = ω + γε ( R R ) M f + δσ + ε 1 + ηi 1ε 1 + ( R R ) M f 1 + ηi 1ε 1 + θσ + ssmb 1 θσ + hhml 1 + δσ + ε (3) (4) where σ is he condiional variance of he winner, loser and momenum porfolios, δσ measures he ime-varying risk premium, γ. η relaes o he lagged squared error erm and measures he impac of recen news on volailiy, η also measures any asymmeric response of volailiy o bad and good news (commonly aribued o as leverage effec), I 1 = 1 if ε 1 < 0 (bad news, also called negaive reurn shock) and I 1 = 0 oherwise, θ relaes o he lagged condiional volailiy and measures he impac of old news on volailiy. Wihin he framework of sysems (3) and (4), he following wo hypoheses can be esed. Firs, he coefficiens on condiional volailiy indicae how news impacs he volailiy of he winners and of he losers. In paricular, we analyze he speed of he response of he winners and losers o news and es for he presence of any asymmery in he response of he winners and losers volailiies o good and bad news. Second, he sign and significance of α in he mean equaions of sysems (3) and (4) indicae wheher he momenum reurns are a uses momenum o explain he reurns from oher sraegies. Thus, in our sudy, momenum is he explained variable whereas in he Carhar model, i is an explanaory variable. 5

8 compensaion for marke risk, he risks associaed wih size and book-o-marke value and ime-varying, unsysemaic risk. We also es wheher he momenum profis can be explained by a simplified version of he above models in he sandard GARCH(1,1)-M framework. This specificaion models he ime-varying risk premium as in (3) and (4) bu does no allow for asymmeric response of volailiy o good and bad news. Pracically, his breaks down o esimaing he following sysems of equaions R = α + β σ = ω + γε R = α + β σ = ω + γε ( R R ) M 1 + θσ f 1 ( R R ) M 1 + θσ f 1 + δσ + ε + ssmb + hhml + δσ + ε (5) (6) 3. Empirical Resuls Table 1 presens summary saisics for he winner, loser and momenum porfolios. The rows represen he ranking periods (J = 3, 6 and 1 monhs) and he columns represen he holding periods (K = 3, 6 and 1 monhs). I is clear from his able ha he winners sysemaically ouperform he losers a he 1% level. Across sraegies, he momenum porfolios earn an average reurn of a monh, wih a range from for he 3-3 sraegy o for he 6-6 sraegy. 9 These resuls corroborae hose of Liu e al. (1999) and Ellis and Thomas (003) for he UK. Table 1 also repors he monhly sandard deviaions and reward-o-risk raios of each porfolio reurns. Consisen wih raional expecaions, he momenum porfolios wih higher reurns have also more risk. For insance, he 6-6 sraegy earns he highes average reurn (0.0193) and, wih a sandard deviaion of , i is also he second mos volaile sraegy. Wih a reward-o-risk raio of , he 1-6 sraegy generaes he highes average reurn in risk-adjused erms, while he 3-3 sraegy offers he lowes risk-adjused reurn (0.195). The conribuion of he aricle is wih regards o he ime-varying unsysemaic risk of he winner and loser porfolios and he impac ha i may have on momenum profis. Wih his in 9 Noe ha all figures in his sudy refer o monhly proporion reurns raher han percenage reurns, unless oherwise saed. 6

9 mind, we firs analyze he performance of he winner, loser and momenum porfolios wihin he sandard marke and Fama and French models and hen allow for ime-varying unsysemaic risk hrough differen specificaions of he GARCH(1,1) model. While doing his, we will also analyze he impac of recen news, old news and bad news on he volailiy of he winners and losers Saic marke and Fama and French models Table repors he OLS esimaes of he marke and Fama and French models (1) and () for he winner, loser and momenum porfolios. 10 In line wih previous research (Jegadeesh and Timan, 1993; Fama and French, 1996; Karolyi and Kho, 004), he resuls indicae ha radiional versions of he marke and Fama and French models fail o explain momenum profis. Regardless of he model, of he ranking period, and of he holding period, he α coefficiens of he momenum sraegies in equaions (1) and () are posiive and significan a he 1% level. The momenum profis esimaed from he marke model range from (3-3 sraegy) o (6-6 sraegy), wih an average reurn a a monh. According o he Fama and French model, he winners ouperform he losers by on average, wih a range of (3-3 sraegy) o 0.0 (1-6 sraegy). While sysemaic risk explains mos of he over-performance of he winners, i fails o accoun for he under-performance of he losers. Irrespecive of he ranking period, of he holding period and of he risk model considered, he losers indeed have negaive alphas ha are significan a he 1% level. As in Hong e al. (000), he momenum profis are herefore driven by he losers. The facor loadings on R M, SMB and HML in (1) and () sugges ha he winner and loser porfolios end o pick small capializaion socks (s>0) wih high marke risk (β>0). The winners have growh characerisics (h<0) and he losers have value characerisics (h>0). The momenum sraegies are predominanly marke-neural (β=0) and size-neural (s=0) and have negaive loadings on HML. These resuls are consisen wih hose previously repored, including he sudies by Chan e al. (1996) and Liu e al. (1999). 10 Engle (198) s ARCH-LM es provides srong evidence of heeroscedasiciy in he OLS residuals of he marke and Fama-French models. Hence, we use Whie s heeroscedasiciy-robus sandard errors. 7

10 3.. GARCH(1,1) versions of marke and Fama and French models Table 3 repors esimaes of he marke and Fama and French models (3) and (4) ha include a GJR-GARCH(1,1)-M erm. To faciliae exposiion, he averages across ranking and holding periods of he coefficien esimaes are discussed in he following secion for he winner, loser and momenum porfolios, bu are no repored direcly in he Table due o space consrains. The esimaion mehod is Maximum Likelihood wih Bollerslev-Wooldridge robus sandard errors. We firs analyze how news, wheher i is recen, disan or negaive, impacs he volailiy of he winners and he losers. We subsequenly es for wheher he ime-varying unsysemaic risk common o he winners and losers explains he profiabiliy of he momenum sraegies. The paern of condiional volailiy The coefficiens γ and η in sysems (3) and (4) relae o he lagged squared error erm and, herefore, o he impac of recen news on volailiy. The average γ+η/ of he condiional marke model equals for he winners and 0.16 for he losers. 11 The average γ+η/ of he condiional Fama and French model is for he winners and for he losers. Clearly, recen news impacs he volailiy of he winners more han i impacs ha of he losers. Wih only one excepion (he 3-3 winner in he Fama and French model), he conclusion holds hroughou in Table 3, irrespecive of he ranking period, of he holding period and of he model considered. The coefficien θ in sysems (3) and (4) reflecs he effec of lagged condiional variance and capures he impac of old news on volailiy. The resuls of he condiional marke model in Table 4 indicae ha he average θ coefficien of he winners (0.5785) is lower han ha of he losers (0.7911). The same conclusion applies o he condiional Fama and French model, for which he winners have an average θ coefficien of and he losers an average θ coefficien of I is clear herefore ha old news has more impac on he volailiy of he losers han on he volailiy of he winners. Looking a he esimaes of θ in Table 3, i appears ha he conclusion holds for he vas majoriy of he porfolios, he 1-1 winner in he marke model being he only excepion. 11 While he parameers in he condiional variance equaion of a symmeric GARCH model are usually required o be posiive, when he GJR form of he model is used, i is possible for he parameer on he asymmery erm (η in our noaion) o be negaive. More specifically, if E(I ) = ½, hen provided ha γ > η/, he negaive parameer would no lead he condiional variance o be negaive. We have checked his condiion and i is saisfied for all models esimaed in his sudy. 8

11 The asymmeric coefficiens (η) in Table 3 sugges ha bad news has differen impacs on he volailiy of he winners and on he volailiy of he losers. For he losers, he mean of he η coefficiens is for he condiional marke model and 0.03 for he condiional Fama and French model. Wih only a few excepions, hese coefficiens are significan a he 5% level in Table 3. Clearly, herefore, bad news increases he volailiy of he losers. For he winner porfolios, however, he average η coefficien equals for he condiional marke model and for he condiional Fama and French model, wih 14 ou of 18 coefficiens ha are insignifican a he 5% level in Table 3. I follows ha he announcemen of bad news does no have any noiceable impac on he volailiy of he winners. I may be he case ha socks whose recen performance has already been poor are hi much harder by furher bad news han socks recenly performing well, which are able o absorb bad news more easily. The evidence of Table 3 hus far indicaes ha, wih relaively few excepions, he losers have higher η and θ, and lower γ, han he winners. Table 3 also repors he persisence in volailiy of he winners and losers, measured as γ+η/+θ. The volailiy of he losers appears o be more persisen han ha of he winners. Indeed, he average γ+η/+θ of he losers (winners) equals (0.8339) for he condiional marke model and (0.7885) for he condiional Fama and French model. For he condiional marke model, his convers ino volailiy half-lives of 3 monhs and 18 days for he winners and 8 monhs for he losers. The volailiy half-lives esimaed from he condiional Fama and French model equal monhs and 0 days for he winners and 18 monhs for he losers. Clearly and wih only one excepion ou of 18 regressions, 1 he volailiy persisence of he losers exceeds ha of he winners. The impac of ime-varying firm specific risk on momenum profis Table 3 also repors, hrough δ, he impac of condiional volailiy on he reurns of he winners, losers and momenum porfolios. An increase in condiional volailiy decreases he reurn of boh he winners and he losers, bu increases he momenum reurns. The δ coefficiens of he momenum porfolios from he condiional marke model range from (1-6 sraegy) o (6-3 sraegy) (Table 3) wih an average a (9) coefficiens ou of 9 are significan a he 5% (10%) level. Similar resuls are repored for he condiional Fama and French model, for which δ equals on average, wih 6 (8) coefficiens ou of 9 ha are significan and posiive a he 5% (10%) level. This suggess ha 1 The excepion is for he 1-1 winner in he condiional marke model (Table 3). 9

12 here is a posiive relaionship beween ime-varying risk and momenum reurn: A 1% increase in condiional volailiy leads, on average, o a 0.43% increase in monhly momenum reurns. The facor loadings on R M, SMB and HML for he condiional volailiy model in Table 3 indicae ha he winners and he losers have value characerisics (h>0) and are iled owards small-capializaion socks (s>0) wih high marke risk (β>0). The laer wo characerisics appear o corroborae he evidence from he uncondiional Fama and French model (Table ). As he loadings of he losers on R M, SMB and HML are ypically higher han hose of he winners, he momenum porfolios have coefficiens on he hree Fama and French facors ha are predominanly negaive. The main conribuion of his paper is o es wheher he momenum profis are a compensaion for ime-varying unsysemaic risk as described by he GJR-GARCH(1,1)-M model. If his is indeed he case, hen he α coefficiens of he momenum sraegies should be saisically indisinguishable from zero when hese erms are incorporaed ino he risk aribuion model. This conjecure is suppored uniformly a he 5% level for boh he condiional marke and Fama and French models. The GJR-GARCH(1,1)-M marke model is able o explain he momenum reurns, since he alpha esimaes are reduced boh in magniude and in saisical significance. The alphas indeed range from (1-3 sraegy) o (1-6 sraegy), wih a mean a The GJR-GARCH(1,1)-M Fama and French model does a good job of explaining he momenum profis oo, wih an average alpha of and a range of (1-3 sraegy) o (6-6 sraegy). Clearly, he resuls of Tables and 3 sugges ha adding a GJR-GARCH(1,1)-M srucure o he models radiionally used o measure performance is crucial o explaining he abnormal reurn of momenum sraegies. Ineresingly, he considerable reducion in relaive price srengh reurns afer allowing for ime-varying risk seems o sem from an increase in he performance of he loser porfolios. This suggess ha he underperformance of he losers idenified in Table is in par due o heir sluggish and asymmeric reacion o bad news. Analysis of resuls Proponens of he efficien markes hypohesis could argue ha our resuls are an indicaion of momenum profis being merely a compensaion for unsysemaic risks common o he winners and losers bu sronger for he former han he laer. This line of hough would herefore conclude ha our findings are consisen wih raional pricing in efficien markes. 10

13 Thus, fuure research could seek o deermine why unsysemaic risk is imporan. I may be he case, for example, ha he ime-varying risks are relaed o indusry effecs, where cerain indusries ha may be over-represened in he momenum porfolios become relaively more or less risky in a cyclical fashion over ime as hey go ino and ou of invesmen favour. However, our resuls are also consisen wih a behavioural explanaion along he lines of Hong, Lim and Sein (000), where informaion on he loser socks akes longer o be fully refleced in prices. The srong impac of old news idenified for he losers and he persisence in heir volailiy are in suppor of he saemen of Hong e al. ha bad news ravels slowly. When a firm wih no or low analys coverage receives bad news, is managers are likely o wihhold ha news as disclosing i would pu downward pressure on price. Since losers are more likely han winners o si on bad news, hey are also more likely o wihhold informaion. For he losers, his convers ino higher volailiy persisence (or higher volailiy half-lives) and higher sensiiviy of volailiy o disan news. The resuls in Table 3 also give credence o he conjecure ha, for winners, good news ravels fas. Managers of no or low coverage firms have srong incenives o disclose good news he minue i arrives as his simulaes he share price. Since winners are, by definiion, more likely han losers o receive good news, hey are more eager o disclose informaion. This convers in our seing ino a higher sensiiviy of winners volailiy o recen news and less volailiy persisence (or lower volailiy half-lives). We idenify anoher ineresing paern in he volailiy of he winner and losers. Relaive o he volailiy of he winners, he volailiy of he losers clearly shows an asymmeric response o good and bad news: bad news subsanially increases he volailiy of he losers, while i does no impac ha of he winners. This is in line wih he predicion of he behavioural argumen pu forward above. Since, relaive o winners, losers have a higher probabiliy of disclosing bad news, negaive reurn shocks increase heir volailiy more han hey increase ha of he winners. The asymmeric response of losers o negaive reurns shocks could be explained as follows. Relaive o winners, he probabiliy ha losers disclose bad news is far greaer. Thus he announcemen of a bad piece of news does no aler he volailiy of winners (as bad news is expeced o be ransiory only) while i pushes up ha of losers. When losers do disclose bad news, invesors inerpre his as a sign ha heir beliefs were correc, leading hem o sell he losers. As a resul, heir volailiy increases and becomes more persisen. A failure o explicily model he asymmeric response of he losers and winners o bad news migh herefore lead us o under-esimae he volailiy of he losers, and consequenly heir 11

14 performance, following a price drop or o over-esimae he volailiy of he winners, and consequenly heir performance, following a price rise. This moivaes he hypohesis ha he momenum profis migh, a leas in par, be a compensaion for an asymmeric response of winners and losers o negaive reurn shocks. Robusness of he resuls o he specificaion of he GARCH(1,1) model In his secion, we es wheher he momenum profis can be explained by a simplified version of he condiional models. Table 4 repors he parameer esimaes of sysems (5) and (6) for he winner, loser and momenum porfolios. Table 4 herefore assumes ha he reurn and condiional volailiy of he momenum porfolios are beer described by a GARCH(1,1)- M model. 13 The omission of he leverage effec in Table 4 does no aler he main conclusions of Table 3 wih regards o he paern of volailiy for he winners and he losers. For example, Table 3 and 4 documen ha he volailiy of he winners (W) is more sensiive o recen news han he volailiy of he losers (L); namely, γ > γ. Similarly, he impac of old news on volailiy in W L Tables 3 and 4 is sronger for he losers; namely, θ > θ. Finally, volailiy in boh ables is found o be more persisen for he losers; namely, γ + θ > Table 3 and θ > γ L + L W W L W γ + η + θ in L + η L L W W W γ + θ in Table As a resul, he average volailiy half-lives are much smaller for he winners han for he losers. Across GARCH specificaions, ranking periods, and holding periods, he volailiy half-life of he winners is 3 monhs and 5 days on average, while ha of he losers is 4 monhs and 13 days. The omission of he leverage effec however has a direc impac on he significance of he ime-varying risk parameer δ in Table 4. Ou of he 18 δ coefficiens esimaed for he momenum sraegies in Table 3, 17 were significan a he 10% level. When, as in Table 4, he impac of news on volailiy is assumed o be symmeric, he number of significan δ coefficiens drops o 3. As a resul, he marke and Fama and French models wih GARCH(1,1)-M erms are less able o explain he momenum profis. Though largely insignifican in Table 4, he average abnormal reurns of he momenum sraegies equal 13 We also examined a pure GJR-GARCH(1,1) ha is, a model wihou a condiional volailiy erm in he mean equaion. However, unsurprisingly, i did no explain he observed momenum profis since in such a model, here is no linkage beween he ime-varying volailiy and he reurns. Therefore, he esimaes from his model are no included in he paper, bu are available from he auhors on reques. 14 Again here are a few excepions ( γ L > γ W for he 3-3 and 3-1 winners of he condiional Fama and French model), bu hese are exremely rare. 1

15 0.015 a monh for he GARCH(1,1)-M marke model and for he GARCH(1,1)-M Fama and French model. These average α coefficiens are in excess of he and average abnormal reurn for he GJR-GARCH(1,1)-M marke model and he GJR- GARCH(1,1)-M Fama and French model, respecively. To summarize, he evidence in Tables 3 and 4 suggess ha i is boh he asymmeric response of he losers o good and bad news and he condiional risk premium ha explain he profiabiliy of he momenum sraegies. Neiher he leverage effec, nor he condiional risk premium in isolaion can explain he abnormal performance of he momenum sraegies. I is he ineracion beween wo ha drives he momenum reurns. To judge he relaive meris of models (1) o (6), he Akaike informaion crierion (AIC) is calculaed for he winners, losers and momenum porfolios. AIC rades off beer model fi for greaer numbers of parameers, and hus a preferred model is one wih he lowes value of he crierion. The resuls are repored in Table 5 for differen specificaions of he marke and Fama and French models. These specificaions include he saic models (1) and (), he GJR- GARCH(1,1)-M models (3) and (4), and he GARCH(1,1)-M models (5) and (6). For a given specificaion of he risk-reurn relaionship, he daa always favor he Fama and French model over he marke model. This indicaes ha he size and book-o-marke value risk facors add explanaory power o he models over and above ha provided by he marke reurn. More perinen o our sudy, he daa evidenly prefer he GJR-GARCH(1,1)-M models o he saic approaches. The GJR-GARCH(1,1)-M marke and Fama and French models have he lowes AIC in he vas majoriy of he cases, and never rank las in erms of AIC. These resuls for he GJR-GARCH (1,1)-M models compare favorably o he AIC of he GARCH(1,1)-M. Irrespecive of he ranking and holding periods, he saic versions of he marke and Fama and French model sand ou as having he highes values of he AIC. This suggess ha ou of he hree specificaions of he marke and Fama and French models, he saic versions provide he wors accoun of he reurns of he winner, loser and momenum porfolios, while he ime-varying condiional volailiy models allowing for asymmeries provide he bes. Robusness of he resuls o he marke examined In order o deermine wheher he abiliy of he asymmeric condiional volailiy model o explain he resuls of he momenum porfolios resuls from some specific feaure of he UK 13

16 marke, or wheher i is likely o be more general, we repea he enire analysis above on winner, loser and momenum porfolios formed from US socks. The US daa cover he period January 1978 o December 001, and were obained from Daasream. The mean reurns of he winner, loser, and momenum porfolios, formed in an idenical way o ha described above for he UK marke, are presened in Table 6. There is ample evidence of momenum effecs, wih he winner porfolio average reurns saisically significanly exceeding hose of he losers for all nine porfolio formaion and holding periods examined. While he sizes of he momenum effecs are slighly smaller for he US, hey are of he same order of magniude as hey were for he UK. For example, for he (1, 1) horizon, he average monhly reurn for he UK was 1.43%, and for he US i is 1.1%. For he laer marke, profiabiliy is highes a 1.71% per monh for he (1, 3) sraegy, whereas i was maximized a 1.93% for he (6, 6) sraegy in he UK. Table 7 repors he parameer esimaes for he saisic marke and Fama-French models using he US daa. I is eviden ha he 3-facor model is no more able o explain he profiabiliy of relaive srengh porfolios for his marke han i was for he UK. For all nine (eigh of he nine) combinaions of porfolio formaion and holding periods examined, he momenum profis are sill posiive and saisically significan a he 5% (1%) level. Finally, Table 8 shows he parameers for he condiional marke and Fama and French models wih a GJR-GARCH (1, 1)-M erm esimaed using US Daa. While he imporance of ime-varying unsysemaic risk appears more uniformly high whaever combinaion of porfolio formaion and holding period are used in he UK conex han for he US, in he laer case, he momenum profis are again largely explained by he incorporaion of he unsysemaic risk erms ino he equaions. This leads boh he sizes of he esimaed alphas and heir levels of saisical significance o reduce. For insance, when he ime-varying unsysemaic risk erms are included in he model, he alpha for he (6,6) momenum sraegy of when he marke model is used is reduced by 40% o and i is reduced by 5% o 0.01 when he Fama-French model is used. For he augmened marke models presened, only one, he (6,1) sraegy, is significanly profiable a he 5% level, and none are profiable a he 1% level; similarly, only one combinaion of formaion and holding periods leads o significan alpha when he Fama-French model is augmened by he inclusion of he condiional unsysemaic risk erm. 15 Our oher major findings concerning he speed of 15 Again, similar resuls are found, bu no repored here, for he symmeric GARCH-M formulaion. 14

17 adjusmen of volailiy and he asymmeric response of volailiy o good and bad news for he winners relaive o he losers sill holds. 4. Conclusions This aricle considers wheher he widely documened momenum profis are a compensaion for ime-varying unsysemaic risk as described by he family of auoregressive condiionally heeroscedasic models. The moivaion for esimaing a GJR-GARCH(1,1)-M model sems from he fac ha since losers have a higher probabiliy han winners o disclose bad news, one canno assume a symmeric response of volailiy o good and bad news. Neiher can we presuppose ha he speed of adjusmen of volailiy o news is he same for he winners and he losers. Such a suggesion is consisen wih he findings of Hong e al. (000) ha for firms wih no or low analyss coverage, bad news ravels slower han good news and hus, he volailiy of he losers may respond more slowly o news han ha of he winners. Our resuls sugges ha he ime-varying unsysemaic volailiy of he winners indeed differs from ha of he losers. For example, he volailiy of he winners is found o be more sensiive o recen news and less persisen han ha of he losers. The converse, ha he volailiy of he losers is found o be more sensiive o disan news and more persisen han ha of he winners, also holds. Fuure research may ascerain he precise cause of hese findings, and in paricular wheher hey are bes aribued o raional, risk-based or behavioral causes. Bu we conjecure ha he ime-varying risk of companies wih no or low analys coverage depends on he naure of he informaion ha is been disclosed: Good news is disclosed earlier, and impacs volailiy sooner, han bad news. Relaive o he volailiy of he winners, ha of he losers also clearly shows a more asymmeric response o good and bad news. As losers have a higher propensiy o disclose bad news, negaive reurn shocks increase heir volailiy more han hey increase ha of winners. Mos imporanly, we also documen ha he GJR-GARCH(1,1)-M models explain much of he profiabiliy of he momenum sraegies, and cerainly have more descripive power han he commonly used size and value risk facors. Ineresingly, neiher he GJR-GARCH(1,1) nor he GARCH(1,1)-M specificaions alone could accoun for he abnormal reurn of he relaive-srengh porfolios. I is herefore a combinaion of he asymmeric response of he losers o good and bad news, he sluggish response of losers o bad news and he condiional risk premium embedded in he GARCH(1,1)-M model, ha explain he profiabiliy of he relaive-srengh porfolios. 15

18 References Andersen, T. G., Bollerslev, T., Chrisoffersen, P. F., Diebold, F. X Volailiy and correlaion forecasing. In G. Ellio, C. W. J. Granger and A. Timmermann Eds.. Handbook of Economic Forecasing Amserdam: Norh-Holland. Barberis, N, Schleifer, A., Vishny, R., 1998, A model of invesor senimen. Journal of Financial Economics 49, Bauwens, L., Lauren, S., Rombous, J. V. K., 006. Mulivariae GARCH models: A survey. Journal of Applied Economerics 1, Carhar, M. (1997) On Persisence in Muual Fund Performance. Journal of Finance 5, Chan L. K. C., Jegadeesh, N., Lakonishok, J., Momenum sraegies. Journal of Finance 51, Chen, H-L., De Bond, W., 004. Syle momenum wihin he S&P-500 index. Journal of Empirical Finance 11, Chen, N-F., Roll, R., Ross, S. A., Economic forces and he sock marke. Journal of Business 59, Chordia T, Shivakumar, L., 00. Momenum, business cycle and ime-varying expeced reurns. Journal of Finance 57, Daniel, K, Hirshleifer, D., Subrahmanyam, A., Invesor psychology and securiy marke underand overreacions. Journal of Finance 53, Ellis, M., Thomas, D. C., 003. Momenum and he FTSE 350. Journal of Asse Managemen 5, Engle, R. F., 198. Auoregressive condiional heeroscedasiciy wih esimaes of he variance of UK inflaion. Economerica 50, Fama, E. F., French, K., Common risk facors in he reurns on socks and bonds. Journal of Financial Economics 33, Fama, E. F., French, K., Mulifacor explanaions of asse pricing anomalies. Journal of Finance 51, Glosen L. R., Jagannahan, R., Runkle, D. E., On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance 48, Griffin, J. M., Ji, X., Marin, S., 003. Momenum invesing and business cycle risk: Evidence from pole o pole. Journal of Finance 58, Grundy B. D., Marin, J. S., 001. Undersanding he naure of he risks and he source of he rewards o momenum invesing. Review of Financial Sudies 14, 1, Hong, H., Lim, T., Sein, J., 000. Bad news ravels slowly: Size, analys coverage, and he profiabiliy of momenum sraegies. Journal of Finance 55, Hong, H., Sein, J., A unified heory of underreacion, momenum rading and overreacion in asse markes. Journal of Finance 54,

19 Jegadeesh, N., Timan, S., Reurns o buying winners and selling losers: Implicaions for sock marke efficiency. Journal of Finance 48, Jegadeesh, N., Timan, S., 001. Profiabiliy of momenum sraegies: An evaluaion of alernaive explanaions. Journal of Finance 56, Karolyi, A., Kho, B-C., 004. Momenum sraegies: Some boosrap ess. Journal of Empirical Finance 11, Lesmond, D. A., Schill, M. J., Zhou, C., 004. The illusory naure of momenum profis. Journal of Financial Economics 71, Liu W., Srong, N., Xu, X., The profiabiliy of momenum invesing. Journal of Business Finance and Accouning 6, Moskowiz, T. J., Grinbla, M., Do indusries explain momenum?. Journal of Finance 54, Nagel, S., Lewellen, J., 006. The condiional CAPM does no explain asse pricing anomalies. Journal of Financial Economics forhcoming. Nelson D. B., Condiional heeroskedasiciy in asse reurns: A new approach. Economerica 59, Okunev, J., Whie, D., 003. Do momenum-based sraegies sill work in foreign currency markes?. Journal of Financial and Quaniaive Analysis 38, Poon, S. H., Granger, C. W. J., 003. Forecasing volailiy in financial markes: A review. Journal of Economic Lieraure 41, Rabemananjara R., Zakoian, J. M., Threshold ARCH models and asymmeries in volailiy. Journal of Applied Economerics 8, Sadka, R., 006. Momenum and pos-earnings-announcemen drif anomalies: The role of liquidiy risk. Journal of Financial Economics forhcoming. Sharpe, W. F., Capial asse prices: A heory of marke equilibrium under. condiions of risk. Journal of Finance 19, Wang, K. Q., 003. Asse pricing wih condiional informaion: A new es. Journal of Finance 58, Wu, X., 00. A condiional mulifacor model of reurn momenum. Journal of Banking and Finance 6,

20 Table 1 Summary saisics of he reurns of he winner, loser and momenum porfolios Winner (Loser) is an equally-weighed non-overlapping porfolio conaining he 10% of socks ha performed he bes (wors) over a given ranking period. Momenum is a porfolio ha buys he winner porfolio and sells he loser porfolio shor. Reurns are measured as proporions raher han percenages. Reward-o-risk raio is he raio of he monhly mean o he monhly sandard deviaion. The p-values in parenheses are for he significance of he mean. They are based on heeroscedasiciy and auocorrelaion robus (Newey-Wes) sandard errors. Holding period of 3 monhs Holding period of 6 monhs Holding period of 1 monhs Winner Loser Momenum Winner Loser Momenum Winner Loser Momenum Panel A: Ranking period of 3 monhs Mean (0.0) (0.) (0.04) (0.01) (0.08) (0.00) (0.01) (0.09) (0.01) Sandard deviaion Reward-o-risk raio Panel B: Ranking period of 6 monhs Mean (0.00) (0.05) (0.00) (0.00) (0.0) (0.00) (0.00) (0.08) (0.00) Sandard deviaion Reward-o-risk raio Panel C: Ranking period of 1 monhs Mean (0.00) (0.13) (0.00) (0.00) (0.04) (0.00) (0.00) (0.11) (0.00) Sandard deviaion Reward-o-risk raio

21 Table Saic marke and Fama and French models The able repors coefficien esimaes for equaions (1) and () for he winner, loser and momenum porfolios. Winner (Loser) is an equally-weighed non-overlapping porfolio conaining he 10% of socks ha performed he bes (wors) over a given ranking period. Momenum is a porfolio ha buys he winner porfolio and shor sells he loser porfolio. α measures he porfolio s abnormal performance, β measures he marke risk of he porfolio, s and h are he porfolio loadings on he size and book-o-marke value facors as measured by Fama and French (1993). MM refers o he marke model and FFM refers o he Fama and French model. Whie s heeroscedasiciy robus - saisics are in parenheses. Holding period of 3 monhs Holding period of 6 monhs Holding period of 1 monhs Winner Loser Momenum Winner Loser Momenum Winner Loser Momenum MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM Panel A: Ranking period of 3 monhs α (-0.87) (-.33) (-3.81) (-6.4) (3.46) (4.11 ) (0.03) (-0.71) (-4.59) (-7.80) (4.8) (6.05 ) (-0.8) (-1.01) (-4.68) (-8.49) (4.60) (6.47 ) β (11.79) (6.08 ) (1.15) (13.64 ) (-1.6) (-1.73) (11.35) (.83 ) (1.06) (13.77 ) (-1.11) (-1.39) (16.03) (9.46 ) (1.4) (17.58 ) (0.3) (-0.49) s (17.45 ) (11.98 ) (-0.89) (16.45 ) (1.7 ) (-1.05) (17.3 ) (14.98 ) (-0.69) h (-1.43) (1.69 ) (-.05) (-1.58) (.3 ) (-.36) (-.10) (4.3 ) (-4.04) Panel B: Ranking period of 6 monhs α (1.05) (1.34 ) (-4.78) (-7.89) (5.67) (7.03 ) (1.8) (1.75 ) (-5.4) (-9.03) (6.60) (8.13 ) (0.07) (-0.39) (-4.77) (-8.3) (4.89) (6.64 ) β (11.8) (.13 ) (11.78) (13.31 ) (-1.9) (-1.64) (11.87) (.39 ) (11.77) (13.77 ) (-0.80) (-1.1) (15.39) (31.08 ) (1.09) (15.50 ) (-0.39) (-0.94) s (16.84 ) (1.04 ) (-1.49) (17.19 ) (1.87 ) (-1.31) (17.65 ) (13.59 ) (-0.99) h (-.47) (.57 ) (-.89) (-.6) (.79 ) (-3.03) (-.1) (4.7 ) (-3.9) Panel C: Ranking period of 1 monhs α (1.69) (.68 ) (-4.46) (-7.06) (5.63) (7.0 ) (.00) (3.3 ) (-5.01) (-8.66) (6.68) (8.69 ) (0.64) (0.76 ) (-4.59) (-8.53) (5.48) (7.73 ) β (13.71) (4.79 ) (10.89) (11.54 ) (0.7) (-0.19) (13.6) (4.14 ) (10.48) (11.58 ) (0.7) (-0.40) (16.15) (35.74 ) (1.47) (16.48 ) (0.51) (-0.46) s (16.97 ) (10.73 ) (-0.84) (17.08 ) (1.35 ) (-1.73) (18.87 ) (14.96 ) (-1.71) h (-.97) (3.35 ) (-4.07) (-.57) (4.56 ) (-4.71) (-.43) (5.13 ) (-4.91) 19

22 Table 3 Condiional marke and Fama and French models wih a GJR-GARCH (1, 1)-M erm The able repors coefficien esimaes for sysems (3) and (4) for he winner, loser and momenum porfolios. Winner (Loser) is an equallyweighed non-overlapping porfolio conaining he 10% of socks ha performed he bes (wors) over a given ranking period. Momenum is a porfolio ha buys he winner porfolio and shor sells he loser porfolio. α measures he porfolio s abnormal performance, β measures he marke risk of he porfolio, s and h are he porfolio loadings on he size and book-o-marke value facors as measured by Fama and French (1993), δσ is he ime-varying risk exposure. The condiional variance of he porfolio reurns follows a GJR-GARCH(1,1) srucure as σ = ω + γε 1 + ηi 1ε 1 + θσ 1, where ω, γ, η and θ are esimaed parameers and I -1 akes a value of 1, when ε -1 is negaive and a value of 0, oherwise. MM refers o he marke model and FFM refers o he Fama and French model. Bollerslev-Wooldridge robus -saisics are in parenheses. Holding period of 3 monhs Holding period of 6 monhs Holding period of 1 monhs Winner Loser Momenum Winner Loser Momenum Winner Loser Momenum MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM MM FFM Panel A: Ranking period of 3 monhs α (-0.17) (.48) (-1.41) (1.67) (-0.44) (-0.10) (1.55) (.15) (.64) (1.74) (-0.19) (0.48 ) (0.7) (3.11) (10.0) (.14) (0.1 ) (0.45 ) β (14.87) (34.9) (15.71) (15.89) (-1.67) (-3.61) (.31) (8.14) (19.30) (1.94) (-0.90) (-4.) (4.08) (7.37) (45.65) (38.61) (.19 ) (0.61 ) s (1.85) (18.48) (-3.56) (1.36) (.51) (-4.91) (5.19) (3.53) (-.38) h (1.07) (4.43) (-3.3) (.4) (6.9) (-4.34) (.10) (8.39) (-3.87) δ (-0.08) (-.66) (-0.07) (-3.31) (1.65 ) (1.49 ) (-1.11) (-.05) (-3.34) (-4.33) (1.63 ) (1.78 ) (-0.0) (-3.17) (-7.91) (-3.39) (.09 ) (1.91 ) ω (.90) (.4) (1.1) (1.45) (1.63 ) (1.4 ) (3.0) (3.48) (1.69) (1.48) (1.64 ) (1.51 ) (.00) (1.58) (-0.1) (.7) (1.48 ) (1.4 ) γ (3.09) (.49) (1.73) (1.4) (.75 ) (.56 ) (.5) (.65) (-.63) (1.3) (3.45 ) (3.97 ) (.7) (.15) (-0.53) (-0.56) (.89 ) (.94 ) η (-.15) (0.0) (1.11) (.06) (-.41) (-.6) (-0.41) (-0.4) (3.4) (.63) (-.46) (-.60) (-0.83) (-1.31) (6.97) (3.1) (-.3) (-1.88) θ (5.71) (10.06) (1.84) (7.88) (6.79 ) (7.84 ) (4.1) (3.18) (16.1) (1.95) (5.7 ) (7.39 ) (8.68) (4.0) (31.56) (11.98) (8.44 ) (10.03 ) γ +η /+θ

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