Forecasting Cross-Section Stock Returns using The Present Value Model. April 2007

Transcription

1 Forecasing Cross-Secion Sock Reurns using The Presen Value Model George Bulkley 1 and Richard W. P. Hol 2 April 2007 ABSTRACT We conribue o he debae over wheher forecasable sock reurns reflec an unexploied profi opporuniy or raionally reflec risk differenials. We es wheher agens could earn excess reurns by selecing socks which have a low marke price compared o an esimae of he fundamenal value obained from he presen value model. The crierion for sock picking is one which could acually have been implemened by agens in real ime. We show ha saisically significan, and quaniaively subsanial, excess reurns are delivered by porfolios of socks which are cheap relaive o our esimae of fundamenal value. There is no evidence ha he under priced socks are relaively risky and hence excess reurns canno easily be inerpreed as an equilibrium compensaion for risk. Keywords: Excess reurns, Trading rule, Efficien markes, Presen value model, Sock prices JEL classificaion: G12; G14 1 Universiy of Exeer, Exeer, EX4 4RJ, UK. 2 Universiy of Edinburgh, Edinburgh, UK, Richard.Hol@ed.ac.uk

2 1 INTRODUCTION I is now widely acceped ha cross-secion sock reurns can be forecas by he raio of he curren sock price o a number of accouning variables such as he raio of he marke value o book value of asses, Fama and French (1992), he price dividend raio, Elon e al. (1983) and he raio of cash flow o marke value of equiy, Lakonishok e al. (1994). Even he well known size effec, Banz (1981), falls ino his class, since size is usually measured by sock price muliplied by he number of ousanding shares. We examine wheher he evidence ha price scaled accouning measures can forecas reurns indicaes a genuine profi opporuniy or insead raionally reflecs risk differences beween socks. This laer inerpreaion follows from he fac ha risky socks, which offer higher expeced reurns, ineviably have relaively low marke prices relaive o accouning variables like book value, curren dividends, curren earnings and he number of ousanding shares. 3 The inerpreaion ha forecasable reurns represen an economic profi opporuniy follows from he hypohesis ha here are irraional swings in marke senimen, see Shiller (1989). Lakonishok e al. (1994) argue ha curren accouning variables serve as a proxy for fundamenal value and hence we can idenify under priced socks as hose wih a low price relaive o hese variables. They presen evidence o sugges ha his is why hese price scaled variables have forecasing power. These very differen heories of sock pricing imply he same empirical relaionships in he daa because curren price is measured relaive o a proxy for he fundamenal 3 For he price dividend raio his follows from he Gordon growh model. Berk (1996) demonsraes ha in an efficien marke size and marke o book will be risk proxies and hence will forecas reurns.

3 which is independen of company risk. I is ineviable in his approach ha risky firms will on average appear relaively under priced. In his paper we aemp o overcome he problem of observaional equivalence by consrucing an esimae of he fundamenal value which reflecs he risk of he sock. We es wheher reurns can be forecas by he raio of curren price o an esimae of he fundamenal value for he sock price calculaed from a model which implicily allows for risk. So risky socks need no have a low marke price relaive o his benchmark and herefore if under priced socks are found o deliver high reurns his canno so easily be explained away as an equilibrium compensaion for risk. The fundamenal value for a sock is esimaed as follows. We assume ha he dividend process for individual firms is saionary in firs differences. The presen value model hen implies ha prices and dividends are co-inegraed, Campbell and Shiller (1987). We esimae he parameers of his co-inegraing regression for each firm in our sample. Our esimae of he fundamenal sock price a any dae is hen obained by subsiuing he sock s curren dividend ino he esimaed co-inegraing regression. 4 The sock s risk premium is refleced in our measure of fundamenal value because socks wih high required raes of reurn will, on average, have a low price relaive o heir dividend. Thus our esimae of he fundamenal value will reflec he risk adjused discoun rae which he marke is acually observed o apply o each sock over he sample period. We assume ha however risk is measured and priced, he relaive risk of a company is fixed wihin he sample period. 5 4 This approach may be compared o he convenional curren dividend yield crierion. This laer approach ess for excess reurns on porfolios of socks which have a high dividend yield relaive o oher socks a he same dae. Our approach will end o selec socks which have a high yield no relaive o oher socks bu relaive o heir own pas dividend yield. 5 I is possible for boh he risk free rae and he average equiy risk premium o be ime varying, since hese aggregae flucuaions will no affec he relaive ranking of socks using our mispricing measure.

4 To ensure ha he explanaory variables are public informaion a he beginning of he period over which reurns are forecas, we use a measure of fundamenal value obained from a co-inegraing regression which is esimaed using only he daa available up o ha same dae ha he forecas is made. In his way we are esing for he exisence of a profiable rading rule which could have been employed by agens rading in real ime. We work wih a sample of socks lised on he New York Sock Exchange and es wheher porfolios of he under priced socks seleced on he above crierion deliver excess reurns over a one o en year ime horizon. This allows us o sudy he whole life cycle of mispricing and hus relae our resuls o he evidence of negaive serial correlaion in long erm reurns repored by Debond and Thaler (1985). We firs compare he absolue size of reurns over one o en years on porfolios of socks which are over priced o reurns on porfolios of under priced socks. Finding a subsanial difference in he size of he realized reurns on hese porfolios we nex repor ess of saisical significance. We evaluae he saisical significance of he ou-performance of he under priced socks. Finally, and nowihsanding he above argumens ha our benchmark raio should no be correlaed wih risk, we examine wheher here is any evidence ha he profis which he sock picking crierion delivers can be explained away by differences in risk. In Secion 2 we discuss our an esimae of he fundamenal value of individual firms. Nex we describe he daa and repor our resuls, and in Secion 4 examine wheher he excess reurns which we find may simply reflec increased risk.

5 2 AN ESTIMATE OF THE FUNDAMENTAL VALUE OF STOCKS The fundamenal value for firm i a dae, * P i,, is he expeced value of fuure dividends,, discouned by he firm specific rae δ. Suppressing he i subscrip: D i, + j i * j P = δ E [ D ] j= 0 + j (1) * Subrac δp from boh sides of (1) and re-arrange o obain P * = δ D ]. If dividends are I(1) hen δ j [ D ] + δ E [ Ε 1 δ 1 δ j= 1 + j ~ µ, where µ is he uncondiional mean of dividend growh and D + j + D+ j ~ D + j is he deviaion of dividend growh from µ. Then: P [ D ] * 2 j ~ = ΘD + Θ + Θ δ Ε + j j= 1 µ (2) where [ ] = δ δ Θ 1, Θ 2 µ is he consan in he coinegraing regression, and he hird erm on he righ hand side of (2) is a saionary residual. The coinegraing vecor for each firm, [ 1, Θ, Θ] 2 µ, reflecs boh he firm specific discoun rae, δ, and is long erm dividend growh rae. This ensures ha when we calculae our esimae of he fundamenal appropriae allowance for risk is made.

6 We esimae (2) a each dae when we wish o calculae a fundamenal value using only price and dividend daa up o and including ha same dae. This allows us o es wheher socks are mispriced condiional on he informaion ses of agens who do no know he rue parameers of he model, bu have o esimae hem from public daa which is available a he dae when he fundamenal value mus be calculaed. Our esimae of he fundamenal value of a sock, ˆ P i,, is hen obained by subsiuing he curren dividend ino he co-inegraing relaionship esimaed a ha dae: we calculae ˆ 2 = Θˆ D + Θˆ µ, where he subscrips i, indicae a parameer P ˆ i, i, i, i, i, esimaed a dae for sock i. In an efficien marke price may differ from he co-inegraing price as a consequence of raionally forecas emporary differences beween dividend growh in he shor run and is long run value, as capured by he hird erm on he righ hand side of (2). Bu his difference is public informaion and hence canno forecas reurns. Under he alernaive hypohesis a fad arises as an addiional erm on he righ hand side of (2). The raio of marke price o coinegraing price will hen be a measure of sock mispricing and hence should forecas reurns if mispricing is emporary. 3 DATA AND RESULTS Daa on prices, dividends, Bea, and size were aken from CRSP (Cenre for Research in Securiy Prices, Universiy of Chicago). The dividend yield is measured by oal annual dividend paymens in he calendar year ending December 31 s divided by he sock price on December 31 s. Size is marke capialisaion. The coinegraing regressions are esimaed using a price and dividend series for each sock which are

7 adjused for all forms of capial re-organisaions, using daa provided by CRSP. The sample of firms sudied in any year is all hose which have been coninuously lised on he New York Sock Exchange from December 1945 o ha same dae and paid dividends in a leas 10 of hose years. 6 The sample consiss of 580 firms in 1960, which falls o 348 in In view of he large number of firms, and daa ses of differen spans, we do no repor he ime series properies of he individual dividend series, or resuls for he coinegraing regressions. The assumpion ha for he ypical firm dividends are firs difference saionary and ha prices and dividends are coinegraed is a necessary condiion for our procedure o successfully idenify under priced socks on average. We will, of course, make misakes in idenifying under priced socks and one reason for his may be ha hese assumpions are false for some firms. The criical issue is wheher here is sysemaic mispricing which can be idenified by a sraegy which requires hese assumpions o be saisfied for he average firm. We measure mispricing by he raio of curren marke price o esimaed fundamenal price. All socks are ranked by his raio each year and hen are assigned o Decile porfolios. The op and boom Deciles are sub-divided ino wo. All firms in a paricular porfolio are iniially given equal weighing and hereafer here is no rebalancing of porfolios. Cumulaive real porfolio reurns for individual porfolios are calculaed assuming ha dividends are re-invesed in he same sock. Cumulaive porfolio reurns are he average of he cumulaive reurns for individual socks. There is no problem of survivorship bias since we race reurns on all firms irrespecive of 6 This sample choice reflecs a rade-off beween he number of firms included and he lengh of daa available on each o esimae he co-inegraing regression. We chose o begin afer World War II as a naural definiion of he sample o avoid any suspicion ha he resuls were obained by daa mining.

8 heir survival hisory afer he sample selecion year. Where firms disappear from he daa se for any reason we assume final paymens o sockholders are reinvesed in he same porfolio in he same proporions as he remaining firms occupy a ha dae. We firs repor reurns on five porfolios consising of he full sample of firms, FS, he 5% mos under-valued firms, UN1; he nex 5% mos under-valued firms, UN2; he 5% mos over-valued firms, OV1 and he nex 5% mos over-valued, OV2. Porfolios are consruced saring from December 31 s 1960 and hen annually on he same dae unil December 31 s For each sar dae, he reurns on porfolios were calculaed for he subsequen decade. Table 1 repors he average reurns across sar daes of hese porfolios for horizons of one o en years. In Table 1 i can be seen ha on average UN1 and UN2 porfolios boh bea he full sample for all horizons. UN1 also beas UN2 for all horizons. The expeced excess reurns are subsanial. For example, he UN1 porfolio delivers an average cumulaive excess reurn of 36.1% over five years, compared o he full sample. The UN1 porfolio beas he OV1 porfolio on average by 45.7% cumulaive over 5 years, or approximaely 9% per annum for 5 consecuive years. Boh of he over-valued porfolios, OV1 and OV2, on average under-perform he full sample average for all horizons, excep he firs year in he case of OV1. The one somewha surprising resul is ha OV1 delivers higher reurns han OV2 for he firs six years. 7 7 This is because he mos over-valued porfolio conained a number of zero dividend firms. Repeaing he whole exercise omiing any firms which paid no dividends in he welve monhs preceding he dae of porfolio formaion overurns his resul. I is ofen argued ha zero dividends are a sign of financial disress so ha he well documened fac ha zero dividend firms deliver higher reurns can be explained as a risk premium, Chrisie (1990).

9 Tracing reurns for a decade conribues evidence on he long-run life cycle of mispricing and allows us o relae our resuls o he work which has repored negaive serial correlaion in long run reurns, for example DeBond and Thaler (1985). In order o focus on he life cycle of reurn we repor in Table 2 he average incremenal reurns from holding each porfolio for one more year. The evidence in Table 2 is consisen wih, bu suggess a qualificaion o, he evidence repored by DeBond and Thaler ha performance over a five year horizon is negaively correlaed wih subsequen reurns. I is consisen in ha an over-valued sock mus have previously delivered high reurns in he course of becoming over-valued and in his case high pas reurns will be followed by low subsequen reurns. However, high reurns which are a resul of he unwinding of earlier under pricing are no followed by unusually low reurns. An undervalued porfolio, which consisenly delivers subsanial excess reurns over a five year horizon, performs marginally beer han he full sample over he following five years, wih a small cumulaive excess reurn of only 4.9%. Five years over performance is followed by five years of approximaely average reurns. This implies ha he underlying facor which explains excess reurns is no underperformance in he pas bu under pricing in he presen. Pas underperformance is an imperfec proxy for curren under valuaion. Table 2 also highlighs an ineresing asymmery in he ime profile of he excess reurns for under and over valued porfolios. The UN1 porfolio delivers a cumulaive excess reurn of 31.6% over he firs five years. On he oher hand he 5% mos overvalued porfolio, OV1, only under-performs in he firs five years by a cumulaive 9.6%. However, in he second half of he decade i underperforms he full sample by a cumulaive 30.3%, wih paricularly pronounced under-performance in he las hree

10 years of he decade. This suggess ha under-valued shares bounce back o fundamenal values more quickly han glamour socks are exposed for heir rue worh. The marke spos a bargain quicker han i loses faih in a favourie. 8 On our bes esimae i is possible o earn subsanial profis using his rading rule. We urn nex o he quesion of wheher hese profis are saisically significan. We regress reurns on he raion of marke price o he heoreical sock price. We argued above ha his raion should no be correlaed wih he risk of he sock providing he firm risk is consan over ime. If his assumpion is valid hen a properly specified reurns regression requires ha we also include regressors which do explicily capure risk. We use hree variables o measure risk, Bea, size and dividend yield. Under he CAPM covariance wih he marke, i.e. Bea, measures risk. Fama and French (1992), and ohers, show ha marke capialisaion is imporan in explaining cross-secion reurns. Berk (1995) shows why size will serve as a risk measure. Finally, Ball (1978) argued ha dividend yield should serve as a cach all measure of risk in an efficien marke. Even if he researcher has no assumed he correc model of risk pricing hen he variance of oal reurns due o risk will be proxied by he dividend yield, since he dividend yield is a componen of he oal reurn. 9 This gives he specificaion of our cross-secion regression, esimaed a dae : R i, 0 1 ( Pi Pˆ, i, ) + γ 2Beai, + γ 3Sizei, + 4 Di, Pi, ui, = γ + γ log γ + (3) 8 Moreover, he average probabiliy of a company which was in OV1 one year re-appearing in OV1 he following year was 53.8%. The average probabiliy of a UN1 company re-appearing he following year was 43.5%. Mispricing is highly persisen, boh for over and under-valued firms. However, he persisence of over valuaion is raher greaer han ha of under valuaion. This is consisen wih he evidence in Table 2. 9 The only assumpion required for his argumen is ha if dividends are paid on sock hen he fracion of oal reurns which are delivered by dividends is no iself sysemaically relaed o risk.

11 The null hypohesis of efficien markes implies ha γ 0 : deviaions of marke 1 = price from he heoreical price reflec expeced emporary deviaions in he dividend growh rae from is long run average and are uncorrelaed wih subsequen reurns. Since firms are drawn from single economy we would expec a macro componen o unexpeced reurns common o all firms a a single dae, so saisic from he crosssecion regression (3) will be biased because we would no expec cov(, ) 0 u, i u j i j. We follow Fama and Mcbeh (1973) (FM) proposed soluion o his problem which recognises ha he esimaes of he coefficiens hemselves are no biased, despie he presence of cross-secion dependence in he error erm. Therefore regression (3) is esimaed for periods and he sample disribuion of γˆ k is consruced for each coefficien, k = 0, The sample sandard deviaion of he esimaed coefficiens provides an esimae of he sandard error of he disribuion of γ k. We hen have he saisic ( γ k ) γ k σ γ k = 1 k, T = T 1 ˆ 2 2 T ˆ γ γ and ˆ = ( T 1) ( ˆ γ ˆ γ ) k σ γ = k 1 k, ˆ = ˆ ˆ for k = 0, 4, where k 2 and T is he number of esimaed cross-secion regressions. A disadvanage of his approach is ha we canno sudy long horizon reurns, as FM esimaion requires non-overlapping daa. In order o run a reasonable number of independen cross-secion regressions we only consider reurns for horizons of up o hree years. We reain he same sar dae, The cross-secion regression is esimaed each year for one year reurns; every second year for wo year reurns and every hree years for hree year reurns. This gave 15 wo year horizon regression and 10 hree year horizon regressions. We repor he resuls in Table 3.

12 Table 3 shows ha for reurn horizons of wo and hree years we can rejec he implicaion of he efficien markes hypohesis ha γ = 1 0. The signs and saisics on he remaining variables are all largely in line wih resuls obained in previous research. The size variable is significan and of he expeced sign in our sample. The dividend yield coefficien is of he expeced sign for mos periods. Alhough i is no saisically significan, we do no aach any imporance o his in view of he few years available from which o calculae he saisic. For he same reason, he posiive, bu only marginally significan coefficien on Bea is unsurprising. The fac ha we could run so few regressions o build up he sample disribuion of he coefficiens in he case of wo and hree year reurns implies ha our bes es had low power o rejec he null hypohesis. I is herefore he more sriking ha we were able o do so. I is ineresing o compare he saisic on our mispricing variable wih he evidence which our es delivers on he role of size. I is widely acceped ha size is an imporan deerminan of reurns over his same ime period, Fama and French (1992). However, he saisic on size in ours es is sill smaller han ha on he mispricing variable for boh wo and hree year reurns. 4. IS OUR CRITERION SIMPLY SELECTING RISKIER STOCKS? The excess reurns on he under priced porfolios appear o be large and saisically significan. However, i migh ye be argued ha he above assumpion, ha firms perceived relaive risk a any ime is he hisoric average up o ha dae in he sample, is invalid. In his case socks which are of emporarily high risk will appear underpriced. One answer o his is ha in he above regression we conrolled for risk using he conemporaneously measured size and dividend yield and Bea measured

13 over he previous five years. Anoher perspecive is provided by looking a he average value of size and Bea for he under and over-valued porfolios, and hese saisics are repored in Table 4. I can be seen in Table 4 ha alhough he average value of Bea for he 5% mos under-valued porfolio is larger han for he 5% mos over-valued, he size of he difference is bo large enough o accoun for all he excess reurn. If he equiy premium over he safe rae of ineres is aken o be he hisoric average of approximaely 8%, hen he difference in Bea beween UN1 and OV1 requires a yield difference of approximaely 9% per annum over he firs five years. Furhermore he average value of Bea for UN2 is less han for OV1, and ye on average for every horizon UN2 beas OV1. The average company size in boh of he under-valued porfolios is greaer han in boh of he over-valued porfolios. Since here is overwhelming evidence, Banz (1981), Fama and French (1992), ha on average smaller firms deliver higher reurns i is sriking ha UN1 and UN2 boh bea OV1 and OV2 despie he fac ha he laer are comprised on average of significanly smaller firms. However many measures of risk we repor i can sill be argued ha excess reurns may be raionalised by some oher model of risk. An alernaive approach is o no focus on ex ane risk measures bu o examine insead wheher as a maer of fac he under priced shares were, when combined ino porfolios, riskier ex pos. Tha is, raher han comparing jus he mean of he resuling reurns disribuion for under

14 priced / over priced porfolios, as in he previous secion we compare he whole disribuions. We repor resuls on he disribuion of five year reurns in Table 5. We examine wheher a risk-averse individual would sricly prefer he disribuion of reurns from he under priced porfolio o he disribuion of reurns from he over priced porfolio. We use he crierion of second order sochasic dominance (SSD). A probabiliy densiy funcion, f, is said o exhibi SSD over a densiy g if a y F y ( x) dx G( ) a x dx, y wih sric inequaliy for some y, where F, G denoe cumulaive densiy funcions associaed wih densiy funcions f and g respecively. If f exhibis SSD wih respec o g, hen f is preferred o g by all risk averers, see Rohschild and Sigliz (1972). We find ha he Decile of mos under-valued shares exhibied second order sochasic dominance over he Decile porfolio of mos over-valued shares. Indeed he under valued socks only fail o exhibi firs order sochasic dominance by a iny margin. Table 5 reveals ha for every sar dae he five year reurns of he mos under-valued Decile were greaer han he five year reurns on he mos over-valued Decile, wih he rivial excepion of 1964 where he cumulaive difference beween he wo was 0.3% over 5 years. To inerpre he dominance of our under priced porfolios over five years as consisen wih raional asse pricing one would have o argue ha he excess reurn on he under priced socks refleced a sronger posiive covariance wih shocks o aggregae wealh of a kind ha was no observed in over hiry years. Finally, consider he relaive proecion offered by he under and over priced porfolios o crashes. The larges annual fall in our sample, wih average welve

15 monh reurns of -29.7%, was in In all cases which overlap his period he mos under valued porfolio 5 year reurns sill bea he mos over valued porfolio. The under priced porfolio bough in December 1969 los, by he end of 1974, 54.2% agains a loss of 49.7% on he corresponding over priced porfolio. We conclude ha i seems hard o accoun for he profis delivered by he rading rule as equilibrium rewards o holding riskier socks. V Summary and Conclusions We found suppor for he hypohesis ha here is a componen of forecasable sock reurns which is driven by irraional swings in marke senimen. We employed a number of disinc approaches o he quesion of wheher he reurns forecasabiliy we uncovered could be raionalised as an equilibrium reward o risk. Firsly, we worked wih a benchmark price raio which did no imply ha risky shares would necessarily appear under priced, unlike hose price scaled accouning variables which have previously been shown o forecas reurns. Secondly we used a number of approaches o evaluae wheher he porfolios of socks which appear under priced were more risky. Each approach gave he same negaive answer. This evidence suggess ha one would have o have a very srong prior belief in efficien markes o argue ha he sock picking rule we have used does no generae economic profis. Our resuls provide a framework in which he evidence for negaive serial correlaion in long run reurns repored by DeBond and Thaler (1985) can be inerpreed. Undervalued socks mus have delivered negaive reurns in he course of becoming

16 undervalued and hence he use of pas reurns will be a proxy for a direc measure of curren mispricing. However, i is an imperfec proxy since we showed ha excess reurns which resuled from an unravelling of earlier mispricing were no correlaed wih subsequen reurns. An ineresing exension of his work would be o ry o find variables o explain he mispricing of equiies as measured by ˆ P i, Pi, REFERENCES Ball, R., (1978), Anomalies in Relaionships Beween Securiies Yields and Yieldsurrogaes, Journal of Financial Economics, 6, Banz, R. W., (1981), The Relaionship Beween Reurn and Marke Value of Common Socks, Journal of Financial Economics, 9, Berk, J. B., (1995), A Criique of Size Relaed Anomalies, Review of Financial Sudies, 8, Campbell J. Y. and R. J. Shiller, (1987), Co-inegraion and Tess of Presen Value Models, Journal of Poliical Economy, 95, Chrisie, W. G., (1990), Dividend Yield and Expeced Reurns: The Zero-Dividend Puzzle., Journal of Financial Economics, 28,

17 DeBond, W. F. M. and R. H. Thaler, (1985), Does he Sock Marke Overreac? Journal of Finance, 40, Elon, E., M. Gruber and J. Renzler, (1983), A Simple Examinaion of he Empirical Relaionship Beween Dividend Yields and Deviaions from he CAPM, Journal of Banking and Finance, 25, Fama E. F. and J. D. MacBeh, (1973), Risk, Reurn and Equilibrium: Empirical Tess, Journal of Poliical Economy, 81, Fama E. F. and K. R. French, (1992), The Cross Secion of Expeced Sock Reurns, Journal of Finance, 47, Lakonishok, J., A. Shleifer and R. Vishny, (1994), Conrarian Invesmen, Exrapolaion and Risk, Journal of Finance, 49, Rohschild M., and J. E. Sigliz, (1972), Increasing Risk II: Is Economic Consequences, Journal of Economic Theory, 3, Shiller, R. J. (1989), Marke Volailiy, MIT Press, Cambridge MA.

18 Table 1. Average Cumulaive Porfolio Reurns Expressed as Percenage. Holding Period, yrs Full Sample FS UN1 UN2 OV2 OV Table 2. Average Incremenal Porfolio Reurns Expressed as Percenage Holding Period yrs Full Sample FS UN1 UN2 OV2 OV

19 Table 3. Averaged Coefficiens for Reurns Regressions R i, 0 1 ( Pi Pˆ, i, ) + γ 2Beai, + γ 3Sizei, + 4 Di, Pi, ui, = γ + γ log γ + Reurn γ 0 γ 1 γ 2 γ 3 γ 4 Horizon, yrs (1.76) (-0.34) (1.60) (-2.25) (0.21) (2.74) (-2.58) (0.66) (-1.99) (-0.17) (3.71) (-2.39) (0.94) (-1.45) (1.05) Table 4 Average Value of Bea and Company Size Full Sample UN1 UN2 OV2 OV1 Bea Size 21,072 15,427 17,636 15,000 9,960

20 Table 5. Five Year Reurns on Decile Porfolios Ranked by ˆ P i, Pi, Sar Dae Mos Undervalued Decile Mos Overvalued Decile

21