Does Noise Create the Size and Value Effects?

Transcription

1 Dos Nois Cra h Siz and Valu ffcs? Robr Arno Rsarch Affilias, LLC Jason Hsu Rsarch Affilias, LLC and Univrsiy of California, Irvin Jun Liu Univrsiy of California, San Digo Harry Markowiz Univrsiy of California, San Digo Firs Draf: March 005 Currn Draf: March 5, 007 Plas do no quo wihou prmission. W would lik o hank Andrw Ang, John Hughs, Fifi Li, Jing Liu, Allan Timmrmann, and Rossn Valkanov for hlpful discussions. W would lik o hank Viali Kalsnik for rgrssion analysis on h dividnd yilds of individual socks. lcronic copy of his papr is availabl a: hp://ssrn.com/absrac=9367

2 Absrac Dos Nois Cra h Siz and Valu ffcs? Black 986 and Summrs 986 suggs ha hr is nois in sock prics in a sns ha h pric of a sock can b randomly diffrn from is inrinsic valu. Such nois can aris from conomic modls.g., Grossman and Sigliz 980 and D Long, Shlifr, Summrs, and and Robr J. Waldmann 990, mark microsrucur.g., Sambaugh 983 and Roll 983, among ohr sourcs. In his papr, w show ha whn hr is nois in h pric of a sock, is xpcd rurn condiional on h pric or h pric-dividnd raio dcrass wih h pric or h pric dividnd-raio. Ths highr xpcd rurns associad wih lowr pric or pric-dividnd raios ar no compnsaion for risk, bu ar gnrad bcaus a sock wih a low pric or a pric-raio is mor likly o hav a ngaiv pric nois hus o b undrvalud. Fama and Frnch 99 us h marix of xpcd rurns condiional on siz-valu dcils as a dmonsraion of siz and valu ffcs. This marix can b compud in closd form using our modl and, for plausibl paramrs, is similar o is mpirical counrpar Tabl V of Fama and Frnch. In our modl, small and valu socks hav slighly highr bas and posiiv alphas. Our sudy suggss ha nois cras h siz and valu ffc. lcronic copy of his papr is availabl a: hp://ssrn.com/absrac=9367

3 Inroducion Many conomiss would agr ha h mark pric of a sock may mporarily dvias from is fundamnal valu. In fac, Blum and Sambaugh 983, Roll 93, Black 986, and Summrs 986, among many ohrs, suggs ha nois may play an imporan rol in financial marks. Howvr, i is no asy o dc h prsnc of hs mporary dviaions, as poind by Summrs 986, Fama and Frnch 988, and Porba and Summrs 988. A h sam im, h cross scion of xpcd rurns prdicd by conomic horis dos no mach ha obsrvd in h daa. In paricular, socks wih a low pric mark capializaion and/or pric-o-fundamnal raio hav highr xpcd rurns, as summarizd by h marix Fama and Frnch 99, Tabl V of xpcd rurn condiional on siz and valu dcils. In his papr, w dmonsra ha nois, a mporary random dviaion of sock prics from hir fundamnals, would produc cross-scional variaions in xpcd rurns. W show ha h marix of xpcd rurns condiional on siz and valu dcils compud using our modl is similar o h marix of Fama and Frnch 99. Thrfor, w suggs ha pric nois cras and manifss islf hrough h siz and valu ffc. Spcifically, wih a simpl and parsimonious modl, whr h valu procss is assumd o b a random walk and h nois is a man-rvring AR procss, w compu xplicily h uncondiional xpcd rurn and show ha nois inroducs xpcd rurns dpndnc on h dividnd yild and idiosyncraic volailiy, in addiion o ba. Th cross scional variaion in uncondiional xpcd rurns is gnrad by variaions in paramrs such as ba, idiosyncraic volailiy, dividnd-pric raio, and volailiy in nois. Mor imporanly, w show ha h cross-scional variaions in condiional xpcd rurns ar gnrad by random ralizaion of h pric nois wihou any paramr variaion. Th marix of Fama and Frnch 99 dmonsras ha h xpcd rurn, condiional pric and pric raio, dcrass wih pric and pric raio. W compu xplicily h xpcd rurn condiional on pric and pric raio and w show ha h condiional xpcd rurn dcrass wih pric and pric raio. Wih plausibl paramrs for nois, whr h condiional volailiy of nois is abou 6%, h marix of xpcd rurn condiional on siz and valu dcils prdicd by our modl is similar o ha of Fama and Frnch 99. In our modl, h siz and valu ffcs hav h sam sourc nois. Th inuiion is h following. A sock wih a posiiv nois should hav a lowr xpcd rurn. Alhough nois is unobsrvabl, hy can b infrrd from prics: nois for a sock is mor likly posiiv if is pric is high. Th sam inuiion applis for pric-book as wll as a variy of ohr pric-fundamnal raios. Throughou his papr, w assum ha firms hav only on sock shar ousanding. Thrfor, w can us pric, mark capializaion, and mark quiy as in Fama and Frnch 99 inrchangably. In his papr, w us valu o man h fundamnal or raional valu of a sock and us i in valu ffcs. Hopfully, which usag of h rm will b clar from conx.

4 Our modl prdics ha small and valu socks ar on avrag riskir, in h sns ha boh sysmaic and idiosyncraic risks ar highr. Th avrag ba 3 and h avrag idiosyncraic volailiy wih nois ar a fw prcns highr han h avrags wihou nois, givn h paramrs calibrad o US mark daa. Howvr, h highr xpcd rurns in small and valu socks canno b accound for by slighly highr sysmaic risks. Thy ar drivn mosly by pricing nois in h sock mark. Our rsul suggs ha valu socks ar, indd, mor likly o b undrvalud. W should rmark ha i is possibl ha highr xpcd rurns of small and valu may no prsis ovr im. On h ohr hand, hy may prsis ovr im du o limi of arbirag, associad wih ihr risks of small and valu socks or ransacional coss. W should poin ou ha boh h nois and h valu procss ar xognously givn in our papr. Th valu procss, which is a Gaussian random walk in h papr, is usd in many acadmic sudis and can b gnrad in an quilibrium modl. This spcificaion is usful for closd-form soluion for h siz and valu sprad. In gnral, h valu procss from ass pricing horis may no hav h xac form w assumd, howvr h inuiion sill applis. Th nois, which dscribs dviaions from quilibrium, is xognously spcifid as a man rvring procss. Our spcificaion of h nois is qui inuiiv and plausibl and is usd xnsivly in liraur Summrs 986, Porba and Summrs 988, Fama and Frnch 988, and Campbll and Kyl 993, o nam a fw. To ndogniz h nois procss, a modl of off-quilibrium is ndd, which is byond h scop of his papr. Our papr is organizd as follows. In Scion, w rviw h rlad liraur brifly. In Scion 3, w formally inroduc h modl of nois and spcify h paramrs of h modl. In Scion 4, w xplor h implicaion on uncondiional xpcd sock rurns in h prsnc of pricing nois. W show ha socks wih grar nois arn highr rurns, on avrag. In Scion 5, w giv h inuiion for h xpcd rurns condiional pric and pric raios. In Scions 6 and 7, w show ha h nois producs h siz and valu ffcs. In scion 8, w compu h marix of xpcd rurn condiional on siz and valu simulanously. W compu h marix of xpcd rurns, ba, and alpha condiional on siz and valu dcils. In Scion 9, w compu xpcd rurns condiional on ihr on rurn or on h full hisory of prics. Finally, Scion 0 concluds. Liraur Rviw Nois is usd in raional financ modls. Blum and Sambaugh 983 and Roll 983, 984 argu ha obsrvd pric is ihr h bid or h ask, no h valu, hus pric is diffrn from valu by a random nois rm. 4 In rm srucur modls, whr h numbr of shocks is usually smallr han h numbr of 3 Lakonishok, Shlifr, and Vishny 994 found ha h ba of h valu socks is abou 0. highr han h ba of h growh socks. 4 Thr ar subsqunly many sudis in mark microsrucur liraur on nois in prics. S for xampl, Danil, Hirshlifr, and Subrahmanyam 00 and Chordia, Roll, and Subrahmanyam 005. Howvr, nois considrd in his papr is lss likly du o mark microsrucur.

5 indpndn scuriis, i is assumd ha h mark prics for bonds ar diffrn from h modl drivd fair valus by a nois. Thorically, in, for xampl, Grossman and Sigliz 980 and D Long, Shlifr, Summrs, and and Robr J. Waldmann 990, pric nois is gnrad by an xognously-spcifid dmand of nois radr. Th origin of mispricing could b du o slownss o incorpora informaion. vn sudis suggs ha i aks abou wks for informaion on mrgrs o b impoundd in h pric. Pric can b diffrn from valu if invsors undr- or ovr-rac. Wih random ralizaion of posiiv or ngaiv nws, ovr- or undr- racion prsumably should gnra nois random dviaion from valu. No ha ovr- or undr-racion is diffrn from opimism or pssimism, which w xpc o gnra biasd dviaions from h valu. In bhavioral financ liraur, pricing rror can aris from invsor ovrracion, as suggsd by Shillr 98, DBond and Thalr 985, 987, Lakonishok, Vishny, and Shlifr 994. In Campbll and Kyl 993 valu is drmind ndognously, bu h pric is diffrn from valu by a man-rvring nois ha is xognously spcifid. Thy show ha his modl can xplain h volailiy and prdicabiliy of h US sock rurns. Black 986 proposs ha financial marks ar noisy ha prics ar diffrn from fair valus du o rading by invsors wihou informaion. H blivs ha nois causs h mark o b somwha infficin bu y prvn popl from aking advanag of infficincis. Summrs 986 argus ha prics ar noisy, bu h powr of h sandard conomric ss ar simply oo wak o ihr dc nois or rjc h fficin Mark Hypohsis. Summrs argus ha h nois is difficul o discrn using varianc raios and auocorrlaions. Our rsuls suggs ha nois manifss islf hrough xpcd rurns in siz and valu ffcs. Fama and Frnch 988 and Porba and Summrs 988 sudy man-rvrsion in prics and poin ou ha on of h possibl xplanaion for man rvrsion is h dviaion of pric from h fficin mark valu. Thy infr h xisnc and propris of nois from h auocorrlaion of rurns. Th siz and valu ffcs hav spurrd spirid dbas sinc Banz 98 and Ringanum 98 documnd ha smallr capializaion socks nd o ouprform on a risk-adjusd basis, and Saman 980 and Rosnbrg, Rid and Lansin 985 documnd ha high book-mark socks also ouprform. Similarly, ohr raios such as arnings-pric, documnd by Basu 977 and dividnd yild, documnd by Razff 984, Shillr 984, Blum 980 and Kim 985, also prdic fuur prformanc. Thr ar many xplanaions for h obsrvd siz and valu ffcs. Fama and Frnch 99 show ha siz and valu, along wih mark ba, capur wll h cross-scional variaion in sock rurns and subsum h xplanaory powrs of ohr financial variabls. Thy propos ha h siz and valu prmia ar compnsaion for risk. Lakonishok, Shlifr and Vishny 994 argu ha h siz and valu prmia ar du o invsor ovrracion rahr han o risk. Goms, Kogan, and Zhang 003 Zhang 006 argus 3

6 ha h valu ffc can b xplaind in a producion conomy. Yogo 006 proposs ha h siz and valu ffcs can b xplaind by invsor prfrncs ha ar non-sparabl in nondurabl and durabl consumpion. Blum and Sambaugh 983 suggs h random bounc bwn bid and ask prics as on sourc of nois and hy us i o xplain h siz ffc. Thy show ha h uncondiional xpcd rurns incrass wih h varianc of h nois. Howvr, hy did no compu h xpcd rurns condiional on h pric. Furhrmor, h bid-ask bounc is usful for xplaining ffcs in daily rurns bu is lss likly h caus for ffcs ha occur a quarrly or annual horizons and h siz ffc is obsrvd in hs horizons. Brk 995, 997 suggss ha nois as a sourc of siz and valu ffcs. H poins ou ha hr is a on-o-on corrspondnc bwn pric and xpcd rurn hus bwn pric and ba. If h xpcd rurn is corrcly spcifid, afr conrolling for ba, hr is no pric dpndnc in xpcd rurns. Howvr, if h xpcd rurn is misspcifid, h pric dpndnc of h missing ba shows up as pric dpndnc of h xpcd rurn. In Brk 995, 997, small socks hav highr xpcd rurns bcaus hy hav highr sysmaic risk. Whras in our papr, h highr xpcd rurn of valu sock is mainly du o h fac ha hy ar likly o b undrvalud. 5 An mpirical vidnc ha disinguishs Brk modl from our modl would b whhr small and valu socks ar xposd o significanly highr sysmaic risks. Arno, Hsu, and Moor 005 and Arno 005a, b also propos ha nois as a likly sourc for siz and valu ffcs. Hsu 006 shows ha mispricing prmium may xis bcaus hr ar invsors wih liquidiy nds. Arno and Hsu 006 show ha man-rvring mispricing can lad o small cap and valu sock ouprformanc; hy prdic ha siz and valu migh b wo manifsaions of on ffc, pricing nois. Brnnan and Wang 006 also sudy, mpirically as wll as horically, h ffc of mispricing on uncondiional xpcd rurns for a largr class of modls, whr mispricings can b du o slownss in adjusmn of pric and sysmaic mispricing in addiion o random nois. Thy did no sudy condiional xpcd rurns which ar our focus. 3 Nois In his scion, w discuss ky assumpions and chnical assumpions of h papr. Th following is h ky assumpion of h papr. Assumpion vry sock has a valu, which is drmind by conomic hory. Th pric of a 5 Th following xampl illusra h diffrnc bwn our modl and ha of h Brk. In an conomy whr h sock rurns ar idnically-disribud bu ar corrlad hrough common facors, h xpcd rurn will b indpndn of h prics undr Brk 995, 997 whil socks wih a lowr pric ar mor likly o hav a highr xpcd rurn undr our modl. 4

7 sock dvias from is valu by a nois. Spcifically, =, whr is indpndn of V s for all and s and is h uncondiional xpcaion of. Th dividnd D of h sock is also indpndn of s, for all and s. In assumpion, h hory ha drmins h valu is unspcifid and can b consumpion-basd ass pricing modls, CAPM, or APT, jus o nam a fw. Th valu is h pric if hr wr no nois and has all h nic propris, for xampl, h xpcd rurn compud using is drmind by risk and hus h cross scion of xpcd rurns compud using is drmind by ba only if h ass pricing modl is APT. For our purpos, i is no ncssary o dfin how h mark arrivs a his valu. Howvr, i may b convnin o hink of h discound cashflow valuaion quaion whr = s= µs D s, whr µ is h discoun ra and D s is h dividnd a im s. Assumpion implis ha =. Tha is, h pric for a sock is a noisy proxy for is valu, which w assum is unobsrvabl, and h pric is, on avrag, righ. Th assumpion on dividnd D is ncssary for drawing conclusion on rurns sinc dividnd D + is par of h cashflow for +, in addiion o h pric +. Wihou loss of gnraliy, w will assum ha = 0. Black 986 also argus ha hr migh b a diffrnc bwn h pric and h fair valu of a sock bu h dos no prsn a form analysis. Summrs 986 assums an addiiv form, = +. Summrs assrs, This assumpion of pricing nois clarly capurs Kyns s noion ha marks ar somims drivn by animal spiris unrlad o conomic aciviis. I, also, is consisn wih h xprimnal vidnc of Tvrsky and Kahnman ha subjcs ovrrac o nw informaion in making probabilisic judgmns. Th formulaion considrd hr also capurs Robr Shillr s suggsion ha financial marks display xcss volailiy and ovrrac o nw informaion. W rmark ha h nois in Assumpion is spcifid in muliplicaiv form, which is usd in Blum and Sambaugh 983 and Fama and Frnch 988 s also Hsu 006. Th addiiv form of Summrs 986 implis ha h nois bcoms ngligibl ovr im as grows, if is saionary as Summrs assums. Aboody, Hughs, and Liu 00 also assum an addiiv form. Campbll and Kyl 993 rcogniz his problm and us an addiiv form wih d-rndd dividnds. Such a problm dos no aris from h muliplicaiv form. Many of h qualiaiv rsuls of h papr follows from his assumpion. W will mak mor chnical assumpions for quaniaiv rsuls. Assumpion Th nois saisfis, whr ɛ ar indpndn sandard normals. + = ρ + ɛ ɛ +, 3 5

8 Whn ρ <, is man-rvring and saionary. This implis ha a nois + a im +, on avrag, will lad o smallr nois a im. Th man rvrsion of owards zro capurs h inuiion ha informaion is slowly impoundd ino prics. Whn ρ = 0, h nois is indpndn and idnically disribud IID. If ρ =, h nois + will b qual o on avrag. In his cas, h nois is infinily prsisn and pric lvls do no prdic rurns R + P 0... = R +. Whhr nois is man rvring or no is an mpirical qusion. To avoid cubrsom noaions, h rs of h papr will assum ha ρ <. Prsumably, h mark ss pric o b is bs sima of, hrfor should rvr owards valu, as nw informaion bcoms known. Howvr, mos of h drivaion in h papr gos hrough wih minor changs if ρ =. W assum ha ɛ is a consan. This assumpion may b a lil rsriciv sinc ɛ could b sa dpndn. For xampl, nois during conomic xpansions may hav a diffrn volailiy from nois during rcssions. Similar spcificaions of h nois follow from Blum and Sambaugh 983, Summrs 986, Fama and Frnch 988, Aboody, Hughs, and Liu 00, Arno and Hsu 006, Hsu 006, and Brnnan and Wang 006. For as of xposiion, w dno h logarihm of by v and logarihm of by p, = v ; = p. 4 quaion can hn b wrin as p = v + ln. 5 W call ++D + h valu rurn R v +, which is dicad by som ass pricing modl. W call ++D + h rurn R +. W will us d = ln D o dno h logarihm of h im dividnd D. W mak h following assumpion on h valu and h valu-dividnd raio. Assumpion 3 Th valu v is a random walk, Th valu-dividnd raio saisfis v + = µ + v + r ɛ r+. 6 v + d + = ρ x x v + ρ x v d + ɛx ɛ x+. 7 Furhrmor, v is indpndn of v s d s for all and s. Assumpion implis ha, if hr is no dividnd, µ is h man of h log-valu-rurn v + v and r is h volailiy. According o Assumpion 3, h valu-o-dividnd raio v d has a man of x v and condiional volailiy of ɛx, and is man rvring wih cofficin ρ x. quaions 6 and 7 in Assumpion 3 ar usd in h liraur on prdiciv rgrssions, s for xampl, Sambaugh 999 and Valkanov and Torous No ha hr is no pric nois in hs sudis, hus h valu-dividnd raio is h pric-dividnd raio. 6

9 Tabl : Summary of Paramrs µ r ɛ ρ x v ρ x ɛx 3% 30% 6% % Th calibraion of hs paramrs ar dscribd in Scion 3. Ass pricing modls ypically drmin h valu-o-dividnd raio from prfrncs of h invsors. For xampl, in h consumpion-basd ass pricing modl whr h rprsnaiv agn has consan rlaiv risk avrsion cofficin and h dividnd growh is indpndn and idnically disribud IID ovr im, h valu-o-dividnd raio is consan. Howvr, in mos modls, h valu-o-dividnd raio is sochasic and saionary. Th abov spcificaion is an approximaion and a simplificaion o a saionary valu-o-dividnd raio. Wih h valu procss and valu-dividnd raio procss spcifid as abov, h dividnd growh procss is implicily drmind. S Ang and Liu 006 for a discussion on rlad issus. Assumpions and 3 ar ndd o obain closd-form infrnc on nois from prics and pric raios. Wih ohr non-gaussian spcificaions, i is no asy o compu in closd form h infrnc abou h nois, bu h sam inuiion applis. Th indpndnc assumpion bwn v and v d is mad o simplify h xprssion. Closd-form infrnc sill obains if h corrlaion is a non-zro consan. Whn hr ar mulipl socks, h shocks ɛ +, ɛ r+, and ɛ x+ could all hav sysmaic componns as wll as idiosyncraic componns. As w will show lar, our rsuls in lar scions sill apply wih a rinrpraion of paramrs whn h corrlaion bwn socks ar inroducd hrough common sysmaic facors. W calibra h abov spcificaion as follows, wih all h paramrs summarizd in Tabl. Th paramr µ only affcs h ovrall magniud of h xpcd rurn. W ak µ o b 0%. Sinc h man and volailiy of h pric-dividnd raio ar small, h volailiy of h sock rurn is largly du o pric flucuaions. No ha, from Assumpions,, and 3, p + p = v + v + + = µ + ρ + r ɛ r+ + ɛ ɛ +, hus, h varianc of h rurn is h sum of h varianc r of h valu rurn v + v and h condiional varianc of h nois +. W will ak r = 5% and ɛ = r /3 5%. Th raio of r / ɛ = 3 givs a raio bwn varianc of h nois and oal varianc of h sock rurn of 0%. Frnch and Roll 986 suggs ha bwn 4% and % of h daily rurn variancs is causd by nois. Fama and Frnch 988 sima ha prdicabl variaion du o man rvrsion is abou 35 prcn of 3-5 yar variancs and hy suggs, following Summrs 986, ha h man-rvrsion may b du o mark infficincy. In his calibraion xrciss, Summrs 986 uss h valus of r ha is of h sam ordr of magniud as. Th valu of ρ can b infrrd from man-rvrsion in prics, assuming h man rvrsion is du o nois. Fama and Frnch 988 shows ha hr ar significan man-rvrsion in prics for holding-priod 7

10 horizons largr han yar. Summrs 986 uss valus of ρ bwn 0.75 o and Porba and Summrs 988 us valus bwn 0 and W will considr a rang of ρ, as Summrs and Porba and Summrs. Howvr, h valu and siz ffc is no ovrly snsiiv o ρ, as long as 0 < ρ <. Th calibraion of paramrs for valu-dividnd raio ar basd on h sudis of Sambaugh 999 and Valkanov and Torous 005 on h prdiciv rgrssion of h mark porfolio. Thy found ha h man dividnd raio is abou 3%, h AR cofficin is abov 0.9 and h condiional volailiy is lss han %. Bcaus nois largly avrags ou in h mark porfolio, 7 w xpcd h man and AR cofficin for h valu-raio procss should b in h nighborhood of hir simas for for h mark, hus w s x v = 4 and ρ x = 0.9. W will s ɛx = 0%. 4 Uncondiional xpcd Rurns In his scion, w sudy h implicaions of nois on uncondiional xpcd rurns. W show ha nois can gnra cross-scional variaions in uncondiional xpcd sock rurns. From quaion and by h saionariy of, w hav + = V + + = L D dno h dividnd of h sock a im. W assum ha i is indpndn of h nois. Thn Th uncondiional xpcd rurn is, P+ + D + = D + = D +. 9 V+ + + D +. 0 Proposiion If Assumpion holds, h xpcd rurn is highr han h xpcd valu rurn. Proof By saionariy, + =, hrfor, + = 0. By Jnsn s inqualiy, + + =. 3 quaion 0 hn givs, P+ = V+ + V Campbll and Kyl 993 sudy pric nois of h mark porfolio. Thir papr suggs ha hr ar sysmaic componns in h pric nois of individual socks. 8

11 Furhrmor, D+ D+ = = D+ Combining inqualiis in 4 and 5, w conclud ha D+. 5 P+ + D + V+ + D +. 6 Blum and Sambaugh 983 suggs ha bid-ask sprads lad o a nois of h form = + ɛ, whr ɛ is man zro and indpndn across h im. Thy show ha h nois incrass h uncondiional xpcd rurn for ρ = 0 and D = 0 cas of h abov Proposiion. Proposiion only rquirs ha h nois is indpndn of h valu and h dividnd. Wih h addiional assumpion ha h nois is an AR procss, w can sablishd an xac rlaionship bwn h uncondiional xpcd rurn and uncondiional xpcd valu rurn. Proposiion If Assumpions and hold, h xpcd rurn is givn by P+ + D + = V+ which is highr han h xpcd valu rurn V+ + holds, hn P+ + D + = µ+ ɛ r +ρ + xv+ Proof Whn quaion 3 holds, w hav ɛ D+ ɛ +ρ + ρ, 7 D+. Furhrmor, if Assumpion 3 also ɛx ρ + x ɛ ρ. 8 ρ + = ρ ɛ ɛ ɛ + = ρ ɛ = ρ, = +ρ ; noing ɛ P+ +D + ɛ ρ, w conclud ha is h uncondiional varianc of ρ. Sinc +ρ and V+ +D +. Whn Assumpion 3 holds, quaion 8 is provd by noing ha V+ D+ = µ+ r ; = V+ D+ + Th uncondiional xpcd rurn in h absnc of nois is µ+ r + x v+ ɛx = µ+ r x v+ ɛx ρ x. ρ x, which should b drmind by ass pricing horis hus should dpnd only on ba undr CAPM or APT. Proposiion and hold wihou any spcificaions of ass pricing hory and hus ar valid qui gnrally. 9

12 Cross-scion variaions in uncondiional xpcd rurns can b gnrad by nois, according o Proposiion. Wih nois, h uncondiional xpcd rurn givn in quaion 8 dpnds also on idiosyncraic volailiy, h volailiy ɛ and AR cofficin ρ of nois and h paramrs x v, x, ρ x of h pric-dividnd raio, in addiion o ba. Tha is, givn wo socks wih ihr diffrn nois varianc or man pric-dividnd raio, h uncondiional xpcd rurns can b diffrn, vn if hy hav h sam sysmaic risk. In ohr words, cross-scional variaions can b gnrad by variaions in hs paramrs. I is no vry saisfacory ha h cross-scional variaion has o b xognously spcifid hrough spcificaion of paramr variaions. On h ohr hand, i is no ru ha on can always gnra cross-scional variaions in xpcd rurns wih paramr variaions. For xampl, in sandard ass pricing modls such as CAPM and APT, variaions in idiosyncraic volailiis do no gnra cross-scional variaions in xpcd rurns. From h abov Proposiion, h ffc of nois on uncondiional xpcd rurns is a h ordr of. Wih a valu of 6% for, givn in Tabl, h chang in uncondiional xpcd rurns is abou 36 basis poin. Howvr, if = 0%, which is no unrasonabl for som socks, h chang will b %. Th diffrnc bwn h uncondiional xpcd rurn and uncondiional xpcd valu rurn is du o Jnsn s inqualiy, which is drivn by h varianc of h random variabl. Thrfor i is only naural ha h diffrnc bwn h xpcd rurn and valu rurn incrass wih for ρ <. Proposiion and ar mor gnralizd vrsions of h rsul prsnd in Hsu 006. Brnnan and Wang 006 also driv similar rsuls. Blum and Sambaugh 983 compu h uncondiional xpcd rurn for ρ = 0 and D = 0 cas of Proposiion. Thy show ha h siz ffc obsrvd in daily rurns can b xplaind by h nois hy suggsd. Brk 997 compus uncondiional cross-scion corrlaion bwn pric and h rurn. As in our modl, h cross scional variaion in uncondiional xpcd rurns in Brk 997 nds o b gnrad from variaions in paramrs. On implicaion of our papr is ha, cris paribus, a lss ransparn sock on ha is mor likly o b mispricd and hrfor has a highr ɛ will hav a highr uncondiional xpcd rurn. This is consisn wih rcn mpirical findings whr h cos of capial for a firm, conrolling for ba, is highr whn h firm is lss ransparn. Hughs, Liu, and Liu 006 argu ha hs mpirical findings may no b xplaind by risk. Th proposiions suggs ha nois could provid a ponial xplanaion for his mpirical finding. Shillr 98 poins ou ha h rurn varianc for a sock, in a world wih IID dividnd growh and CRRA rprsnaiv prfrnc, should b qual o h varianc of is dividnd growh. Howvr, mpirically, h varianc in sock dividnd growh is lowr han h varianc in rurn, giving ris o Shillr s xcss-volailiy puzzl. In our modl, h varianc of h rurn is h sum of h varianc of h 0

13 valu rurn and h varianc of h nois. This ponially offrs a prhaps indlica xplanaion for h xcss-volailiy puzzl, as suggsd in Campbll and Kyl 993. In lar scions, h condiional xpcd rurn will b compard wih quaion 8. 5 Th Inuiion for Condiional xpcd Rurns In his scion, w prsn h inuiion for why xpcd rurns dpnd on pric or pric raios whn hr is nois in pric. L us firs assum ha h nois is obsrvd. In his scion and his scion only, for h noaional simpliciy, w will us h addiiv form of nois: = +. I hn follows ha Th facor + + D + = + + D D + = is h rlaiv mispricing a im, ++D rurn wihou nois, and + + rurn saisfis h following rlaion is h valu rurn, which is h is du o nois a im +. To b spcific, w will assum ha h valu + + D + = R f + βλ + βf + + r ɛ r+, which is ru undr ihr CAPM or APT. Th gross risk-fr ra is R f, h facor is F +, h facor risk prmium is λ, idiosyncraic risk is givn by ɛ r+, and h idiosyncraic volailiy is r. W can wri + + D + = R f + R f + βλ + βf + + r ɛ r This quaion implis ha h ba and volailiy of h rurn is scald by a facor of prmium is also scald by h sam facor. Thus, R f + +. Th risk V + βλ + βf + + r ɛ r+ is a fair rurn wih horically corrc compnsaion. Th rm + R f rprsns h xra rurn sprad ha is no associad wih risk bu is associad wih mispricing gnrad by nois. Whn < 0, h sock is undr-valud and h sprad is posiiv. No ha in his cas, boh sysmaic risk and idiosyncraic risk ar highr. Furhrmor, + + = ρ + ɛ ɛ + +. Whn h AR cofficin ρ of h nois is no zro, h pricing rror produc by nois a im will b prsisn and lad o an avrag pricing rror of ρ + a im +, hus lading o an xra rm

14 ρ + in xpcd rurn. Puing all rms oghr, h rurn is + + D + = R f ρ + R f + βλ + βf + r ɛ r+ + ɛ ɛ = R f ρ + R f + βλ + βf + r ɛ r+ + ɛ ɛ +. Accordingly, suppos ha hr is a ngaiv pricing rror a im, < 0, h idiosyncraic risk will b highr bcaus boh r > r and hr is an xra risk associad wih nois a im +, h ba hus h risk prmium associad wih h facor risk will b highr. In addiion, hr is an alpha rm, R f ρ, which is du o h fac ha h sock is undr-valud. In raliy, w do no obsrvd h nois. Howvr, w can sill infr from h pric or pric raios. Th lowr h pric or h pric raios, h mor likly is ngaiv and h sock is undr-valud. Undr h Gaussian sing spcifid in Assumpions -3, h infrnc can b prcisly compud. In h rs of h papr, w will compu h avrag givn or pric raios and hus h xpcd rurn condiional on or pric raios. No ha in Brk 995, 997, highr xpcd rurns for low-pricd socks ar du o highr sysmaic risks, which is diffrn from ours. 6 Th Siz ffc In his scion, w sudy h xpcd rurn, condiional on h currn pric. W show ha h condiional xpcd rurn dcrass wih. W also compu h xpcd rurn condiional on pric dcils. No ha h rurn is, + + D + = D +. 0 W ar inrsd in h xpcd rurn, condiional on h currn pric, P+ + D +. As w nod prviously, h valu rurn ++D + is drmind by pricing modls and may hav sysmaic as wll as idiosyncraic componn; for our purpos, i is no ncssary o spcify his. Similarly, may also hav sysmaic componns, as in Campbll and Kyl 993. Th sysmaic componns will no affc h infrncs on individual nois in an conomy wih a larg numbr of socks, as w shown in h appndix. No ha p = v + ln. To draw infrnc of nois from pric p, w nd o know h join disribuion of v and. I is naural o assum ha h disribuion of is is saionary disribuion, which has man of 0 and varianc of ɛ ρ. Sinc v is no saionary, hr is no naural choic of disribuion for v. W will assum ha v is normal wih man v and varianc v. From Assumpions,, 3, v and ar indpndn.

15 Proposiion 3 Suppos Assumpions,, and 3 hold. Furhrmor, assum ha h disribuion of is is uncondiional disribuion and h disribuion of v is normal wih man v and varianc v. Thn h xpcd rurn condiional on is P+ + D + p whr γ = ɛ ρ v + ɛ. = µ+ r +ρ P ργ P ργ + x v+ ɛx ρ + ɛ P γ x ρ P γ, Th proof is givn in h appndix. I is clar ha h xpcd rurn, condiional on, dcrass wih. Th rsuls from h proposiion is inuiiv. Considr h cas whr h nois is indpndn ovr im ρ = 0. In his cas, + = + +. Th xpcaion of + condiional on is indpndn of whn ρ = 0. Thus, h xpcd rurn will b dcrasing in. If hr is a ngaiv nois, h sock is undr-valud, so ha h subsqun rurn is high on avrag. Clarly, w do no obsrv ; howvr, w can infr informaion on from obsrving. Tha is, h pric can b a noisy signal for h nois. Rcall, p = v + ln. 3 Thrfor, h highr h p, h highr h probabl pricing rror, on avrag, and h lowr h nx priod rurn. In his papr, w do no assum ha ρ = 0, hus + nd no b indpndn of. This is plausibl sinc som forms of pricing rror may rquir monhs or yars o b idnifid and corrcd by h mark. Whn 0 < ρ <, h ffc of nois on rurn should b rducd. In his cas, a posiiv ralizaion of nois a im implis on avrag a posiiv raliaion a +, alhough h i will b smallr. Suppos, for xampl, h nois is prsisn; in his cas, ρ approachs, and is a random walk. If his is h cas, alhough h nois affcs h mark pric, i dos no affc h rurn bcaus h rror dos no corrc ovr im; an undr-valud sock rmains undr-valud. W should rmark ha in Proposiion 3, h paramr µ is assumd o b a consan. This implis ha h xpcd valu rurn is indpndn of valu v, which is ru in many ass pricing horis, such as Capial Ass Pricing Modl CAPM and h Arbirag Pricing Thory APT and can b obaind mor or lss undr homohic prfrnc. Howvr, his assumpion dos no always hold. For xampl, Black and Lirman 99 assum ha h risk prmium of a sock should b proporional o is mark cap which is pric, which is an asy way o clar h mark. In his cas, µ dpnds linarly on v. Dpnding on rlaiv magniud of h cofficin of his linar dpndnc and h γ, h condiional xpcd rurn may dcras or incras wih. 3

16 Fama and Frnch 99 provid an informaiv illusraion of h siz ffc as follows. Socks ar classifid ino dcils according o hir mark capializaion and h avrag rurn for ach dcil is compud. W will rm hs avrags h xpcd rurn condiional on dcils. Ths xpcd rurns dmonsra h cross-scional variaions in xpcd rurn condiional on siz. Th siz sprad is dfind o b h diffrnc bwn h xpcd rurn condiional on h 0h dcil and s dcil, which is mor coars masur of siz ffc. Boh xpcd rurn condiional on dcils and siz prmium can b compud in our modl. L δ i by h following quaion Nδ i = i, i =,..., 9, 0 whr N is h cumulaiv probabiliy disribuion funcion of h sandard normal random variabl, δ 0 =, and δ 0 = +. A im, p is normally disribud wih man p and varianc p = v + ɛ ρ. Thrfor, p i = p δ i + p, i = 0,,..., 9, 0, divid p -spac ino dcils. Proposiion 4 Siz ffc Suppos ha h assumpions of Proposiion 3 hold, hn h xpcd rurn condiional on dcil is µ+ ɛ r Nˆp i Nˆp +ρ i + x v+ ɛx ρ + ɛ Nˇp i Nˇp x 0. ρ i, 4 0. whr ˆp i δ i + ργ p and ˇp i δ i + γ p, i =,..., 9. Th siz sprad is givn by µ+ r Nˆp 9 + Nˆp +ρ 0. + x v+ ɛx ρ x + Th proposiion can b provd from Proposiion 3 by ingraion. ɛ Nˇp ρ 9 + Nˇp Whn ɛ = 0, h condiional xpcd rurn is indpndn of, and h rurn sprads bwn wo pric dcils porfolios ar zro. Similarly, as v incrass, h sprad dcrass, bcaus a highr v is quivaln o a lowr ɛ. For calibraion, w us paramrs givn in Tabl. In addiion, w nd o spcify v. Sinc v is no saionary, hr is no naural choic for v and v. Forunaly, v dos no affc h p dpndnc. W choos v o b a h sam ordr of magniud r. Wih hs paramrs, h siz sprad is abou 3%. Th mor prsisnc h nois xhibis, h lss ffc i has on h sprad. Thus, h sprad dcrass wih ρ for small ρ. Howvr, for a givn Σ ɛ, h highr ρ lads o a highr uncondiional varianc of, which is assumd o b h prior disribuion of, hus highr sprad. This ffcs dominas for ρ nar. Thus, h sprad has an U-shapd dpndnc and hus a minimum, his faur maks i rlaivly asir o gnra highr sprads han lowr sprads. So far, w hav xamind a singl sock; w hav no considr nois in a muli-ass framwork. If hr ar mulipl asss, w nd o considr h corrlaions bwn h valu rurns and h corrlaions bwn nois. W argu in h appndix ha our rsuls on pric dpndnc sill hold. Spcifically, w can sill xamin h pric dpndnc of xpcd rurns on a sock-by-sock basis, if h corrlaions ar 4

17 inroducd hrough a facor srucur and h numbr of ass is larg. 8 Roughly spaking, in his cas, w hav infinily many signals on a fw facors. As such, h facors will b complly rvald and h infrnc problm rducs o ha wihou sysmaic facors. Arno, Hsu, and Moor 005 and Arno 005a propos nois as a likly sourc for siz and valu ffcs. Hsu 006 shows ha mispricing prmium may xis bcaus hr ar invsors wih liquidiy nds. Brk 997 and Arno 005b suggs ha siz and valu ar highly inrrlad and may b proxis for a shard risk. Arno and Hsu 006 show ha man-rvring mispricing can lad o small cap and valu sock ouprformanc; howvr, hy prdic ha siz and valu migh subsum ach ohr. Brnnan and Wang 006 also us a similar modl o xplor ass pricing implicaion associad wih mispricing. Similar o Hsu 006, hy driv a rurn prmium associad wih mispricing. Spcifically hy argu ha common liquidiy masurs in financ may b proxis for mispricing and ha simad liquidiy prmium is likly mispricing prmium. 7 Th Valu ffc Many mpirical sudis analyz xpcd rurns condiional on pric-fundamnal raios, such as pricdividnd raio, pric-book raio, and pric-arning raios. In his scion, w xamin h pric-dividnd raio dpndnc of xpcd rurns whn nois is prsn. Concpually, h analysis applis in h sam way o any pric-fundamnal raio dpndnc. Sinc w hav o spcify dividnd-pric raio for compuing rurn alrady, w choos h pric-o-dividnd raio insad of ohr raios o avoid addiional paramrs. In his scion, w us h pric-dividnd raio X P D = p d o draw infrnc on h nois. W will us x o dno ln X = p d. Rcall, whn hr is nois, p = v + ln. 6 Th rror also works islf ino h pric-dividnd raio, p d = v d + ln. 7 Thus, a high pric-dividnd raio can b a signal for a high nois. This sam logic applis qually for pric-book, pric-arnings, and ohr pric-fundamnal raios. Th spcificaion of valu-dividnd raio givn in quaion 7 implis h following rlaionship for h pric-dividnd raio, p + d + = ρ x x v ρ x ln + ρ x p d + ρ ρ x + ɛx ɛ x+ + ɛ ɛ No ha his is h assumpion ndd for APT o hold. 5

18 Dnoing x = p d, w hav, x + = ρ x x + ρ x x + ρ ρ x + ɛx ɛ x+ + ɛ ɛ +, 9 whr x = x v ln is h man of x. W mak h sandard assumpion ha valu-dividnd raio is saionary, which mans ha x + is saionary, hus ρ x <. Th abov quaion implis ha h log pric-dividnd raio x is a signal on h nois. This implis ha pric-dividnd raio and ohr pric-fundamnal raios could provid infrnc on h nois. Sinc x is saionary, w can us is uncondiional disribuion as h prior disribuion for infrnc. Proposiion 5 Suppos ha Assumpions,, and 3 hold. Furhrmor, assum ha h disribuion of, x is hir uncondiional disribuion. Thn h xpcd rurn condiional on x is P+ + D + x X ργ + x v+ ɛx ρ + ɛ X ρ xγ ρ x x ρ = µ+ r +ρ X ργ X ρ xγ ρ x, whr γ = ρ x ɛ ρ x ɛ + ρ ɛx. Th proof is givn in h Appndix. Th inuiion for h x dpndnc is h sam as h inuiion for h p dpndnc xplord in in Scion 6. A high pric-dividnd raio implis a high nois, on avrag, hus a low xpcd rurn. Proposiion 5 also implis ha h rurn is prdicd by h dividnd yild vn hough h valu rurn is no. This is no surprising bcaus hr is a on-o-on corrspondnc bwn xcss volailiy and dividnd yild prdicabiliy. Tha is, whil rurn xhibis xcss volailiy rlaiv o dividnd variaion, valu rurn dos no, and whil dividnd yild prdics rurn, i dos no prdic valu rurn. No ha boh h xcss volailiy and dividnd yild prdicabiliy puzzl rsuls from nois insad of a raional quilibrium. W can also compu h xpcd rurn condiional on valu dcils, following Fama and Frnch 99. A im, x is normally disribud wih man x and varianc ɛx + ɛ. Thrfor, x ρ x ρ i = + ɛ δ ρ x ρ i + x, i = 0,,..., 9, 0, divids x -spac ino dcils. W will rm h diffrnc in h xpcd rurns bwn s and 0h dcil h valu sprad. Proposiion 6 Valu ffc Suppos assumpions in Proposiion 5 hold. Thn h xpcd rurn condiional on valu dcil is µ+ ɛ r Nˆx i Nˆx +ρ i + x v+ ɛx ρ + ɛ Nˇx x 0. ρ i Nˇx i, whr ˆx i δ i + ργ + ɛ and ˇx ρ x ρ i = δ i + ρ x γ + ρ x + ɛ, i =,..., 9. Th ρ x ρ valu sprad is givn by µ+ ɛ r Nˆx 9 + Nˆx +ρ + x v+ ɛx ρ + ɛ Nˇx x 0. ρ 9 + Nˇx

19 Th proposiion can b provd from Proposiion 5 by ingraion. For h paramrs givn in Tabl, h valu sprad is abou 6%. Th dpndnc on ρ is mor snsiiv for h valu sprad, primarily du o h fac ha h volailiy x of pric-dividnd raio x is much smallr han ha of h volailiy v of h valu v. 8 Th Siz-Valu ffc So far, w hav sudid h xpcd rurn condiional on ihr h pric or h pric-dividnd raio alon. W now compu h xpcd rurn condiional on h pric and pric-dividnd raio simulanously. In our modl, h siz and valu ffcs ar boh drivn by h sam sourc: h nois in h pric. Convrsly, boh pric p and pric-dividnd raio p d ar noisy signals of. W assum ha h corrlaion bwn v and v d is zro, howvr, hr is an imprfc corrlaion bwn p and p d inducd by h nois. Whn p is low, i is likly ha is ngaiv, bu w ar no sur, bcaus h valu v is no obsrvd. Whn boh p and p d ar low, i is mor likly ha is ngaiv. Thus p and p d ar corrlad bu no a subsiu of ach ohr. Using boh of hm simulanously givs us mor prcis informaion abou. Proposiion 7 Suppos Assumpions,, and 3 hold. Furhrmor, assum ha h disribuion of, x is hir uncondiional disribuion and h disribuion of v is normal wih man v and varianc v. Thn h xpcd rurn condiional on p and x is, P+ + D + x, p P ργ 3 X ργ 4 + x v+ ɛx ρ + ɛ P ρ xγ 3 x ρ X ρ xγ 4 ρ x whr γ 3 = = µ+ r v +ρ v and γ + ρ x 4 = + ρ P ργ 3 X ργ 4 v ρ x. + ρ x + ρ P ρ xγ 3 X ρ xγ 4 ρ x Th proof is givn in h Appndix. W assum ha h corrlaion bwn v and v d is zro for noaional simpliciy. Incorporaion of a non-zro corrlaion is sraighforward. Fama and Frnch 99 us h marix of xpcd rurn condiional on siz and valu dcils o dmonsra h siz and valu ffcs. Nx w compu hs condiional xpcd rurns using our modl. W firs divid p, x spac ino clls of 0 dcils by 0 dcils. No ha p and x ar join normal wih variancs v + and x + ρ ρ and corrlaion ˆρ = Following x ρ ρ ɛx p ρ x. + ɛ ρ Fama and Frnch, w will firs us p i o dividd p spac ino 0 dcils. For i-h siz dcil, w furhr divid x spac ino 0 dcils, using x i,j = x + ρ δ x ρ i,j + x, whr δ i,j can b solvd numrically. L fz z z dno h xpcaion of fz for z bwn z and z for a sandard normal random variabl z., 7

20 Proposiion 8 Siz-Valu ffc Suppos ha assumpions in Proposiion 7 hold. Thn h xpcd rurn condiional on i, j dcil of p, x spac is, ˆp N i+ ˆρz ˆp N i ˆρz ˆx i,j+ µ+ r ɛ ˆρ ˆρ ˆx i,j +ρ x v+ ɛx whr ˆp i δ i + ρ γ 3 p + ˆργ 4 δ i + ρ x γ 3 p + ˆργ 4 ρ + ɛ x ρ + ɛ ρ x ρ ɛx ρ x N + ɛ ρ i =,..., 9, and z is a sandard normal random variabl. ˇp i+ ˆρz N ˆρ 0.0 ˇp i ˆρz ˆρ ˇx i,j+ ˇx i,j,, ˆx i,j δ i,j + ρ γ 4 + ɛ + ˆργ ρ x ρ 3 p, ˇp i, and ˇx i, = δ i,j + ρ x γ 4 +ρ x + ɛ + ρ ρ x ρ x ˆργ 3 p, Th proof is givn in h appndix. L us considr h cas whr hr ar many socks wih corrlaions bwn sock rurns. W show ha, in h appndix, if h corrlaions in h rurns as wll as nois is inroducd hrough a facor modl, h infrnc on is h sam as if hr is no facor. This mans ha, Proposiions 3 8 hold whn h corrlaions ar hrough facors, providd w rplac h varianc paramrs by hir idiosyncraic componns. Suppos h rurns of all socks ar givn by a facor modl and all hav h sam ba and sam idiosyncraic volailiy. Thn h cross-scion avrag ar h sam as populaion avrag, hus can b compud using Proposiions 3-8. So, hs proposiion imply cross-scional variaions in condiional xpcd rurns, vn in h absnc of paramr variaion. Th variaion in his cas is gnrad by random ralizaion of h pric nois. Of cours, paramr variaions in raliy, such as variaions in bas and idiosyncraic volailiy, lad o addiional cross-scional variaions in xpcd rurns. Nx w will show ha hs variaions ar consisn wih hos obsrvd in h US daa, wih plausibl paramrs. For h calibraion xrcis, w us paramrs spcifid in Tabl. W prsn xpcd rurns condiional on boh siz and valu in Tabl. Th inuiion for h abl is simpl. Dcil xpcd rurns ar rally xpcd rurns condiional on pric inrvals or pric-raio inrvals, which dcrass wih pric and/or pric-raios, as shown in h abl. W assum ha socks ar indpndn draws from h sam disribuion. I is inrsing o compar Tabl wih Tabl V of Fama-Frnch 99, which ar sampl avrag of rurns condiional on siz and pric-o-book dcils. As w poind ou arlir, w choos pric-dividnd dcils mainly o avoid xra paramrs. W xpc h diffrnc in using pric-dividnd raio and pricbook raio o b small. Th xpcd rurns our Tabl ar similar o hos of Tabl V of Fama and 8

21 Tabl : xpcd Annual Rurns Condiional on Siz and Valu Dcils Dividnd-o-Pric Raio All All Small-M M M M M M M M M Larg-M This abl prsns annual xpcd rurns, in prcnag, condiional on pric M and dividnd-o-pric dcils. Ths xpcd rurns ar compud using Proposiion 8 wih h paramrs givn by Tabl. Th ba in h absnc of nois is assumd o b. Tabl 3: Ba Condiional on Siz and Valu Dcils Dividnd-o-Pric Raio All All Small-M M M M M M M M M Larg-M This abl prsns ba of pric M and dividnd-o-pric dcils. Th paramrs ar givn by Tabl. Frnch 99, whn annualizd. Th xpcd rurns ar monoonic as a funcions of dcils whil h monooniciy is no sric in Tabl V of Fama and Frnch 99, prsumably bcaus of masurmn rrors in h sampl avrags. I is imporan o drmin whhr small and valu socks hav highr xpcd rurns bcaus hy hav highr sysmaic risks. In Tabl 3, w prsn h ba marix for siz-valu dcils. Assuming ha ba in h absnc of nois is, small and valu socks hav a slighly highr ba. Socks in h smalls dcil hav a ba of.0 whil hos in h largs dcil has a ba of Similarly, Socks in h lows dividnd-pric raio dcil hav a ba of 0.98 whil hos in h highs dcil has a ba of.03. This finding is consisn Lakonishok, Shlifr, and Vishny 994 who find ha h bas of valu porfolios wih rspc o h valu-wighd indx nd o b abou 0. highr han h bas of h glamour porfolios. Assuming an annual riskfr rurn of.04, w can compu h abnormal rurn alpha, ha is, h risk-adjusd xcss xpcd rurn for ach siz and valu dcil wih bas givn in Tabl 3. W prsn 9

22 Tabl 4: Alpha Condiional on Siz and Valu Dcils Dividnd-o-Pric Raio All All Small-M M M M M M M M M Larg-M This abl prsns annual alpha, in prcnag, of pric M and dividnd-o-pric dcils. Th paramrs ar givn by Tabl and h gross riskfr rurn is assumd o b.04. alpha in Tabl 4. Small and valu socks hav posiiv alpha whil h larg and glamor socks hav ngaiv alpha. Socks in h smalls dcil hav an alpha of.67% whil hos in h largs dcil hav an alpha of -.8%. Similarly, socks in h lows dividnd-pric-raio dcil hav an alpha of -0.98% whil hos in h highs dividnd-pric-raio dcil hav an alpha of.47%. Ths wo abls show ha, in our modl, small and valu socks hav highr xpcd rurns bcaus hy ar undr-valud du o ngaiv pric nois, no bcaus hr hav highr bas. On migh wondr if hs alphas prsis ovr im. On h on hand, i is possibl ha alphas may b liminad ovr im. On h ohr hand, i is possibl ha hy will prsis ovr im bcaus of limis o arbirag, associad wih ihr ransacion coss or risks in h sragis o xplor hs alphas. As a modl for h cross scion of xpcd rurn, our papr is diffrn from Brk 995, 997. Th hrogniy of xpcd rurn is mainly drivn by h random ralizaion of h nois, whil i is spcifid in rms of h hrogniy of h ba. Suppos ha sock rurns ar idnically disribud bu corrlad hrough sysmaic facors. In his cas, hr is no cross-scion variaion in xpcd rurns and h corrlaion bwn pric and h xpcd rurn will b zro, undr Brk. By conras, undr our framwork, a sock wih a lowr pric sill has a highr xpcd rurn. On h ohr hand, on can hav an xampl whr hr is corrlaion bwn pric and rurn bu no condiional sprads. Th xpcd rurns condiional on h pric dcils in Proposiions 4, 6, and 8 ar sa indpndn. I is possibl ha h siz and valu ffcs may b sa dpndn, for xampl, hr ar mpirical sudis documning ha h siz and valu sprads ar diffrn bwn booms and rcssions. Th mos naural way o inroduc h sa dpndnc in our modl is hrough h sa-dpndnc of h condiional varianc of nois. This can b ponially usd o accommoda h dpndnc on businss cycls of siz and valu ffcs. Summrs 986 argus ha h daa in conjuncion wih currn mhods provid no vidnc agains h viw ha financial mark prics dvia widly and frqunly from raional valuaions. W would 0

23 lik o argu ha h siz and valu ffcs ar vidnc for h viw ha financial mark prics dvia from valus. 9 Condiioning on Pas Prics In prvious scions, w hav sudid h xpcd rurn, condiional on currn prics and/or pric raios. In his scion, w will sudy h xpcd rurns condiional on boh currn and pas prics. W can also compu h xpcd rurn condiional on pas pric-raios as wll; w choos prics o b h condiioning variabls for noaional simpliciy. W firs considr h xpcd rurn condiional on pas rurn r. Tha is, w ar inrsd in h man of + condiional on h prvious priod rurn 9 P = r. A high rurn r implis a high and low on avrag, hus lowr xpcd rurn for +. This is h rurn rvrsal ffc. Proposiion 9 Condiioning on Rurn If Assumpions,, and 3 hold, h xpcd rurn a im + condiional on rurn R is, P+ + D + r = µ+ ɛ R r ργ 5 +ρ + x v+ ɛx ρ + ɛ R γ 5 x P ρ, R ργ 5 R γ 5 whr γ 5 = +ρ ɛ r ɛ + r+ ρ ɛ ρ. Th condiional xpcd rurn dcrass wih r for ρ <. Th proof is givn in h Appndix. According o Proposiion 9, a man-rvring nois lad o rurn rvrsal. Tha is, h xpcd rurn, condiional on pas rurn, dcrass wih h pas rurn. In h US mark daa, rurn rvrsal is obsrvd for horizons grar han yars DBond and Thalr 985, 987 and Chopra, Lakonishok and Rir 99. Howvr, rurn momnum, which mans ha h xpcd rurn incrass wih h pas rurn, is obsrvd for horizons lss han yar Jgadsh and Timan, 993, 00. Thus h obsrvd xpcd rurn condiional pas rurn canno b xplaind by man-rvring nois, a las for horizon lss han yar. No ha condiioning on rurn / is diffrn from condiioning on pas prics and sparaly, which w urn o nx. So far, w hav condiiond on currn prics or pric raios o produc siz and valu ffcs and on pas rurn P o produc momnum and rvrsal ffcs. Howvr, i is obvious ha on should us h full pric hisory. W now considr h im + xpcd rurn condiional on pas mark prics, P s, for s =,,..., 0. Our analysis can b xndd o includ pas pric-raios. W only prsn h cas for pas prics for as of xposiion. 9 Sricly spaking, h prvious-priod rurn should b +D. Howvr, w do no hav h closd form soluion for h infrnc of. Nvrhlss, h inuiion sill applis.

24 W would lik o compu, P+ + D + {p s } P 0, 3 whr 0. Tha is, h xpcd rurn from im o + condiional on prics from 0 o. W will nd o addiionally spcify h prior disribuion for v 0 and 0. W assum ha 0 is drawn from h uncondiional disribuion of, which has a man of 0 and varianc of ɛ ρ. W assum ha v 0 drawn from a normal disribuion wih a man v 0 and v 0. W assum ha v 0 and 0 ar indpndn in h prior disribuion. Proposiion 0 Condiioning on Currn and Pas Prics Suppos Assumpions,, and 3 hold. Furhrmor assum ha ρ v = 0 ɛ, whr = + +ρ 4 r r ρ +ρ ρ r. Thn h xpcd rurn a im condiional h prics from 0 o is P+ + D + r = µ+ P h p r ɛ s= 0 + P h s h h p s P h 0 ρ h 0 0 +ρ P h p h h p s P h 0 ρ h x v+ ɛx ρ + ɛ x ρ whr h p = ɛ ρ ρ r+ ɛ + ρ, h = s= 0 + P h s P hp s= 0 + P h s h h p s P h p s= 0 + P h s h h p s P h 0 h 0 0 P h 0 h 0 0 ρr+ ɛ r+ ɛ + ρ, and h 0 = h ρ h p. ɛ is, 33 Again, w includ h proof in h Appndix. W us h convnion for h produc opraor j s=i ha h produc is if h uppr indx j is smallr han h lowr indx i. According o quaion 33, h condiional xpcd rurn dcrass wih h currn pric bu incrass wih pas prics. No ha h < ; pas prics ar discound by powrs of h in h condiional xpcd rurns, h furhr away in h pas, h highr h discoun and h lowr h rlvanc o nx priod rurn. In gnral, h varianc of condiional on pas prics dpnds on 0. Howvr, whn 0, his varianc gos o a consan, which can b shown o b. Th chnical condiion a h bginning of h proposiion implis ha h condiional varianc rachs a im 0 and is assumd only o simplify h noaion. In h Appndix, w show rsuls for h gnral cas. 0 Conclusion In his papr, w propos ha nois as a sourc for cross-scional variaions in xpcd rurns. Whn hr is nois in mark pric, h uncondiional xpcd rurn dpnds no only on ba bu also on h

25 idiosyncraic volailiy, pric-o-dividnd raio, and volailiy of h nois. Mor imporan, w show ha random ralizaions of nois gnra cross-scional variaions in xpcd rurn condiional on pric and pric raios. In paricular, wih plausibl paramrs, such as a nois volailiy of 6% pr annum, h marix of xpcd rurn condiional on siz and valu dcils is similar o ha of Fama and Frnch 99. Sinc h diffrnc in ba for diffrn siz and valu dcils is small in our modl, small and valu socks hav highr xpcd rurn bcaus hy ar undr-valud du o pric nois, no bcaus of highr sysmaic risk. Thus our rsuls suggs ha nois cra siz and valu ffc. Black argus ha nois should always b prsn bcaus invsors ar risk avrs and ar no sur whhr informaion is jus pur nois. According o Black 986, nois cras h opporuniis o rad profiably, bu a h sam im maks i difficul o rad profiably. If Black is righ, siz and valu ffcs ar likly o coninu o prsis. In classic fficin marks, h fuur prospcs of an invsmn acily ris and fall wih shar pric, so ha h inrnal ra of rurn IRR of an invsmn will no b advanagd by a drop in pric or disadvanagd by an incras. Our assumpions, for his simplisic xampl, sand in sark conras-whn prics ris h subsqun IRR will fall and whn prics fall h IRR will ris. This rsuls in sock pric rvrsion owards valu, prurbd by a sady flow of nw nois. Givn h volailiy of shar prics, i is unlikly ha ihr posiiv or ngaiv srial corrlaion, id o rvrsion owards h unknowabl discound ru fair valu, will b vidn in any saisically significan fashion. Th signal-o-nois of his paricular par of h rurn would b so low as o b vry difficul o as ou of h daa xcp in aggrga daa across many sampls and many yars of daa. Isn his prcisly h parn ha has bn obsrvd im and again in mpirical sudis, spanning many im inrvals and marks? On araciv faur of his modl is ha i can b sd mpirically. By accping h principl of dcoupling pric from valu, wih a man-rvring rror, w can mpirically masur h paramrs of his modl. For xampl, a narrow cas of our modl applis if w assum ha h fuur is fixd and ha pric is mrly h marks currn sima of a drminisic valu. Tha is, if w hav a crysal ball which allows us o s h fuur, w can discoun i back o a currn N Prsn Valu, which riss wih h passag of im wih zro varianc. On can, for xampl, ak all socks in xisnc n, wny, hiry or fory yars ago, and all subsqun cash flows using h currn pric as a proxy for rmaining fuur cash flows, and compu h original valu and nois rm, and, basd on subsqun rurns, obsrv h hisorical man rvrsion and volailiy of h pricing rror. Boh may wll by im-varying, no saic. Our modl assums ha nois is indpndn of h valu and h dividnd. On could xamin h implicaions of rlaxing ha assumpion. Indd, for crain forms of dpndnc, w would xpc ha h valu and siz ffcs should disappar. mpirical vidnc clarly dos no suppor his form of h 3