Labor Income and Predctable Stock Returns Tano Santos Graduate School of Busness Columba Unversty and NBER Petro Verones Graduate School of Busness Unversty of Chcago, NBER, and CEPR March 29, 2005 Abstract We propose a novel economc mechansm that generates stock return predctablty n both the tme seres and the cross-secton. Investors ncome has two sources, wages and dvdends, that grow stochastcally over tme. As a consequence the fracton of total ncome produced by wages fluctuates dependng on economc condtons. We show that the rsk premum that nvestors requre to hold stocks vares wth these fluctuatons. A regresson of stock returns on lagged values of the labor ncome to consumpton rato produces statstcally sgnfcant coeffcents and large adjusted R 2 s. Tests of the model s cross-sectonal predctons on the set of 25 Fama-French portfolos sorted on sze and book-to-market are also met wth consderable support.. We thank Fernando Alvarez, Nck Barbers, John Y. Campbell, John H. Cochrane, George Constantndes, Kent Danel, Gene Fama, Lars P. Hansen, John Heaton, Mchael Johannes, Martn Lettau, Sydney Ludvgson, Lubos Pastor, Zenyu Wang and semnar partcpants at the 2000 NBER Summer Insttute n Asset Prcng, Columba Unversty, Northwestern Unversty and the Unversty of Chcago. We thank Lor Menzly for excellent research assstance. All errors are our own. Address correspondence to Petro Verones, Unversty of Chcago GSB, 5807 South Woodlawn Avenue, Chcago IL 60637; emal: fpverone@gsb.uchcago.edu. 1
Researchers, at least snce Mayers (1972), have long recognzed the mportance of accountng for labor ncome, and, more generally, human captal, n asset prcng tests. Indeed, labor ncome consttutes around 75% of consumpton and human captal s a sgnfcant component of wealth. In ths paper we propose a mnmal extenson of the standard consumpton asset prcng model where consumpton s funded by sources other than fnancal, labor ncome n partcular, and that allows for tractable and nterpretable formulas for prces and returns. The model shows that allowng for ths alternatve source of consumpton mmedately yelds mplcatons for the dynamcs of asset prces. In partcular, we show, theoretcally and emprcally, that fluctuatons n the fracton of consumpton funded by labor ncome results n stock return predctablty both n the tme seres and the cross-secton. In order to llustrate the role of labor ncome on the predctablty of stock returns, we consder a general equlbrum model where a representatve agent receves ncome from two sources, fnancal and nonfnancal (human), and where the mx between the two sources of ncome vares over tme. We frst show that changes n the fracton of consumpton funded by labor ncome nduce fluctuatons n the expected excess return of the market portfolo. The ntuton for ths result s straghtforward. If, say, most of consumpton s funded by labor ncome, fnancal assets consttute a small fracton of consumpton and thus covary lttle wth t. For ths reason nvestors requre a low premum n order to hold them. It follows then that the rato of labor ncome to consumpton should forecast stock returns at the aggregate level. Notce that the tme seres predctablty of aggregate returns stems solely from fluctuatons n the fracton of consumpton funded by labor ncome. No other ngredent s requred to generate ths predctablty, whether t be habt persstence, as n Campbell and Cochrane (1999), house money effects as n Barbers, Huang, and Santos (2001) or, learnng effectsasn Tmmermann (1993) and Verones (2000). Our model then provdes an alternatve source of predctablty so far unexplored n the lterature. We test ths mplcaton by regressng aggregate market returns on the labor ncome to consumpton rato and fnd that ths varable s a strong predctor of long horzon returns. Ths result s robust to alternatve constructons of ths rato and the ncluson of the dvdend yeld. In addton, when runnng a smlar regresson n model smulated data the share of labor ncome to consumpton s also a sgnfcant predctor of long horzon returns. The ablty of the labor ncome to consumpton rato to forecast long horzon returns suggests that ths varable s also a useful predctor of the cross-secton. The falure of the CAPM to explan the cross-secton of stock returns stands as a central fndng n emprcal 2
asset prcng and many have argued for a role for labor ncome n tests of the cross-secton. For nstance, a common concern, famously emphaszed by Roll (1977), s that the return on the value weghted portfolo of assets lsted n major US stock exchanges s typcally taken as a proxy for the return on the market portfolo and that ths proxy may not be good enough. Buldng on ths nsght, Jagannathan and Wang (1996) and Campbell (1996) nclude a measure of the return on human captal as part of the returns on aggregate wealth. Recently as well Lettau and Ludvgson (2001a,b) have shown that a varable that measures devatons of consumpton from ts stable relaton wth wealth, that ncludes both human and non human (fnancal) wealth, has remarkable predctve power both n the tme seres and the cross-secton of stock returns. In our framework, the CAPM wth respect to the total wealth portfolo holds condtonally f agents have log utlty. In general, although the condtonal CAPM does not hold for relatve rsk averson dfferent than one, we show that t holds approxmately. We obtan the asset s beta n closed form for the log utlty case and show that t s a functon of two varables. The frst one s related to the current sze of the asset s contrbuton to consumpton. If ths contrbuton s sgnfcant and covares postvely wth consumpton growth, the asset wll command a hgher premum than an otherwse dentcal asset wth a lower contrbuton to consumpton. The second varable s, agan, the labor ncome to consumpton rato whch captures changes n the overall covarance between the return on the total wealth portfolo and fnancal assets. It follows then that takng nto account labor ncome n asset prcng requres more than updatng the defnton of the market portfolo (the total wealth portfolo.) It requres the use of condtonng nformaton as well. In addton, our model also allows us to assess the effect that sortng procedures based on prces have on tests of the cross-secton. In our setup sortng by the prce of the securty, normalzed by dvdends, s akn to sortng by expected dvdend growth, whch s the source of cross-sectonal dfferences n expected excess returns. Stll, we show that the sortng per se s not enough to capture the cross-sectonal dsperson of average returns and that condtonng by the labor ncome to consumpton rato s key to fully capturng ths dsperson. We test these predctons n the set of 25 portfolos sorted by sze and book-to-market ntroduced by Fama and French (1993) and fnd that the condtonal CAPM performs consderably better than ts uncondtonal counterpart when usng the labor ncome to consumpton rato as a condtonng varable. We also run extensve smulatons of the model to show that t can reproduce both the poor performance of the uncondtonal CAPM and the better one of 3
ts condtonal verson. To do so we smulate fnancal data for multple assets whch we then sort nto portfolos accordng to ther prce dvdend ratos. We then test the uncondtonal and the condtonal CAPM n ths set of fcttous test portfolos. Our smulatons show that the model can reproduce the flat relaton between the average and the excess returns predcted by the uncondtonal CAPM as well as the sgnfcant prcng errors that are usual n tests of the CAPM. Instead, n our condtonal specfcaton ftted and average returns lne up ncely along the 45 degree ray and the prcng errors are not sgnfcantly dfferent than zero. In our framework then the value spread puzzle s less of a puzzle. Our model s related to a number of recent papers that attempt to model multple securtes n a general equlbrum settng. 1 The paper closest to ours n what concerns the cash flow model s Menzly, Santos and Verones (2004, MSV henceforth.) However, there are several key dfferences wth respect to that paper. Frst, the cash flow model proposed here s slghtly more general than the one n MSV. Our prcng formulas then apply to ths more general class of cash flow models. Second, the economc mechansm at work n each of these two models s very dfferent: Whle MSV focuses on the nteracton between tme varyng rsk preferences and fluctuatons n expected dvdend growth to explan the relaton between valuaton ratos and asset returns, ths paper has constant rsk preferences. By constructon, we shut down n ths model the man source of varaton n MSV, and thus the two models are exact oppostes n terms of the economc mechansm that matter for changes n expected returns. As already mentoned, we focus here on the varaton n the labor ncome / fnancal ncome mx to explan the varaton over tme of asset returns. Thrd, and fnally, MSV focuses ts emprcal exercses on ndustry portfolos whereas here we concentrate on a set of prcesorted portfolos, such as the sze and book-to-market sorted portfolos. Our model offers a framework where the sortng procedure can be easly nterpreted and where the role of labor ncome n prcng ths partcular set of test portfolos can be cleanly nvestgated. Our paper places tself at the ntersecton of two strands of the lterature on asset prcng. On the one hand the paper contrbutes to the body of work that documents the tme seres predctablty of the aggregate market returns. The early predctablty lterature documents the forecastng power of prces scaled by ether dvdends or earnngs and of varous nterest rate measures. 2 More recently Lettau and Ludvgson (2001a) manpulate the budget constrant 1 See Menzly, Santos and Verones (2004), Cochrane, Longstaff and Santa Clara (2004), and Longstaff and Pazzes (2004). 2 Campbell and Shller (1988a,b), Fama and French (1988a,b), Hodrck (1992), and Lamont (1998) document 4
to show that the consumpton to total wealth rato contans nformaton about stock returns. Our paper adds to ths lterature by provdng yet more evdence on the predctablty of stock returns. A crtcal dfference between our work and prevous emprcal research though s the fact that our predctve varable s nether a verson of the stock prce scaled by ether dvdends or earnngs nor some other fnancal varable lke the term premum, but rather a pure macroeconomc varable. Furthermore, gven that t s drectly observable, t does not need to be estmated, as n Lettau and Ludvgson (2001a,b). Fnally t s mportant to emphasze that our testable mplcaton does not result from basc manpulatons of ether the defnton of returns (Cochrane (2001), page 395-6) or the budget constrant. The present paper also adds to the growng body of emprcal research that concentrates on testng the condtonal versons of the CAPM. For nstance, researchers lke Cochrane (1996), Jagannathan and Wang (1996), Ferson and Harvey (1999), and Lettau and Ludvgson (2001b) use condtoned down versons of the CAPM (or the Consumpton CAPM) to obtan mprovements n the ablty of these models to descrbe the cross-secton of stock returns. Others, lke Bollerslev, Engle, and Wooldrdge (1988), use GARCH methods to explore the role of changng betas n the context of the condtonal CAPM. They fnd that the condtonal covarances of each return wth the market return vary substantally and are a sgnfcant determnant of tme-varyng rsk prema. Fnally, Ferson and Harvey (1991) and Fama and French (1997), among others, parameterze both market rsk prema and factor loadngs as functons of both aggregate varables and ndvdual asset characterstcs. They fnd substantal varaton n both prema and loadngs. 3 Few of these papers, though, derve the expresson of beta from a full-fledged general equlbrum model and dentfy the varables that should proxy for the nvestors nformaton set from theoretcal consderatons. It s then one of the objectves of ths paper to place tests of condtonal models on frmer theoretcal ground. In summary, our model hopes to provde a coherent vew of the tme seres and crossthe predctve power of prces scaled by dvdends and earnngs. Campbell (1987), Fama and French (1989), Hodrck (1992), and Kem and Stambaugh (1986) show the forecastng power of nterest rate measures. 3 The lterature on the varaton n betas s large. See Bollerslev, Engle, and Wooldrdge (1988), Ng (1991), Bodurtha and Mark (1991), Evans (1994), Braun, Nelson, and Suner (1995), and Cho and Engle (1999). Ball and Kothar (1989) argue that varaton n betas and market prema can explan the performance of wnners and losers. Ferson, Kandel and Stambaugh (1987), Campbell (1987), Ferson (1989), Harvey (1989), and Shanken (1990) develop tests of asset prcng models wth tme varyng rsk prema and betas. Franzon (2001) uses rollng regressons to document changes n the betas of growth and value stocks. For a crtcsm of the ablty of tme varaton n betas to address puzzles n the cross-secton see Lewellen and Nagel (2003). 5
sectonal predctablty of returns. Indeed, snce Merton (1973) t has been understood that varables that predct market returns are natural condtonng varables for tests of the crosssecton. 4 Our contrbuton here s to show how takng nto account labor ncome generates these two forms of predctablty and to clarfy the lnk between them. In addton, and as Lettau and Ludvgson (2004) have recently emphaszed, these fndngs hghlght the fact that nformaton contaned n consumpton and labor ncome may be relevant for the long run valuaton of fnancal assets. The rest of the paper s organzed as follows. Secton 1 presents the model. Secton 2 derves the mplcatons of the model for prces and returns. Secton 3 presents the emprcal results and 4 concludes. 1. THE MODEL Preferences - We assume the exstence of a representatve consumer whose preferences over aggregate consumpton C t are represented by the nstantaneous utlty functon e φt C 1 γ t / (1 γ) f γ 6= 1 U (C t,t)= e φt log (C t ) f γ =1, (1) where γ s the coeffcent of relatve rsk averson and φ s the subjectve dscount rate. Endowment - We assume that consumpton s funded by labor ncome, w t, and the proceeds from an nvestment n n 1 addtonal fnancal securtes, whose nstantaneous dvdend streams we denote by Dt,for =2,...,n. For notatonal convenence let Dt 1 = w t. Modelng choces concernng Dtª n =1 and C t cannot be made ndependently as market clearng mposes C t = w t + P n =2 D t. Thus, partcular assumptons on the processes governng ª D n t nduce n turn a specfc aggregate consumpton process. Ths s the essence of the =1 dffculty when studyng models wth multple assets, namely how to model dvdends and consumpton so that they are emprcally plausble, mutually consstent and, at the same tme, tractable enough to yeld nterpretable formulæ for prces and returns. 5 It s useful 4 See Cochrane (1996), Ferson and Harvey (1999), and Lettau and Ludvgson (2001b) for recent contrbutons n ths drecton. For example Ferson and Harvey (1999) state that smple proxes for tme varaton n expected returns, based on common lagged nstruments, are also sgnfcant cross-sectonal predctors of returns." 5 See also Bossaerts and Green (1989). Cochrane, Longstaff and Santa-Clara (2004) were recently able to obtan closed form formulas for prces n the specal casewherethereareonlytwoassets whose dvdends are log-normal, and agents have log utlty. Ther model, however, mples that n the long run, one of the two assets would domnate the economy wth probablty one. 6
then to brefly revew the nature of these dffcultes n order to better motvate our specfc assumptons. Let D t = Dt 1,...,Dt n be the vector of dvdends. Assume for nstance that dvdends Dt are gven by ddt Dt = µ D (D t ) dt + ν 0 db t (2) for some drfts µ D (D t), whereν s an n 1 constant vector and db t s an n 1 vector of Brownan motons. The process for aggregate consumpton C t cannowbewrttenas dc t C t = µ c (s t ) dt + σ c (s t ) 0 db t where s t = s 1 t,...,s n 0 t = Dt 1 /C t,...,dt n 0 /C t are the share of consumpton produced by dvdends (or labor ncome, for =1), and µ c (s t ) = σ c (s t ) = nx s tµ D (3) =1 nx s tν. (4) =1 One dffculty n obtanng tractable and nterpretable formulas for prces s the dependence of the drft and volatlty of the consumpton process on the shares s t = s 1 t,...,s n 0 t. Stll, by makng judcous but economcally motvated choces for both the consumpton and share processes, closed form solutons can be obtaned for both prces and returns. The next two assumptons summarze the essence of our cash-flow model. Assumpton 1. The aggregate consumpton process s gven by dc t C t = µ c (s t ) dt + σ 0 c db t (5) where µ c (s t )=µ c + s 0 t θ (6) where θ = θ 1,...,θ n 0,andσc =(σ c,1, 0,..,0) 0.Thespecfcaton of θ s explaned below. Assumpton 2. The vector of consumpton shares s t follows a contnuous tme, vector autoregressve process ds t = Λ 0 s t dt + I (s t ) Σ (s t ) db t (7) 7
where Λ s an (n n) matrx wth the property λ j 0 for 6= j, andλ = P j6= λ j, and I (s t ) s a dagonal matrx wth -element gven by s t,andσ (s t ) s an (n n) matrx whose th row s nx σ (s t )=ν 0 s j t ν0 j. (8) The cash-flow model (7) s extremely convenent whle retanng a natural economc nterpretaton. The restrctons on the matrx Λ as well as the choce for the functonal form of the volatlty functon (8) guarantee that both s t 0 and P n =1 s t =1for all t. Thus, total ncome always equal consumpton, dvdends are never negatve, and, for a generc matrx Λ, no asset ever comes to fully fund consumpton. In addton, fluctuatons n the share of one asset may n turn nduce varaton n the fractons that other assets contrbute to total consumpton. Fnally, cash-flow shocks naturally correlate across dfferent assets and the dffuson term n (7) s flexble enough to capture a rch pattern n the covarance structure. 6 The cash flow model mpled by (7) generates an ntutve model for dvdend growth. Indeed, an applcaton of Ito s Lemma shows that the process for dvdends, D t, s gven by j=1 dd t = I (µ c + θ)+λ 0 D t dt + I (D t ) Σ D (s t ) db t (9) where I (µ c + θ) s a dagonal matrx wth µ c + θ n ts poston, the th element of Σ D (s t ) s σ D (s t)=σ c + σ (s t ),and θ = ν 0 σ c. That s, dvdends follow a vector autoregressve process whose growth rates depend on the long term growth of consumpton tself, µ c, as well as a parameter θ that regulates the nstantaneous covarance between consumpton growth and shares: µ ds cov t t s, dc nx t = θ s j t t C θj (10) t j=1 Thus, share processes that have a hgher covarance wth consumpton growth wll mply dvdend streams wth a hgher growth n average. Notce that the constants θ are not dentfed as we can add a constant to all of them wthout changng any of the covarances. For ths reason we can renormalze them so that nx θ j s j =0. (11) j=1 6 For nstance, f the assets are nterpreted as ndustry portfolos, t s only natural that cash-flow shocks n a partcular ndustry contan nformaton about cash-flow growth n related ndustres. 8
From Assumpton 1, ths condton mples that the uncondtonal expected consumpton growth s E nx µ c,t = µc + θ j s j = µ c. In addton, ths structure mples that the model s j=1 nternally consstent. Applyng the general formula for expected consumpton growth, (3), to (9) we fnd dct nx nx E t = s C tµ D (s t )= s 1 t µc t D + θ Dt + Λ 0 nx D t = µc + s tθ =1 =1 whch equals (6) n Assumpton 1. 7 We note that n the data, θ turnouttobeverysmall. Thus, the condtonal expected consumpton growth s n fact essentally constant. In our smulatons, we fnd that the mnmum and maxmum condtonal expected consumpton growth are just 2.25% and 2.39%, respectvely, showng that the term P n =1 s tθ has essentally no mpact on E t [dc t /C t ]. On the other hand, ths specfcaton for expected consumpton growth allows us to obtan closed form solutons for stock prces. 2. RESULTS =1 2.1 Equlbrum prces Gven the consumpton stream of the representatve agent, the standard asset prcng formula s Z Pt U c (τ,c τ ) = E t t U c (t, C t ) D τdτ. (12) The appendx then proves the followng proposton: Proposton 1: Under assumptons 1 and 2, the prce of asset s gven by P t = b 0 D t (13) where b 0 s the -th row of the matrx ³ ³ b 0 = I eφ Λ 0 1 ³ and I eφ s the dagonal matrx wth th element gven by (14) eφ = φ (1 γ) µ c + 1 2 γ (1 γ) σ0 cσ c (1 γ) θ 7 Here, we used the convenent fact that P n =1 s t/d t [Λ 0 D t] = 1 C P n =1 [Λ0 D t] =0, the latter equalty stemmng from the restrctons on the Λ matrx. 9
Ths formula s very general. Because of the autoregressve nature of the cash flow model, and the fact that consumpton growth depends on the shares as well, as t should n a general equlbrum model (see equaton (3)), the prce of asset does not depend only on ts own dvdend Dt, but also on the dvdend level of all assets. Stll, the effects of the asset s own dvdends are frst order n the determnaton of ts prce. Ths can be seen from the dvdend process tself (9), as the th element of the autoregressve model has the addtonal drft component µ c + θ, whch determnes the long term propertes of the dvdend process tself. In addton, numercal examples show that the b elements of the matrx b are an order of magntude larger than all the other entres. Proposton 1 has an mmedate mplcaton for the value of the market and the total wealth portfolo, as they are smply Pt M = P n =2 P t and Pt TW = P n =1 P t respectvely. In ths case we fnd that Pt M = b 0 M D t and Pt TW = b 0 TW D t (15) where b 0 M = P n =2 b0 and b0 TW = P n =1 b0. From (13) or (15) t s also mmedate to compute the expected excess stock return. The excess stock returns of asset, drt,aredefned as: drt = dp t + Dtdt Pt r t. The followng result then apples to both ndvdual assets and to the market portfolo: Proposton 2: The expected excess return on asset s µ E t dr t = γ σ 0 cσ c + b0 I (s t) (θ 1 n s 0 t θ) b 0 s t (16) where I (s t ) s the dagonal matrx wth s t n ts th entry. In the case of the market portfolo, the formula holds substtutng b M for b throughout. Note that n a standard model wth..d. consumpton and where consumpton equals dvdends, the equty premum would be gven smply by γσ 0 cσ c. The model proposed here s not only able to generate a tme varyng equty premum as the shares move over tme, but also a more szable equty premum than that n a standard..d settng, thereby partly addressng the Mehra and Prescott (1985) equty premum puzzle. 2.2 An Example 2.2.1 Assumptons and dscusson To sharpen ntutons about the role of labor ncome n asset prcng tests t s useful to specalze the model further. Frst, we restrct the utlty functon to the log case (γ =1)as n 10
the log CAPM of Rubnsten (1976). Second, we use a smple verson of the cash-flow model where the share of each asset depends only on ts own past share value, and not on those of other assets. In addton, we assume that all fnancal assets are uncondtonally dentcal, that s all fnancal assets have the same covarance between share and consumpton growth. The next assumpton summarzes the essence of ths smpler cash-flow model. Assumpton 3. (a) Let λ j = as j for j 6= where P n =1 s =1. Then the process governng s t s gven by ds t = a s s t dt + s t σ (s t ) db t (17) where σ (s t ) s defnedn(8). (b) θ = θ for =2,...n. 8 Assumpton 3 has a natural nterpretaton for the cash-flow process: Asset contrbutes afractons t to overall consumpton and, n the presence of shocks to s t, t mean reverts to a long run value s, whch s the asset s steady-state contrbuton to consumpton. Second, the relatve share s /s t proxes for expected dvdend growth. In fact, specalzng equaton (9) to ths case, we mmedately obtan dd µ E t s t Dt = µ c + θ + a s 1 (18) t Thrd, fnancal assets have dentcal cash-flow rsk, that s, the covarance of share and consumpton growth s the same across these assets: µ ds cov t t s, dc t =(θ θ w ) s w t for =2,..,n, (19) t C t where θ w = θ 1. Ths assumpton means that, uncondtonally, there wll be no cross-sectonal dfferences n prces-dvdend ratos or n average returns. Stll, as we wll show below, crosssectonal dsperson n expected dvdend growth, as measured by s /s t, generates nterestng condtonal cross-sectonal varaton. Assumpton 3 then s useful to nvestgate exactly how labor ncome nteracts wth dsperson n expected dvdend growth to generate patterns n the cross-secton of average returns and to understand the tests of the condtonal CAPM where 8 Part (a) of Assumpton 3 mples that λ = P j6= asj = a 1 s. Ths smpler model s the one used by Menzly, Santos and Verones (2004), though they allow for cross-sectonal varaton n the covarance between share and consumpton growth and n the speed of mean reverson, a. ThscasesalsoconsderednSantos and Verones (2001). 11
labor ncome s shown to play an mportant role. Fnally, assumpton 3 together wth (10) mples that µ ds w cov t t s w, dc t = (θ θ w )(1 s w t ) (20) t C t 2.2.2 Prce dvdend ratos The next proposton follows mmedately from Proposton 1. Proposton 3: The prce of asset s gven by Pt µ µ 1 1 Dt = φ + a φ φ + a Defne next P M t = P n =2 P t and Dt M = P n =2 D t.then Pt M Dt M µ s µ µ µ 1 1 1 s w = φ + a φ φ + a 1 s w = t Fnally, the prce of the total wealth portfolo, Pt TW, s s t. (21) µ 1 ψ(s w t ) (22) φ P TW t C t = 1 φ. (23) Naturally, the prce dvdend rato of an asset s an ncreasng functon of the relatve share. A hgh relatve share s /s t mples a hgh expected dvdend growth (see equaton (18)) and thus the hgh prce dvdend rato. As expresson (22) shows, the result extends to the case of the market portfolo and t has a strong ntutve appeal. The frst term of (22), 1/φ, s the prce-dvdend rato of the total wealth portfolo (see equaton (23)). The second term, ψ(s w t ), corrects for the presence of an alternatve source of ncome other than dvdends from the market portfolo. Notce that ψ(s w t )=1only f s w = s w t. That s, an economy n ts steady state yelds a prce dvdend rato that s no dfferent than the usual one. Devatons from ths steady state generate movements n the prce dvdend rato of the market portfolo. For nstance, f s w <s w t then the prcedvdend rato s hgher than t s long-run level, 1 φ. There are two reasons for ths. Frst, f sw t s relatvely hgh nvestors are less exposed to fluctuatons n the stock market, and hence they requre a lower compensaton to hold t; ths, n turn, translates nto hgher prces. Second, a hgh share of labor ncome to consumpton sgnals that future aggregate dvdend growth s gong to be above that of consumpton as s w t wll mean revert to s w. Ths further renforces the postve effect on the prce-dvdend rato. 12
2.2.3 Expected excess returns Proposton 4: The expected excess returns of asset and the market portfolo are respectvely gven by E t dr t = σ 0 c σ c + θ θw s w t (24) and 1+ a φ ³ s µ E t dr M t = σ 0 c σ c +(θ θ w s w t (1 s w t ) ) φ (1 s w t )+a(1 sw ) To understand equaton (24) recall frst that n the log economy the expected excess return of the total wealth portfolo, E t dr TW t, s gven by σ 0 c σ c. The expected excess return of asset s above or below E t dr TW t dependng on whether the covarance between share and consumpton growth, (θ θ w ) s w t, s postve or negatve (see equaton (19)). To provde some evdence on ths matter, Table 1 shows the correlaton between consumpton growth and the share of labor ncome to consumpton, where ths rato s constructed usng dfferent measures. As can be seen, ndependently of the defnton used, the correlaton between consumpton and share growth s negatve and hence (θ θ w ) > 0. It follows then that under Assumpton 3, cov ds t/s t,dc t /C t > 0 and that an asset s contrbuton to consumpton grows precsely when consumpton does, whch makes the asset rsky. For ths reason fnancal assets command a premum over that of the total wealth portfolo, σ 0 cσ c. The expected excess return of asset s also determned by the expected dvdend growthasproxedbytherelatveshares /s t (see (18)). Anassetwthalargecurrentsharecommands a hgher premum than an otherwse dentcal asset wth a lower share, as t s a larger fracton of consumpton and thus rsker. Thedegreetowhchchangesns w t affect E t dr t depends also on the value of s t. If s t 0, changes n s w t do not affect the requred return, as asset does not contrbute to consumpton and hence does not covary wth ts growth. Notce that s /s t s hgh for stocks that pay n the future, whch could be termed growth stocks. These assets then wll have both low expected returns and a relatvely lower senstvty to changes n s w t. Equaton (24) shows that, n the context of the present model, whatever cross-sectonal dsperson n average returns that s observed emprcally can only sprng from the condtonal dsperson n expected excess returns. The model generates ths condtonal cross-sectonal dsperson through temporary shocks to expected dvdend growth. The role of labor ncome as a useful varable n tests of the condtonal CAPM can now be easly llustrated. For s t (25) 13
nstance, sortng portfolos by the prce dvdend rato s, n ths model, equvalent to sortng by the relatve share, s /s t (see equaton (21)). Ths sortng captures the source of (condtonal) dsperson n the cross-secton of returns. Stll theszeofthsdspersonsgovernedbythe share of labor ncome to consumpton, s w t. If s w t s very small, whatever dsperson there s n prce dvdend ratos, does not translate nto a large cross-sectonal dsperson of returns. Conversely, a small dsperson n prce dvdend ratos can translate nto a large dsperson of returns when s w t s large. It follows that the ncluson of labor ncome, normalzed by consumpton n ths case, n tests of the cross-secton can help algn portfolos, partcularly when these are sorted accordng to some valuaton rato. As for the market portfolo equaton, (25) shows that the nstantaneous expected return depends non-lnearly on the fracton of consumpton produced by labor ncome s w t = w t /C t. Its functonal form shows that expected returns are equal to σ 0 cσ c both when s w t =0and when s w t =1. Indeed, when s w t =0,thenC t = P n j=2 Dj t = DM t and we revert to an economy wth no other endowment than the rsky assets. In ths case E t dr M t = σ 0 c σ c, the standard equty premum n the log economy. More puzzlng perhaps at frst s that ths model mples that E t dr M t = σ 0 c σ c when s w t =1. In ths case we must have s t =0for all =2,..,n and thus P M t = C t a φ (a + φ) (1 sw ). SncenthscasewealsohavethatC t = w t, the prce s perfectly correlated wth wages (and hence consumpton), yeldng the result. Clearly, these two cases are extreme, gven that, as shown n Fgure 1, s w t les comfortably n the nterval (0.7, 0.95) n the postwar sample. In ths case, what s the relatonshp between E t dr M t and s w t? In equaton (25) the denomnator of the second term s always postve, hence the behavor of expected stock returns depends solely on the sgn of θ θ w,whchs postve, as dscussed earler. Ths s also economcally ntutve: If wages are much smoother than dvdends, an ncrease n dvdends s accompaned by an ncrease n consumpton and hence a decrease n s w t = w t /C t. Ths nduces a negatve covarance between consumpton growth and changes n s w t whch, from (20), mples θ θ w > 0. Ths yelds n turn a negatve relaton between expected returns and the labor share s w t when s w t s n the relevant range (0.7, 0.95). The economc ntuton of ths result s also clear: As s w t ncreases, consumpton becomes fueled by labor ncome only, decreasng the covarance between consumpton growth and dvdend growth. Ths n turn translates nto a lower covarance between consumpton growth 14
and returns, generatng a lower rsk premum. Thus, a hgh labor ncome to consumpton rato should forecast low future excess returns. 2.2.4 The CAPM representaton The log utlty case also has the advantage of provdng smple and ntutve formulas for the CAPM representaton of expected returns. As shown n Proposton 3, Pt TW = φ 1 C t. Ths mples that ths asset s perfectly correlated wth the stochastc dscount factor and thus, from standard results (e.g. Duffe (1996)), a CAPM representaton holds. Specfcally, we have Proposton 5: Expected excess returns on ndvdual securtes are gven by: µ E t dr t = β TW s,s w t E t dr TW t, (26) s t where β TW µ s s,s w t t = cov t dr t,drt TW =1+ θ θ w var t dr TW t σ 0 cσ c ³1+ φ a ³ s s t s w t. (27) Equaton (26) s the CAPM wth respect to the total wealth portfolo. β TW depends on both a common factor, whch s the labor ncome to consumpton rato, and an asset s characterstc, the relatve share. Thus, the presence of non-fnancal sources of ncome nduces tme varaton n the asset s beta. The ntuton, of course, for (27) s dentcal to the one above for (24) andweomttnthenterestofspace. Testsof(26) requre observaton of the total wealth portfolo, whch s dffcult. The appendx shows the followng: Proposton 6: The expected excess returns on ndvdual securtes are gven by: E t dr t = β w, (s t )E t [drt w ]+β M, (s t )E t dr M t, (28) where β w, (s t ) and β M, (s t ) are the multple regresson coeffcents: β w, (s t ) β M, (s t ) 0 = Σ wm 1 covt dr t,drt w cov t dr t,dr M t and where Σ wm s the varance-covarance matrx of drt w and drt M. Versons of equaton (28) have been the focus of much research lately. For nstance, Jagannathan and Wang (1996) test a verson of the above equaton where they also extend the defnton of the market portfolo to nclude returns to human captal and where ther 15
condtonng varable s the properly defned default premum, shown to forecast busness cycles. More recently Lettau and Ludvgson (2001b) have tested a smlar equaton n a dfferent set of test portfolos where the condtonng varable s the consumpton to wealth rato, a condtonng varable that they show predcts future market returns. 2.3 Predctablty when γ>1 The ntuton developed n Secton 2.2 carres through to the case where γ > 1. In what concerns the tme seres predctablty t s easy to see from equaton (16) that under Assumpton 3 the expected excess return of any asset s only a functon of s t and s w t. Ths mmedately mples that, as n the log utlty case, the expected excess return on the market portfolo s only a functon of s w t and thus ths latter varable should forecast future excess returns. As for the cross-secton of stock returns, and as was the case wth log utlty, sortng by the prce dvdend rato s akn to sortng by expected excess returns. In fact, t s possble to show that when γ>1astock wth a hgh expected dvdend growth s stll characterzed by a hgh prce dvdend rato and a low expected excess return. As n the log utlty case then, the sortng procedure effectvely removes dfferences n condtonal expected excess returns that are due to cross-sectonal varaton n s /s t. Thus, the remanng varaton n the cross-sectonal dsperson of returns s only drven by s w t, whch acts as a common factor across stocks. Unlke the log utlty case though, when γ>1the condtonal CAPM wth respect to the total wealth portfolo does not hold. The reason s that the correlaton between consumpton shocks and the returns on the total wealth portfolo s not one. Stll, the smulaton exercse performed n secton 3.3.4 shows that ths correlaton s as hgh as 90% and thus lttle s lost by assumng that the condtonal CAPM holds approxmately when γ>1. Indeed, n lne wth ths latter fndng and the ntuton above, we also show n secton 3.3.4 that a verson of the condtonal CAPM that uses the share of the labor ncome to consumpton as a condtonng varable can explan essentally all of the cross-sectonal varaton n average returns whereas the uncondtonal CAPM leaves much of ths varaton unexplaned. 3. EMPIRICAL RESULTS 3.1 Data descrpton We consder returns on the value weghted CRSP ndex, whch ncludes NYSE, AMEX, and NASDAQ, as our measure of fnancal asset returns. Dvdend prce ratos are also obtaned from CRSP and the rsk free rate s the 90-day Treasury bll. For the cross-sectonal tests we 16
use the set of portfolos constructed by Fama and French (1993), formed by ntersectng fve sze sorted portfolos wth fve other portfolos sorted by book-to-market. As for the macroeconomc tme seres they are all obtaned from the Bureau of Economc Analyss. Consumpton (C t )sdefned as non durables plus servces. We use the defnton of labor ncome (w t ) n Lettau and Ludvgson (2001a), whch s, brefly, wages and salares plus transfer payments plus other labor ncome mnus personal contrbutons for socal nsurance mnus taxes. Both the consumpton and labor ncome seres are quarterly and our sample perod s 1948-2001. Wth these two seres we construct our man state varable, the share of labor ncome to consumpton s w t = w t /C t. As a robustness check we also use two alternatve constructons of the share of labor ncome to consumpton. Frst we use Compensaton of Employees (wt ce ) as our defnton of labor ncome, whch s computed by addng Wage and Salary Accruals plus Supplements to Wages and Salares (Employer Contrbutons for Socal Insurance plus Other Labor Income.) The labor ncome to consumpton rato s then computed as the rato of the Compensaton of Employees to the Consumpton of non-durables plus servces, s w t = wt ce /C t. Second, general equlbrum models blur the dstncton between consumpton and ncome. Thus a normalzaton of labor ncome by dsposable ncome, w t /Y t, s also theoretcally sound and we employ t below as well. Fgure 1 plots these three seres for our sample perod. FIGURE 1 ABOUT HERE Table 1 provdes some summary statstcs. As can be seen, our measures of s w t are all hghly persstent and thus cauton has to be exercsed when drawng nferences about the forecastng ablty of these varables. Also, as dscussed n secton 2.2, the mpact of changes n the share of labor ncome to consumpton on the expected excess return of the market portfolo depends crtcally on the sgn of the correlaton between ds w t and dc t. AscanbeseennPanelB, ndependent of the partcular measure of the share of labor ncome to consumpton employed, ths correlaton s negatve. Thus for the purposes of the nterpretaton of the emprcal results below n lght of our fndngs n secton 2.2, θ θ w > 0. TABLE 1 ABOUT HERE Panel C reports results of an Augmented Dckey Fuller (ADF) test for the presence of a unt autoregressve root n log (s w t )=log(w t ) log (C t ), whch our model assumes s statonary. 17
The results are reported for dfferent choces regardng the lags n log (s w t )=a 0 + a 1 log s w X t 1 + ζ j log s w t j + t (see Hamlton (1994) Chapter 17.) The optmal number of lags s chosen accordng to a sequental procedure descrbed n Campbell and Perron (1991) and n Ng and Perron (1995), and t s denoted by an astersk n the approprate entry n the panel. For the sample 1948-2001 we cannot reject the hypothess that log (s w t ) follows a unt root process at the standard confdence levels. Snce a reasonable concern les n the low power of the ADF test n relatvely small samples, t s useful to ascertan whether the same result obtans when we use a longer sample perod. To ths end, we nterpolate the annual data n the NIPA tables that are now avalable for the perod 1929-1948 to obtan a quarterly seres of s w t forthslongersample perod. The results are contaned n Panel D. In ths case we can reject the null of a unt root at the 10% level for the case where s w t s bult usng ether the labor ncome measure of Lettau and Ludvgson (2001a) or the Compensaton of Employees. Instead when s w t s measured as the laborncometodsposablencomerato,wecanrejectauntrootatthe5%level. Although the evdence s mxed, the power of these tests s, as mentoned, low and the restrcton that log (s w t ) s statonary rests on sold economc ntuton: It s not reasonable to assume that consumpton can grow to be nfntely larger than labor ncome, or, alternatvely, that labor ncome can grow to be several tmes hgher than consumpton. Our model (5) - (8) s smply a tractable way of capturng ths basc economc ntuton. 3.2 Predctablty of aggregate returns 3.2.1 s w t and the predctablty of long horzon returns The man predcton of our model s that a hgh share of labor ncome to consumpton, s w t, predcts low future returns. Panels A and B of Fgure 2 gve a vsual mpresson of the behavor of both the share of labor ncome to consumpton and the log dvdend yeld versus long horzon returns, measured by the four year cumulatve returns. The share of labor ncome to consumpton rato does ndeed move n the opposte drecton to the long horzon returns, as predcted by theory. As for the log dvdend yeld and long horzon returns, the frst part of the sample shows the famlar pattern n the predctablty lterature: Log dvdend yelds comove wth long horzon returns. The plot also shows the strkng behavor of the market durng the 1990s. Contrary to the pror hstorcal experence, log dvdend prce ratos and long horzon returns move n opposte drecton durng that perod untl the correcton of 2001. j=0 18
FIGURE 2 ABOUT HERE In Table 2 Panel A we report the results of regressons of long horzon excess returns on lagged values of s w t on the log dvdend yeld and on both. That s, we run Regresson 1 r t,t+k = α 1 + β 1 (K)s w t + ε t+k (29) µ D M Regresson 2 r t,t+k = α 2 + β 2 (K)log t + ε t+k (30) µ D Regresson 3 r t,t+k = α 3 + β 3 (K)s w M t + β 4 (K)log t Pt M P M t + ε t+k, (31) where r t,t+k s the cumulatve log excess return over K perods. For each regresson, we report the pont estmates of the ncluded explanatory varable, the adjusted R 2 s and, n parentheses, the Newey-West corrected t statstc. As already mentoned, the share of labor ncome s hghly persstent and ths complcates the nference on the sgnfcance of the estmated coeffcent. For ths reason we also report, n brackets, the t statstc obtaned usng the standard errors proposed by Hodrck (1992). 9 Ang and Bekaert (2001) show that these standard errors have better small sample propertes than alternatve ones and thus we compute them throughout. TABLE 2 ABOUT HERE Turnng frst to the regresson of long horzon returns on s w t, Panel A of Table 2 shows that ths varable s a statstcally sgnfcant predctor of returns for horzons of two years and longer when the Newey-West t statstc s consdered. The sgnfcance remans at horzons of three and four years when usng the Hodrck t statstcs, beng only margnally so for the two year horzon. Consstent wth the theory and the negatve correlaton between changes n shares and consumpton growth, the sgn of the coeffcent s negatve: Postve nnovatons n s w t lead to low future returns. The explanatory power s also hgh, rangng from 16% for the two year regresson to 42% for the four year regresson, dsplayng the famlar ncreasng pattern wth respect to the forecastng horzon. The dvdend yeld does not forecast returns as well and t s sgnfcant only at the one year horzon, precsely where s w t was not. When both regressors are ncluded, as n (31), the share of labor ncome to consumpton becomes sgnfcant at all horzons, ndependently of 9 Hodrck (1992) refers to the standard errors he proposes as standard errors (1B) (see page 362.) We follow Ang and Bekaert (2001) n referrng to them as Hodrck standard errors. 19
whether one consders the Newey-West t statstc or the more strngent Hodrck t statstc. The sgnfcance of the log dvdend yeld mproves nstead only margnally. Fnally, the R 2 s ncrease consderably over the case where only the share of labor ncome to consumpton s ncluded and reaches a remarkable 57% at the four year horzon. 3.2.2 Robustness to alternatve defntons of s w t and sample perods The share of labor ncome to consumpton forecasts returns at long horzons whereas the dvdend yeld, contrary to receved wsdom from the predctablty lterature, does much more poorly. Indeed, as already mentoned and strkngly documented n Fgure 2, the late 1990s saw hgh returns and low dvdend yelds and ths has serously dmnshed the ablty of the dvdend yeld to forecast long horzon returns. The queston then remans whether the predctablty of the share of labor ncome s somehow related to ts performance durng ths pecular perod. To address ths queston Table 2 Panel B reports the results of regressons (29) (31) but now run n a shorter sample that excludes the extraordnary market of the second half of the 1990s (1948-1994). The log dvdend yeld s now a strongly sgnfcant predctor of stock returns, ndependently of whether one draws the nference consderng the Newey-West or the Hodrck t statstcs. The lack of predctablty of the dvdend yeld n the complete sample s an artfact of the stock market performance durng the late 90s. Notce as well that the R 2 has ncreased consderably to the levels that are usual n the predctablty lterature. The sgnfcance of s w t as a strong predctor of long horzon returns remans n ths shorter sample and s robust to the ncluson of the dvdend yeld, though t s slghtly lower f the Hodrck t statstcs are consdered to draw nferences. As for the constructon of s w t, our defnton of labor ncome and consumpton s standard (see for example Lettau and Ludvgson (2001a)) but, clearly, gven that theory s slent on the specfcs of ths constructon, t s useful to know whether ts forecastng ablty s robust to the two alternatve measures of s w t ntroduced n secton 3.1. TABLE 3 ABOUT HERE Frst, Panel A of Table 3 uses s w t = wt ce /C t as our defnton of the share of labor ncome to consumpton, where recall wt ce s defned as Compensaton of Employees, rather than Labor Income. The results are very smlar to the prevous ones f not stronger. s w t s statstcally sgnfcant at the 5% level at all horzons, ndependent of the standard errors used to compute 20
the t statstcs. The R 2 s are of very smlar magntudes to those n Table 1. Once agan, the sgnfcance of s w t s there even when controllng for the log dvdend yeld. We also run regressons 1 and 3 usng s w t = w t /Y t, that s normalzng labor ncome by dsposable ncome. As shown n Panel B of Table 3, the sgnfcance of s w t s unaffected, except when one consders the one year horzon and computes the t statstc wth the Hodrck standard errors, where the sgnfcance s only at the 10% level. These results extend to the case where both s w t and log Dt M /Pt M are ncluded though now the sgnfcance of the log dvdend prce rato s weaker f the Hodrck standard errors are used. 10 3.2.3 Smulaton results Detals of the smulaton procedure: In ths secton we report results from smulatng the model descrbednsecton1. Specfcally, we generate 10,000 years of artfcal quarterly data for the consumpton growth process as well as for the shares of 200 dentcal assets. 11 We use these fcttous assets to construct both the market portfolo and a set of prce-sorted portfolos to test, n artfcal data, both the tme seres and cross-sectonal mplcatons of our model and compare them to ther emprcal counterparts. Fgure 3 Panel A shows the hstogram of the generated process for the share of labor ncome to consumpton rato. Notce that the bulk of the mass concentrates n the hstorcal range and, as the thn left hand sde tal shows, s w t wanders as low as.5 only very rarely. FIGURE 3 ABOUT HERE 10 As dscussed n Santos and Verones (2001), we performed addtonal robustness checks. Frst, we used Monte Carlo smulatons to compute standard errors that are robust to spurous regresson. We found that s w t s sgnfcantly negatve at the 5% level n all samples and for all constructons. Second, we also tested drectly whether the covarance between consumpton growth and returns s negatvely related to s w t.weused the Vech-Garch model of Bollerslev, Engle and Woodrdge (1988) to compute the tme seres of ths condtonal covarance, and then regress t on s w t. We found a negatve slope coeffcent, as predcted by our model. The coeffcent though s not sgnfcant f the Consumpton Deflator s used to obtan real consumpton, whereas t s sgnfcant f the CPI s used. Usng dfferent data and monthly frequency, Duffee (2004) fnds nstead an nsgnfcant postve relaton. We conclude that drect evdence on the condtonal covarance between return and consumpton growth s weak, a result that s to be expected gven the nose n the real consumpton growth data seres (especally at the monthly frequency). 11 The fact that all assets are ex ante dentcal mples that only the propertes of the wages/consumpton rato s w t matter for the market portfolo, and not the number or the characterstcs of ndvdual securtes. To see ths, notce that for any two assets, j 2, wehaveb = b j j and bk = b k j for k 6=, j. Thus, t s easy to see that we can wrte Pt M = P ³ n =2 P t = C ebm t + ³b 1 M e b M s w t for two constants e b M and b 1 M. 21
Table 4 Panel A contans the parameters for the consumpton growth and share processes, (5) and (17) respectvely. It also reports our choces for the preference parameters, γ and φ. We set γ =60for, as t s well known, a hgh level of rsk averson s needed to match the equty premum n consumpton based models even n those that depart from the tradtonal CRRA preferences. For nstance, the habt formaton model of Campbell and Cochrane (1999), a model that s able to match many features of the data, mples a rsk averson level around 80. Arelatverskaversonequalto60mayappeartoohgh evenhgherthantheoneusedby Mehra and Prescott (1986). Recall that n the standard benchmark case where consumpton growth s..d. and consumpton equals dvdends the equty premum s gven by γσ 2 c.thus, f the volatlty of consumpton growth equals 1% (see Table 1 Panel A), even γ =60mples an equty premum of just 0.6%. TABLE 4 ABOUT HERE Snce the am of ths paper s not to address the equty premum puzzle, we choose to use the volatlty of consumpton growth measured over the longer sample 1929-2001, whch s approxmately 2.8%. Even ths hgher volatlty of consumpton growth s not suffcent to match the hstorcal equty premum n the benchmark case wth..d. consumpton growth and where dvdends equal consumpton, as n ths case the mpled equty premum s just 4.7%, stll short of the hstorcal average of 7.8% (see Table 1 Panel A). In contrast, as Panel B of Table 4 shows, the predctablty nduced by the labor ncome to consumpton rato rases the equty premum from 4.7% to a 7.53%, a notable ncrease. Smlarly, n the absence of any predctablty, the volatlty of returns would automatcally be gven by the volatlty of consumpton growth, 2.8%. Instead the varaton n s w t almost trples the volatlty of the returns on the market portfolo to 6.21%. Clearly, ths ncrease n the volatlty of returns s not enough to match the hstorcal volatlty of about 16% and ths produces a Sharpe rato that s hgher n the calbraton than ts emprcal counterpart. In addton notce that the level of the rsk free rate s a reasonable 3% and wth a very small volatlty. 12 Panel B also shows the mean and standard devaton of the share of labor ncome to consumpton. The average value of the share s dentcal to s w whereas the volatlty and the correlaton of ds w t 12 As s known, under power utlty the rsk-free rate s r f t = φ + γe t [dc t /C t ].5γ(1 + γ)σ c σ 0 c. The almost zero volatlty of rsk free rate notwthstandng a γ =60just reflectsthefactthatnthemodel,e t [dc/c] s essentally constant, as dscussed earler. 22