Habit Formation, the Cross Section of Stock Returns and the Cash-Flow Risk Puzzle

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1 Hab Formaon, he Cross Secon of Sock Reurns and he Cash-Flow Rsk Puzzle Tano Sanos Columba Unversy, CEPR and NBER Pero Verones Unversy of Chcago, CEPR and NBER Aprl 6, 2010 Absrac Non-lnear exernal hab perssence models, whch feaure promnenly n he recen equy premum asse prcng and macroeconomcs leraure, generae counerfacual predcons n he cross-secon of sock reurns. In parcular, we show ha n he absence of crossseconal heerogeney n frms cash-flow rsk, hese models produce a growh premum, ha s, socks wh hgh prce-o-fundamenal raos command a hgher premum han socks wh low prce-o-fundamenal raos. Ths mplcaon s a odds wh he well-esablshed emprcal observaon of a value premum n he cross-secon of sock reurns. Subsanal heerogeney n frms cash-flow rsk yelds boh a value premum as well as mos of he sylzed facs abou he cross-secon of sock reurns, bu generaes a cash-flow rsk puzzle : Quanavely, value socks have o have oo much cash-flow rsk compared o he daa o generae emprcally plausble value premums. We hank semnar parcpans a Carnege Mellon, UCLA, Prnceon Unversy, The Federal Reserve Bank of New York, Columba Busness School, London Busness School, London School of Economcs, and he Graduae School of Busness of he Unversy of Chcago for her commens and Gene Fama, Lars Hansen, John Heaon, and We Jang for valuable suggesons. Errors, of course, reman our own. Ths paper has crculaed prevously under he le Cash-flow Rsk, Dscoun Rsk and he Value Premum

2 1 Inroducon The equy premum and he value premum puzzles consue wo of he focal pons of he asse prcng leraure. As s well-known, he sarng pon of he frs s he nably of sandard consumpon models o raonalze he observed level of he equy premum, he volaly and predcably of reurns, and he low and sable rsk-free rae (see Panels A and B of Table 1.) The value premum puzzle s nsead concerned wh he falure of he Capal Asse Prcng Model (CAPM) o explan he cross-secon of average reurns of porfolos sored accordng o book-o-marke (see Panel C of Table 1 and Fg. 1.) Surprsngly, hese wo puzzles are, for he mos par, suded separaely. Ths s unforunae because, as we argue here, he wo puzzles canno be ackled ndependenly: Any economc mechansm proposed o address one of hem mmedaely has general equlbrum mplcaons for he oher. In hs paper, we focus on one mporan mechansm, hab perssence, whch has feaured promnenly boh n he asse prcng leraure 1 and n he real busness cycle leraure. 2 In parcular, we nvesgae a non-lnear exernal hab formaon model a là Campbell and Cochrane (1999), a framework parcularly successful n addressng he equy premum puzzles descrbed above, and nvesgae he mplcaons of hs model for he esablshed facs n he cross-secon of sock reurns. We show ha hese mplcaons are problemac and ha for hs reason, he success of he non-lnear hab formaon mechansm has o be pu on hold. In parcular, we show ha when, mporanly as we shall see, frms dffer only n her expeced dvdend growh, hab perssence models counerfacually generae a growh premum raher han a value premum, a pon also recenly made by Leau and Wacher (2007). The reason s a he hear of he hab formaon model: The varaon over me of he marke prce of consumpon rsk, whch s responsble for he success of he model o explan he properes of he marke porfolo, neracs wh he mng of cash-flows o generae a erm premum. Indeed, asses ha pay far n he fuure are more sensve o shocks 1 Sundaresan (1989), Consanndes (1990), Abel (1990), Ferson and Consanndes (1991), Deemple and Zapaero (1991), Danel and Marshall (1997), Campbell and Cochrane (1999), Heaon (1993,1995), L (2001), and Wacher (2000). All hese papers focus exclusvely on he me-seres properes of he marke porfolo and do no nvesgae mplcaons for he rsk and reurn properes of ndvdual secures. 2 For example Boldrn, Chrsano, and Fsher (2001) use he hab perssence model of Consanndes (1990) o nvesgae wheher hs mechansm can be conssen wh boh key asse prcng and real busness cycle facs. See also, among ohers, Bovn and Gannon (2005), Chrsano, Echenbaum, and Evans (2005), Gannon and Woodford (2004), Ravn, Schm-Grohé, and Urbe (2006), and Smes and Wouers (2003,2004). 1

3 n he sochasc dscoun facor han asses wh fron-loaded cash-flows. For hs reason he former are rsker and command a premum over he laer. We show ha growh socks are precsely hose ha pay far n he fuure and hus, hey command a counerfacual premum over value socks. In order o solve hs growh premum puzzle nduced by hab formaon preferences, we nroduce ex ane heerogeney n frms cash-flow rsk, ha s, frms dffer from each oher no only n her expeced dvdend growh, bu also n he covarance of her cash flows wh consumpon growh self. In hs case, we fnd ha he sandard sorng procedure used n he leraure o allocae socks o porfolos accordng o her prce-o-fundamenal rao endogenously selecs as value socks hose wh hgher cash-flow rsk, an mplcaon emprcally confrmed by a seres of recen papers. 3 Ths hgher cash-flow rsk of value socks naurally ranslaes no hgher expeced reurns, as nvesors requre a premum o hold socks whose cash flows fall a he same me as he aggregae consumpon. Usng smulaons, we show ha f he heerogeney n frms cash-flow rsk s suffcenly large, hen ndeed socks wh low prce-dvdend raos, value socks, do command a subsanal premum compared o hgh prce-dvdend rao socks. 4 Tha s, a value premum appears. We show ha, under hese condons, he model hen no only maches he properes of he marke porfolo, as Campbell and Cochrane (1999) fnd, bu s acually able o replcae mos, f no all, of he sylzed facs abou he cross-secon of sock reurns, ncludng (a) he falure of he CAPM and hus, he value premum puzzle, (b) he beer performance of he condonal CAPM, and (c) he beer prcng ably of he Hgh-Mnus-Low (HML) facor as n Fama and French (1993). In addon, he model also yelds a large varaon of he value premum over he busness cycle, an addonal sylzed fac ha s well-documened n he daa. 5 Obvously he remanng queson s hen a quanave one: s he cash-flow rsk requred o mach he cross-secon of sock reurns conssen wh he daa? Unforunaely no. The 3 See Cohen, Polk, and Vuoleenaho (2009), Campbell, Polk, and Vuoleenaho (2005), Bansal, Dmar, and Lundblad (2005), Parker and Jullard (2005), and Hansen, Heaon, and L (2008). Also Lew and Vassalou (2000) and Vassalou (2003) show ha news abou forecass of Gross Domesc Produc (GDP) growh correlae wh value sock reurns. 4 In our model, book value s no well defned and so we use prce-dvdend raos n leu of marke-o-book raos hroughou (see Sanos and Verones, 2006; and Leau and Wacher, 2007). Fama and French (1996, Table II), Fama and French (1998, Table III), and Leau and Wacher (2007, Table I) show ha sorng by earnngs-o-prce or cash-flow o prce generaes as szable a value premum as sorng by book-o-marke. 5 See for example Leau and Ludvgson (2001) and Cohen, Polk, and Vuoleenaho (2003). 2

4 exernal hab perssence model nvesgaed here succeeds n reproducng he mes seres and cross-seconal properes of asse daa a he expense of an mplausble hgh (low) level of cash-flow rsk for value (growh) socks when compared o he daa. Tha s, hab preferences nduce a cash-flow rsk puzzle. The nuon s relavely smple: Snce hab preferences end o generae a growh premum when socks do no dffer n cash-flow rsk, a large crossseconal dfference n cash-flow rsk s needed n order o undo he growh premum. An exensve smulaon exercse hghlghs he severy of he problem. Oulne of he paper. The arcle develops as follows. Secon 2 nroduces a general equlbrum model wh mulple secures whch s solved for prces and reurns n Secon 3. Ths model generalzes he seng pu forward n Menzly, Sanos, and Verones (2004), MSV henceforh, n order o have more flexble preferences and a more manageable process for frms cash flows whle reanng he closed-form soluons ha are such an mporan analycal advanage when dealng wh mulple secures. Secon 4 nvesgaes qualavely he mplcaons of he model for he cross-secon of sock reurns whereas Secon 5 does he same quanavely. I s n Secon 5.3 where we nroduce he cash-flow rsk puzzle. In Secon 6 we use our model o shed new lgh on sandard asse prcng models ess. In parcular, we use our model o consruc an HML facor as n Fama and French (1993) and provde an economc foundaon for s success as a cross-seconal predcor. In hs secon, we also show ha he model s able o mach o a surprsng degree he me-seres varaon n he value premum, whch s a he hear of he recen neres n he condonal CAPM. Secon 7 concludes. Relaed leraure. Our work ouches on several recen papers n he leraure on he crosssecon of sock reurns, bu dffers from hem n several respecs. Frs, Sanos and Verones (2006) and Leau and Wacher (2007) nvesgae he effec ha cross-seconal dfferences n cash-flow duraon (as defned by he expeced dvdend growh) have on he cross-secon of expeced reurns. They boh fnd ha asses wh low duraon have hgh expeced reurns and low prce-dvdend raos whereas he oppose s rue for hgh duraon asses, ha s he value premum. Our work depars from hese wo papers n wo crucal aspecs. Frs, boh papers make assumpons o avod he naural growh premum ha comes wh he varaon n he dscoun rae, whch s necessary o mach he properes of he marke porfolo: Sanos and Verones (2006) fal o mach he volaly of he aggregae sock 3

5 reurns and Leau and Wacher (2007) assume away general equlbrum resrcons, and nsead assume ha he varaon n he dscoun facor s unprced by marke parcpans. These assumpons ensure ha dfferences n duraons generae a value premum. Duraon effecs are also presen n hs paper, bu he presence of he srong dscoun effecs mpled by he Campbell and Cochrane (1999) model make, as explaned above, cross-seconal dfferences n duraon generae a growh premum raher han a value premum. Sanos and Verones (2006) don have dscoun effecs and Leau and Wacher (2007) smply assume ha hey go unprced. 6 Here, we don ake hs sand, bu raher, and reasonably n our vew, assume ha dscoun effecs are boh presen and prced and some oher ngreden s needed o generae he value premum. Cross seconal dfferences n cash-flow rsk are such an ngreden. Second, he combnaon of cross-seconal dfferences n cash-flow rsk and dscoun effecs generaes he emprcally documened me-seres varaon n he value premum, a regulary for whch, o he bes of our knowledge, here s no exan heorecal explanaon. Neher Sanos and Verones (2006) nor Leau and Wacher (2007) address hs ssue. Our paper also ouches on he recen leraure emphaszng cross-seconal dfferences n long-run rsk across asse classes. For nsance, Parker and Jullard (2005) and Bansal, Dmar, and Lundblad (2005) use cross-seconal dfferences n he long-run covarance beween reurns, consumpon growh, and dvdend growh, o offer a characerzaon of cross-seconal dfferences n one-perod reurns. Clearly, he long-run componens of cash-flow rsk are bu one conrbuon o one-perod reurns; ransen componens may also be very mporan. Recognzng hs, Hansen, Heaon, and L (2008) offer a characerzaon of he long-run rade-off beween rsk and reurn. Ths long-run rade-off s key because ransen componens, whch may be frs-order for one-perod reurns, are neglgble n he long-run. 7 In conras, our defnon of cash-flow rsk s enrely unrelaed o low-frequency componens n consumpon growh, whch are (mosly) absen from our paper, and raher emphaszes conemporaneous covarances of consumpon and dvdend growh. More mporanly, he dscoun effecs ha 6 Ths s he fundamenal reason why duraon effecs are enough o generae a value premum n hese wo papers. All asses have dencal, and posve, cash-flow rsk, bu some have her dvdends more fron-loaded han ohers. Ths has wo consequences. Frs, and by assumpon, he more fron-loaded he dvdends, he lower expeced dvdend growh and hus, he lower he prce-dvdend rao. Second, he more fron-loaded he dvdend, he rsker he asse as would consue a larger fracon of curren consumpon and hus, he hgher he premum. Thus, hese asses are value and a value premum arses n hese wo papers. 7 More precsely, for hese auhors, value socks are rsker because her cash-flow growh process loads relavely more han growh socks on low-frequency componens of consumpon growh. 4

6 play so promnen a role n our calbraon are enrely absen n hs leraure so ha, once agan, he me-seres varaon of he value premum canno be generaed n he conex of hese models. Fnally, none of hese papers offers an negraed vew of he me-seres and he cross-secon of sock reurns bu ry o relae he properes of he marke porfolo wh he cross-seconal properes of porfolos ha add up o he marke self. We argue here, n conras wh he cash-flow leraure, ha hese wo sdes of he asse prcng puzzles have o be jonly consdered, oherwse he nferences on cash-flow parameers are msleadng. 8 Campbell and Vuoleenaho (2004) decompose shocks o marke reurns no shocks o expeced dscoun raes and shocks o expeced dvdend growh raes. They show ha value and growh load on hese shocks dfferenly and hs, combned wh he marke prce of rsk assocaed wh hese shocks, generaes a value premum and s correspondng puzzle. Dfferenly from us, however, hey neher connec he me-seres properes of he marke porfolo wh he magnudes of he cash-flow rsk needed o generae a value premum, nor do hey address he me-seres varaon of he value premum. Our paper also relaes o he leraure on condonal asse prcng. For nsance, Leau and Ludvgson (2001) show ha emprcally a condonal verson of he CAPM ouperforms s uncondonal counerpar. Ther resuls provde emprcal evdence supporng our model s mplcaon ha condonng varables capure he me-seres varaon n he value premum. In our seup, as n he daa, he condonal CAPM performs beer han he uncondonal CAPM. Imporanly, hough, n our model he condonal CAPM s a msspecfed asse prcng model, and so, wh enough daa, can also be rejeced. The presen paper s obvously relaed o MSV, bu here are also many dfferences wh ha paper. Frs, our model s more general han he one n MSV and he addonal flexbly s nsrumenal n he emprcal performance of he model. In parcular, whle MSV only consder he log-uly case and have approxmae formulas for he cross-secon of prces, n hs paper we solve for he power uly case and oban exac soluons. Second, whereas MSV are concerned wh he me-seres predcably of ndusry porfolos, he presen paper focuses on he cross-seconal predcably of value-sored porfolos. Ths focus allows us also o shed lgh on he vas leraure on cross-seconal predcably, somehng MSV dd 8 See also Brennan, Wang, and Xa (2004) and Brennan and Xa (2006) for a paral equlbrum model ha es he me-seres o he cross-secon of sock reurns. An nvesmen-based general equlbrum model of he cross-secon s also pu forward by Gomes, Kogan, and Zhang (2003) who buld on he paral equlbrum model of Berk, Green, and Nak (1999). See also Zhang (2005). 5

7 no ouch upon and, n parcular, he presen paper s afer a quanave assessmen of he cash-flow rsk effecs needed o generae a plausble value premum. 2 The model We consder an endowmen economy wh n fnancal asses. Each asse has an nsananeous dvdend sream denoed by D, for = 1,.., n. The consumpon good s mmedaely pershable and non-sorable, whch yelds he equlbrum resrcon C = n D. (1) =1 Ths add-up, general equlbrum resrcon s mporan n our seng, as breaks he heorecal valdy of he CAPM, as dscussed below. 9 Unforunaely, even relavely smple processes for D mply aggregae consumpon processes ha are dffcul o work wh and resrcve assumpons need o be made for racably (see dscusson n MSV; Sanos and Verones, 2006; Cochrane, Longsaff, and Sana-Clara, 2008; and Marn, 2008). We follow MSV and Sanos and Verones (2006), and make assumpons abou aggregae consumpon C, and he jon dynamcs of he shares of aggregae consumpon produced by each asse, denoed by Assumpon 1. Aggregae consumpon s gven by s = D C. (2) dc C = µ c (s ) d + σ c db, where B s an n 1 vecor of Brownan moons, and Above, s = ( s 1,..., s n θ CF s explaned below. µ c (s ) = µ c + µ c,1 (s ) and µ c,1 (s ) = s θ CF. (3) ), θcf = ( θ 1 CF,..., ), θn CF and σc = (σ c, 0,..., 0). The specfcaon of As n Campbell and Cochrane (1999), we assume consumpon growh has consan volaly. Unlke hem, however, we assume expeced consumpon growh has a predcable 9 Several recen arcles have emphaszed he mporance of marke clearng condons n fnance, such as Sanos and Verones (2006), Johnson (2006), Cochrane, Longsaff, and Sana-Clara (2008), and Marn (2008). None of hese papers combnes hab formaon wh mulple rees, nor nvesgaes he properes of he crosssecon of sock reurns, excep for Sanos and Verones (2006), already dscussed n he nroducon. 6

8 componen ha depends on he dsrbuon of shares. We make hs assumpon for four reasons: Frs, follows naurally from he general equlbrum resrcon (1) n any model ha has dvdend processes as prmves (see Eq. (29) n he Appendx A). 10 Second, hs assumpon s conssen wh he recen long-run rsk leraure, whch shows a small perssen predcable componen n consumpon growh (see, e.g., Bansal and Yaron, 2004; Hansen, Heaon, and L, 2008). In our model, such a predcable componen s also small. 11 Thrd, he specfc assumpon (3) allows us o oban analycal formulas for asse prces, an mporan propery gven our focus on he cross-secon of sock reurns wh many asses. Fnally, n our model he me varaon of expeced consumpon growh breaks he heorecal valdy of he CAPM, boh condonally and uncondonally, a propery ha we explo o provde nsghs on he economc meanng of he ess of he CAPM provded n he leraure. Assumpon 2. For each, he share s follows he mean reverng process where ds = φ ( s s ) d + s σ (s ) db, (4) σ (s ) = ν n s j ν j. (5) The cash-flow model (4) mposes a srucure on he relave sze of frms, where sze s measured as he fracon of oal oupu produced by a gven frm. In parcular, mposes he economcally plausble assumpon ha no frm wll ake over he economy, as s > 0 for all. In addon, he volaly σ (s ) n (5) ensures ha n =1 s = 1 for all, whch n urn mples ha (1) s always sasfed. Alhough he form of he volaly σ (s ) n (5) seems ad hoc, can acually be recovered from frs prncples n a model wh mulple dvdend processes each wh consan volaly (see Appendx A). Insead, he key smplfyng assumpon s he mean-reverson componen n he drf rae of (4). 2.1 Expeced cash-flow growh and cash-flow rsk 10 Such models also mply ha he volaly of consumpon s a weghed average of dvdend volales, whch we nsead approxmae o a consan. 11 We show n smulaons below ha expeced consumpon growh flucuaes beween a maxmum of 2.22% and a mnmum of 1.87%, a very mld varaon compared o he 1.5% sandard devaon of consumpon growh ha we assume. Indeed, for our baselne case, whch has he maxmum varaon n expeced consumpon growh, when we regress fuure consumpon growh on ln (P /C ) n arfcal daa we fnd R 2 s ha are puny, beween 0.1% and 0.2% a he hree- and four-year horzons, respecvely. 7

9 Gven Assumpons 1 and 2, we can apply Io s Lemma o D = s C and oban: dd D = µ D,d + σ D (s ) db, (6) where he dvdend drf and volaly are gven by ( s µ D, = µ c + θ CF + φ s ) 1 σ D (s ) = σ c + σ (s ). (8) (7) In hese formulas, θ CF = ν σ c. Two commens are n order: Frs, Eq. (7) shows ha when he asse s relave share, s /s, s hgh and hus, he asse s relave conrbuon o oal consumpon s below s long-erm average, he asse has an expeced dvdend growh hgher han he uncondonal expeced consumpon growh µ c (adjused for a small Io erm θ CF ).12 In addon, he drf rae of dvdends µ D depends on a parameer θ CF, whch s asse specfc and depends on he correlaon of he sock s share s wh consumpon growh, as shown below. Whle echncally θ CF s smply an Io erm obaned from he defnon D = s C, we noe ha quanavely has a mnmal mpac on he average dvdend growh self: As we show n he calbraon secon, θ CF s an order of magnude smaller han he oher wo drf componens.13 Second, n our model he sochasc dscoun facor s only drven by shocks o consumpon growh. Thus, cash-flow rsk s measured by he followng covarance ( dd σ CF, Cov D, dc ) = σ C cσ c + θ CF s θ CF. (9) The condonal cash-flow rsk of asse, σ CF,, wll play a promnen role n hs paper. The erm θ CF s θ CF s paramercally ndeermnae, ha s, addng a consan o all θ CF leaves hs erm unaffeced, as n =1 s = 1. Thus, we can mpose he denfably resrcon n s j θ j CF = 0. (10) 12 MSV fnd srong emprcal suppor for he nverse relaon beween relave share and dvdend growh n ndusry porfolos. 13 In our model, θ CF s are unformly dsrbued around he nerval [ θ CF, θ CF ]. The maxmum level of θ CF = , whch s much smaller han boh he assumed average consumpon growh µ c = 2% and he flucuaons n expeced dvdend growh nduced by he hrd erm n (7), φ(s /s 1), whch s over 10%. 8

10 The expeced covarance beween asse s cash-flow growh and consumpon growh s σ CF = E [ ( σ ] dd CF, = E [Cov D, dc )] = σ C cσ c + θ CF. (11) The parameer θ CF hen regulaes he relave cash-flow rsk of ndvdual asses. Noce ha he benchmark level of rsk of an asse s he rskness of aggregae consumpon: An asse s rsky (safe) f s cash-flows are more (less) rsky han aggregae consumpon. Ths s a general equlbrum resrcon as, by defnon, he varance of consumpon growh mus be a weghed average of s covarances wh ndvdual dvdend growh. Throughou we refer o eher σ CF or θ CF as cash-flow rsk as here s a one-o-one mappng beween hem. We conclude hs secon by emphaszng ha he presen framework can be generalzed o nroduce more realsc feaures bu, clearly, a he cos of addonal complexy. nsance, we assume for smplcy ha frms are nfnely lved and ha agens know he long-erm average sze s. A plausble generalzaon s one n whch s s unknown, and agens learn abou over me as hey observe dfferen dvdend and consumpon realzaons. In hs case, he prcng funcon, Eq. (20) below, wll depend on he expeced long-erm share E [s ] raher han on s. Ths exenson hough would largely leave he resuls unaffeced. Indeed, sandard flerng resuls mply ha he varaon n E [s ] would be ndependen of consumpon shocks, as consumpon does no yeld any addonal nformaon on s ha s no already n he shares hemselves (see also Pasor and Verones, 2003). I follows ha hs addonal varaon would be unprced and hus, would have no mpac on frms expeced reurns. Second, snce expecaons move more slowly han sgnals, he rao E [s ]/s would sll end o move nversely wh s, exacly as n he case n whch s s known. Snce he rao s /s, as s formally shown n Proposons 2 and 3, s he key varable affecng he frm s expeced dvdend growh, s prce/dvdend rao, and s expeced reurn, follows ha he cross-seconal relaon beween prce/dvdend raos and expeced reurns would no change f we were o assume ha s was unknown. Fnally, he learnng model also parally addresses our assumpon of an nfnely lved frm: assumng s are randomly seleced a me zero, some frms would hen converge o very low szes, effecvely dsappearng from he economy. 14 In summary hen, addng learnng o he model o accoun for he fac ha he agens are no lkely o know he long-run conrbuons of he dfferen frms o he overall 14 We also solved he model assumng exponenal dsrbued ex mes (frm deah), whch lead o he usual ncrease n he me dscoun. The resuls reman he same, bu he model becomes more challengng as o keep saonary, we mus have a flow of frms enerng he economy. For 9

11 economy subsanally complcaes he analyss whou largely affecng he relaon beween prce/dvdend raos and expeced reurns, whch s he focus of hs paper. 2.2 Preferences and Hab Dynamcs There s a represenave nvesor who maxmzes [ ] E u (C, X, ) d, (12) where he nsananeous uly funcon s gven by In Eq. 0 u (C, X, ) = e ρ (C X ) 1 γ. (13) 1 γ (13), he varable X denoes an exernal hab level and ρ denoes he subjecve dscoun rae. In Campbell and Cochrane (1999) he fundamenal sae varable drvng he audes owards rsk s he surplus-consumpon rao, S = (C X ) C 1. To oban closedform soluons for prces when here are mulple secures, MSV use a log hab model and specfy nsead he nverse surplus S 1 as a mean-reverng process. MSV s modelng devce hough canno be appled when γ > 1 and, moreover, hey only oban approxmae formulas for he case θ CF 0. The presen paper offers a generalzaon of MSV ha allows us o handle a large class of models. The key ngreden n hs generalzaon s o focus on he process ( ) γ C G = = S γ. (14) C X To oban a plausble, ye racable, model for he dynamcs of G, consder frs he mplcaons for G under he sandard assumpon ha X s an exponenally weghed average of pas consumpon levels, as n Consanndes (1990) and Deemple and Zapaero (1991), X = λ e λ( τ) C τ dτ. An applcaon of Io s Lemma o (14) yelds he process dg = [ µ G (G ) σ G (G ) µ c,1 (s ) ] d σ G (G ) σ c db 1, (15) where µ G (G ) and σ G (G ) > 0 are complcaed funcons of G, provded n Eq. (31) and (32) n Appendx A. Eq. (15) shows ha a hgher expeced consumpon growh µ c,1 (s ) mples a lower drf rae of G. Inuvely, an ncrease n he expeced growh rae of consumpon mples a hgh fuure level of consumpon relave o he curren hab X and hus, a hgher surplus consumpon rao S, and, gven (14), a lower expeced G. As n MSV and Campbell and Cochrane (1999), we make specfc assumpons on µ G (G ) and σ G (G ) n (15) o oban a more manageable process. In parcular, we assume µ G (G ) = k ( G G ) and σ G (G ) = α (G λ). (16) 10

12 The frs componen of he drf of G s a mean-reverson componen and capures he basc dea of hab perssence models, namely ha he hab X evenually caches up wh C. The second componen, as dscussed above, lnks he drf rae of G o µ c,1 (s ). As for he dffuson componen, and as n MSV, λ 1 bounds G from below a λ and α > 0 ransms he nnovaons n consumpon growh, db 1, o he convexy of he uly funcon. 15 Noe ha MSV s model s a specal case of (15) and (16) and obans when γ = 1 and consumpon growh s..d., whch s acheved by seng µ c,1 (s ) = 0. 3 Equlbrum asse prces and reurns 3.1 The oal wealh porfolo The nex proposon generalzes he resuls n MSV o he presen model n wha concerns he oal wealh porfolo, ha s, he clam o he aggregae consumpon process. Proposon 1. The prce-consumpon rao, he expeced excess reurn, and dffuson erms of he oal wealh porfolo are, respecvely: P T W C = α T W 0 (s ) + α T W [ ] E dr T W 1 (s ) S γ (17) = (γ + α (1 λs γ )) S γ α (1 λsγ ) n f1 T W (s ) + S γ σ 2 c + wj T W σ j CF, (18) R, = Sγ α (1 λsγ ) n f1 T W (s ) + S γ σ c + wj T W σ j D (s ), (19) σ T W { where α T 0 W (s ), α T 1 W (s ), f1 T W (s ) and w T W j } are gven n Appendx B. As n Campbell and Cochrane (1999) and MSV, he prce-consumpon rao of he oal wealh porfolo s ncreasng n he surplus-consumpon rao S : A hgh S mples a low local curvaure of he uly funcon, a less rsk-averse aude of he represenave agen, and hus, a hgher prce-consumpon rao. Unlke Campbell and Cochrane (1999) and MSV, he prce-consumpon rao now depends on he enre vecor of shares s. Inuvely, hs effec sems from our assumpon abou consumpon growh predcably (see he dscusson afer Assumpon 1). The funcons α T W 0 (s ) and α T W 1 (s ) are ypcally decreasng n expeced 15 Clearly, he assumpons (16) hen mply ha hab X s no longer he weghed average of pas consumpon, as above, bu a more complcaed non-lnear funcon of pas consumpon shocks. See Campbell and Cochrane (1999) for a dscusson. See also Hansen (2008) for a dscusson of he rsk-reurn mplcaons of our hab model proposed above. 11

13 consumpon growh because n our seup, he elascy of neremporal subsuon s less han one. Thus, hs componen mples ha an ncrease n µ c (s ) resuls n lower prces. 16 As for he expeced excess reurns, (18), and he volaly of reurns, (19), we pospone he dscusson of he nuon of hese expressons unl Secon 5.2, when we assess quanavely he mplcaons of he model. 3.2 Prces and reurns for ndvdual secures The nex proposon delvers closed-form soluons for ndvdual sock prces: Proposon 2. The prce of asse s gven by P ( ) = α 0 + α 1S γ s + α 2 (s ) + α 3 (s ) S γ D s ( s s ), (20) where α 0, α 1 are posve consans and α 2 (s ) and α 3 (s ) are posve lnear funcons of he share vecor s gven n Appendx B. As before, a hgher surplus-consumpon rao S, whch mples lower rsk averson, or a hgher expeced dvdend growh, as measured by he relave share s /s (see (7)), resul naurally n hgher prce-dvdend raos. The las erm n (20) shows ha shocks o he surplus-consumpon rao have a sronger effec on he prce-dvdend rao he hgher he asse s expeced dvdend growh. Ths s lnked o he duraon effec ha plays so promnen a role n wha follows. Fnally, as was rue for he oal wealh porfolo and for he same reason, he prce of each ndvdual asse also depends on funcons of he vecors of shares α 2 (s ) and α 3 (s ). The nex proposon presens a characerzaon of expeced excess reurns. The nuon and mplcaons of Proposon 3 are gven n deph n Secon 4. Proposon 3. The expeced excess reurn of asse s gven by E [ dr ] = µ DISC, + µ CF,, 16 To revew he economc reasonng, a low elascy of neremporal subsuon mples a ase for consumpon smoohng. An ncrease n expeced consumpon growh yelds a hgher desre for curren consumpon, and hus, lower savngs. Because socks and bonds are less desrable now for he represenave consumer, prces have o drop n order o encourage hm o hold hem, resulng, for example, n a decrease of he prce-consumpon rao of he oal wealh porfolo. 12

14 where µ DISC S γ, = (γ + α (1 λs γ )) ( ) α (1 λs γ f1 s, s s + S γ ) σ2 c (21) µ CF, = (γ + α (1 λs γ )) 1 ( ) + η 1 + f2 (S, s ) s σ CF, + η jσ j CF,, (22) j wh f 1 ( s /s ) α, s = 0 + α 2 (s ) ( s /s ) α 1 + α 3 (s ) ( ) s /s > 0 and f2 (S, s ) = α 2 (s ) + α 3 (s ) S γ α 0 + α 1 Sγ s > 0, and η j s gven n expresson (39) n Appendx B. 4 Growh versus value premums The key emprcal observaon n he cross-seconal leraure s ha growh asses, whch are hose wh hgh prces relave o fundamenals, say prce-dvdend raos, have on average lower reurns han asses wh low prce-dvdend raos, value socks. In hs secon we nvesgae wha s requred of he model o generae qualavely hs fac. For hs we make use of he resuls n boh Proposons 2 and 3 above. 4.1 Dscoun rsk effecs and he growh premum We sar by focusng on he componen of he premums µ DISC, n (21), whch s he par of he premum ha s drven by varaon of he aggregae dscoun proxed by S γ. To nerpre hs erm furher, noce frs ha P /P S γ /Sγ S γ = ( f1 s /s, s ) + S γ s he elascy of prces o shocks n he varable drvng he aggregae dscoun, whch s S γ. The volaly of hese dscoun shocks s (23) α (1 λs γ ) σ c, (24) whch s he dffuson componen of ds γ /Sγ, he nverse of our sae varable G, as follows from a basc applcaon of Io s Lemma o (15). Clearly, only he componen of hese shocks ha covares wh he shocks o he sochasc dscoun facor s prced. From Eq. (33) n Appendx B, he dffuson erm of he hab sochasc dscoun facor s σ m = [γ + α (1 λs γ )] σ c. (25) 13

15 The componen of he asse s premum ha s lnked o dscoun effecs s hen he produc of (23), (24), and (25), whch s expresson (21) n Proposon 3. Cross-seconal varaon n he dscoun effecs can only be drven by dfferences n he prce elascy (23), whch s n urn drven by he behavor of he funcon f 1 ( s /s, s ). We have been unable o oban a general characerzaon of hs funcon, bu for parameer values ha are emprcally relevan we fnd ha ( s /s ), s f 1 ( ) s /s < 0, and hus, asses wh a hgher expeced dvdend growh, as measured by he relave share s /s, dsplay sronger dscoun effecs. The nuon s sraghforward: socks wh a hgh expeced dvdend growh pay he bulk of her proceeds far n he fuure. Thus, mnor varaons n he aggregae dscoun rae hrough he rsk averson of he represenave nvesor resul n large percenage varaons of he prce of he asse. Ths varaon s naurally prced and hus, he hgher requred premum of asses wh hgh relave shares The growh premum We can now relae hese fndngs o he observaon ha when only dscoun effecs are presen, a growh premum arses. For hs s useful o urn o Fg. 2, where we plo µ DISC,, as gven by (21), as a funcon of our proxy for expeced dvdend growh, s /s, for he case n whch all frms have dencal cash-flow rsk, ha s, θ CF 0 for all. To generae hs plo, he level of surplus S s se o s seady sae value S and he parameers used are hose of he calbraon exercse dscussed n deal n Secon 5.1. When all frms have dencal cash-flow rsk, expresson (20) mples ha sorng asses accordng o her prce-dvdend rao (P/D) s akn o sorng hem on expeced dvdend growh, s /s. Snce low prce-dvdend rao socks are hose wh low relave shares s /s, value socks are hose locaed on he lef-hand sde of Fg. 2 and hus, have low expeced excess reurns. Smlarly, hgh prce-dvdend rao socks are hose wh hgh s /s and hus, growh socks are on he rgh-hand sde of Fg. 2 and have hgh expeced excess reurns. Thus, f cross-seconal dfferences n cash-flow rsk are small, so ha E [ dr ] µ DISC, han value socks, ha s, a growh premum s obaned. for all socks, growh socks command a hgher premum To renforce hs pon, we conduc an exensve smulaon, ha we descrbe n deal below, o reproduce he sorng procedure ha s sandard n he leraure on he cross-secon of sock reurns. Our purpose s o replcae Fg. 1, where we plo average (log) marke-obook of value-sored porfolos versus average excess reurns. The equvalen n smulaed daa 14

16 for he case n whch frms have homogenous cash-flow rsk s repored n he op panel of Fg. 3. The fgure clearly shows ha socks wh hgh average prce-dvdend raos yeld a hgher average reurn, n conras wh he daa n Fg. 1. To summarze hen, f dscoun effecs were o be he only ones presen, he cross-secon of excess reurns would dsplay a growh premum raher han he value premum ha s observed emprcally. 4.2 Cash-flow rsk effecs and he value premum The source of premums relaed o cash-flow shocks, µ CF, gven n (22), has wo componens o. The frs s relaed o shocks n he asse s dvdends and he second s relaed o shocks n he dvdends of he res of he asses n he economy, whch, as shown n (20), affec he prce of asse as well. The logc for he sources of he premums lnked o cash-flow shocks s he same as n he dscoun effecs case. Frs, can be easly shown ha he elascy of he prce wh respec o shocks o s own dvdends s, P /P D /D 1 = 1 + f2 (S, s ) ( s s ) + η. Recall also ha we denoe σ CF, = cov ( dd /D, dc /C ) (see Eq. (9)). The frs erm of µ CF, s hen he componen of he dvdend shocks ha covares wh shocks o he sochasc dscoun facor mulpled by he effec ha hese shocks have on he prce of asse, as measured by he prce elascy. As for he second erm n µ CF,, can be shown ha P /P D j /Dj = η j for j. As before, hs componen of he premum resuls from he produc of hs (cross) elascy and he prced componen of he shock o asse j s dvdends, σ j CF,. How does he curren level of expeced dvdend growh, as measured by s /s, affec he cash-flow rsk componen of expeced sock reurns? Gven he condonal covarance of he dvdend of asse wh aggregae consumpon, σ CF,, he frs erm of (22) s unambguous: Snce f2 (S, s ) > 0, f he asse s rsky, ha s, f σ CF, > 0, hen a hgh expeced dvdend growh ranslaes n a lower premum semmng from curren dvdend volaly. The nuon s also clear: a sock ha pays more n he fuure han oday has a relavely low dvdend compared o he fuure. Thus, he rsk embedded n curren dvdends, σ CF,, has a relavely low mpac on he oal rsk of he sock. In he lm, f he sock does no pay any dvdend oday, canno have any cash-flow rsk, as here s zero curren covarance of dvdends wh consumpon. If nsead he asse s dvdends covary negavely wh consumpon growh 15

17 (σ CF, < 0), hen a hgh expeced dvdend growh ncreases he rsk premum. The argumen, of course, s he converse of he prevous one. The effec ha he curren expeced dvdend growh of asse has on he second erm of he cash-flow rsk componen of sock reurn (22) s more dffcul o ell. However, we found numercally ha, on average, he cash-flow componen of expeced reurn s ncreasng n σ CF, alhough varaon n shares s generae small devaons from hs ncreasng paern. Fnally, we noe ha n our model, he rsk premum only depends on he relave share s /s and no on he seady sae dvdend share s per se. The reason s ha here are wo forces a play when consderng he effec of s on requred premums. Frs, a sock wh a hgh average dvdend share s s more exposed o consumpon rsk, on average, bu second, also has a hgher average prce. Ths hgher prce mples a lower percenage sensvy of he sock o consumpon shocks. Snce rsk premums depend on percenage reurns, hese wo forces exacly offse each oher n our model. 17 The curren level of dvdends s nsead key n deermnng he curren cash-flow rsk, as s he covarance beween consumpon growh and curren dvdends ha has a drec bearng on he rskness of he sock The value premum We showed n Secon 4.1 ha he sole presence of dscoun effecs generaes a counerfacual growh premum. To see wheher cash-flow effecs can produce he desred value premum nsead, we urn o Fg. 4. The frs wo panels repor, respecvely, he dscoun, µ DISC,, and he cash-flow rsk componen, µ CF,, of expeced reurns. Panel C adds up boh componens o oban E [ dr ]. Le us sar wh Panel A, whch repors he same quany as n Fg. 2, µ DISC,, bu for he case n whch θ CF dffer across frms. Ineresngly, we see ha hgher cash-flow rsk ncreases he level of he dscoun componen of he expeced reurn. The reason s ha a hgher cash-flow rsk decreases he prce of he asse, on average. Thus, shocks o he aggregae dscoun (S γ ) have a larger percenage mpac on he sock prce, and hus, mply a hgher rsk. Noneheless, for gven cash-flow rsk level, θ CF, he relaon beween µ DISC, and expeced dvdend growh s /s s posve, as dscussed n he prevous secon. Panel B plos he cash-flow componen o expeced reurn, µ CF,, as a funcon of expeced 17 Tha expeced reurns are ndependen of s s also a feaure of he sandard asse prcng model. Indeed, consder he case where he represenave consumer has he sandard power uly funcon, s = s for all and, fnally, le C follow a smple geomerc Brownan moon. In hs case, P = s C K where K s a consan. Snce he rsk premum s µ = γcov `dp /P, dc /C and dp /P s ndependen of s, he wo asses have dencal rsk premums, ndependenly of s. Fnally noce ha he asse wh a hgher s has a hgher prce. 16

18 dvdend growh, as proxed by s /s, for varous levels of θ CF each correspondng o a lne n he plo. As explaned n he prevous secon, he cash-flow rsk componen of expeced excess reurns s decreasng n he expeced dvdend growh for socks wh hgh cash-flow rsk. Panel C repors he oal expeced reurn for each asse obaned by addng he cash-flow rsk componen µ CF, o he dscoun rsk componen, µ DISC,. Value socks (asses wh low P/D rao) have, on average, hgh rsk (σ CF ) and low expeced dvdend growh (s /s ). Ths combnaon corresponds o he area around he op-lef corner of he plo n Panel C, ha s, o hgh expeced excess reurn. Conversely, growh socks (asses wh hgh P/D raos) mus have a combnaon of low σ CF and hgh s /s. Ths combnaon can be found on he boomrgh corner of he plo n Panel C, ha s, low expeced reurn. As can be seen hen, value socks wll command a hgh premum and growh socks a low (and even negave) premum. Thus, f cross-seconal dfferences n cash-flow rsk are large, hen value socks have hgher expeced excess reurns han growh socks and a value premum s obaned. To beer llusrae hs pon, he op panel of Fg. 5, as was he case wh Fg. 3, agan plos he average prce-dvdend raos of prce-dvdend sored porfolos agans her average excess reurns n smulaed daa, bu now for he case n whch frms have heerogeneous cashflow rsk. Our purpose s o assess o wha exen he model can reproduce Fg. 1, whch s obaned from hsorcal daa. As n Fg. 1 and n conras wh Fg. 3, he presence now of heerogeney n cash-flow rsk generaes a value premum: Low prce-dvdend rao socks, value socks, are hose ha earn a hgh average excess reurn. The model s hus, able o generae a value premum, alhough, clearly, he queson s wheher can do so wh a reasonable cross-seconal dsperson of cash-flow rsk. 4.3 Condonal versus uncondonal value premums A novel heorecal mplcaon of our framework s ha he presence of dscoun rsk effecs, whch are assocaed wh he me-seres varaon n rsk preferences, affecs he dynamcs of he value premum, a feaure for whch here s already some emprcal evdence (Cohen, Polk, and Vuoleenaho, 2003, Table V). Essenally, dscoun rsk effecs nerac wh he cross-seconal dsperson n cash-flow rsk o nduce flucuaons n he value premum, as shown n Fg. 6. Ths fgure plos he expeced excess reurns of hree asses agans he surplus-consumpon rao, S. The doed lne shows he expeced excess reurn of he marke porfolo; he sold lne corresponds o he expeced excess reurn of a represenave value sock wh hgh cash-flow rsk and low expeced dvdend growh; fnally he dashed lne corresponds o he premum of a represenave growh sock wh low cash-flow rsk and hgh expeced 17

19 dvdend growh. As can be seen, when he surplus-consumpon rao s low (hgh), he value premum s hgh (low): Asses wh a hgh value of θ CF are parcularly rsker when he represenave agen s hghly rsk-averse whch occurs whenever adverse consumpon growh shocks depress he surplus-consumpon rao, ncreasng n urn he marke premum and s dvdend yeld. Thus, n our model, he value premum has a srong predcable componen, beng hgh (low) when he marke premum s hgh (low). 5 Quanave mplcaons of he model In hs secon, we conduc a smulaon sudy o evaluae he exen o whch he model can mach he sandard reurn momens boh n he me-seres and he cross-secon, whch can be found n Table 1. The emprcal daa se s sandard and s brefly descrbed n he legend o Table 1. Panel A shows he mean and sandard devaon for he reurns on he marke porfolo and he rsk-free rae. Panel B shows he predcably regressons of Fama and French (1988) and Campbell and Shller (1988) for wo dfferen sample perods, whch are mean o emphasze he sensvy of hese resuls o he parcular perod under consderaon. Panel C shows he sandard sascs for he cross-secon of book-o-marke sored (decle) porfolos. In parcular, we repor average excess reurns for he en value-sored porfolos for wo sample perods, (Panel C-1) and (Panel C-2). The value premum s 5.50% n he sample, whch s very smlar o he correspondng one n he longer sample. For he shorer sample, we also repor he average marke-o-book, he Sharpe (1964) rao, and he prce-dvdend rao, he laer beng he varable along whch we are gong o be sorng porfolos n smulaed daa as he model lacks a counerpar for he book value. Noce a srong feaure of he daa: Value socks have hgher Sharpe raos han growh socks and ndeed, from he hghes marke-o-book porfolo o he lowes, he Sharpe rao almos doubles An mporan regulary concerns he CAPM alphas and beas of hese porfolos. For he poswar sample, here s a fla f no slghly negave correlaon beween he CAPM beas and average reurns, whch s a he hear of he value premum puzzle. Indeed, he alphas of value socks are posve and sascally sgnfcan and he exreme growh porfolo s negave and also sascally sgnfcan. The evdence s somewha dfferen for he prewar sample. Panel C-2 repors he annualzed monhly average excess reurns for he en value-sored porfolos for a sample perod gong 18

20 back o 1926 as calculaed by Ang and Chen (2007, Table 1, Panel A.) Relave o he earler sample, he CAPM beas correlae posvely wh average reurns, raher han negavely, and hs gves some hope for he CAPM o address he value premum Deals of he smulaon We smulae 10,000 years of quarerly daa for 200 frms ha we hen sor no en porfolos accordng o her prce-dvdend rao (see foonoe 4) n an effor o mmc he sandard procedure used n he cross-seconal leraure. Table 2 conans he parameer values ha are used hroughou. We se he average consumpon growh a 2% and s sandard devaon a 1.5%, whch should be measured agans he value n he poswar sample of 1.22% and he one for he longer sample sarng n 1889, whch s 3.32% (see Campbell and Cochrane, 1999, Table 2.) We choose γ = 1.5, whch s beween he values used by MSV, γ = 1, and Campbell and Cochrane (1999), γ = 2. Ths choce mples a seady sae value of he local curvaure of he uly funcon of γs 1 = 48, hgher han he already hgh value of Campbell and Cochrane (1999) whch s 35. The mnmum value of hs local curvaure s Fnally he parameers k and α are smlar o he values chosen by MSV. As for he share process, all our resuls depend on he rao s /s and no he level s, and so we se s = 1/200 = 0.005, whou loss of generaly. The cross-secon of sock reurns s sensve hough o paramerc choces of oher cash-flow parameers, θ CF, φ, and ν. To avod parameer prolferaon, we resrc he share volaly ν = (ν,0, 0,..., 0, ν,, 0,..). Gven a value for he cash-flow rsk parameer, θ CF, he frs enry by defnon mus be ν,0 = θ CF /σ c. The second enry he dosyncrac par s chosen consan across all asses accordng o he formula, ν 2, = ν2 max(ν 2,0 ), where ν s a chosen parameer. In oher words, ν s he maxmum share volaly across asses. We repor frs he resuls for our benchmark case where φ = 0.07, whch s he value ha MSV (Table I) esmae for he marke porfolo, and ν = 0.55, and θ CF = 0.345%, whch as we wll show shorly are values ha allow us o mach he momens repored n Table 1. Secon 5.3 conans a horough dscusson of he economc sgnfcance of hese laer assumpons, focusng especally on her mpac on he man rade-off we hghlgh n hs paper: he enson beween dscoun effecs and cash-flow rsk effecs n wha concerns he cross-secon of sock reurns. In wha follows, we refer o θ CF as he cash-flow rsk parameer, bu he reader should keep n mnd ha s he suppor of he cash-flow rsk parameers of 18 On hs pon, see also Campbell and Vuoleenaho (2004) and Fama and French (2006), who also show ha nowhsandng he evdence above, he CAPM s sll rejeced n he longer sample. 19

21 ndvdual asses. 5.2 Smulaon of he model Table 3 s he counerpar o Table 1 bu n smulaed daa. 19 As can be seen he model does a reasonable job a capurng he man paerns of he emprcal sample. Panel A shows ha he model generaes a szable equy premum, hough a b low compared o oher smulaons of exernal hab perssence models, and hgh volaly of sock reurns. As n MSV, he model yelds a low rsk-free rae hough wh volaly somewha hgher han s emprcal counerpar. Panel B shows ha he model produces he predcably a long horzons, hough he R 2 s are lower han he ones observed n he emprcal daa. 20 Why s he equy premum lower n our model han n oher exernal hab perssence models, such as Campbell and Cochrane (1999) and MSV? Ths s he resul of a feaure of our model ha s absen from Campbell and Cochrane (1999) and MSV, namely, he small predcable varaon n expeced consumpon growh due o general equlbrum resrcons (see he dscusson afer Assumpon 1.) To gauge he nuon of hs resul, s useful o reurn o Proposon 1. Consder he case where here s a posve shock o consumpon growh. Ths mmedaely ranslaes no a hgher prce, whch s now a clam o a larger dvdend. In hab perssence models, hs posve consumpon shock gves a second posve jol o prces hrough he ncrease n he surplus-consumpon rao, S, whch lowers he represenave agen s rsk averson. Because hs makes socks more volale and rsker, hey command a larger premum. Ths s he sandard effec n hab perssence models and corresponds o he frs erms n (18) and (19) for he expeced reurn and volaly, respecvely. In our framework hough, shocks o consumpon growh and shocks o expeced consumpon growh are posvely correlaed. Thus, on average, n he presence of a posve consumpon growh shock, expeced consumpon growh s also hgh and hs makes he oal wealh porfolo less desrable o a represenave consumer wh a srong preference for neremporal smoohng, as s sandard n hab perssence models. 21 Ths s a negave force on prces whch parally undoes he posve effecs dscussed above. As a resul, he volaly 19 We do no repor -sascs of smulaon resuls, because he large sample (40,000 quarers) makes hem meanngless. We ake he smulaed values as populaon momens and compare hem wh her emprcal counerpars. 20 Sll, he rskless rae volaly s vasly lower han he one of radonal hab perssence models, such as Abel (1990) and Boldrn e al. (2001), who repor a rskless rae volaly of 17.87% and 24.6%, respecvely. 21 Yang (2007) proposes a model ha combnes he Epsen-Zn uly framework wh hab perssence precsely o allow for a more flexble specfcaon of he neremporal elascy of subsuon. 20

22 s lower and so s he requred premum when compared o he sandard hab model wh..d. consumpon growh. Ths s he second erm n boh (18) and (19). Noce ha hs effec s also bound o affec he srengh of he predcably regresson. I s mporan o noe, however, ha our model does no generae oo much predcably of consumpon growh, as already dscussed n foonoe 11. Panel C of Table 3 repors he quanave mplcaons of he model for he crosssecon. The model generaes a szable value premum of a b above 5%, hough he average reurns of ndvdual porfolos are off by as much as he equy premum s. Ineresngly, Sharpe raos n smulaed daa show he same ncreasng paern when movng from growh o value as n he emprcal sample: Value socks are good deals accordng o hs merc boh n he daa and n he model. The model s able o generae no jus he value premum bu he value premum puzzle as well. Indeed, noce ha he CAPM alphas are negave for wo growh porfolos (porfolos 1 and 2) and posve for he res of he porfolos, whch maches surprsngly well he emprcal sample n Table 1. Imporanly hough, he CAPM beas covary posvely wh average reurns n he cross-secon, a paern ha s conssen wh he sample (see Panel C-2 of Table 1), bu no he poswar sample. Why does he CAPM fal n our seng? As menoned already, our model feaures a mld me varaon n expeced consumpon growh hrough he erm µ c,1 (s ) = s θ CF n (3). The prce/consumpon rao of he oal wealh porfolo n (17) s a non-lnear funcon of boh he surplus-consumpon rao S and expeced consumpon growh, P T W C = α T W 0 (s ) + α T 1 W (s ) S γ. Ths resul shows ha general equlbrum resrcons mply ha reurns on he oal wealh porfolo P T W are deermned by wo ypes of shocks. Frs, consumpon shocks dc, whch affec he prce P T W hrough he level of consumpon self, he surplusconsumpon rao S, and her mpac on he sysemac componen of he varaon n shares s. The second source of varaon of he oal wealh porfolo P T W s he componen of he varaon n expeced consumpon growh ha s orhogonal o consumpon shocks. Ths s nduced by he dosyncrac componens of shares s. As hese vary, expeced consumpon growh changes and so does he prce P T W, bu hs varaon s no prced by he habbased sochasc dscoun facor. In essence, he dosyncrac componen of he varaon n expeced consumpon growh breaks he perfec correlaon beween he oal wealh porfolo reurn and he hab sochasc dscoun facor, whch n urn nvaldaes he CAPM, boh condonally and uncondonally. We emphasze ha, as dscussed afer Assumpon 1, n mul-asse models he varaon n expeced consumpon growh resuls from he general 21

23 equlbrum resrcon (1). I follows ha, n general, we should no expec he CAPM o hold n a model wh mulple rees. 22 In concluson, our model s able o capure o a surprsng degree he man characerscs of he reurn dsrbuon boh n he me-seres and he cross-secon and hus, seems a useful lens hrough whch o draw nferences abou cash-flow parameers. Specfcally, s he value of θ CF needed o mach hese momens hgh or low? We urn o hs queson nex. 5.3 The cash-flow rsk puzzle In hs secon, we assess wha our choce of he cash-flow rsk parameer θ CF means for he properes of he cash-flow process. Evaluang he assumed magnude of he cash-flow rsk parameer s no easy, as requres observng asse dsrbuons ha accrue o consumers, and hen calculae he hard-o-esmae correlaon wh consumpon growh. Here, we ge a hs queson by measurng nsead he model mpled cash-flow beas, as defned and esmaed n emprcal daa by Cohen, Polk, and Vuoleenaho (2009), and compare he model-mpled values o her emprcal counerpars. Specfcally, usng daa from 1928 o 1999, Cohen, Polk, and Vuoleenaho (2009) regress dfferen measures of frms cash-flows on he correspondng measures of marke cash-flows as n, for example, R 1 j=0 ρ j CP V dp +j,j+1 = βp CF,0 + βp CF,1 R 1 j=0 ρ j CP V dmk +j + ε p +R 1 (26) for each me and each porfolo p = 1,..., 10. Here, d p +j,j+1 s he dvdend growh a me + j of he porfolo p whch was formed j + 1 years earler, ha s, a 1. Smlarly, d mk +j s he dvdend growh of he marke a me +j. Fnally, ρ j CP V = 0.95 s a dscoun, and R s he number of years over whch he average growh rae s compued. They call he regresson coeffcen β p CF,1 he cash-flow bea, as measures essenally how porfolo cash-flows covary wh aggregae cash-flows. The emprcal esmaes of Cohen, Polk, and Vuoleenaho (2009) are repored n Table 4 Panel A. Noce frs ha emprcally, rrespecve of he cash-flow measure used, value socks have hgher cash-flow beas han growh socks, hough magnudes dffer across measures. If eher (accumulaed) reurn on equy, 4 j=0 ρj ROE p +j,j+1, or (accumulaed) dvdend growh 22 We expressed he ssue n erms of shares, bu hs fndng s rue more generally. Indeed, n he general se up n Appendx A we may assume v = [v 1, 0,..., 0, v,...]. Shocks o consumpon hen depend only on db 1, he frs Brownan moon, as he ohers are dversfed away. However, expeced consumpon growh depends on he shares, µ (s ) = P s µ, whose movemen does depend on an aggregaon of he dosyncrac shocks. 22