Jonathan W. Lewellen. Submitted in Partial Fulfillment. of the. Requirements for the Degree. Doctor of Philosophy. Supervised by Professor Jay Shanken

Transcription

1 On he Predicabiliy of Sock Reurns: Theory and Evidence by Jonahan W. Lewellen Submied in Parial Fulfillmen of he Requiremens for he Degree Docor of Philosophy Supervised by Professor Jay Shanken William E. Simon Graduae School of Business Adminisraion Universiy of Rocheser Rocheser, New York 2000

2 ii Curriculum Viae The auhor was born in Wes Lafayee, Indiana on May 1, He sudied economics and finance a Indiana Universiy from 1990 o 1994, and graduaed wih a Bachelor of Science degree in He came o he Universiy of Rocheser in he Fall of 1994 and began graduae sudies in applied economics and finance a he Simon Graduae School of Business Adminisraion. He received financial suppor from he Simon School from 1994 o 1999 and received an Olin Fellowship from 1997 o He pursued his research, specializing in he pricing of financial asses, under he direcion of professors G. William Schwer, Jay Shanken, and Jerold Warner. He earned a Maser of Science in applied economics in 1997.

3 iii Acknowledgemens I graefully acknowledge he advice and suppor of my disseraion commiee, Bill Schwer, Jay Shanken, and Jerry Warner. Their eaching and encouragemen are ulimaely responsible for his disseraion. I owe a special deb o Jay Shanken, whose guidance has simulaed my research in counless ways and whose friendship is ye more valuable. I have also learned much from my colleagues a he Simon School, especially Kaharina Bibus, Andreas Ginschel, Jarrad Harford, Chrisoph Hinkelmann, Michelle Lowry, Susan Shu, Denis Suvorov, and Peer Wysocki. This paper has benefied from he commens of Greg Bauer, Ken Daniel, Ken French, S.P. Kohari, John Long, René Sulz, and Jiang Wang. I hank he Simon Graduae School of Business Adminisraion and he Olin Foundaion for financial suppor.

4 iv Absrac Empirical sudies find ha sock reurns are predicable boh cross-secionally and over ime. Broadly speaking, his disseraion invesigaes wheher he empirical paerns in sock reurns are consisen wih an efficien capial marke. The paper consiss of wo essays. In he firs essay, I invesigae he abiliy of firms book-o-marke raios o predic reurns, which has been documened exensively in cross-secional ess. To help undersand he source of his predicabiliy, I examine he ime-series relaions among expeced reurn, risk, and book-omarke. Consisen wih raional pricing, book-o-marke capures significan ime-variaion in risk, bu provides no incremenal informaion abou expeced reurns. In he second essay, I explore he effecs of esimaion risk, or invesor uncerainy abou he parameers of he cashflow process, on he behavior of prices and reurns. I show ha, wih esimaion risk, he observable properies of prices and reurns can differ significanly from he properies perceived by raional invesors. As a consequence, esimaion risk can generae reurn predicabiliy in ways ha resemble irraional pricing. Simulaion evidence suggess he effecs of esimaion risk can be economically significan.

5 v Table of Conens Chaper 1 Inroducion. 1 Chaper 2 The Time-Series Relaions among Expeced Reurn, Risk, and Booko-Marke Disinguishing beween Characerisics and Risk Daa and Descripive Saisics The Predicabiliy of Porfolio Reurns Expeced Reurns, Characerisics, and Risk: Empirical Resuls Summary and Conclusions.. 40 Chaper 3 Esimaion Risk, Marke Efficiency, and he Predicabiliy of Reurns The Model Capial Marke Equilibrium The Time-Series Properies of Prices and Reurns The Cross-Secion of Expeced Reurns Informaive Priors, Seady Sae, and Simulaions Summary and Conclusions.. 89 References 93 Appendix A.. 98 Appendix B.. 104

6 vi Lis of Table s Chaper 2 Table 2.1 Summary saisics for indusry, size, and book-o-marke porfolios. 15 Table 2.2 Summary saisics for facors. 18 Table 2.3 Predicabiliy of indusry reurns. 21 Table 2.4 Predicabiliy of size and book-o-marke porfolio reurns 24 Table 2.5 Uncondiional hree-facor regressions: Indusry porfolios 27 Table 2.6 Condiional hree-facor regressions: Indusry porfolios 29 Table 2.7 Uncondiional hree-facor regressions: Size and book-o-marke porfolios.. 32 Table 2.8 Condiional hree-facor regressions: Size and book-o-marke porfolios.. 35 Table 2.9 Three-facor regressions wih indusry-neural HML: Indusry porfolios.. 39 Chaper 3 Table 3.1 Predicabiliy in seady sae 87

7 vii Lis of Figures Chaper 3 Figure 3.1 Equilibrium price of he risky asse 61

8 1 On he Predicabiliy of Sock Reurns: Theory and Evidence Chaper 1 Inroducion Over he pas 20 years, we have accumulaed much evidence ha sock reurns are predicable. A he aggregae level, Fama and Schwer (1977), Keim and Sambaugh (1986), Fama and French (1989), and Kohari and Shanken (1997) show ha ineres raes, he yield spread beween low- and high-grade deb, aggregae dividend yield, and aggregae book-omarke predic ime-variaion in expeced reurns. Furher, LeRoy and Porer (1981) and Shiller (1981) argue ha he volailiy of sock prices is oo high o be explained by a model wih consan discoun raes, providing indirec evidence ha expeced reurns change over ime. A he firm level, Fama and French (1992) conclude ha size and book-o-marke ogeher explain much of he cross-secional variaion in average reurns. Jegadeesh and Timan (1993) also show ha pas reurns conain addiional informaion abou expeced reurns. In sum, here seems lile doub ha expeced sock reurns vary boh cross-secionally and over ime. 1 The inerpreaion of predicabiliy, however, is more conenious. The empirical paerns in reurns are poenially consisen wih eiher marke efficiency or irraional mispricing. In general erms, marke efficiency implies ha prices fully reflec all available informaion. To formalize his idea for empirical esing, Fama (1976) disinguishes beween he probabiliy disribuion of reurns perceived by he marke, based on whaever informaion invesors view as relevan, and he rue disribuion of reurns condiional on all informaion. The marke is said o be informaionally efficien if hese disribuions are he same. As an obvious consequence, marke efficiency implies ha invesors correcly anicipae any cross-secional or ime-variaion in rue expeced reurns. While Fama s definiion ignores poenially imporan issues like heerogeneous beliefs, i provides a useful framework for hinking abou a broad se 1 Clearly, his lis of empirical papers and predicive variables is no mean o be exhausive, and a considerable amoun of subsequen research exends, confirms, and criiques hese findings. See Fama (1991) for a more complee survey of he evidence.

9 2 of asse-pricing quesions. This paper conains wo essays which, broadly speaking, aemp o undersand wheher he empirical resuls are consisen wih an efficien capial marke. In he firs essay, I invesigae he abiliy of book-o-marke raios o predic reurns. An exensive lieraure shows ha he raio of a firm s book value o marke value of equiy in shor, book-o-marke explains significan cross-secional variaion in expeced reurns. The inuiion is ha book value in he numeraor conrols for he size of he firm (he size of expeced cashflows), while marke value in he denominaor capures informaion abou discoun raes. Boh efficien-marke and mispricing sories have been offered o explain he evidence. In his paper, I examine he imeseries relaions among expeced reurn, risk, and book-o-marke o help undersand he source of he predicabiliy. As discussed in Chaper 2, he ime-series analysis can help disinguish beween he raional- and irraional-pricing sories. In he second essay, I invesigae he impac of esimaion risk on he behavior of asse prices. In he finance lieraure, esimaion risk refers o invesor uncerainy abou he parameers of he reurn or cashflow process. In oher words, esimaion risk exiss whenever invesors do no have perfec informaion abou some imporan feaure of he economy. Alhough i represens purely subjecive uncerainy, esimaion risk can have imporan consequences for asse pricing because i affecs invesmen decisions. In Chaper 3, I presen a simple model of capial marke equilibrium, and explore he consequences of esimaion risk for reurn predicabiliy and ess of marke efficiency. I also presen simulaion evidence o give an indicaion of he economic significance of he resuls. Again, he fundamenal goal is o undersand wheher esimaion risk migh help explain he ime-series and cross-secional evidence described above.

10 3 On he Predicabiliy of Sock Reurns: Theory and Evidence Chaper 2 The ime -series relaions among expeced reurn, risk, and book-o-marke Empirical research consisenly finds a posiive cross-secional relaion beween average sock reurns and he raio of a firm s book equiy o marke equiy (B/M). Saman (1980) and Rosenberg, Reid, and Lansein (1985) documen he associaion beween expeced reurns and B/M, which remains significan afer conrolling for bea, size, and oher firm characerisics (Fama and French, 1992). The explanaory power of B/M does no appear o be driven enirely by daa snooping or survival biases; i is found in sock markes ouside he Unied Saes (Chan, Hamao, and Lakonishok, 1991; Haugen and Baker, 1996) and in samples drawn from sources oher han Compusa (Davis, 1994). As a whole, he evidence provides considerable suppor for he cross-secional explanaory power of B/M. A leas wo explanaions have been offered for he empirical evidence. According o asse-pricing heory, B/M mus proxy for a risk facor in reurns. The significance of B/M in compeiion wih bea conradics he capial asse pricing model (CAPM) of Sharpe (1964), Linner (1965), and Black (1972), or more precisely, he mean-variance efficiency of he marke proxy. However, he evidence migh be consisen wih he ineremporal models of Meron (1973) and Breeden (1979). In hese models, he marke reurn does no compleely capure he relevan risk in he economy, and addiional facors are required o explain expeced reurns. If a mulifacor model accuraely describes sock reurns, and B/M is cross-secionally correlaed wih he facor loadings, hen he premium on B/M simply reflecs compensaion for risk. A posiive relaion beween B/M and risk is expeced for several reasons. Chan and Chen (1991) and Fama and French (1993) sugges ha a disinc disress facor explains common variaion in sock reurns. Poorly performing, or disressed, firms are likely o have high B/M. These firms are especially sensiive o economic condiions, and heir reurns migh be driven

11 4 by many of he same macroeconomic facors (such as variaion over ime in bankrupcy coss and access o credi markes). In addiion, following he argumens of Ball (1978) and Berk (1995), B/M migh proxy for risk because of he inverse relaion beween marke value and discoun raes. Holding book value consan in he numeraor, a firm s B/M raio increases as expeced reurn, and consequenly risk, increases. Alernaively, B/M migh provide informaion abou securiy mispricing. The mispricing view akes he perspecive of a conrarian invesor. A firm wih poor sock price performance ends o be underpriced and have a low marke value relaive o book value. As a resul, high B/M predics high fuure reurns as he underpricing is eliminaed. Lakonishok, Shleifer, and Vishny (1994) offer a raionale for he associaion beween pas performance and mispricing. They argue ha invesors naively exrapolae pas growh when evaluaing a firm s prospecs. For example, invesors end o be overly pessimisic abou a firm which has had low or negaive earnings. On average, fuure earnings exceed he marke s expecaion, and he sock does abnormally well. Thus, he mispricing argumen says ha B/M capures biases in invesor expecaions. Fama and French (1993) provide evidence of a relaion beween B/M and risk. Using he ime-series approach of Black, Jensen, and Scholes (1972), hey examine a mulifacor model consising of marke, size, and book-o-marke facors, where he size and book-o-marke facors are sock porfolios consruced o mimic underlying risk facors in reurns. If he model explains cross-secional variaion in average reurns, he inerceps will be zero when excess reurns are regressed on he hree facors. Fama and French find, as prediced by he risk-based view, ha he model does a good job explaining average reurns for porfolios sored by size, B/M, earnings-price raios, and oher characerisics. Furher, hey documen a srong associaion beween a sock s B/M raio and is loading on he book-o-marke facor. More recenly, Daniel and Timan (1997) argue in favor of a characerisics-based model, consisen wih he mispricing view. They sugges ha he hree-facor model does no direcly

12 5 explain average reurns. Insead, he model appears o explain average reurns only because he facor loadings are correlaed wih firms characerisics (size and B/M). To disenangle he explanaory power of he facor loadings from ha of he characerisics, Daniel and Timan consruc es porfolios by soring socks firs on B/M raios and hen on facor loadings. This soring procedure creaes independen variaion in he wo variables. Consisen wih he mispricing sory, Daniel and Timan find a sronger relaion beween expeced reurns and B/M han beween expeced reurns and facor loadings. Daniel and Timan conclude ha firm characerisics, in paricular B/M, and no covariances deermine expeced sock reurns. In his essay, I provide furher evidence on he risk- and characerisics-based sories. In conras o Fama and French (1993) and Daniel and Timan (1997), I focus on he ime-series relaions among expeced reurn, risk, and B/M. Specifically, I ask wheher a porfolio s B/M raio predics ime-variaion in is expeced reurn, and es wheher changes in expeced reurn can be explained by changes in risk. Recenly, Kohari and Shanken (1997) and Poniff and Schall (1998) find ha B/M forecass sock reurns a he aggregae level, bu he predicive abiliy of B/M for individual socks or porfolios has no been explored. The ime-series analysis is a naural alernaive o cross-secional regressions. An aracive feaure of he ime-series regressions is ha hey focus on changes in expeced reurns, no on average reurns. The mispricing sory suggess ha a sock s expeced reurn will vary over ime wih B/M, bu i says lile abou average reurns if mispricing is emporary. Crosssecional regressions, however, can pick up a relaion beween average reurns and B/M. The ime-series regressions also highligh he ineracion beween B/M and risk, as measured by ime-variaion in marke beas and he loadings on he Fama and French (1993) size and booko-marke facors. Furher, I can direcly es wheher he hree-facor model explains imevarying expeced reurns beer han he characerisics-based model. These resuls should help disinguish beween he risk and mispricing sories. The empirical ess iniially examine B/M s predicive abiliy wihou aemping o conrol

13 6 for changes in risk. I find ha a porfolio s B/M raio racks economically and saisically significan variaion in is expeced reurn. An increase in B/M equal o wice is ime-series sandard deviaion forecass a 4.6% (annualized) increase in expeced reurn for he ypical indusry porfolio, 8.2% for he ypical size porfolio, and 9.3% for he ypical book-o-marke porfolio. The average coefficien on B/M across all porfolios, 0.99, is approximaely double he cross-secional slope, 0.50, found by Fama and French (1992, p. 439). B/M explains, however, only a small fracion of porfolio reurns, generally less han 2% of oal volailiy. Reurn predicabiliy indicaes ha eiher risk or mispricing changes over ime. Of course, we canno disinguish beween hese explanaions wihou some model of risk. Following Daniel and Timan (1997), I examine B/M s explanaory power in compeiion wih he Fama and French (1993) hree-facor model. The mulifacor regressions employ he condiional asse-pricing mehodology of Shanken (1990), which allows boh expeced reurns and facor loadings o vary over ime wih B/M. In hese regressions, ime-variaion in he inerceps measures he predicive abiliy of B/M ha canno be explained by changes in risk. The mispricing view suggess ha he inerceps will be posiively relaed o B/M; he risk-based view implies ha changes in he facor loadings will eliminae B/M s explanaory power, assuming he Fama and French facors are adequae proxies for priced risk in he economy. Empirically, he facors absorb much of he volailiy of porfolio reurns, which permis relaively powerful ess of he compeing sories. I find ha B/M explains significan imevariaion in risk, bu does no provide incremenal informaion abou expeced reurn. In general, he loadings on he size and book-o-marke facors vary posiively wih a porfolio s B/M raio, and saisical ess srongly rejec he hypohesis of consan risk. The resuls for marke beas are more difficul o characerize: across differen porfolios, B/M predics boh significan increases and significan decreases in bea. Overall, B/M conains subsanial informaion abou he riskiness of sock porfolios. In conras, he inerceps of he hree-facor model do no vary over ime wih B/M. For

14 7 he indusry porfolios, he average coefficien on B/M (ha is, variaion in he inercep) has he opposie sign prediced by he overreacion hypohesis and is no significanly differen from zero. Across he 13 porfolios, eigh coefficiens are negaive and none is significanly posiive a convenional levels. The resuls are similar for size and book-o-marke porfolios: he average coefficiens are indisinguishable from zero, and roughly half are negaive. Imporanly, he inferences from he mulifacor regressions are no driven by low power. For all hree ses of porfolios, saisical ess can rejec economically large coefficiens on B/M. In shor, he hree-facor model measures risk sufficienly well o explain ime-variaion in expeced reurns. 2 As an aside, I find ha he book-o-marke facor, HML, explains common variaion in reurns ha is unrelaed o is indusry composiion. Daniel and Timan (1997) argue ha HML does no proxy for a disinc risk facor, bu explains reurn covariaion only because similar ypes of firms become mispriced a he same ime. For example, a bank wih high B/M will covary posiively wih HML simply because he facor is weighed owards underpriced financial firms. The ime-series regressions provide evidence o he conrary. As an alernaive o HML, I esimae he regressions wih an indusry-neural book-o-marke facor. This facor is consruced by soring socks on heir indusry-adjused B/M raios, defined as he firm s B/M minus he indusry average, so he facor should never be weighed owards paricular indusries. The resuls using he indusry-neural facor are similar o hose wih HML. Thus, HML s explanaory power does no appear o be driven by indusry facors in reurns. The remainder of he essay is organized as follows. Secion 2.1 inroduces he ime-series regressions. Secion 2.2 describes he daa o be used in he empirical ess. Secion 2.3 esimaes he simple relaion beween expeced reurns and B/M, and Secion 2.4 ess wheher 2 I also replicae he empirical ess using size in place of B/M, wih similar resuls. There is some evidence ha size and expeced reurns are negaively relaed in ime series. In condiional hree-facor regressions, size capures significan ime-variaion in risk, bu does no conain addiional informaion abou expeced reurns. Deails are available on reques. I hank Ken French for suggesing hese ess.

15 8 he predicive abiliy of B/M can be explained by changes in risk, as measured by he Fama and French (1993) hree-facor model. Secion 2.5 summarizes he evidence and concludes Disinguishing beween characerisics and risk Book-o-marke explains cross-secional variaion in average reurns afer conrolling for bea. Fama and French (1993) provide evidence ha B/M relaes o common risk facors in reurns. In conras, Daniel and Timan (1997) argue ha he Fama and French facors appear o be priced only because he loadings are correlaed wih firm characerisics, like B/M. This secion inroduces he ime-series mehodology used in he curren paper and discusses, more generally, asse-pricing ess of he risk and mispricing sories Time-series mehodology The empirical ess iniially examine he simple relaion beween expeced reurns and B/M. The explanaions ha have been offered for he cross-secional evidence also sugges ha expeced reurns will vary over ime wih B/M. According o he risk-based view, B/M should capure informaion abou changes in risk, and consequenly, expeced reurn. The mispricing view says ha B/M is relaed o biases in invesor expecaions, and will conain informaion abou under- and overpricing. Thus, boh explanaions predic a posiive slope coefficien in he regression R i () = γ i0 + γ i1 B/M i (-1) + e i (), (2.1) where R i is he porfolio s excess reurn and B/M i is is lagged book-o-marke raio. Noe ha eq. (2.1) specifies a separae ime-series regression for each porfolio, wih no consrain on he coefficiens across differen porfolios. The regressions focus only on he ime-series relaion beween expeced reurns and B/M, and do no pick up any cross-secional relaion. Eq. (2.1) makes no aemp o undersand he source of ime-varying expeced reurns. According o radiional asse-pricing heory, a posiive slope in eq. (2.1) mus be driven by an

16 9 associaion beween B/M and risk. I follows ha he predicive power of B/M should be eliminaed if he regressions conrol adequaely for changes in risk. The characerisics-based sory, on he oher hand, suggess ha B/M will capure informaion abou expeced reurns ha is unrelaed o risk. To help disinguish beween he wo explanaions, I examine he predicive power of B/M in compeiion wih he Fama and French (1993) hree-facor model. The mulifacor regressions employ he condiional ime-series mehodology of Shanken (1990). Roughly speaking, hese regressions combine he hree-facor model wih he simple regressions above. Fama and French esimae he uncondiional model R i () = a i + b i R M () + s i SMB() + h i HML() + e i (), (2.2) where R M is he excess marke reurn, SMB (small minus big) is he size facor, and HML (high minus low) is he book-o-marke facor. Uncondiional, here, refers o he implici assumpion ha he coefficiens of he model are consan over ime. If his assumpion is no saisfied, he esimaes from eq. (2.2) can be misleading. The uncondiional inerceps and facor loadings could be close o zero, bu migh vary considerably over ime. The condiional regressions allow boh expeced reurns and facor loadings o vary wih B/M. Suppose, for simpliciy, ha he coefficiens of he hree-facor model are linearly relaed o he firm s B/M raio, or a i = a i0 + a i1 B/M i (-1), b i = b i0 + b i1 B/M i (-1), s i = s i0 + s i1 B/M i (-1), h i = h i0 + h i1 B/M i (-1). (2.3) Subsiuing hese equaions ino he uncondiional regression yields a condiional version of he hree-facor model: R i = a i0 + a i1 B/M i + (b i0 + b i1 B/M i )*R M + (s i0 + s i1 B/M i )*SMB + (h i0 + h i1 B/M i )*HML + e i, (2.4) where he ime subscrips have been dropped o reduce cluer. Muliplying he facors hrough gives he regression equaion for each porfolio. Thus, he condiional regressions conain no

17 10 only an inercep and he hree facors, bu also four ineracive erms wih he porfolio s lagged B/M. 3 Basically, eq. (2.4) breaks he predicive power of B/M ino risk and non-risk componens. The coefficien a i1, he ineracive erm wih he inercep, measures he predicive abiliy of B/M ha is incremenal o is associaion wih risk in he hree-facor model. A non-zero coefficien says ha changes in he facor loadings, capured by he coefficiens b i1, s i1, and h i1, do no fully explain he ime-series relaion beween B/M and expeced reurn. Thus, raional asse-pricing heory predics ha a i1 will be zero for all socks, assuming ha he facors are adequae proxies for priced risk. The mispricing, or characerisics-based, view implies ha B/M will forecas reurns afer conrolling for risk and, consequenly, a i1 should be posiive Discussion The condiional regressions direcly es wheher he hree-facor model or he characerisic -based model beer explains changes in expeced reurns. To inerpre he regressions as a es of raional pricing, we mus assume, of course, ha he Fama and French facors capure priced risk in he economy. This assumpion could be violaed in wo imporan ways (see Roll, 1977). Firs, an equilibrium mulifacor model migh describe sock reurns, bu he Fama and French facors are no adequae proxies for he unknown risks. In his case, B/M can predic ime-variaion in expeced reurns missed by he hree-facor model if i relaes o he rue facor loadings. Forunaely, his problem will no be a concern for he curren paper because he hree-facor model will, in fac, explain he predicabiliy associaed wih B/M. Unforunaely, he assumpion can also be violaed in he opposie way: mispricing migh explain deviaions from he CAPM, bu he size and book-o-marke facors happen o absorb 3 Similar regressions appear in previous sudies. Fama and French (1997) esimae regressions in which only he facor loadings on HML vary wih B/M. He e al. (1996) esimae a model in he spiri of eq. (2.4), bu hey consrain he inerceps and book-o-marke coefficiens o be he same across porfolios. Given previous cross-secional evidence, he B/M coefficien will be non-zero in he absence of ime-varying expeced reurns.

18 11 he predicive power of B/M. This possibiliy is a concern paricularly because he facors are empirically moivaed. Daniel and Timan (1997), for example, argue ha he consrucion of HML, which is designed o mimic an underlying risk facor in reurns relaed o B/M, could induce spurious correlaion beween a porfolio s B/M raio and is facor loading. HML is weighed, by design, owards firms wih high B/M. If similar ypes of firms become mispriced a he same ime, hen we should expec ha a firm will covary more srongly wih HML when is B/M is high. As a resul, apparen changes in risk migh help explain B/M s predicive abiliy even under he mispricing sory. In defense of he ime-series regressions, i seems unlikely ha changes in he facor loadings would compleely absorb mispricing associaed wih B/M. More imporanly, Daniel and Timan s argumen canno fully accoun for he relaion beween B/M and risk. The argumen suggess ha he loadings on HML will end o vary wih B/M, bu i does no say anyhing abou he loadings on he marke and size facors. We will see below, however, ha B/M capures significan ime variaion in marke beas and he loadings on SMB. Furher, I provide evidence in Secion 2.4 ha he ime-series relaion beween B/M and he facor loadings on HML is no driven by changes in he indusry composiion of he facor. I esimae he condiional regressions wih an indusry neural facor, which prevens HML from becoming weighed owards paricular indusries. When his facor is used in place of HML, we will coninue o see a srong ime-series relaion beween B/M and he facor loadings. Finally, i is useful o noe ha many indusries have large uncondiional facor loadings on HML, which suggess ha HML does no simply capure mispricing in reurns. Inuiively, Daniel and Timan s argumen suggess ha a given sock will someimes vary posiively and someimes negaively wih HML. Depending on he ype of firms ha are currenly under- and overpriced, HML will be relaed o consanly changing micro- and macroeconomic facors. For example, HML will be sensiive o ineres rae and inflaion risk when i is weighed owards underpriced financial firms, bu will be negaively relaed o hese risks when financial

19 12 firms are overpriced. Corresponding o he changes in HML, a sock will end o covary posiively wih HML when similar firms are underpriced, bu negaively when similar firms are overpriced. Over ime, however, a firm s average facor loading on HML should be close o zero under he mispricing sory, unless firms are persisenly under- and overpriced (which seems unreasonable). This inuiion can be formalized. Suppose ha emporary overreacion explains deviaions from he CAPM, and ha HML, because of is consrucion, absorbs his mispricing (ignore he size facor for simpliciy). To be more specific, assume ha he proxy for he marke porfolio, M, is no mean-variance efficien condiional on firms B/M raios. However, HML is consruced o explain he deviaions from he CAPM, and R M and HML ogeher span he condiional angency porfolio. Appendix A proves ha, in he ime-series regression R i () = a i + b i R M () + h i HML() + e i (), (2.5) he uncondiional facor loading on HML, h i, will equal zero if asses are correcly priced on average over ime. 4 This resul reflecs he idea ha emporary mispricing should no explain uncondiional deviaions from he CAPM. As noed above, however, many indusries have large uncondiional loadings on boh SMB and HML, which herefore suggess ha he facors do no simply capure mispricing in reurns. In summary, he mulifacor regressions es wheher he hree-facor model or he characerisic -based model explains ime-variaion in expeced reurns. The inerpreaion of he regressions, like he resuls for any asse-pricing es, is limied by our need o use a proxy for he unobservable model. Neverheless, he regressions should help us undersand wheher he risk or mispricing sory is a beer descripion of asse prices. 4 The resul also requires ha ime-variaion in b i and h i is uncorrelaed wih he facors expeced reurns. This assumpion seems reasonable since I am ineresed in he facor loadings changing over ime wih firm-specific variables, like B/M, no wih macroeconomic variables (he appendix provides a numerical example). I is also consisen wih he empirical evidence presened in Secion 2.4.

20 Daa and descripive saisics The empirical analysis focuses on indusry porfolios. These porfolios should exhibi cross-secional variaion in expeced reurns and risk, so he ess can examine a diverse group of porfolios. Indusry porfolios are believed a priori o provide variaion in expeced reurns and facor loadings, while soring by oher crieria is ofen moivaed by previous empirical evidence. Hence, indusry porfolios are less suscepible o he daa-snooping issues discussed by Lo and MacKinlay (1990). As a robusness check, I also examine porfolios sored by size and B/M. In cross-secional sudies, differen ses of porfolios ofen produce vasly differen esimaes of risk premia. Of course, he ime-series regressions in his paper migh also be sensiive o he way porfolios are formed. Size porfolios have he advanage ha hey conrol for changes in marke value, which has been shown o be associaed wih risk and expeced reurns, ye should be relaively sable over ime. The book-o-marke porfolios allow us o examine how he expeced reurns and risk of disressed, or high-b/m, firms change over ime. The porfolios are formed monhly from May 1964 hrough December 1994, for a ime series of 368 observaions. The indusry and size porfolios consis of all NYSE, Amex, and Nasdaq socks on he Cener for Research in Securiy Prices (CRSP) apes, while he book-omarke porfolios consis of he subse of socks wih Compusa daa. Socks are sored ino 13 indusry porfolios based on wo-digi Sandard Indusrial Classificaion (SIC) codes as repored by CRSP. For he mos par, he indusries consis of consecuive wo-digi codes, alhough some excepions were made when deemed appropriae. 5 The size porfolios are formed based on he marke value of equiy in he previous monh, wih breakpoins deermined by NYSE deciles. To reduce he fracion of marke value in any single porfolio, he larges wo porfolios are furher divided based on he 85h and 95h perceniles of NYSE socks, for a oal of 12 porfolios. Finally, he book-o-marke porfolios are formed based on he raio of book equiy 5 Deails available on reques.

21 14 in he previous fiscal year o marke equiy in he previous monh. Again, he breakpoins for hese porfolios are deermined by NYSE deciles. The lowes and highes deciles are furher divided using he 5h and 95h perceniles of NYSE socks, for a oal of 12 porfolios. For all hree ses of porfolios, value-weighed reurns are calculaed using all socks wih CRSP daa, and value-weighed B/M raios are calculaed from he subse of socks wih Compusa daa. 6 To ensure ha he explanaory power of B/M is predicive, I do no assume ha book daa become known unil five monhs afer he end of he fiscal year. Also, o reduce he effec of poenial selecion biases in he way Compusa adds firms o he daabase (see he discussion by Kohari, Shanken, and Sloan, 1995), a firm mus have hree years of daa before i is included in any calculaion requiring book daa. The ime-series regressions use excess reurns, calculaed as reurns minus he one-monh T-bill rae, and he naural logarihm of B/M. Table 2.1 repors summary saisics for he porfolios. The average monhly reurns for he indusry porfolios range from 0.83% for uiliies and elecommunicaions firms o 1.28% for he service indusry (which includes enerainmen, recreaion, and services), for an annualized spread of 6.1%. Coincidenally, hese indusries also have he lowes (3.67%) and highes (6.78%) sandard deviaions, respecively. The size and book-o-marke porfolios also exhibi wide variaion in average reurns and volailiy. Average reurns for he size porfolios vary from 0.80% for he larges socks o 1.24% for he smalles socks, and he sandard deviaions of reurns decrease monoonically wih size, from 6.68% o 4.17%. Average reurns for he book-o-marke porfolios range from 0.76% for he second decile hrough 1.46% for he socks wih he highes B/M. Ineresingly, he sandard deviaion of reurns are U-shaped; hey decrease monoonically wih B/M unil he sixh decile, which has a sandard deviaion of 4.42%, and increase hereafer, o 6.86% for porfolio 10b. The saisics for B/M, like hose for reurns, reveal considerable cross-secional 6 The socks included in he calculaion of B/M are a subse of hose included in he calculaion of reurns, and we can inerpre he esimae of B/M as a proxy for he enire porfolio. The inferences in his paper are unchanged when porfolio reurns are based only on hose socks wih Compusa daa.

22 Table 2.1 Summary saisics for indusry, size, and book-o-marke porfolios Each monh from May 1964 hrough December 1994, value-weighed porfolios are formed monhly from all NYSE, Amex, and Nasdaq socks on CRSP. Firms mus also have Compusa daa for he book-o-marke porfolios. Book-o-marke (B/M) is calculaed as he raio of book equiy in he previous fiscal year o marke equiy in he previous monh for all socks wih Compusa daa. The indusry porfolios are based on wo-digi SIC codes. The size porfolios are based on he marke value of equiy in he previous monh, wih breakpoins deermined by NYSE deciles; porfolios 9 and 10 are furher divided using he 85 and 95 perceniles of NYSE socks. The book-o-marke porfolios are based on B/M in he previous monh, wih breakpoins deermined by NYSE deciles; porfolios 1 and 10 are furher divided using he 5 and 95 perceniles of NYS63E socks. Reurn (%) Book-o-marke Number of firms Porfolio Mean Sd. dev. Mean Sd. dev. Auocorr. Adj. R 2 a May 1964 Dec Panel A: Indusry porfolios Na. resources Consrucion Food, obacco Consumer producs Logging, paper Chemicals Peroleum Machinery, equipmen ,222 Transporaion Uiliies, elecom Trade Financial ,747 Services and oher Panel B: Size porfolios Smalles ,

23 Table 2.1. Coninued. Reurn (%) Book-o-marke Number of firms Porfolio Mean Sd. dev. Mean Sd. dev. Auocorr. Adj. R 2 a May 1964 Dec a b a Larges Panel C: Book-o-marke porfolios Lowes b a Highes a Adjused R 2 from regressing he porfolio s B/M raio on he value-weighed B/M raio of all socks ha mee boh CRSP and Compusa daa requiremens. 16

24 17 differences in porfolio characerisics. Average B/M doubles from 0.40 for chemical firms o 0.82 for he ransporaion indusry. A similar spread is shown for size porfolios, wih B/M ranging from 0.51 for he larges socks o 1.03 for he smalles socks. The book-o-marke porfolios, of course, have he greaes cross-secional variaion, wih average B/M ranging from 0.15 for he low-b/m porfolio o 2.66 for he high-b/m porfolio. The sandard deviaions over ime are also reasonably high, reflecing he volailiy of sock reurns. The ime-series sandard deviaion of B/M is, on average, 0.20 for he indusries, 0.24 for he size porfolios, and 0.29 for he book-o-marke porfolios. Variaion in B/M will be necessary for he ime-series regressions o have power disinguishing beween he compeing hypoheses. Table 2.2 repors summary saisics for he Fama and French (1993) facors, which are described fully in Appendix A. The marke facor, R M, is he excess reurn on he CRSP valueweighed index, and he size and book-o-marke facors, SMB and HML, are zero-invesmen porfolios designed o mimic underlying risk facors in reurns. The average monhly reurn of R M is 0.39%, of SMB is 0.30%, and of HML is 0.38%. The risk premium for each facor is measured by is mean reurn, so hese averages imply posiive compensaion for bearing facor risk. As noed by Fama and French, he procedure used o consruc SMB and HML appears o successfully conrol each facor for he influence of he oher, as demonsraed by he low correlaion beween he facors, equal o Also, SMB is posiively correlaed wih R M (correlaion of 0.36), while HML is negaively correlaed wih R M (-0.35). Thus, he reurns on he size and B/M facors are no independen of he marke reurn, reflecing he fac ha heir consrucion did no conrol for differences in he beas of he underlying socks. The CAPM and mos empirical sudies examine he relaion beween simple-regression marke beas and expeced reurns. To enhance comparison wih cross-secional sudies, I use size and B/M facors ha are orhogonal o R M. These facors, SMBO and HMLO, are consruced by adding he inerceps o he residuals when SMB and HML are regressed on a consan and he excess marke reurn. From regression analysis (e.g., Johnson, 1984, p. 238),

25 18 Table 2.2 Summary saisics for facors The facors are calculaed monhly from May 1964 hrough December R M is he reurn on he CRSP value-weighed index minus he one-monh T-bill rae. SMB is he reurn on a porfolio of small socks minus he reurn on a porfolio of big socks. HML is he reurn on porfolio of high- B/M socks minus he reurn on a porfolio of low-b/m socks. SMBO and HMLO are orhogonalized versions of SMB and HML, consruced by adding he inerceps o he residuals in regressions of SMB and HML on a consan and R M. All reurns are repored in percen. Correlaion Facor Mean Sd. dev. Auocorr. R M SMB HML SMBO HMLO R M SMB HML SMBO HMLO he coefficiens in he hree-facor model will be unaffeced by he change in variables, excep ha marke beas will now be he simple-regression beas of he CAPM. Table 2.2 shows ha he average reurn on he book-o-marke facor increases from 0.38% o 0.47%, bu he reurn on he size facor decreases from 0.30% o 0.21%. The correlaion beween he size and booko-marke facors, 0.07, remains close o zero The predicabiliy of porfolio reurns This secion invesigaes he simple ime-series relaion beween expeced reurns and B/M. The simple regressions help evaluae he economic imporance of B/M, wihou regard o changes in risk or mispricing, and provide a convenien benchmark for he condiional hreefacor model. In addiion, he analysis complemens recen sudies which find ha B/M forecass aggregae sock reurns (Kohari and Shanken, 1997; Poniff and Schall, 1998). As discussed above, he risk and mispricing views boh sugges ha B/M will predic porfolio reurns. For each porfolio, I esimae he ime-series regression R i () = γ i0 + γ i1 B/M i (-1) + e i (), (2.6)

26 19 where R i is he porfolio s excess reurn and B/M i is he naural log of is lagged book-o-marke raio. The slope coefficien in his regression is expeced o be posiive. Several complicaions arise in esimaing eq. (2.6). Firs, he appropriae definiion of B/M is unclear. Cross-secional sudies sugges ha a porfolio s B/M relaive o oher firms could be imporan. Thus, B/M i (-1) migh be defined as eiher he porfolio s acual B/M raio or is B/M raio minus an aggregae index. The laer varies primarily wih marke-adjused sock reurns, and would be a beer measure if common variaion in B/M is unrelaed o mispricing. 7 Asse-pricing heory provides lile guidance. The conclusions in his paper are no sensiive o he definiion of B/M, and for simpliciy I repor only resuls for raw B/M. Also, o ease he inerpreaion of he resuls, B/M is measured as deviaions from is ime-series mean for he remainder of he paper. As a consequence, when B/M i equals zero in he regressions, B/M is acually a is long-run average for he porfolio. Second, Sambaugh (1999) shows ha conemporaneous correlaion beween reurns and B/M will bias upward he slope coefficien in eq. (2.6). Suppose ha B/M follows he AR(1) process B/M i () = c i + p i B/M i (-1) + u i (). (2.7) The bias in he esimae of γ i1 is approximaely E γ i1 γ i1 [ $ ] [cov(e i, u i ) / var(u i )] [-(1+3p i ) / T], (2.8) where T is he lengh of he ime series. The residuals in eqs. (2.6) and (2.7), e i and u i, are negaively relaed because a posiive sock reurn decreases he porfolio s B/M. Also, Table 2.1 shows ha B/M is highly persisen over ime, wih auocorrelaions ranging from 0.96 o 0.99 a he firs lag. Togeher, he correlaion beween e i and u i and he persisence in B/M 7 Kohari and Shanken (1997) and Poniff and Schall (1998) show ha aggregae B/M predics marke reurns during he period 1926 hrough 1992, which could reflec aggregae mispricing. Their resuls for he period 1963 hrough 1992 are much weaker. For he curren paper, preliminary ess indicae ha aggregae B/M has lile power o forecas he marke, size, and book-o-marke facors.

27 20 impar a srong upward bias in he esimae of γ i1. In a relaed conex, for marke reurns regressed on aggregae B/M, Kohari and Shanken (1997) boosrap he disribuion of he slope and find ha Sambaugh s formula is empirically valid. The ess below adjus for his bias Indusry porfolios Table 2.3 repors resuls for he indusry porfolios. The evidence provides some suppor for a posiive associaion beween expeced reurns and lagged B/M, bu he high volailiy of sock reurns reduces he power of he ess. The bias-adjused slopes range from for food and obacco firms o 1.75 for he naural resources indusry, and 10 of he 13 coefficiens are greaer han zero. The average esimae is posiive, 0.58, alhough i is only abou one sandard error, 0.62, from zero (he sandard error reflecs cross-secional correlaion in he esimaes). Sronger evidence of predicive abiliy is provided by he χ 2 es of he slope coefficiens. This es rejecs a he 5% level he hypohesis ha B/M does no capure any varia ion in expeced reurns. The average coefficien, 0.58, is similar o he cross-secional slope, 0.50, esimaed by Fama and French (1992). Economically, he average coefficien is reasonably large. Consider, for example, he effec ha a change in B/M equal o wo sandard deviaions would have on expeced reurns. For he average indusry porfolio, he ime-series sandard deviaion of B/M is An increase in B/M wice his large maps ino a 0.38% change ( ) in expeced reurn for he ypical porfolio, or 4.67% annually. On he oher hand, he predicive power of B/M is low as measured by he adjused R 2 s. Lagged B/M explains a mos 1% of he oal variaion in porfolio reurns. This resul is consisen wih previous sudies a he marke level, which generally find ha pre-deermined variables explain only a small fracion of monhly reurns (e.g., Fama and French, 1989). In addiion o he ordinary leas squares (OLS) esimaes jus described, Table 2.3 repors seemingly unrelaed regression (SUR) esimaes of he equaions. OLS reas he regression for

28 Table 2.3 Predicabiliy of indusry reurns R i () = γ i0 + γ i1 B/M i (-1) + e i () The indusry porfolios are described in Table 2.1. R i is he porfolio s monhly excess reurn (in percen) and B/M i is he naural log of he porfolio s book-o-marke raio a he end of he previous monh. The able repors boh ordinary leas squares (OLS) and seemingly unrelaed regression (SUR) esimaes of he slope coefficiens. The OLS bias-adjused slopes correc for small-sample biases using eq. 2.8 in he ex. The bias correcion for he SURs, as well as he covariance marix of he bias-adjused esimaes, is obained from boosrap simulaions. OLS Porfolio γ i1 Bias-adj γ i1 Sd. err. Adj. R 2 γ i1 Sd. err. Bias-adj γ i1 Sd. err. Na. resources 2.56 * Consrucion Food, obacco Consumer producs * Logging, paper Chemicals Peroleum 2.33 * * Mach., equipmen Transporaion Uiliies, elecom Trade Financial 1.93 * * Services, oher * Average 1.43 * * 0.17 (sd. err.) (0.62) (0.62) (0.20) (0.23) χ 2 a * * 9.83 (p-value) (0.142) (0.015) (0.048) (0.707) SUR a χ 2 = c Σ -1 c, where c is he vecor of coefficien esimaes and Σ is he esimae of he covariance marix of c. Under he null ha all coefficiens are zero, his saisic is asympoically disribued as χ 2 (d.f. 13). * Denoes coefficiens ha are greaer han wo sandard errors from zero or χ 2 saisics wih a p-value less han

29 22 each porfolio separaely, and ignores ineracions among he equaions. The residuals across porfolios are correlaed, however, because indusries excess reurns are driven by many of he same macroeconomic facors. SUR uses his informaion o esimae he sysem of equaions more efficienly (Zellner, 1962). Alhough SUR requires an esimae of he residual covariance marix, he efficiency gain is likely o be large because (1) he error erms are highly correlaed across porfolios (see Greene, 1993, p. 489), and (2) he dimension of he covariance marix (13 13) is small relaive o he lengh of he ime series (368 monhs). Indeed, Table 2.3 shows ha he average sandard deviaion of he SUR slopes is 0.40, compared wih 0.86 for OLS. While he sandard deviaions are esimaed wih error, he large decrease suggess ha SUR is subsanially more efficien. I was noed above ha OLS slope esimaes are biased upward. I am no aware of any research ha explores he bias in SUR esimaes, and here is lile reason o believe ha i is idenical o ha of OLS. Wihou an analyical esimae, I rely on boosrap simulaions o assess he sampling disribuion of he SUR slopes. The simulaion procedure, described in Appendix A, randomly generaes ime series of reurns and B/M, imposing he resricion ha expeced reurns and B/M are unrelaed. Since he rue coefficien in he simulaion equals zero, he mean of he disribuion represens he bias in SUR esimaes. Furher, he sandard deviaion of he disribuion provides an esimae of he SUR sandard error. 8 Table 2.3 shows ha he bias-adjused SUR esimaes end o be smaller han heir OLS counerpars. The coefficiens range from for he machinery and equipmen indusry o 0.96 for peroleum firms, and eigh of he 13 esimaes are posiive. The average coefficien on B/M, 0.17, is posiive, alhough i is under one sandard error, 0.23, from zero. In addiion, he χ 2 saisic canno rejec he hypohesis ha all slope coefficiens are zero. The simulaions 8 I also simulae he disribuion of he OLS slope esimaes and find ha he analyical esimae of he bias is reasonably accurae. The average bias from he simulaions is 0.92 compared wih 0.85 from eq. (2.8). The sandard errors from he simulaion, however, end o be larger han he OLS esimaes. For example, he sandard deviaion of he average coefficien is 0.76, compared wih he OLS sandard error of 0.62.

30 23 indicae ha he average bias in he SUR esimaes, 0.43, is abou half he bias in he OLS regressions, The magniude remains significan, however, and he average SUR coefficien decreases by wo-hirds, from 0.60 o 0.17, afer correcing for bias. In sum, he evidence in Table 2.3 is consisen wih a posiive relaion beween B/M and expeced reurns, bu B/M explains, a mos, a small fracion of reurns. Afer adjusing for bias in he regressions, only he χ 2 saisic for he OLS slope coefficiens is significan a convenional levels. We will see below ha he power of he ess is much greaer in he condiional hree-facor regressions, because he facors absorb much of he volailiy of reurns. In addiion, he size and book-o-marke porfolios reveal a considerably sronger relaion beween B/M and fuure reurns. As a final observaion, i is useful o keep in mind ha he regressions canno rejec economically meaningful coefficiens on B/M. A ypical confidence inerval around he average esimae, for eiher OLS or SUR, would include reasonably large coefficiens. Moreover, low explanaory power does no imply ha B/M is necessarily unimporan. For example, Kandel and Sambaugh (1996) show ha predicive variables wih low explanaory power can have a large impac on asse allocaion decisions. I suspec a similar resul would hold a he porfolio level: he opimal porfolio held by a risk-averse, Bayesian invesor is probably sensiive o predicive variables which have low saisical significance Size and book-o-marke porfolios Table 2.4 shows resuls for he size and book-o-marke porfolios. For simpliciy, I repor only he SUR esimaes, along wih he bias-adjused esimaes, since he evidence above indicaes ha SUR increases he precision of he slope esimaes. The able shows ha B/M predics saisically reliable variaion in reurns for boh he size and book-o-marke porfolios. Afer correcing for bias, four coefficiens for he size porfolios and nine coefficiens for he book-o-marke porfolios are more han wo sandard errors above zero. All 12 esimaes are