Habit, Production, and the Cross-Section of Stock Returns

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Habit, Production, and the Cross-Section of Stock Returns Andrew Y. Chen 2014-103 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Habit, Production, and the Cross-Section of Stock Returns AndrewY.Chen FederalReserveBoard andrew.y.chen@frb.gov ∗ November21,2014 Abstract Solutionstotheequitypremiumpuzzleshouldinformusaboutthecrosssectionofstockreturns.Anexternalhabitmodelwithheterogeneousfirms reproducesnumerousstylizedfactsaboutboththeequitypremiumandthe valuepremium.Theequitypremiumislarge,time-varying,andlinkedwith consumptionvolatility. Thecross-sectionofexpectedreturnsislog-linear inB/M,andtheslopematchesthedata.Theexplanationforthevaluepremium lies in the interaction between the cross-section of cash flows and the time-varying risk premium. Value firms are temporarily low productivityfirms,whichwilleventuallyexperiencehighcashflows. Thepresent value of these temporally distant cash flows is sensitive to risk premium movements. Thevaluepremiumistherewardforbearingthissensitivity. Empiricalevidenceverifiesthatvaluefirmshavehighercash-flowgrowth. Thedataalsoshowthatvaluestockreturnsaremoresensitivetoriskpremiummovements,asmeasuredbyconsumptionvolatilityshocks. ∗ IwouldliketothankRenéStulzandLuZhangforhelpfulcomments,aswellasJuliaThomas andAubhikKhanforthetrainingthatmadethisworkpossible. IwouldalsoliketothankLars KuehnandOliverBoguthformakingtheirdataavailable. Theviewsexpressedhereinarethose oftheauthoranddonotnecessarilyreflectthepositionoftheBoardofGovernorsoftheFederal ReserveortheFederalReserveSystem.

1. Introduction In the decades since the publication of Mehra and Prescott (1985)’s equity premium puzzle, economics and finance have produced a handful of models thatrationalizethehighaveragereturnsandvolatilityoftheaggregatestockmarket. Whetherthesesolutionsalsohelpusunderstandthecross-sectionofstock returnshasreceivedlessattention. Exploringthisquestionisimportantbecause what we ultimately want is not merely a solution to the equity premium puzzlebutaframeworkforunderstandingassetpricesingeneral. Amodelthatexplains the return on the aggregate stock market but cannot address any other assetleavesmuchtobedesired. In this paper, I show that one solution to the equity premium puzzle, habit formation, provides insight into the cross-section of stock returns. I embed a variantofCampbellandCochrane(1999)externalhabitpreferencesinamodel withheterogeneousfirms. Themodelreproducesnumerousstylizedfactsabout thevaluepremium. Bothinthemodelandinthedata,expectedreturnsarelinearinthelogofbook-to-market(B/M).Moreover,themodelmatchestheslope oftherelationship.Theslopeonlog(B/M)isapproximatelyfive,indicatingthata 20%higherB/Mimpliesa100b.p.increaseinexpectedreturnsoverthenextyear. These cross-sectional predictions come with predictions about the equity premiumandbusinesscyclethatareconsistentwiththedata. Themodelmatches the first two moments of aggregate excess returns, the risk-free rate, consumption,output,andinvestment,aswellasexcessreturnanddividendpredictability regressions. The model’s equity premium dynamics are critical to understanding the cross-section of stock returns. The equity premium is time-varying and persistent,resultinginacross-sectionalequitytermpremium: stockswithtemporally distant cash flows earn higher returns. To understand this, note that when the equitypremiumrises,allstockpricesfall,sincecashflowsarediscountedmore aggressively. Stocks with temporally distant cash flows are hit harder, however. This is because the rise in discount rates is persistent. Thus, the effects of the rise are compounded for distant cash flows. Since the equity premium rises in bad times, investors demand high returns in exchange for these negative price reactions. This link between time-varying equity premia and an equity term premium 1

isfoundinanumberofmodels(SantosandVeronesi(2010),Chen(2012)). This modeldiffersinthatvaluefirms(highB/M)endogenouslyhavetemporallydistant cash flows, and thus higher expected returns. Cash flows are the result of investmentandproductionbut,forthebasicintuition,onecansimplyexamine productivity.Valuefirmshavelowproductivitytodayand,asaresult,producelittlecashflow. Productivityeventuallyrecovers,however,aswillthecashflowsof valuefirms. Thus,valuefirmshavetemporallydistantcashflows. Growthfirms (low B/M) follow the opposite pattern: high productivity and cash flow today, normal productivity and cash flow later, and thus temporally close cash flows. Theexistenceofanequitytermpremiumthenleadstoavaluepremium. This explanation of the value premium is consistent with three empirical facts about value firms: compared to growth firms, value firms (1) have lower returnonequity(FamaandFrench(1995)),(2)highercash-flowgrowth(Lakonishok,Shleifer,andVishny(1994),Chen(2012)),and(3)morenegativediscount ratebetas(CampbellandVuolteenaho(2004),Campbell,Giglio,Polk,andTurley (2012)). Thefirstempiricalfactiswellestablished. Thesecondandthirdarenot, andsoIpresentadditionalevidenceshowingtheirrobustness. The second empirical fact, that value firms have higher cash-flow growth, is controversial. This fact conflicts with the basic intuition that “growth” firms should grow, and thus value firms should not. In the value premium literature, however, the terms “value” and “growth” refer to high and low B/M. Low B/M stocksmaynotgrowmuchatall,andindeedtheempiricalevidenceforlowB/M stocks having high cash-flow growth comes entirely from two studies of equity duration: Dechow, Sloan, and Soliman (2004) and Da (2009). The trouble with duration studies is that for equities, duration is very difficult to measure. Measuringdurationrequiresestimatingbothadiscountrateandterminalvalue,both ofwhichareverydifficulttoidentifyatthefirm-level. Measuring cash-flow growth, on the other hand, is straightforward. As recently emphasized by Chen (2012), by a number of measures, value firms have highercash-flowgrowth,andthustemporallydistantcashflows.1 Ipresentadditionalempiricalevidencethatvaluefirmshavehighcash-flowgrowth. Existing papersfocusondividends,butthereareother,arguablymoreappropriate,measuresofcashflows. Ialsoconsiderearningsbeforeextraordinaryincome,earningsafterextraordinaryincome,earningsplusdepreciation,andcashflowsfrom 1See also Lakonishok, Shleifer, and Vishny (1994), Bansal, Dittmar, and Lundblad (2005), Hansen,Heaton,andLi(2008),Chen,Petkova,andZhang(2008). 2

operatingandinvestingactivities. Iexaminebuy-and-holdportfolios,whicharguablyareabettermatchforthemodel’smechanics.Ifindthat,regardlessofthe definitionofcashflow,valuefirmshavehighercash-flowgrowth. Theremainingempiricalfactisthatvaluefirmshavemorenegativediscount rate betas. Campbell and Vuolteenaho (2004) and Campbell, Giglio, Polk, and Turley (2012) find evidence for this view, but they use vector autoregression frameworks which may be sensitive to specification (Chen and Zhao (2009)). Since discount rates are not directly observable, one cannot entirely avoid this critique, but one can provide supporting evidence using an entirely different framework. To this end, I examine consumption volatility betas of value and growthfirms. Unlikestandardexternalhabitmodels,thediscountrateshocksof thismodelarecloselylinkedwithtime-varyingconsumptionvolatility.2 Tomeasureconsumptionvolatility,IuseBoguthandKuehn(2013)’sMarkovchainofthe componentsofconsumptiongrowth. Ifindthatvaluefirmshavemorenegative betaswithrespecttomovementsinconsumptionvolatility. Thisrelationshipis seen both in the consumption volatility betas of book-to-market portfolios, as wellasinfirm-levelregressions. Thepaperproceedsasfollows. IntheremainderofthissectionIdiscussrelatedliterature. Becausetherelationshipbetweenvalueandcash-flowgrowthis controversial, Ithenpresentempirical evidence onvalue andcash-flow growth in Section 2. Section 3 presents the model, solution method, and calibration. Section 4 shows that the model addresses numerous equity premium facts as well as facts about the value premium. Section 5 inspects the mechanism for generatingthevaluepremiumandprovidesempiricalevidenceaboutvalueand discountrateshocks. Section6concludes. Related Literature The model builds off of Santos and Veronesi (2010), who alsostudythecross-sectionofstockreturnsinanexternalhabitmodel. Thispaper can be considered an extension of their model into a production economy. Adding production reverses the value – expected return relationship. Without production,valuestocksarecharacterizedbyahighdividendpriceratio. Mean reversionimplieslowdividendgrowthandlowexposuretodiscountrateshocks. In contrast, a model with production characterizes value with book-to-market. Value firms are then low productivity firms, and mean reversion implies high 2ThisresultisduetothepresenceofproductionandisanalyzedinChen(2014). 3

cash-flowgrowthandhighexposuretodiscountrateshocks. InadditiontoSantosandVeronesi(2010),thispaperiscloselyrelatedtoChen (2012).Chen(2012)showsthatanymodelwithtime-varyingriskpremiawillalso have a cash-flow growth premium, and that this cash flow premium is empirically related to the value premium. While Chen (2012)’s theoretical results are reducedformandqualitative,themodelofthispaperisstructuralandquantitative.Thus,thispapercanbeconsideredaformalgeneralequilibriumfoundation fortheearlierresults. AnumberofpapersalsogenerateavaluepremiuminaQ-theoreticalframework(Zhang(2005),Carlson,Fisher,andGiammarino(2005),Cooper(2006)). In thesepapers,thecashflowsofvaluefirmsaremorecyclicalasaresultofoperatingleverageorcostlyreversibility. Thesetechnologicalfeaturesareabsentfrom the model of this paper. Indeed, in this model, the relationship between cash flowsandaggregatestatevariablesdoesdependsignificantlyonB/M.Thus,the mechanismunderlyingthevaluepremiuminthispaperisdistinctfromexisting Q-theorymodels.Empirically,thecashflowriskchanneldoesfindsomesupport (Cohen, Polk, and Vuolteenaho (2009), Santos and Veronesi (2010)). This channelcanbeaddedtothemodelwiththeinclusionofoperatingleverageorcostly reversibilitybutisbeyondthescopeofthispaper. A handful of papers model the equity premium and the cross-section in a long-runrisksetting. Thesepapersfindthatthelong-runriskframeworkisconsistentwithseveralfactsaboutthecross-section. Avramov,Cederburg,andHore (2011)findsize,value,andmomentumeffectsinanendowmenteconomy. Favilukis and Lin (2011) and Ai and Kiku (2012) find value effects in production economies. While these papers successfully generate a large equity premium andvolatileexcessreturns,theyusetheversionofthelong-runriskmodelwhich doesnotproducetime-varyingriskpremia. Thispaperovercomesthisissueby using the external habit framework, which does produce time-varying risk premia. Gabaix(2012)time-varyingdisastermodelproducestime-varyingriskpremia and is qualitatively consistent with the value premium, but Gabaix (2012) doesnotprovideaquantitativeanalysis. 4

2. Empirical Evidence: Value Firms Have High Cash- Flow Growth Themodel’smechanismreliesonvaluefirmshavinghighercash-flowgrowth than growth firms. In this section, I discuss the existing empirical evidence regardingthemechanism,andpresentnewevidenceinsupportofthemechanism. The conventional view is that value firms have low cash-flow growth. Evidenceforthisviewcomesexclusivelyfromequitydurationstudies,namelyDechow,Sloan,andSoliman(2004)andDa(2009).Bothpapersfindthatvaluefirms haveshortdurations,andthus,lowcash-flowgrowth. Equityduration,however, isverydifficulttomeasure. Generallyspeaking,thedurationofanassetissomethinglike ∞ Duration= (cid:88) PV(CF t ) t. P t=0 0 Notethatmeasuringdurationrequiresbothadiscountrateandaterminalvalue. Whilethesetwoarebothdirectlyobservableforbonds,theyareextremelydifficulttomeasureforequities. Dechow, Sloan, and Soliman (2004) and Da (2009) rely on identifying assumptionswhichbiasthemtowardfindingthatvaluefirmshavelowdurations. Inparticular,Dechow,Sloan,andSoliman(2004)assumethattheterminalvalue isequaltothemarketvalue,lesssomepresentvalueofthenext10yearsofcash flow. Since value firms have low market value compared to current cash flows, this leads to a low terminal value and, thus, a low duration. Da (2009) assumes thattheterminalROEisequaltothemeanofROEforthefirstsevenyearsafter portfolioformation.SincevaluefirmshavelowROEatportfolioformation(Fama andFrench(1995)),thisassumptionleadstoalowterminalvalueand,thus,low duration. ThesebiasesarepointedoutbyChen(2012). Measuringcash-flowgrowthismuchmorestraightforward. Indeed, anumberofpapersfocusedonotherissueshappentoprovidesummarystatisticsregarding value and cash-flow growth (Lakonishok, Shleifer, and Vishny (1994), Bansal, Dittmar, and Lundblad (2005), Hansen, Heaton, and Li (2008)). These papersuniformlyfindthatvalueportfolioshavehighercash-flowgrowth. Unfortunately, even cash-flow growth offers multiple methods of measurement. Forexample, thepreviouslymentionedpapersuseportfoliosthatarere- 5

balanced periodically using various methods. Chen (2012) provides a detailed examination of cash-flow growth and value in buy-and-hold portfolios. For the majorityofhismethodsofanalysis,hefindsthatvaluefirmshavehighcash-flow growth. This section presents additional empirical evidence regarding value and cash-flowgrowth. Inparticular,Chen(2012)focusesondividendsandearnings before extraordinary income. Other definitions of cash flow are arguably more appropriateempiricaltargetsformodels. Existingmodels, includingtheonein this paper, abstract from dividend policy (Miller and Modigliani (1961)). As a result, model dividends are equal to net income plus depreciation less net investment (cash flow from operating and investing activities). Both sides of the equation can be considered as cash flow, and the right-hand side is arguably a moreappropriateempiricaltargetsinceitcapturesthereal(nonfinancial)activitiesofthefirm. I look at four different notions of cash flow. The first is a common notion of cash flow, earnings before extraordinary income (ib). This measure is stable and reflects the ongoing activities of the firm, but investors must face the consequencesofextraordinaryincome,andthusthestockpriceshouldreflectthese items. Thus, I also look at earnings (ni), which includes extraordinary income. Earnings, however, reflect depreciation charges, which are not represented in cash flows in the model. I thus also examine is earnings plus depreciation (ni +dp). Lastly,cashflowsinthemodelcanalsocomefromselling/buyingcapital. ThusIalsoexamineearningsplusdepreciationlessnetinvestment(ni+dpcapx+sppe). Thislastmeasureisclosestinspirittothecashflowsofthemodel. I use tercile book-to-market sorted portfolios and CRSP and COMPUSTAT data from 1971–2011. The portfolios are buy-and-hold portfolios. Cash flows from delisted stocks are reinvested in the remaining stocks, following Chen (2012)’sprocedure. Thechoiceoftercilesisduetotheuseofnetinvestment. Net investment is quite volatile, and the use of large portfolios averages out much of this volatility and paints a clearer picture of the typical cash flow dynamics. Therelativelyshortpost-1971sampleisduetothelimitedavailabilityofsalesof plant, property, and equipment (sppe) data. The relatively modern sample periodisuseful,however,inthatoneoftheonlymeasurementswhereChen(2012) finds that value firms have low cash-flow growth is when he looks at dividend growth for buy-and-hold portfolios in the post-1963 sample. I will show that, duringasimilarsampleperiod,manyotherdefinitionsofcashflowpresentthe 6

oppositepictureforbuy-and-holdportfolios. Table 1 shows cash flow levels. It shows the cash flow from a $1 investment invalueorgrowthportfolios,averagedacrossportfolioformationyears.Thefirst thingthatjumpsoutfromthetableisthatvaluestocksdonothavesignificantly lower cash flows than growth stocks. In the first year after portfolio formation, valuestockspay7centsperdollarinvestedwhilegrowthstockspay6cents,with respecttoearningsbeforeextraordinaryincome. Infact,netofextraordinaryincome and investment, value firms pay much less. Using this definition, in the firstyearvaluepayshalfacentperdollarwhilegrowthpaysanorderofmagnitude more. The second pattern which emerges from the table is that value has highercash-flowgrowth. Thereislittleactioninthecashflowsofgrowthfirms, butthevaluecashflowsexhibitapparentgrowth. [Table1abouthere] Table2showsgrowthratesofthecashflowsfromthepreviousfigure. Italso considers two additional definitions of cash flow: earnings (after extraordinary income) and earnings plus depreciation. By all definitions of cash flow, value portfolioshavemuchhighercash-flowgrowththangrowthportfoliosinyeartwo. Indeed, using the definition closest in spirit to the model (earnings plus depreciationlessnetinvestment),valueexperiencesahuge544%growthincashflow between years one and two, while growth gets a meager 5% growth. Cash-flow growthisalsomonotonicallyincreasinginB/Musingalldefinitions. Cash-flow growth of value exceeds that of neutral which exceeds that of growth. An additionalpatternwhichisseeninTable2isthatthegrowthratesmeanrevert. Value beginswithstrikinglyhighcash-flowgrowthinyeartwo,butgrowthslowsdown quickly. Growthportfoliosfollowtheoppositepattern. Boththehighcash-flow growthofvalueportfoliosanditssubsequentmeanreversionwillbeseeninthe model. [Table2abouthere] Analyzingcash-flowgrowthforthelongertermfacesdatalimitations. Forty yearsofdataprovidesonly10nonoverlappingfour-yearperiods.Thus,itisprobablybesttofocusontheyeartwoandyearthreegrowthrates. Nevertheless,the factthatthecashflowpatternsarecommonacrossmultipledefinitionsofcash flowisreassuring. 7

3. AGeneralEquilibriumModelwithHeterogeneous Firms Havingestablishedthatthekeyelementofthemechanismisconsistentwith the data, I now present the model. The model is an rel business cycle model with external habit formation, capital adjustment costs, and idiosyncratic firm productivity. Itisdesignedtohavetheminimalfeaturesforbothanequitypremiumandanendogenouscross-sectionoffirms. Itisessentiallyaheterogenous firmextensionofChen(2014),andthusdiffersfromtraditionalhabitmodelsby featuringatime-varyingconsumptionvolatilitychannel. Marketsarecomplete,andtimeisdiscreteandinfinite. Fortheremainderof thepaper,lowercasedenoteslogs–thatis,c ≡logC . t t 3.1. RepresentativeHousehold A unit measure of identical households j ∈ [0,1] chooses asset holdings to maximizelifetimeutility (cid:40) ∞ (C −H )1−γ(cid:41) (cid:69) (cid:88) βt j,t t , (1) 0 1−γ t=0 where H , the level of habit, is determined by aggregate consumption and is t takenasexternalbythehousehold. I specify the evolution of habit using surplus consumption, rather than the levelofhabititself. Thatis,let C −H S ≡ t t , (2) t C t bethesurplusconsumptionratio,whereC isaggregateconsumption.Thensurt plusconsumptionfollowsanAR1-processinlogs s t+1 ≡(1−ρ s )s¯+ρ s s t +λ(c t+1 −c t ). (3) This approach leads to a simple stochastic discount factor and makes for ease of comparison with the existing literature on external habit (Campbell and Cochrane(1999),andWachter(2006),amongothers). 8

The habit process differs in that the conditional volatility λ is a constant. Thisisasubstantialdeviationfromtheliteratureonexternalhabitmodelsinendowmenteconomies,whichspecifythisconditionalvolatilityasastime-varying and countercyclical (e.g. Campbell and Cochrane (1999), Menzly, Santos, and Veronesi(2004)). InChen(2014),Ishowthattheintroductionofproductionresultsincountercyclicalconsumptionvolatility,whichisquantitativelyverysimilar to the assumed countercyclical volatility of surplus consumption typical of endowment economy models. For comparability with Campbell and Cochrane (1999),Ifixλattheirsteadystatevalue 1 λ= −1. (4) S¯ Thefactthatmarketsarecompleteandexternalhabitspecificationmeanthat thehouseholdsideofthemodelboilsdowntoasimplestochasticdiscountfactor (cid:181) (cid:182)−γ M t,t+1 =β C t+1 S t+1 . (5) C S t t 3.2. HeterogeneousFirms There is a unit measure of heterogeneous firms, indexed by i ∈ [0,1]. The firmsproduceconsumptionusingcapitalK i,t Π(K ,B ,A )=A B K α , (6) i,t i,t t t i,t i,t whereaggregateproductivity A andidiosyncraticproductivityB arebothAR1 t i,t processesinlogs: a t+1 =ρ a a t +σ a (cid:178) a,t+1 (7) b i,t+1 =ρ b b i,t +σ b (cid:178) b,i,t+1 , (8) where(cid:178) a,t+1 and(cid:178) b,i,t+1 areindependentstandardnormalrandomvariables. Allheterogeneityinthemodelsoriginatesfromtheidiosyncraticproductivity process (8). This approach is used for three reasons. The first is thatit is a very simplewayofintroducingacross-sectionoffirms. Thesecondisthatalargeliterature documents substantial heterogeneity in productivity (Syverson (2011)). Thethirdisthatthisapproachisthestandardwayofmodelingfirmheterogene- 9

ity in both macroeconomics and finance (Hennessy and Whited (2005), Zhang (2005),KhanandThomas(2008),Bloom(2009)). Wewillsee, however, thatthis approachhasdifficultiesmatchingthetremendousheterogeneityinassetvaluations that is seen in the data. Matching the heterogeneity in the data with additionalsourcesofheterogeneityisaninterestingquestionforfutureresearch— however,itisbeyondthescopeofthispaper. Capitalaccumulatesaccordingtotheusualcapitalaccumulationrule, K i,t+1 =I i,t +(1−δ)K i,t , (9) andfirmsfaceaconvexcapitaladjustmentcost φ(cid:181) I (cid:182)2 Φ(I ,K )= i,t −δ K . (10) i,t i,t i,t 2 K i,t I assume that the adjustment costs are a pure loss. They do not represent payments to labor. Adjustment costs are included because production economies produceacounterfactuallysmoothTobin’sQunlessoneincludesaninvestment friction. Quadraticcostsarechosenforsimplicity,butarichermodelwouldincorporate investment frictions by modeling an investment goods sector, as in Boldrin, Christiano, and Fisher (2001), or would feature heterogeneous plants withnon-convexcostsofadjustment,asinKhanandThomas(2008). Becauseofcompletemarkets,thefirm’sobjectiveisstandard: (cid:189)∞ (cid:190) max (cid:69) (cid:88) M [A B K α −I −Φ(I ,K )] . (11) 0 0,t t i,t i,t i,t i,t i,t {Ii,t,Ki,t+1} t=0 Itchoosesinvestmentandcapitaltomaximizefuturedividends,discountedwith thehousehold’sSDF. 3.3. RecursiveCompetitiveEquilibrium Equilibrium is defined recursively. Thus, in the remainder of this section I drop the time subscripts and represent the next period with primes. Let µ representthedistributionoffirmsovercapitalK andidiosyncraticproductivityB . i i Theaggregatestateisthetripleofthedistributionoffirmsµ,surplusconsumptionS,andaggregateproductivity A. Therecursivecompetitiveequilibriumislawsofmotionforthedistribution 10

offirmsΓ(µ,S,A)andaggregateconsumptionC(µ,S,A), acapitalpolicyforthe firmG(K ,B ;µ,S,A)andvaluefunctionforthefirmV(K ,B ;µ,S,A)suchthat i i i i 1. Firmoptimalityholds:G(K ,B ;µ,S,A)andV(K ,B ;µ,S,A)solve i i i i (cid:189) V(K ,B ;µ,S,A)=max Π(K ,A,B )−Φ(I,K ) i i i i i (cid:48) {I,K } i (cid:90) ∞ (cid:90) ∞ + dF((cid:178)(cid:48) ) dF((cid:178)(cid:48) )M(A (cid:48) ;µ,S,A) a b −∞ −∞ (cid:190) ×V(K (cid:48) ,B (cid:48) ;µ(cid:48) ,S (cid:48) ,A (cid:48) ) , (12) i i wheretheproductivityprocessesaregivenby(7)and(8),capitalaccumulation is given by (9), the SDF is the household’s intertemporal marginal rateofsubstitution (cid:181) C(µ(cid:48) ,S (cid:48) ,A (cid:48) )S (cid:48)(cid:182)−γ M(A (cid:48) ;µ,S,A)=β , (13) C(µ,S,A)S S (cid:48) evolvesaccordingto(3),µ(cid:48) isgivenbyΓ(µ,S,A),andF((cid:178)(cid:48) )isthestandard a normalcumulativedistributionfunction. 2. Firmdecisionsareconsistentwiththelawofmotionforconsumption: (cid:90) C(µ,S,A)= dµ(K ,B ) (cid:169)Π(K ,B ,Z)−Φ(I(K ,B ;µ,S,A),K ) (cid:170) (14) i i i i i i i whereI(K ,B ;µ,S,A)=G(K ,B ;µ,S,A)−(1−δ)K . i i i i i 3. Firmdecisionsareconsistentwiththelawofmotionforthedistributionof firms—thatis,let(cid:66)betheBorelalgebrafor(cid:82)2. Thenµ(cid:48)=Γ(µ,S,A)isgiven + by ∀(K ,B )∈(cid:66), 1 1 (cid:90) (cid:90) µ(cid:48) (K ,B )= dµ(K ,B ) dF((cid:178)(cid:48) ). (15) 1 1 i i b {(Ki,Bi)|G(Ki,Bi;µ,S,A)∈K1 } (cid:110) (cid:178)(cid:48) b | exp(ρ b b+σ b (cid:178)(cid:48) b )∈B1 (cid:111) 3.4. Krusell-SmithSolutionMethod I solve the model with a variant of the Krusell and Smith (1998) algorithm, similar to Khan and Thomas (2008). As in Khan and Thomas (2008), I approxi- 11

matethedistributionoffirmsµwiththeaggregatecapitalstockK. Thus,theapproximate aggregate state is a triple of aggregate capital, surplus consumption, andaggregateproductivity: (K,S,A). Idiscretizetheaggregateandidiosyncraticproductivityprocesses(7)and(8) usingtheRouwenhorst(1995)method. Ithenconjecturethatthelawsofmotion foraggregateconsumptionandcapitalfollowthefollowinglog-linearforms: c (cid:48)=logC˜(K,S,A )=θC +θC k+θC sˆ (16) j 0,j 1,j 2,j k (cid:48)=logΓ˜(K,S,A )=θΓ +θΓ k+θΓ sˆ, j 0,j 1,j 2,j where j ∈{A ,...,A }representstheaggregateproductivitystate. Notethatthe 1 NA use of aggregate productivity dependent coefficients allows for a non-linear relationshipbetweenconsumptionandtheaggregatestate. ThegoaloftheKrusell-SmithmethodistofindthecoefficientsθC ,θΓ such i,j i,j that 1. FirmsmaximizevaluegiventhelawsofmotionθC ,θΓ i,j i,j 2. Estimatesof(16)onsimulateddatausingpoliciesfromstep1producecoefficientsclosetoθC ,θΓ ,andR2’sclosetoone. i,j i,j ThemoststraightforwardapplicationofKrusell-Smithsearchesforthisapproximate equilibrium by doing a fixed-point iteration using the firm’s problem defined in (12) and a simulation of a distribution of firms. However, there is no theoremthatsuggeststhatthisfixed-pointiterationwillconverge, andindeedI findthatittypicallydoesnot. Toaidinfindingequilibrium,Iapplythe‘equilibrium-in-simulation’method (Krusell and Smith (1997))—that is, I first solve solve the approximate equilibrium version of (12). I then plug the resulting value function into the following problem: (cid:189) G(K ,B ;K,S,A;C)=argmax Π(K ,A,B )−Φ(I,K ) i i i i i (cid:48) {I,K } i (cid:90) ∞ (cid:90) ∞ + dF((cid:178)(cid:48) ) dF((cid:178)(cid:48) )M ∗ (A (cid:48) ;K,S,A;C) a b −∞ −∞ (cid:190) ×V(K (cid:48) ,B (cid:48) ;K (cid:48) ,S (cid:48) ,A (cid:48) ) (17) i i (cid:181) C˜(K (cid:48) ,S (cid:48) ,A (cid:48) )S (cid:48)(cid:182)−γ M ∗ (A (cid:48) ;K,S,A;C)=β . (18) CS 12

Thisprocedureintroducestoday’saggregateconsumptionasanadditionalstate variable and solves for a new investment policy which accounts for aggregate consumption. I then use this augmented investment policyG(K ,B ;K,S,A;C) i i inthesimulationstep. Thisallowsmetofinda‘market-clearing’C ateachdate inthesimulation.Thatis,ateachdate,IusearootfindertofindtheC thatsolves equation(14). Notethatoncetheequilibriumisfound,aggregateconsumption from the simulation of the firms and that produced by the law of motion are equal,andsoproblem(17)withmarketclearing(14)andproblem(12)produce identicalchoices. Thepresenceofexternalhabitsignificantlycomplicatesthecomputationally demanding Krusell-Smith algorithm. External habit preferences introduce an additional aggregate state variable, surplus consumption, which is completely absentfromthestandardRBCeconomy. Asaresult,usingtheRBCequilibrium asaninitialguessfortheKrusell-Smithalgorithmwillcausethealgorithmtofail. Toaddressthisproblem,Iapplyahomotopymethod. Isolveaseriesofmodels withthefollowingalteredSDF (cid:181) (cid:48)(cid:181) (cid:48)(cid:182)χ(cid:182)−γ C S M (cid:48)=β . (19) C S I begin by solving the model with χ=0. Here the RBC model serves as a good initialguess. Oncetheprogramisfairlyclosetoequilibrium,Iincreaseχby0.1 andusethepreviouslawsofmotionasaninitialguessforthenewmodel.Irepeat thisprocessuntilχ=1.0,whichisequivalenttothemodelpresentedin(3). Surplus consumption also adds the difficulty that it is an endogenous state variable that is not predetermined. As a result, the habit process equation (3) must be solved at every date in the simulation step of the algorithm. Note that thesimulationstepinvolvessimulatinganentiredistributionoffirms,andsoan entire distribution of decision rules must be accounted for in solving equation (3). Thisalsosignificantlyincreasesthecomputationalburdenofthealgorithm. 3.5. CalibrationtoPost-WarU.S.Data The model is calibrated to post-war (post 1947) U.S. data. This sample period is chosen because the World Wars introduce structural changes that may not be captured by the model. In particular, over the long sample (post-1929) HP-filteredoutputandinvestmentareessentiallyuncorrelated. 13

AggregatequantitiesaretakenfromtheBEA.Firm-leveldataaretakenfrom CRSP/COMPUSTAT. B/M-sorted portfolio are taken from Ken French’s website. Aggregateasset-pricemomentsaretakenfromBeelerandCampbell(2009). Table3showsthecalibration. Preferenceparametersarechosenasmuchas possibletofitunconditionalmomentsofassetprices. Sincetimepreferenceβis reflected in risk-free assets, I choose it to fit the mean 30-day T-bill return. The modelanddataT-billreturnsmatchnicelyatabout1%peryear.Thepersistence ofsurplusconsumptionρ affectsthepersistenceofassetprices. ThusIchoose s ρ to approximately match the annual persistence of the CRSP price/dividend s ratioof0.87. Thetworemainingpreferenceparameters,thesteadystatesurplus consumption S¯ and utility curvature γ, jointly control risk aversion. Thus, it is difficult to identify these parameters separately. For ease of comparison with theliteratureonexternalhabit,Ichooseγ=2tomatchCampbellandCochrane (1999)’s value, and then choose S¯ to fit the mean Sharpe ratio of the CRSP index. Themodeldoesagoodjobmatchingthedatahere: bothSharperatiosare roughly0.40. [Table3abouthere] I choose aggregate technology parameters to fit moments of the real economy. The production curvature α and depreciation rate δ are chosen to fit the capital-outputratioandthemeangrowth-adjustedinvestmentrate. Thesemomentsmatchnicelyat0.40and0.08respectively. Thevolatilityofaggregateproductivityσ ischosentofitthevolatilityofHP-filteredlogGDP(Iuseasmootha ingparameterof6.25,asarguedbyRavnandUhlig(2002)). Thedataandmodel match well in this dimension, producing a volatility of about 1.5%. The persistence of aggregate productivity is chosen to fit the persistence of the Solow residual with constant labor. A critical parameter of the production technology is the quadratic adjustment cost parameter φ. I choose this parameter value to hit the volatility of aggregate consumption growth (non durables and services). The data and model match well here, producing a volatility of about 1.5% per year. Firm-leveltechnologyparametersarechosentofitthecross-sectionalmeans of time-series moments. I target non-micro-cap firms (firms with market equityofmorethan600million). Thepersistenceofidiosyncraticproductivityρ b is chosen to fit the persistence of firm-level ROE. The volatility of idiosyncratic productivityσ ischosentomatchthevolatilityofindividualstockreturns. b 14

4. Quantitative Results I begin by showing that the model addresses equity premium puzzles (Section4.1).ThissectionmostlyverifiestheresultsofChen(2014),sothediscussion willbebrief. Section4.2containsthemainquantitativeresults. Itshowsthatthe modelreproducesthevaluepremiumdescribedbyfirm-levelregressions. 4.1. MatchingtheDataonEquityPremiumPuzzles Table 4 shows aggregate asset-price moments. The model produces a large and volatile equity premium, and a low and smooth risk-free rate. The large and volatile equity premium is typical of habit models (Jermann (1998), Campbell and Cochrane (1999)). The smooth risk-free rate comes from time-varying consumptionvolatilitythatcounterbalancesthepowerfulintertemporalsubstitution effect of habit models. Time-varying volatility, in turn, comes from the “precautionary volatility” channel of Chen (2014). In bad times, the household isunsureofhowmuchprecautionarysavingsitneedsnextperiod. Thismotive andproductionleadtocountercyclicalconsumptionvolatility. [Table4abouthere] Table 5 shows regressions of future dividend growth and excess returns on theprice-dividendratio. Becausetheprice-dividendratiomustpredictdividend growth or returns, these regressions form a nice characterization of the drivers of asset-price fluctuations (Campbell and Shiller (1988), Cochrane (2011) and Koijen and Van Nieuwerburgh (2010)). The table shows that, as in the data, the price dividend ratio has little predictive power for future dividend growth, and has strong predictive power for future excess returns. These results show that themodelcapturesthenatureofstockmarketfluctuations. Thesetime-varying excessreturnsalsocomefromendogenoustime-varyingconsumptionvolatility (seeChen(2014)). [Table5abouthere] Table6showsbasicbusinesscyclemoments. Asintendedbythecalibration, the model produces low consumption volatility. As in the data, investment is much more volatile than output and consumption is much less volatile. The modelalsoreplicatestheco-movementofconsumption,investment,andGDP. 15

[Table6abouthere] Taken together, Tables 4, 5, and 6, show that the model does a good job of addressingtheequitypremiumpuzzles.Themodelgeneratesalargeandvolatile equitypremium,lowandsmoothrisk-freerate,andassetvaluationsthataretied toexpectedexcessreturns.Thesedata-likeasset-pricedynamicscomewithgood predictionsaboutbasicbusinesscyclevariables. 4.2. MatchingtheDataontheCross-SectionofStockReturns Wehaveseenthatthemodelisabletoaddresstheequitypremiumpuzzles. External habit combined with production produces a large and volatile equity premium, a low and smooth risk-free rate, and asset price fluctuations that are linkedtoexcessreturnsratherthantheprice-dividendratio.Thisbringsustothe mainquestionofthepaper. Isexternalhabitconsistentwiththecross-sectionof stockreturns? Table 7 examines this question and shows the main result of the paper. It showsregressionsofnextyear’sreturnsontoday’slogB/M.Theregressionsare firm-levelandfollowtheFamaandMacBeth(1973)method. Inboththemodel and data, the log(B/M) coefficient is positive and highly statistically signifii,t cant. Moreover, the slopes are large and similar in magnitude, with a value of aboutfive. Thisslopemeansthatinboththemodelanddata,a20%higherB/M impliesaroughly100b.p. higherexpectedreturn. [Table7abouthere] Though firm-level regressions provide the most statistical power and offer the simplest quantitative description of the value premium, the literature often examines value-weighted portfolio sorts. Table 8 shows summary statistics on decileB/M-sortedportfolios. Theexpectedreturnscolumnsshowthat, inboth the model and data, expected returns are monotonically increasing in B/M. In themodel,expectedreturnsculminatetoaneconomicallysignificantdecile10- 1returnofabout2.5%peryear. [Table8abouthere] Thedecilevaluepremiumissmallerthanthatofthedata,butitcomeswitha muchsmallerdispersioninB/M.Recallthatallheterogeneityinthemodeloriginates froma single AR1 idiosyncraticproductivity process. For parsimony and 16

tomaintainclarityofthemechanism,themodelabstractsfromothersourcesof heterogeneity such as differences in financial frictions or life-cycle effects. As a result,themodelgeneratesadispersioninB/Mthatissignificantlylessthanthe data.Inthemodel,logB/Mdiffersby0.4betweenthehighandlowdeciles.Inthe data,thisdifferenceis1.4. Thisdifferenceinspreadsmeansthatthetraditional high-lowportfolioreturnsofthemodelarenotcomparabletothedata. ReproducingtheenormousB/Mdispersioninthedataisaninterestingquestion,but isbeyondthescopeofthispaper. Amoreeffectivewaytoillustratetheportfoliosortresultsistointerpretthem as a nonparametric regression (Cochrane (2011)). Figure 1 provides this interpretation. Itplotstheaveragereturnsofthe10B/M-sortedportfoliosagainstlog B/M. Returns are equal-weighted in this figure because the functional form in thedatareflectsequal-weighting,thatis,value-weightedreturnsdonotresultin a log linear pattern in the data. The figure shows that the model captures this log-linearform.Thematchisnotonlyqualitativebutquantitativetoo.Theslope ofthetherelationshipbetweenexpectedreturnsandlogB/Missimilarinboth model and data, consistent with the results of the firm-level regressions (Table 7). [Figure1abouthere] 5. Inspecting the Mechanism In this section I explain how this value premium works. I begin by showing that value firms have high cash-flow growth (Section 5.1), consistent with the empirical results of Section 2. I go on to illustrate how high cash-flow growth leadstohighexpectedreturnsthroughexposuretodiscountrateshocks(Section 5.2). Ithenprovideempiricalevidenceforthishighexposure(Section5.3). Thelasttwosubsectionsproviderobustness. Section5.4rulesoutotherpotentialmechanisms. Section5.5illustratestheroleofgeneralequilibrium. 5.1. ValueandCash-FlowGrowth To gain a picture of the mechanism, it helps to start with the meaning of value in the model. Figure 2 plots B/M and expected returns against the two firmstatevariables,idiosyncraticproductivityandcapital. Theleftpanelshows 17

that value firms are low productivity firms with high capital. Capital, however, is slow-moving. As a result, value is primarily characterized by low productivity. Therightpanelshowshowvalueisconnectedtoexpectedreturns. Expected returnsdeclinestronglyinidiosyncraticproductivity. Overall, weseethatvalue firmsarelowproductivityfirmsthathavehighexpectedreturns,consistentwith thequantitativeresultsofSection4. [Figure2abouthere] Thelowproductivityofvaluefirmsleadstocash-flowgrowth. ThisrelationshipisillustratedinFigure3whichplotsthecash-flowgrowthofportfoliossorted onB/M.Thedarkestlinesshowvalueportfolios. Soonafterportfolioformation, valueportfolioshavehighdividendgrowth,butthisgrowthslowsdownquickly andeventuallyreachestheaveragegrowthrateofzero(themodelabstractsfrom balancedgrowth).Intuitively,valuefirmshavelowproductivity,butthislowproductivity is temporary. Mean reversion implies that productivity grows, and so cashflowsgrow. Thiscontrastswithgrowthportfolios(thelightestlines),which showthereversepattern. Whengrowthfirmsaredeclaredasgrowth, theyhave very high productivity. Mean reversion then means that their productivity will fall,leadingtolowcash-flowgrowth. Boththisinitialspreadincash-flowgrowth anditssubsequentmeanreversionareconsistentwiththeempiricalevidenceof Section2. Intermsofmagnitudes,thedispersionincash-flowgrowthissimilar to the cash flow measure that includes investment in Table 2, though the measurementofcash-flowgrowthinthedataisquitenoisy. [Figure3abouthere] While mean reversion provides a simple story of the relationship between value and cash-flow growth, the exact relationship is endogenous. Investment also affects cash flows and is not discussed in the simple story above. Indeed, lowproductivitytendstoencouragedisinvestment,increasingcashflowstoday and decreasing cash-flow growth. Whether the productivity channel or the investmentchanneldominatesdependsontheparameterchoices. Infact,themodelwithouthabitproducesthecounterfactualpredictionthat valuefirmshavelowcash-flowgrowth. ThisisseeninFigure4,whichplotscash flowsforB/Mportfoliosinamodelwithouthabit. Aswiththehabitmodel, the adjustmentcostsinthismodelarechosentomatchthevolatilityofconsumption 18

growth. The figure shows exactly the opposite pattern of figure 3, with growth firms (dotted line) beginning with high cash-flow growth and eventually mean reverting to the steady state of no growth. Here, the investment channel dominatestheproductionchannel. Intuitively,removinghabitincreasestheelasticity ofintertemporalsubstitutionofthehousehold,andrequiresthattheadjustment costs be lowered in order to fit the volatility of consumption. With low costs of adjustingcapital,thatis,lowcostsofinvesting,theinvestmentchannelbecomes dominant. [Figure4abouthere] 5.2. Cash-FlowGrowthandExpectedReturns Sofar,Ihaveshownthatvalueischaracterizedbylowproductivityandhigh cash-flowgrowth. Tofinishthestory,Ineedtoshowthathighcash-flowgrowth leads to high expected returns. This last link is due to the large discount rate shocks that drive the external habit model. High cash-flow growth means that cashflowsaredistributedfarintothefuture,andthusaremoreexposedtolarge discountrateshocks. Investorsthendemandhighreturnsinexchangeforbearingthishigherexposure. Anumberofpreviouspapersdiscussthislink(Cornell (1999),LettauandWachter(2007),SantosandVeronesi(2010),Chen(2012)),but hereIprovideanewsketchoftheintuition. ThissketchusessimpleInvestments 101formulasandsostepsoutsideofthegeneralequilibriummodel. Consideragrowingperpetuity D P = 1 , (20) 0 κ −g 0 whereκ isthediscountrateandg isthegrowthrateofcashflows. Nowsuppose 0 that the discount rate gets hit by a completely unexpected shock ∆κ. The price nextperiodisthen D (1+g) P = 1 . (21) 1 κ +∆κ−g 0 Takingafirst-orderTaylorapproximationofthedefinitionofthereturngivesus D +P (cid:181) 1+g (cid:182) R ≡ 1 1 ≈(1+κ )− ∆κ. (22) 1 P 0 κ −g 0 0 19

If discount rates suddenly go up, the stock price takes a hit, and so we have a negativesignonthesecondterm. Noticealsothatthesecondtermisincreasing in the cash-flow growth rate g. Discount rate shocks hit high cash-flow growth assets particularly hard. Intuitively, high cash-flow growth means that most of thecashflowswilloccurinthedistantfuture,andthesedistantcashflowsarehit multipletimesbyapersistentshocktodiscountrates. Informally,thelawofonepriceimpliesthat3 (cid:181) 1+g (cid:182) σ (M ) (cid:69) [R −R ]≈ Cov (−∆κ,−M ) 0 1 (23) 0 1 f κ −g t 1 (cid:69) (M ) 0 0 1 Provided that the discount rate shock is positively correlated with the SDF, this higher exposure to discount rate shocks commands a risk premium. In the model, this correlation is indeed positive. As in most habit models, a negative shock both increases discount rates and increases the marginal utility of consumption (Campbell and Cochrane (1999)). This paper differs though, in that this simultaneous increase is endogenous and the result of precautionary savingsdynamics(Chen(2014)). This theoreticallink betweencash-flowgrowthandexpectedreturnsisvery similartothatwhichcomesfromthestandarddurationformula %∆P ≈−[Duration]∆κ. (24) 1 Highcash-flowgrowthfirmshavelongdurationsandthusaremoresensitiveto discountrateshocks. However,Ichoosetofocusoncash-flowgrowthbecauseit iseasiertodefineandmeasure. 5.3. EmpiricalEvidence: ValueandDiscountRateShocks We’ve already seen some evidence in support of the mechanism in Section 2. Consistentwiththemechanism,valuefirmshavehighcash-flowgrowthbya number of definitions of cash flow. But to complete the empirical verification, we should also see that value firms are more exposed to discount rate shocks. Themodel’sdiscountrateshocksareduetotime-varyingconsumptionvolatility (Chen (2014)). This section shows that both in the model and the data, value 3Formally,thereisnouncertaintyatdate0andsotheexpectedreturnshouldbe1/E [M ].A 0 1 moreformalillustrationwouldinvolveshockstothevolatilityoftheSDFbutprecludestheuse ofsimpleformulas. 20

returnsarebadwhenconsumptionvolatilityrises. To show this I need an empirical measure of consumption volatility. I use Boguth and Kuehn (2013)’s measure, which comes from an estimation of a Markovchainmodelforthefirstandsecondmomentsofconsumptiongrowth. AnadvantageofBoguthandKuehn’smeasureisthattheytakeadvantageofthe informationinthecomponentsofconsumption,whichhelpsalleviateproblems regarding identifying persistent volatility in the short post-war quarterly consumption data. Since the model is annual and lacks the consumption componentsusedinBoguthandKuehn(2013),Icomputeconsumptionvolatilityfrom themodel’slawsofmotionforthesimulatedresults. Figure 5 shows consumption volatility betas for 10 book-to-market sorted portfolios. Consumptionvolatilitybetasareconstructedbyregressingexcessreturns on changes in consumption volatility. The left panel shows that, with a couple exceptions, consumption volatility betas decline monotonically in B/M. Therightpanelshowsbetasfromthemodel.Here,consumptionvolatilityispreciselymeasuredusingthelawsofmotionofthemodelandthebetasareaveraged over numerous simulations. As a result, we get a cleanly declining relationship betweenconsumptionvolatilitybetasandB/M. [Figure5abouthere] Thezig-zagginginthedatapanelisnotsurprisingconsideringtheshortsample of quarterly post-war consumption and the high volatility of portfolio returns. Firm level results provide more statistical power, and are shown in Table 9.ThetableshowsFama-Macbethregressionsofconsumptionvolatilitybetason logB/M.Betasareconstructedbyregressingreturnsonchangesinconsumption volatility in rolling windows. The table shows that in both model and data, the relationship is negative, statistically significant, and similar in magnitude. The tableusesforward-lookingbetas,thatis,thewindowsfordatet runfromdatet to 40 quarters after date t. I use forward-looking betas because theory predicts that it’s the future return covariance that matters. Using the more traditional backward looking windows does not materially affect the data columns, but it does affect the model columns. This result is likely because firms in the model arecharacterizedbystationarystatevariablesandcannotdisplaypermanentdifferences as in the data. The window is long because the model only contains 3 aggregate technology states in order to maintain tractability. Shorter windows 21

showastrongerrelationshipbetweenconsumptionvolatilitybetasandbook-tomarketinthedata. [Table9abouthere] Aweaknessoftheresultsisthattheoverallleveloftheconsumptionvolatility betasdifferssignificantlybetweenthemodelanddata. Thisdeviationisdueto the precise measurement of consumption volatility in the model, as well as the singleshocknatureofthemodel. ThesetwofeaturesmeanthatTFP,consumptionvolatility,andstockpricesofportfoliosmoveinlock-step,leadingtohighly negativeconsumptionvolatilitybetas. Softeningthelevelofthebetaswouldinvolveintroducingadditionalsourcesofaggregateriskandisaninterestingpath forfutureresearch. 5.4. OtherPotentialMechanisms The model does not have fixed operating costs, irreversible investment, asymmetricadjustmentcostsoncapital,orfixedcostsofinvestment. Thus,the mechanism is distinct from the operating leverage channel of Carlson, Fisher, andGiammarino(2005),theinflexibilitychannelofZhang(2005)(seealsoGala (2010)),aswellastherealoptionchannelofAiandKiku(2012). Thereisstilloneimportantchanneltoexclude:thecyclicalityofcashflows.It couldbethatvaluefirmshavehigherreturnsduetothefactthattheircashflows aremore‘procyclical.’ Figure6showsthatthisisnotthecase. [Figure6abouthere] The left panels of Figure 6 plot the cash flows of value and growth firms against the two state variables that represent the business cycle in this model: surplus consumption and aggregate productivity. For both value and growth firms,cashflowsdeclineinsurplusconsumption. Sincehighsurplusconsumption represents a good state, in this respect both value and growth firms have countercyclical cash flows. Regarding the magnitude of the countercyclicality, there is no apparent difference. On the other hand, in terms of aggregate productivity, growth firms are clearly more procyclical. Value firm cash flows are generallyinvarianttoaggregateproductivity,but,growthfirmcashflowsclearly 22

increase. Thecyclicalityofcashflowsitselfwouldthenleadtoavaluediscount, notavaluepremium. The right panels of Figure 6 shows that this result is intuitive. These panels shownetinvestment(investmentnetofdepreciation)forvalueandgrowthfirms. In bad times, that is, in states with low surplus consumption or low aggregate productivity, value firms are disinvesting. These are times when the household reallyvaluesconsumption,andsincevaluefirmsareunproductive,itisefficient for the value firms to discard their capital and provide cash flows to the household. This behavior leads to countercyclical cash flows for value firms. On the otherhand,growthfirmsareinvestinginbadstates. Thehouseholdwantsconsumption,butsincegrowthfirmsaresoproductive,itisefficientforthefirmto give the household less consumption so that it can invest for the future. This behaviorleadstoprocyclicalandriskiercashflowsforgrowthfirms. Of course, the risk of holding a stock is not just the risk of its cash flow next period. Everycashflowintotheinfinitefutureaffectstheriskofthestock. Both thetemporaldistributionandtheshort-termcyclicalityofafirm’scashflowsaffectitsriskandreturn. Innet,thehighcash-flowgrowthofvalueoutweighsthe lowercyclicalityofitscashflows. 5.5. TheRoleofGeneralEquilibrium Themodelisgeneralequilibrium(GE),andGEhasmanyimportantimplicationsfortheresults. Oneimportantroleisthatitpinsdowndifficult-to-observe investmentfrictions. Theseinvestmentfrictionshaveasignificanteffectonthe model’scross-sectionalassetpricingresults. Toshowthis,Iconductapartialequilibrium(PE)experiment. FirstItakethe laws of motion for consumption and aggregate capital (16) and apply parameters values from the calibration (Table 3). Note that these parameter values are calibrated using a GE model. I then plug these laws of motion into the firm’s problem(12)andsolveforfirminvestmentpolicies,butIchangetheadjustment costs for the firm’s problem to be 1/20th of their calibrated value. These lower adjustmentcostsareinlinewithpartialequilibriumestimateswhichuseaconstantSDF(forexample,Whited(1992)). Lastly,Isimulateapaneloffirmsusing thesePEinvestmentpolicies(updatingaggregatesusingtheGElawsofmotion). This procedure mimics that used in the large literature on partial equilib- 23

rium dynamic firm models (for example Zhang (2005), Carlson, Fisher, and Giammarino(2005),HennessyandWhited(2005)). IamconjecturinganSDF,and thensolvingforoptimalfirmbehaviorgiventhisSDF,butIdonotgoontocheck thattheSDFisconsistentwithfirmbehavior. Notethatinthisexamplemarkets will not clear, that is, equation (14) does not hold. Indeed, consumption is not clearly defined since I can calculate consumption either from the conjectured lawofmotionorbyaggregatinginthepanelsimulation. Table10showsthatinthisPEmodel,thevaluepremiumdisappears.Itshows Fama-Macbethregressionsofnextyear’sreturnsontoday’slogB/Mratio. While the GE model matches the data quite nicely, in the GE model, the slope on log B/Mbecomestinyandstatisticallyinsignificant. [Table10abouthere] Figure7explainswhy thevaluepremiumgoesaway. Itshowsthecash-flow growthofvalueandgrowthfirms,comparingtheGEmodeltothePEmodel. In theGEmodel, thereisalargespreadincash-flowgrowth, butinPE,thespread is tiny. Intuitively, a firms do not want to have high cash-flow growth because temporally distant cash flows raise its discount rate and lowers its value. The firmtriestoreduceitsdiscountratebyshiftingitscashflowsfromthefutureto the present, that is, by disinvesting. The low adjustment costs of the PE model reducethecostsofthisdisinvestment,andthusresultinalowervaluepremium. [Figure7abouthere] Notethatthelowelasticityofintertemporalsubstitution(EIS)impliedbyexternal habit preferences and the need to match aggregate consumption volatility are critical to the quantitative effects in this discussion. This low EIS means thatthehouseholdhasastrongdesiretosmoothconsumptionacrosstime,and, throughtheSDF,thefirmhasastrongincentivetosmoothcashflows.Thisstrong smoothing motive combined with the volatility of consumption growth seen in U.S.datathenimplylargeinvestmentfrictions. Thisstandsincontrasttolongrun risk and disaster models, which typically imply a large EIS, and therefore smallinvestmentfrictions. 24

6. Conclusion Ishowthatexternalhabitformationisconsistentwithoneimportantaspect ofthecross-sectionofstockreturns. Arealbusinesscyclemodel,extendedtoincludeexternalhabitpreferencesandidiosyncraticproductivitygeneratesavalue premium that is quantitatively consistent with the data. The value premium arisesasaresultofthetemporaldistributionofcashflows. Valuefirmsaretemporarilylowproductivityfirms,buttheytendtohavehighercash-flowgrowthin thefuture.Thesetemporallydistantcashflowsaremoreexposedtothediscount rate shocks that originate from time-varying consumption volatility. Empirical evidence confirms that value firms have higher cash-flow growth and are more sensitivetoconsumptionvolatilitymovements. 25

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7. Tables and Figures Table1:MeanCashFlowofValueandGrowth ‘EbeforeExtraordinary’isearningsbeforeextraordinaryincome(ib).‘E+Dep-NetInv’ isearningsplusdepreciationlesscapitalexpendituresplussalesofplant,property,and equipment (ni + dp - capx + sppe). ‘Year’ is year after portfolio formation. ‘Growth,’ ‘Neutral,’and‘Value’arebuy-and-holdvalue-weightedtercileportfoliossortedonB/M. Cashflowisaveragedacrossportfolioformationyears.Sampleis1971-2011. MeanCashFlowper$InvestedinYear0 EBeforeExtraordinary E+Dep-NetInv Year Growth Neutral Value Growth Neutral Value 1 0.063 0.084 0.073 0.029 0.029 0.005 2 0.067 0.093 0.098 0.031 0.034 0.030 3 0.072 0.102 0.113 0.035 0.037 0.034 29

Table2:GrowthRatesofMeanCashFlowforValueandGrowth ‘EbeforeExtraordinary’isearningsbeforeextraordinaryincome(ib).‘E’isearnings(ni), ‘Dep’isdepreciation(dp),‘NetInv’iscapitalexpenditureslesssalesofplant,property, and equipment (capx-sppe). ‘Year’ is year after portfolio formation. ‘Growth,’ ‘Neutral,’and‘Value’arebuy-and-holdvalue-weightedtercileportfoliossortedonB/M.The growth rate in year t is the growth rate of mean cash flows between years t and t−1. Sampleis1971-2011. GrowthofMeanCashFlow(%peryear) EBeforeExtraordinary E Year Growth Neutral Value Growth Neutral Value 2 6.8 10.2 34.5 6.4 10.1 50.2 3 6.6 9.4 14.6 6.6 7.0 16.4 4 8.4 7.7 6.9 7.4 11.0 9.8 E+Dep E+Dep-NetInv Year Growth Neutral Value Growth Neutral Value 2 8.9 9.3 17.3 5.1 20.0 544.3 3 8.7 8.3 9.4 12.0 6.4 11.8 4 9.1 10.3 7.2 14.4 22.6 -1.7 30

Table3:Calibration Themodelisannual,andallparametervaluesandempiricalmomentsareannual.Consumption is real non-durable goods and services consumption. The volatility of GDP andrelativevolatilityofconsumptionareloggedandHP-filteredwithasmoothingparameterof6.25. Parameter Value Target Data Model UnconditionalAssetPriceMoments β TimePreference 0.89 Mean30-DayT-billReturn 0.89 0.96 ρ PersistenceofSurplus 0.86 PersistenceofCRSPPrice/Div 0.87 0.88 s Consumption S¯ Steady-StateSurplus 0.06 MeanSharpeRatioof 0.44 0.39 Consumption CRSPIndex Long-RunGrowthMoments α ProductionCurvature 0.35 MeanOutput/Capital 0.41 0.42 δ DepreciationRate 0.08 MeanInvestmentRate 0.07 0.08 UnconditionalBusinessCycleMoments σ VolatilityofTFP 0.03 VolatilityofGDP(%) 1.61 1.70 a ρ PersistenceofTFP 0.92 PersistenceofSolowResidual 0.92 0.87 a φ AdjustmentCost 19 VolatilityofCons.Growth(%) 1.32 1.38 FirmLevelData ρ Persistofidioprod 0.65 PersistenceofFirmROE 0.40 0.48 b σ Volofidioprod 0.80 VolFirmStockReturn(%) 0.35 0.34 b ChosenOutsideoftheModel γ UtilityCurvature 2.00 Foreaseofcomparisonwith Campbell-Cochrane(1999) 31

Table4:UnconditionalAggregateAssetPriceMoments All figures are annual. Data moments are taken from Beeler and Campbell (2009) and correspondto1947-2008. Themodelcolumnsshowmeansandpercentilesfrommany simulationsofthesamelengthastheempiricalsample. USData Model mean 5% 50% 95% CalibratedMoments (cid:69)(r )(%) 0.89 0.96 -3.98 0.34 8.45 f AC1(r ) 0.84 0.83 0.57 0.86 0.99 f (cid:69)(R −R )/σ(R ) 0.44 0.46 0.26 0.46 0.69 m f m UntargetedMoments (cid:69)(r −r )(%) 6.36 7.42 4.95 7.44 9.91 m f σ(r −r )(%) 16.52 18.20 12.25 17.78 25.19 m f AC1(r −r ) 0.08 -0.06 -0.28 -0.06 0.14 m f σ(r )(%) 1.82 3.87 0.55 2.25 12.20 f (cid:69)(p −d ) 3.36 2.57 2.00 2.57 3.04 m m σ(p −d ) 0.45 0.43 0.24 0.41 0.70 m m AC1(p −d ) 0.87 0.88 0.72 0.90 0.95 m m 32

Table 5: Predicting Dividend Growth and Excess Returns with the Price- DividendRatio All figures are annual. Data moments are taken from Beeler and Campbell (2009) and correspondto1947-2008. Themodelcolumnsshowmeansandpercentilesfrommany simulationsofthesamelengthastheempiricalsample. PanelA:PredictingDividendGrowth (cid:80)L j=1 ∆d m,t+j =α+β(p m,t −d m,t )+(cid:178) t+L USData Model L mean 5% 50% 95% 1 0.003 0.009 -0.009 0.008 0.033 βˆ 3 0.012 0.013 -0.015 0.013 0.048 5 0.044 0.016 -0.036 0.017 0.057 1 0.112 0.492 -0.616 0.550 1.399 t-stat 3 0.193 0.669 -0.828 0.754 2.055 5 0.482 0.678 -1.608 0.857 2.408 1 0.000 0.010 0.000 0.006 0.032 R2 3 0.001 0.017 0.000 0.012 0.049 5 0.011 0.026 0.000 0.015 0.080 PanelA:PredictingExcessReturns (cid:80)L j=1 (r m,t+j −r f,t+j )=α+β(p m,t −d m,t )+(cid:178) t+L USData Model L mean 5% 50% 95% 1 -0.12 -0.12 -0.26 -0.11 -0.01 βˆ 3 -0.27 -0.23 -0.47 -0.22 -0.03 5 -0.42 -0.33 -0.64 -0.30 -0.02 1 -2.63 -1.91 -3.30 -1.99 -0.24 t-stat 3 -3.19 -2.50 -4.44 -2.54 -0.40 5 -3.37 -3.13 -6.72 -2.98 -0.31 1 0.09 0.07 0.00 0.07 0.16 R2 3 0.19 0.14 0.01 0.13 0.28 5 0.26 0.19 0.01 0.19 0.39 33

Table6:BusinessCycleMoments All figures are annual. Data moments correspond to 1947-2011. The model columns showmeansandpercentilesfrommanysimulationsofthesamelengthastheempirical sample. USData Model mean 5% 50% 95% CalibratedMoments σ(y )(%) 1.50 1.82 1.30 1.78 2.38 hp σ(∆c)(%) 1.32 1.38 0.91 1.35 1.95 UntargetedMoments σ(c )/σ(y ) 0.49 0.47 0.39 0.47 0.56 hp hp σ(i )/σ(y ) 2.68 3.44 3.04 3.43 3.89 hp hp ρ(y ,c ) 0.84 0.99 0.98 1.00 1.00 hp hp ρ(y ,i ) 0.58 1.00 0.99 1.00 1.00 hp hp AC1(∆c) 0.52 0.04 -0.20 0.04 0.28 (cid:69)(AdjCost/Y)(%) 1.01 0.63 0.96 1.64 (cid:69)(AdjCost/I)(%) 5.92 2.89 5.48 10.61 34

Table7:RegressionsofFutureReturnsonBook-to-Market All figures are annual. Data moments correspond to 1947-2012. The model columns showmeansandpercentilesfrommanysimulationsofthesamelengthastheempirical sample. DependentVar:R i,t+1 USData Model mean 5% 50% 95% intercept 18.62 18.34 13.33 18.03 24.14 t-stat 5.74 3.88 2.99 3.88 4.80 log(B/M) 5.70 5.84 3.58 5.84 7.86 i,t t-stat 4.88 3.34 2.17 3.30 4.57 Table8:SummaryStatisticsfrom10Book-to-MarketSortedPortfolios Allfiguresareannual. Returnsarevalue-weighted. Datamomentscorrespondto1947- 2012. The model columns show means and percentiles from many simulations of the samelengthastheempiricalsample. (cid:69)(R ) σ(R ) port port port USData Model USData Model Lo 7.7 9.6 20.9 19.9 2 8.0 10.3 17.1 21.4 3 8.1 10.6 16.7 22.5 4 8.6 10.9 17.6 22.9 5 9.5 11.1 18.3 23.4 6 9.7 11.3 17.6 23.9 7 9.8 11.5 19.3 24.5 8 11.6 11.7 21.1 24.9 9 11.9 12.0 20.4 25.6 Hi 13.3 12.3 25.7 26.7 35

Figure1:B/MDecileReturnsasaFunctionofB/M.Figuresareannual. Returns areequal-weighted. 36

Figure2: B/MandExpectedReturnsasaFunctionofFirmStates. Figures are annualandcomputedfromthemodelsolution. 37

Figure3: Cash-FlowGrowthofBook-to-MarketSortedPortfolios. Figures are annualandcomputedfrommodelsimulations. Figure 4: Cash-Flow Growth of Book-to-Market Sorted Portfolios: No Habit. Figuresareannualandcomputedfrommodelsimulations. 38

Figure 5: Consumption Volatility Betas of Book-to-Market Sorted Portfolios. Betas are from regressions of returns on changes in consumption volatility. Changesinconsumptionvolatilityarenormalizedbytheirstandarddeviation. 39

Table9:RegressionsofConsumptionVolatilityBetasonBook-to-Market Figuresareannualized.ConsumptionvolatilityinthedataisBoguthandKuehn(2013)’s estimate. Consumptionvolatilityinthemodelisthetrueconsumptionvolatility. StandarderrorsareNewey-Westwith12lags.Regressionsarefirm-levelFama-Macbethusing weightedleastsquareswheretheweightsaretheinverseofthesquaredstandarderror oftheconsumptionvolatilitybetaestimate. Betasareconstructedbyregressingexcess returnsonchangesinconsumptionvolatilityfor40quartersintothefuture. DependentVar:ConsumptionVolBeta USData Model mean 5% 50% 95% log(B/M) -1.02 -2.68 -4.20 -2.64 -1.24 i,t t-stat -2.57 -3.83 -6.08 -3.67 -1.56 40

Figure6: TheCyclicalityofValueandGrowthCashFlows. Valuefirmplotsare calculatedusingthemediancapitalandproductivityoffirmsinthe10thdecile ofB/M-sortedportfolios. Growthfirmplotsarefromthefirstdecile. Netinvestmentisinvestmentnetofdepreciation. Aggregatecapitalisfixedatitsmean. 41

Figure 7: Partial Equilibrium: Cash-Flow Growth of Book-to-Market Sorted Portfolios. ‘GE’usescalibratedparametervaluesfromTable3andequilibrium laws of motion. ‘PE’ uses equilibrium aggregate laws of motion, but firm-level decision rules consistent with adjustment costs which are 1/20th of the value fromTable3. Table 10: Partial Equilibrium Experiment: Regressions of Future Returns on Book-to-Market All figures are annual. Data moments correspond to 1947-2012. The model columns showmeansandpercentilesfrommanysimulationsofthesamelengthastheempirical sample. DependentVar:R i,t+1 USdata GEmodel PEmodel intercept 18.62 18.34 10.82 t-stat 5.74 3.88 3.60 log(B/M) 5.70 5.84 0.53 i,t t-stat 4.88 3.34 0.83 42