PRODUCTION-BASED ASSET PRICING IN A MONETARY ECONOMY: THEORY AND EVIDENCE.

Transcription

1 This version: January 15, 008 PRODUCTION-BASED ASSET PRICING IN A MONETARY ECONOMY: THEORY AND EVIDENCE. By Abraham Lioui a and Parice Ponce b,c,* a Deparmen of Economics, Bar Ilan Universiy, Israel. b PRISM, Faculy of Managemen, Universiy of Paris 1 Panhéon-Sorbonne, France. c Finance Deparmen, ESSEC Business School, France. * Correspondence o: Parice Ponce, ESSEC Business School, Déparemen Finance, Avenue Bernard Hirsch, BP 105, 9501 Cergy Ponoise Cedex, France. Tel: 33 (0) Fax: 33 (0) Ponce@essec.fr

2 PRODUCTION-BASED ASSET PRICING IN A MONETARY ECONOMY: THEORY AND EVIDENCE. Absrac This paper considers capial asse pricing based on he producion side of a moneary economy. Relying on a general version of he radiional Real Business Cycle model wih cash and credi goods, we find ha he variables deermining he mean excess reurns of all financial asses are i) capial growh, ii) he nominal rae and iii) he share of capial in aggregae wealh. Our model is parsimonious in ha he resuls do no lean on any paricular specificaion of he producion funcion nor capial adjusmen coss. Empirical evidence gives srong suppor o he presence of he elicied facors in he cross secion of excess reurns on porfolios sored i) by firm characerisics or ii) by indusry. Boh uncondiional and condiional versions of he model are shown o perform as well as he Fama-French (1993) hree-facor model when he universe of 5 porfolios sored by size and book o marke raio is considered. On an exended porfolio universe, our model is shown o perform slighly beer han he Fama-French model. We also show ha our macro-relaed risk variables have some predicing power for he aggregae dividend yield and equiy marke premium. 1

3 1. INTRODUCTION The equiy premium and value premium puzzles quesion he usefulness of he sandard capial asse pricing (CAPM) and consumpion-based capial asse pricing (CCAPM) models. According o Mehra and Presco (1985), he equiy premium pus forward he inabiliy of he smooh consumpion process ha characerizes he real world o explain he cross secion of asse reurns. As o he CAPM, is failure sems from is inabiliy o explain he cross secion of excess reurns of porfolios sored by firm characerisics (size and book-o-marke raio). In a series of very influenial papers, Fama and French (199, 1993, 1996 and 006) have shown ha he CAPM, even in he long run, is unable o explain he anomaly ha high book-o-marke firms have high expeced excess reurns in spie of having low marke beas. Among all enaive soluions o hese puzzles, he Fama-French hree-facor model (FF3 hereafer) is beyond doub he mos successful and popular. I adds o he marke porfolio facor of he sandard CAPM wo porfolio facors, one aimed a capuring he size effec (SMB) and he oher he value effec (HML). While hese facors have been based on pure empirical consideraions, Fama and French (1996) suggesed ha HML is probably relaed o corporae disress and herefore reflecs he risk premium required by economic agens o inves ino disressed firms. The lieraure on asse pricing followed wo main direcions o cope wih he failure of he sandard asse pricing models. One direcion was o ry and find some inerpreaion o he Fama-French (FF hereafer) facors. Liew and Vassalou (000) for example showed ha HML has some predicive abiliy for he GNP growh rae, and ha SMB and HML convey significan informaion abou fuure GDP growh. This informaion is essenially independen from informaion abou he equiy marke. Recenly, Pekova (006) showed ha he HML may approximae for innovaions in some sae variables in an iner-emporal CAPM framework. Finally, Xing (008) recenly showed ha HML may approximae for he growh rae of capial invesmen: an invesmen growh facor, defined as he difference in reurns beween low invesmen socks and high invesmen socks, conains an informaion similar o HML. Campbell e al. (007) however quesion he original FF inerpreaion of HML and show ha HML does no capure corporae disress. They sugges a way o

4 measure disress by a probabiliy of failure. Firms wih a high probabiliy of failure (hus in deep disress) display lower equiy reurns han firms exhibiing a low risk of failure, alhough he loadings of he former on he marke, SMB and HML facors are higher han he laer. Campbell e al. (007) conclude ha his finding is inconsisen wih he conjecure ha size and value capure compensaion for disress risk. The second direcion followed by he lieraure was o build new general equilibrium models ha exend he basic framework of he CAPM/CCAPM. Some auhors ried alernaive preference specificaions mainly implying ime non-addiiviy. For insance, Epsein and Zin (1991) inroduced a recursive uiliy approach ha allows for he disincion beween risk aversion and iner-emporal elasiciy of subsiuion, and Campbell and Cochrane (1999) applied he exernal habi approach o asse pricing. Ohers relaxed he key assumpion of marke compleeness underlying he sandard CAPMs. For example, Consaninides and Duffie (1996) derived explicily he pricing kernel when idiosyncraic risks are no hedgeable, Brav e al. (00) esed a sochasic discoun facor (SDF) ha is an average of he implici SDFs from individual Euler equaions, and Jacobs and Wang (004) showed ha he cross secion variaion in consumpion growh is a priced facor. Finally, Yogo (006) showed ha disaggregaing consumpion by spliing durable from non durable goods leads o a model ha performs much beer ha he sandard CCAPM. Several of he papers quoed above and some followers insised ha he proposed exensions do solve he aforemenioned puzzles and even someimes oher anomalies such as he negaive relaionship beween level of invesmen and sock reurn. A paricular srand of research providing new equilibrium asse pricing models, which has been iniiaed by Cochrane (1991, 1996), is variously called Produciviy- or Producion- or Invesmen-based asse pricing heory. The basic idea underlying his lieraure is o absrac oally from he consumpion side of he economy and o rely on he producion side o derive asse pricing predicions. Is main empirical implicaion is ha under he assumpion of consan reurns o scale, invesmen reurns and sock reurns should be equal. This heory has been esed a he aggregae level by Cochrane (1991), and across individual socks by Cochrane (1996) and Liu e al. (007). 3

5 A recen series of papers showed ha his heory is able o explain well known financial puzzles or anomalies. According o Zhang (005) and Cooper (006) for example, i can explain he value premium puzzle. Zhang (005) argues ha cosly reversibiliy of invesmen and counercyclical price of risk cause asses in place harder o reduce in bad imes, which makes value socks riskier han growh socks.. He uses a sandard AK model where A is sochasic and driven by boh sysemaic and idiosyncraic risks, K exhibis decreasing reurns o scale and he SDF is given exogenously. The adjusmen cos funcion is quadraic and asymmeric o accoun for parial irreversibiliy. Cooper s (006) findings are similar: as real invesmen is much irreversible, (value) firms who have idle capial benefi when he shock on is demand is posiive (since hey do no have o increase heir capial level) bu suffer when he shock is negaive due o capial invesmen irreversibiliy. Li e al. (007) explain he negaive invesmen/sock reurn relaionship by he fac ha he invesmen o capial raio affecs adversely he reurn on equiy (or, equivalenly, invesmen) since i increases adjusmen coss. Finally, in a recen conribuion, Balvers and Huang (008) develop a produciviy-based asse pricing model and show ha a condiional version of his model performs as well as FF3 in explaining he cross secion reurns on 5 FF porfolios sored by size and BM raio. This srand of lieraure is affeced by wo main drawbacks ha his paper inends o improve on. The firs is relaed o he main empirical predicion of he heory, namely ha reurns on capial invesmen are equal o reurns on equiies. The main moivaion for he producion approach was o avoid he difficul idenificaion of he represenaive invesor s preferences. However, empirical esing of he heory requires very srong assumpions as o he srucure of he producion and he adjusmen cos funcions. Moreover, addiional assumpions have o be made so ha produciviy shocks can be idenified. These are he crucial ingrediens needed o compue he capial invesmen reurns. We know from he macroeconomic lieraure how difficul hese variables are o idenify. And here is no apparen consensus among researchers as o which ype of funcions should be used. Zhang (005) for example invokes decreasing reurns o scale o explain he value premium while mos producion-based models use consan reurns o scale o obain esable implicaions. Overall, he specificaion burden has been shifed from preferences and consumpion 4

6 o producion funcion, adjusmen coss and echnological shocks. This so far does no make he producion roue much beer han he consumpion approach. The second drawback is ha he SDF is lef unidenified. Therefore, he heory is vacuous when i comes o pricing derivaives for insance. When he SDF is specified, i is in a very ad hoc manner. For example, Cochrane (1996) assumes ha i is a linear funcion of reurns on invesmen, Xing (008) builds an invesmen growh porfolio (as seen above) and shows ha i capures he informaion embedded in he HML facor, and Chen and Zhang (007) replace he SMB and HML facors by wo new ones: he reurn on a porfolio which is long socks having low invesmen o asses raios and shor socks wih high invesmen o asses raios; he reurn on a porfolio which is long socks having high earning o asses raios and shor socks wih low earnings o asses raios. Two recen papers by Balvers and Huang (007) and Belo (007) do overcome his criicism by deriving he SDF explicily from he opimal behavior of eiher he consumer or he firm. Balvers and Huang (007) show ha he marginal uiliy of consumpion is affeced by echnological shocks. Belo (007) makes he firm opimize is produciviy level insead of is invesmen level, which leads o an exac idenificaion of he SDF wihou recourse o preferences. Ye, Belo's (007) approach is no immune from he idenificaion issue menioned above. The main purpose of his paper is wofold. Firs, we propose a producion-based macro model of expeced asse reurns where an explici role is given o money and hereby o moneary policy. All he above menioned papers are indeed cas in a purely real producion economy. Our second objecive is o overcome he dichoomy beween consumpion and producion asse pricing and o build a model where he cenral planer (he represenaive agen) is boh a consumer and a producer. Since our analysis is explicily ied o he consumpion and he producion sides of he economy, he SDF characerizaion is immediae and draws on he CCAPM framework. The main advanage of our approach, in addiion o feauring explicily he moneary secor of he economy, is is parsimony. We need no specify he producion and adjusmen cos funcions, nor he naure of produciviy shocks. As far as empirical invesigaion is concerned, his parsimony of our heory is very valuable. 5

7 We inroduce money in our economy using he Lucas and Sokey (1983, 1987) disincion beween cash and credi goods. This choice presens several advanages as compared o he ransacion approach, he pure cash-in-advance approach involving oal consumpion, or he money in-he-uiliy-funcion approach. Firs, he cash-inadvance binds only a fracion of oal consumpion, which is more realisic han assuming is binds oal consumpion. Second, i has been shown recenly (see Yogo (006)) ha using a wider definiion of consumpion (i.e. durables and non durables) improves he qualiy of CCAPMs where consumpion involves non durables and services only. To idenify he SDF, which is bu he marginal uiliy of aggregae consumpion, we proceed in wo seps.. Firs, we find he relaionship a he represenaive agen s opimum beween cash goods and credi goods consumpion. As he raio of marginal uiliies is equal o he raio of prices, we can show ha he consumpion of cash goods can be wrien as a funcion of credi goods consumpion and he nominal ineres rae. In he second sep, we express aggregae consumpion as a funcion of credi goods consumpion and hus show ha he pricing kernel is simply he marginal uiliy of credi goods consumpion. Finally, we make explici use of he fac ha he represenaive agen s wealh is he sum of real capial and real money balances. Since he cash-in-advance consrain is assumed o be binding, real balances are equal o cash goods consumpion, which depends upon he level of real capial and he capial o wealh raio. Given he relaionship beween he consumpion of credi goods and ha of cash goods, he marginal uiliy of credi goods can be wrien as a funcion of real capial, he nominal ineres rae and he capial o wealh raio. The previous analysis is obained in a fairly general seing and needs no specificaion of he producion side of he economy beyond he AK capial accumulaion process. Our main heoreical finding is ha he cross secion of expeced asse reurns can be explained by a hree-facor pricing model where he facors are innovaions in he growh raes of real capial, he nominal ineres rae and he capial o wealh raio. The las wo erms sem from he presence of money and would vanish in a purely real 6

8 producion economy. The nominal rae reflecs he radeoff beween cash and credi goods consumpion, while he capial o wealh raio akes ino accoun he radeoff beween capial and money balances. Considering explicily he producion side of he economy is our main innovaion visà-vis he exan lieraure ha incorporaes money in asse valuaion modeling. The main finding in his lieraure is ha he growh rae of real money is an addiional facor o non moneary models. This resul has been obained wheher money reduces ransacion coss (see Marshal (199) and Balvers and Huang (008)), belongs in he uiliy funcion (see Bakshi and Chen (1996), Basak and Gallemeyer (1999) and Lioui and Ponce (004)), or is due o a cash-in-advance consrain (see Chan e al. (1996) and Balduzzi (007)). Considering he producion side of he economy brings abou a new original facor, namely he capial o wealh raio. In addiion, he fac ha he nominal ineres rae can be subsiued for he growh rae of real balances may help overcome he poenial problem of accuraely defining and hus measuring wha are in pracice real money holdings. Our empirical resuls can be summarized as follows. Based on US daa, we show ha an uncondiional version of our hree-facor model performs as well as he FF3 model on he universe of he 5 FF porfolios sored by size and BM raio and on an exended universe where 17 indusry porfolios have been added. A condiional version of he model, where he facor loadings have been allowed o be ime-varying as funcions of he nominal ineres rae, a erm spread, a defaul spread and he aggregae dividend yield, performs exremely well as compared o he hard-o-bea FF3 model on boh universes. The wo money-relaed facors are sysemaically priced and he srong cross variaion in he loadings for porfolios leads o a naural explanaion of he value premium puzzle and an alernaive explanaion of he negaive invesmen/sock reurn relaionship documened in he lieraure. The res of he paper is organized as follows. Secion describes he economic environmen and derives our pricing kernel and heoreical asse pricing equaions. Secion 3 repors he empirical resuls obained from various compeing models, including he FF3, several consumpion-based and producion-based non-moneary 7

9 models, for he US equiy marke during mos of he pos-war period. Secion 4 concludes.. THEORETICAL FRAMEWORK This secion describes he producion economy in which are cas he pricing kernel and he pricing equaion for he equiy marke porfolio a equilibrium. We will focus on he crucial difference beween a real economy and a moneary economy. Finally, we provide a muli-facor CAPM-like represenaion for he excess reurns on individual financial asses, on which will be grounded our empirical analysis..1. Economic seing and pricing kernel This secion briefly describes our producion moneary economy and provides he derivaion of he pricing kernel. Alhough i is well-known ha he laer is linked o he marginal uiliy of credi good consumpion, we re-derive i o fix he noaion and he model s main assumpions and o ensure ha our presenaion is self-conained. Following Lucas and Sokey (1987), we consider an economy where credi goods can be acquired in exchange for any asse bu cash goods mus be paid by previously accumulaed money balances 1. The represenaive agen has an infinie horizon and maximizes he expeced uiliy of her consumpion sream of cash goods ( c 1, ) and credi goods ( c, ) under her budge consrain. Her consumpion and porfolio decisions hus maximize: (,c ) ds c (1) ρ E e s U 1,, 1 To gain space, we sar direcly in coninuous ime, insead of seing he framework in discree ime and hen ake he limi as he period lengh shrinks o zero. The cash-in-advance consrain is assumed o be binding for cash goods, and credi goods can be ransformed ino cash goods one-for-one. The iming of evens hen is as follows. A ime, consumers-producers hold he real money balances (m) hey need o buy cash goods o be consumed beween and + d. They will also consume credi goods during ha period bu will pay for hem only a ime + d. To ensure ha he wo kinds of goods will sell a he same price, a each insan, he sellers of he cash goods, who canno consume he cash goods hey produce, are assumed o accumulae he cash receips during he period so ha he resuling cash is available for invesing only a ime + d, like in he case of credi goods. 8

10 where ρ is he rae of ime preference, U is assumed o be a hrice coninuously differeniable, increasing and sricly concave uiliy funcion, c 1 and c denoe consumpion of cash goods and credi goods, respecively, E is he expecaion operaor condiional on curren endowmens and he sae of he economy. When maximizing (1), he represenaive consumer respecs a wealh consrain and limis her aenion o admissible conrols only. Various asses are available in his economy. Firs, here are financial asses (in number N) which pay no inermediae dividends, are in zero ne supply and whose prices are assumed o follow diffusion processes, he parameers of which are endogenous o he model. There also exis nominal bonds, also in zero ne supply, which earn an insananeous yield equal o he nominal ineres rae R. Lasly, money is issued by a cenral banker, who arbirarily ses is nominal rae of reurn o zero. On a priori grounds, money hus would be sricly dominaed by he nominal bonds yielding R. However, as he cash-in-advance (CIA) consrain binds cash good consumpion, he represenaive agen will hold money balances in equilibrium. The nominal money supply, denoed by M, follows a diffusion process exogenous o he model. Finally, here is real capial, denoed by k, which accumulaes, depreciaes, and yields an oupu according o he diffusion process: dk = f ( k ; x ) δk c1, c, d + k σk,dzk, where f(.), he producion funcion, depends on k and an arbirary number of sae variables sacked in he vecor x, δ is he capial depreciaion rae, k () σ is he volailiy of he capial growh rae and Z k is a one-dimensional Brownian moion, defined on he appropriae filered probabiliy space, reflecing real (echnological) shocks. This implies he following condiions, where subscrips on U denoe parial derivaives: U > 0,U > 0,U < 0,U < 0,U U U. Also, he Inada condiions are saisfied: ( ) 0 > c1 c c1c1 cc c1c1 cc c1c lim U = lim U =. c 1 c c 1 0 c 0 9

11 Noe ha f(.) can be considered as he oupu ne of adjusmen coss bu before depreciaion. Noe furher ha our formalizaion is very general as, unlike wha is cusomary in producion-based asse pricing heory, no assumpion is needed as o he srucure of he funcion f(.), beyond he neo-classical resricions on he influence of capial k 3, or ha of he implici adjusmen coss. This feaure will prove crucial in he empirical analysis. Finally, he possible influence of non-neural money will be exered hrough he consumpion sream of cash and credi goods. Since financial asses and nominal bonds are in zero ne supply, he represenaive agen s wealh dynamics wries: w = k + m (3) i.e. oal real wealh, w, is equal o real money holdings, m M/P where P denoes he general price level, plus real capial k. The pricing kernel relaes direcly o he marginal uiliy of consumpion 4. The real pricing kernel is defined by Λ = e ρ U ( c ) ( c ) U c ( ( c ),c ( c )) c, where c sands for oal consumpion and 1 U c = is he marginal uiliy of oal consumpion. c From he firs order condiions of he represenaive agen's opimizaion program, he marginal uiliies of consumpion of he wo goods are known o be relaed by: Uc1, 1 + R = (4) U The cos of consuming one-dollar amoun of he cash good is one plus he opporuniy cos R, while he cos of consuming one-dollar worh of he credi good is one. Eq. (4) saes ha he cos raio (1+R )/1 mus a he opimum be equal o he raio of marginal uiliies 5. c, Differeniaing ( c ( c ),c ( )) U wih respec o c yields: 1 c 3 We impose he neo-classical resricions: f k >0, f kk >0, f(0)=0, lim k 0 f k =+, lim k f k =0. The represenaive agen sars a dae =0 wih a posiive k 0. 4 See for insance Cochrane (001). 5 Noe ha in a discree ime seing, he cos of one uni of he cash good - he l.h.s. of Eq. (4) - would be 1 + R /(1+ R ), i.e. R would be discouned. In coninuous ime, his discouning vanishes. 10

12 U c c = ( c ) c ( c ) 1 Uc + U 1 c c c Subsiuing ino he preceding expression yields: U c + c ( c ) = (( 1 R) c 1( c ) + c( c )) Uc Following for insance Lucas and Sokey ( 1987) and Bohn (1991), we define oal consumpion as ( 1+ R ) c ( c ) c ( c ) c = +, R.c (.) being he opporuniy cos induced by 1 paying cash he cash good. I hen follows ha he pricing kernel Λ is also equal o: ρ e Uc ρ e U c 1 ( c, c ) Λ = (5) Since he CIA consrain is assumed o be binding, Eq. (5) can equally be wrien as: Λ = ( m, c, ), which incidenally is wha one would ge from models wih a single consumpion good and money included in he uiliy funcion 6, or from models where money is inroduced o reduce ransacion coss 7. 1,, In addiion, since we have from Eq. (4) : (, c ), (, ) c = ϕ c, one can also wrie Eq. (5) as: 1,, R Λ = ϑ R. Consequenly, our modeling of a moneary economy leads so far o wo-facor asse pricing models, he facors being he consumpion of credi goods and eiher real money holdings or he nominal ineres rae. I is worh noing ha in a pure exchange (or ree ) moneary economy, we would sill have wo-facor models wih credi good consumpion replaced by aggregae income since hen c 1, + c, = y. As we wan o derive producion-based pricing models, and herefore eliminae credi good consumpion as a facor, we inroduce he capial o wealh raio ω k /w. I hen follows from he CIA consrain and Eq. (3): c 1 ω = (6) ( 1 ω ) w = 1, m = k ω Since hen c 1, is a funcion of ω and k, and, from Eq. (4), c, is a funcion of c 1, and R, he pricing kernel given by Eq. (5) rewries, for a given impaience rae ρ: (,R ) Λ = Φ k ω (7), 6 See, for example, Bakshi and Chen (1996), Basak and Gallemeyer (1999) and Lioui and Ponce (004). 7 See, for insance, Marshall (199) and Balvers and Huang (007). 11

13 Therefore, he pricing kernel is driven by hree endogenous key variables: he level of real capial, he nominal ineres rae and he capial/wealh raio. These variables are of course affeced by he sae variables (x) and he echnological shocks Z k. The laer are nooriously very hard o idenify as hey depend on specific assumpions regarding he producion funcion and he adjusmen coss. Therefore, i is a relaive advanage of our framework ha k iself, and no echnological shocks, appears as an argumen of he pricing kernel. In he same vein, anoher advanage of our approach is ha we need no specify he capial adjusmen cos funcion. The laer plays an imporan role as o he abiliy of producion-based models o explain various anomalies (e.g. Zhang (005) and Cooper (006) for he value premium puzzle). However, i is unforunaely difficul o specify as many srucural forms are conceivable on heoreical grounds. As a resul, empirical ess may lead o very differen conclusions. Obviously, in a real economy, he las wo facors on he r.h.s. of Eq. (7) vanish and only he amoun of capial k maers [see for insance Balvers and Huang (007) 8 ]. The inroducion of money hus brings abou he nominal rae, which conrols he rade-off beween cash and credi goods, and he capial/wealh raio as a conrol for he rade-off beween real capial and real balances. The presence of R and ω will in fac prove crucial in empirical ess regarding he equiy marke premium and he aggregae dividend yield... Producion Economies: Real vs. Moneary This secion is inended o highligh he difference beween real and moneary economies. While our empirical analysis will mainly focus on he cross-secional behavior of asse reurns, we show here why a moneary version of our producion economy is very likely also o fare beer empirically han a real version on various oher issues, such as he ime variaion and he predicabiliy of he equiy marke excess reurn or of he dividend yield. 8 Noe ha hey use k as a condiioning variable, no as a direc facor like we do here. This has a bearing on empirical esing since he esimaion procedure is no he same. 1

14 Le p denoe he real price a dae of a financial asse (say, he equiy marke porfolio) ha is a claim on an infi nie sream of cash and credi goods. By definiion of he (real) pricing kernel, we hus have: Λ p = E Λ s ( c1,s + c,s ) ds (8) Le us assume, as is sandard in he lieraure, a separable log uiliy funcion: U ( c1,,c, ) φln c1, + ( 1 φ) ln c, =, wih 0 < φ < 1. (9) Then Eq. (5) becomes: and Eq. (4) yields: Therefore, one has: 1+ R Λ 1, ρ φ = e (10) c 1, φ c, φ 1 = c1, = 1 φ c 1 φ 1+ R c, c + = + φ 1 (11) 1, c, 1 c, 1 φ 1+ R Using Eqs. (10) and (11) for = s, he inegrand presen in Eq. (8) is equal o: Λ I hen follows from Eq. (8) ha: s R s ( ) 1,s +,s = ρs c c e 1 φ 1+ R s p = 1 1 φe 1 φ ρ e ρ ( s ) which, using Eq. (11), yields he pricing equaion: R s 1+ R s ds c, (1) c 1, p + c, 1 ρ( s ) Rs φe e ρ 1+ R = R 1 φ 1+ R s ds (13) In a real economy, c 1 and φ are equal o zero and he classical pricing equaion p = c /ρ is recovered. In a moneary economy, he condiional expecaion presen in he r.h.s. of Eq. (13) makes he price p lower han in he comparable real economy. This resul is very inuiive as his expecaion is nohing bu he sum of he discouned opporuniy coss of holding money. This is reminiscen of he opporuniy cos presen in an 13

15 individual agen s balance shee equaion wrien a dae for all fuure sources and uses of funds. As real balances are needed o buy he sream of cash goods, non-ineresbearing money is required whose opporuniy cos is he nominal rae. Consequenly, all oher hings being equal, an increase in he laer rae decreases he curren value p of he equiy share. The inverse of he l.h.s. of Eq. (13) is he insananeous dividend yield. Our resul implies ha he laer depends on he nominal ineres rae and he vecor (x) of sae variables. The influence of he sae variables is exered hrough he condiional naure of he expecaion E [.]. Consequenly, moneary policy will impinge more han is usually acknowledged on he behavior of financial asse prices. Also, in a real economy (wih c 1 = φ = 0), he dividend yield (c/p ) is he consan ρ. As he empirical evidence hins a a raher volaile dividend yield (see panel B of Table I), inroducing money is likely o generae an empirically more convincing model. Finally, i is readily shown, using Eq. (6) o bring forward he capial/wealh raio ω, ha Eq. (1) rewries: p ρ R e 1+ R 1 = E φρ ( s ) s ds ( 1+ R ) k s 1 ω ω which implies by a sandard applicaion of Iô s lemma ha he expeced reurn on equiy is given by: ( c c ) 1 dp + 1, + E d p, d = µ ( ω,k,r, x ) and is herefore boh im e-varying and (parially) predicable as i depends on curren values of observable variables. This is in accordance wih a now huge empirical lieraure on he predicabiliy of asse reurns. By conras, in a real economy, expeced equiy reurns would depend on k and x only. (14).3. The cross secion of expeced reurns 14

16 Le S j, denoe he real price a dae of he j h financial asse 9, j = 1,, N, µ j, is insananeous expeced reurn, and r() he real rae of ineres. From sandard financial heory, he expeced excess reurn on asse j is such ha: 1 ds r Cov Λ = j, d (15) d S j, Λ µ j, ; Therefore, he dynamics of he real pricing kernel obeys he following sochasic differenial equaion: where: c U c c 1 cc1 µ Λ, = ρ + µ c, + Uc Uc + 1 c U c U U c c c c σ c, ' σ c, + d Λ Λ µ 1 c, 1 c 1 = µ U U c d + Λ, σ Λ, c c c 1 1 σ c, 1 ' σ ' dz c, 1 c1cu + U c c c c 1 σ c, ' σ c, 1 (16) σ Λ, cu = U c c c1u cc1 σc, + c U c σ c1, and Z is a muli-dimensional Brownian moion, denoes a ranspose, and µ ci, and σ c i, (i = 1, ) sand for he drifs and diffusion vecors of he wo consumpion processes, respecively. Noe ha he echnological and moneary shocks are embedded in Z. Eqs. (15) and (16) ogeher imply ha he expeced excess reurn on asse j is equal o: c U 1 c c1 j, c1, c c j, c, µ j, r = σc, + σ 1 c, (17) U c σc, U c σc, σ ' σ 1 or else: mu c m σ j, ' σ m, c U c c σ j, ' σ c, µ j, r = σ m, + σ c, (18) U c σ m, U c σ c, where σ and are he diffusion vecors for he j h asse reurn and he real balances j, σ j, processes, respecively. c U σ ' σ 9 In secion. above, p was inerpreed as he price of he marke porfolio, no a single asse, hence he change of noaion. 15

17 Eq. (18) shows ha real balances risk, such as previously invesigaed for insance by Marshall (199) and Balvers and Huang (007), is priced. Noe ha a represenaion similar o ours has been obained by he laer auhors under he assumpion of a consan opporuniy se (here are no predicors in heir economy). The derivaion of Eq. (18) is known no o require such a resricive assumpion. Anoher represenaion of resul (18) grounded on he FOC for he represenaive agen s opimal decisions wries: µ j, r = η R c U 1 U cc1 c c U + U cc c σ R, σ σ c, ' σ j, R, σ R, + η c, c U 1 U cc1 where he nominal ineres rae is subsiued for real balances, rae elasiciy of he demand for money, and c σ c, σ R ' σ j, c, σc, (19) η denoes he ineres ηc, is elasiciy o credi good consumpion. Since he appropriae definiion of money holdings is in pracice debaable (M1 vs. M for insance) and migh lead o measuremen errors, his represenaion may perform empirically beer han ha of Eq. (18). We herefore will es boh. I is imporan o sress ha he represenaion (17) - (18) - (19) is very general, and holds in paricular for general uiliy funcions. I also holds for insance in a pure exchange ( ree ) economy. In he laer economy, he corresponding equaions are obained by subsiuing oupu for consumpion. Given he characerizaion (7) for he pricing kernel in a producion moneary economy, a hree-facor ( ) (15) can be wrien as: ω,k, R CAPM is easily derived. Applying Io's lemma o Eq. (7), Eq. j, σ ' σ σ ' σ σ ' σ µ r = ξ ξ ξ ω (0) k j, k, σk, R j, R, σr, j, ω, σ ω, where ξ i denoes he elasiciy of he marginal uiliy of consumpion o he variable i, and reflecs he marke price of he corresponding risk. The sign of hese marke prices 16

18 of risk depend on he represenaive agen s preferences and canno be deermined wihou imposing addiional srucure on he model. Our empirical invesigaion will shed ligh on his issue. To summarize, we will es, in boh uncondiional and condiional versions, he model encapsulaed in Eq. (0) agains hree ses of alernaives: (i) a real, non-moneary version where he las wo erms on he r.h.s. of Eq. (0) disappear and where he firs erm involves eiher real capial, as in Eq. (0), or aggregae consumpion or oal oupu; he firs version will be called K-CAPM, he second C-CAPM, and he las Y-CAPM: σ j, ' σc, (C-CAPM ) µ j, r = ξc (1) σ σ j, ' σy, (Y-CAPM ) µ j, r = ξy () σ σ j, ' σk, (K-CAPM ) µ j, r = ξk (3) σ c, y, k, (ii) a moneary pure exchange economy where capial and he capial o wealh raio vanish and he hree-facor ( ω,k, R ) CAPM (0) is replaced by a wo-facor ( c, ) or (, ) y CAPM: R j, c j, c, σc, R j, R, σr, R σ ' σ σ ' σ µ r = ξ ξ (4) or σ ' σ σ ' σ µ r = ξ ξ (5) j, y j, y, σy, R j, R, σr, (iii) he sandard CAPM and is 3-facor exension by Fama and French (1993). 17

19 3. EMPIRICAL EVIDENCE ON EXCESS RETURNS This secion provides empirical ess of he various iner-emporal capial asse pricing models (ICAPM) grounded on he analysis conduced in his paper, i.e. various versions of Eq. (0) or is alernaives. We will perform our ess on U.S. sock porfolios for mos of he pos-war period. We inend o esimae which macroeconomic sources of risk are priced, and hus assess wheher a producion-based CAPM performs beer han a consumpion-based one or he ad hoc Fama-French model, and wheher explicily inroducing money in he economy adds somehing empirically imporan regarding equiy excess reurns Daa We have obained daa from several sources. However, all our daa are quarerly, as monhly daa for real capial are no available. Our sample covers he period from I o 004-II, i.e. 18 observaions. When he heoreical equaions o be esed involve growh raes, no levels, of variables, we will be lef wih 181 observaions. Daa for M1, GDP (oupu), CPI and he hree-monh Treasury bill rae have been downloaded from FRED a he Federal Bank of Sain Louis. Aggregae consumpion (of cash and credi goods) is measured by Personal Consumpion Expendiures of Non-Durable Goods and Services, as compiled by he Bureau of Economic Analysis. The series on Credi Goods is obained by subracing real balances (which proxy cash goods) from aggregae consumpion. Daa on real capial daa has been obained from Paul Gomme [see Gomme and Ruper (007)]. The definiion of capial we have used is consruced from depreciaion rae and invesmen daa, and iniial capial socks given by chainype index convered o year-000 dollars. The capial/wealh raio, ω, has been compued by dividing real capial by real wealh, he laer defined as real capial plus real money balances compued from M1 and he CPI. Daa on excess reurns on he marke porfolio has been downloaded from Prof. Kenneh French s websie. The marke risk premium is compued as he value-weighed reurns on all NYSE, AMEX, and NASDAQ socks (obained from he CRSP files) minus he one-monh Treasury bill rae. Monhly daa have been convered ino quarerly daa by discree compounding. 18

20 The benchmark in esing alernaive asse pricing models is he celebraed hree-facor model pioneered by Fama and French (1993, 1995, and 1996). We will use i as he reference model explaining he cross-secional variaion in porfolio excess reurns. The firs facor is he excess reurn on he marke porfolio ( Marke ), he second is he reurn on a porfolio invesed in small socks and shor in big socks ( SMB for Small Minus Big ), and he hird is he reurn on a porfolio long in high book-o-marke socks and shor in low book-o-marke socks ( HML for High Minus Low ) 10. We have downloaded from K. French s websie hese facors, as well as he 5 Fama- French (FF hereafer) porfolios sored by size and book-o-marke equiy which will consiue our firs universe of porfolios 11. Some auhors, e.g. Lewellen, Nagel and Shanken (006) have argued ha excess reurn ess using hese porfolios are biased favorably due o he facorial srucure inheren o heir consrucion. Consequenly, o reduce his bias, we enlarge our second universe o comprise, in addiion o he 5 FF porfolios, he 17 indusry porfolios also compiled by Fama and French. Acually, he heoreical equaions o be esed make use of growh raes, no levels, of variables. Alhough mos level series are no saionary, change rae series are saionary. Accordingly, Table I repors he raes of change of all aggregae variables, bu repors levels for all variables having he dimension of a rae of reurn, such as he Treasury bill rae or he porfolio reurns. More precisely, Table 1 presens he summary saisics for all he relevan variables, excep for he 5 Fama-French porfolios o save space. Inser Table I abou here 10 The SMB reurn is he average reurn difference beween hree small and hree big sock porfolios (size measured by marke value of equiy, breakpoin a he median) and he HML reurn is he average reurn difference beween wo high and wo low book-o-marke sock porfolios (breakpoins a he 30 h and he 70 h perceniles), where book-o-marke is he raio of he accouning value of equiy o is marke value. 11 The FF porfolios are obained from an independen sor of all NYSE, AMEX and Nasdaq socks ino quiniles based on size and book-o-marke raio (hence here are 5x5=5 porfolios). See Fama and French (1993) for more deails. 19

21 Panel A repors he saisics relaive o he growh raes of aggregae consumpion ( c ), consumpion of credi goods ( c ), real oupu ( y ), real capial ( k ), he capial o wealh raio ( ω ), and o he level of he laer raio ( ω level ). Panel B exhibis he levels of four predicive, reurn-relaed macro-variables: he hreemonh Treasury bill rae ( T-bill ), he erm spread measured as he difference beween he en-year (consan mauriy) Treasury bond yield and he hree-monh Treasury bill rae ( Term ), he defaul spread measured as he difference beween he yield of a Baa- raed bond and ha of an Aaa-raed bond boh having a consan 10 year mauriy ( Def ), and he dividend yield measured as he oal dividends paid off during he las 1 monhs divided by he acual price of he marke porfolio ( Div ). Panel C exhibis he excess reurn on he marke porfolio over he T-bill rae ( Marke ) and he reurns on he Fama-French small minus big size-relaed ( SMB ) and high minus low book-o-marke-relaed ( HML ) porfolios. The average equiy marke premium is in line wih wha is repored in he exan lieraure (4.5% annualized). Panel D repors he correlaion marix for he growh raes of six variables: aggregae consumpion, credi goods, real oupu, real capial, he capial o wealh raio and he hree-monh Treasury bill ( R ). Of paricular relevance o his sudy, he capial-owealh raio ω is no correlaed wih real capial k, which will allow he inserion of boh variables in he regressions. Panel E shows he correlaion marix for he levels of he four predicive variables Tbill, Term, Def and Div. As inuiion suggess, here is a raher srong correlaion beween he level of he T-bill rae and ha of he oher hree variables. Panel F exhibis he firs-order auo-regression coefficiens for he growh raes of he six following variables: aggregae consumpion, credi goods, real oupu, real capial, he capial o wealh raio, and he hree-monh Treasury bill. I also repors he Phillips- Peron es of saionariy for each variable, and is criical values. The large persisence of he capial growh rae ( k ) is worh noing. Alhough heoreically very aracive, 0

22 in paricular because i avoids he nooriously difficul measuremen of produciviy shocks, his variable will hus probably have a hard ime o be empirically a significan variable explaining cross secional porfolio reurns. Panel G repors he average excess reurns (over he 3-monh Treasury bill) on he 5 Fama-French porfolios. As frequenly repored in he lieraure on he value premium [e.g. in Fama-French (006)], i is readily seen ha (i) excep for big size porfolios, here is a clear posiive relaionship beween risk premium and book-o-marke raio, implying ha value socks command a higher excess reurn han growh socks, and (ii) excep for low book-o-marke porfolios, here is a clear inverse relaionship beween excess reurn and size. 3. Esimaion procedure Uncondiional models According o an ICAPM [see Meron (1973)], he excess reurn (over he risk-free rae) on a risky asse (or porfolio) depends on is marke bea and is exposure o random changes in he invesmen opporuniy se. Of course, he economerician has o decide, from a heoreical model such as he one presened in he previous secion, wha sae variables, if any, may affec he invesmen opporuniy se, and is led in general o es various empirical versions of he model. To esimae an uncondiional ICAPM, wo kinds of regressions mus be run, following he Fama and MacBeh (1973) sandard procedure. In a firs pass, he ime-series of excess reurns (over he risk-free rae) on a risky asse or porfolio ( r j, ) is ypically regressed on he excess reurn on he marke porfolio or he change rae of an aggregae such as consumpion, oupu or capial, and possibly on he growh rae of oher explanaory variables such as he nominal rae and he capial/wealh raio, all generically denoed below by Y i,. We hus have: R F j, R rj, = α j + β j,y Y i i, + ε j, j = i=1 1,,.., N (5) 1

23 where F is he number of facors and N he number of asses. In he second pass, we es in cross-secion he hypohesis ha he uncondiional expeced excess reurns on asses or porfolios obey: F ( rj) = λ j + µ R E γ βˆ (6) j where he λ j should be zero and he independen variables βˆ j,. are esimaes obained from regression (5). For example, he sandard CAPM (wih Y being he excess reurn on he marke porfolio) leads o γ YM significan ly posiive and all oher γ Yi equal o zero, while he FF hree-facor model leads o γ YM and wo oher γ Yi (Y 1 he SMB porfolio and Y he HML porfolio) significanly posiive, and all oher γ Yi equal o zero. i= 1 Y i j,y i As an oher example, our model represened by Eq. (0) saes ha he Y i are he growh raes of real capial, of he nominal ineres rae, and of he capial/wealh raio (i.e. i = 3): ( ) = γ βˆ + γ βˆ j k j,k R j,r + γ ω βˆ j ω µ j R E r, Alernaive models represened by (various versions of) Eqs. (1)-(4) give rise o equaions similar o Eq. (7) bu wih i = 1 or only. (7) 3... Condiional models In view of he general failure of uncondiional CAPMs or CCAPMs o explain crosssecional asse reurns, researchers have invesigaed condiional versions where various predicive macro-variables are used in he firs-pass ime-series regressions. The procedure is he one described for he uncondiional models excep ha he ime-series of excess reurns on a risky porfolio (r j, ) is regressed no only on he variables Y i, as above, bu also on he level of sae variables, generically denoed below by X k,. The laer are deemed o represen changes in he invesmen opporuniy se or proxies for he laer, and o ensure a degree of predicabiliy in asse expeced reurns 1. In principle, boh he consan α j and he βˆ j,. in Eq. (5) hus are ime-varying. To avoid undue complexiy, however, we choose o le α j consan. In addiion, he ime- 1 The lieraure is abundan. See for insance Fama and Schwer (1977), Campbell (1987), Campbell and Shiller (1988), Fama and French (1989), and recenly Yogo (006) or Pekova (006).

24 varying βˆ j,. are assumed o depend linearly on he following four predicive variables (X k, ) defined in Panel B of Table I: he shor erm ineres rae T-bill, he erm spread Term, he defaul spread Def, and he dividend yield Div. This choice is dicaed by previous resuls available in he exan lieraure, e.g. in Pekova (006) or Yogo (006). We hus have: R j, F K R rj, = α j + βj,i,0 + β i= 1 l= 1 so ha ulimaely, developing he double sum, we have o esimae, for each porfolio j, (1+K)F = 5F beas where F is he number of variables Y i. For insance, for our j,i,l X l, Y producion-based moneary model, we will have 15 beas. i, + ε j, j (8) In he second pass, we conduc cross-secion regressions o es he hypohesis ha he condiional expeced risk premia on porfolios obey: F F K ( r ) = λ + γ βˆ j j Y j,y + i i µ R E γ βˆ (9) j i= 1 i= 1 l= 1 Y i,l j,y,x where he independen variables ˆβ are esimaes obained from regression (8). j,. i l 3.3. Resuls Preliminaries: predicive regressions Before esing compeing versions of he ICAPM, we briefly invesigae wheher various combinaions of macro-variables help explain he aggregae dividend yield, he excess reurn on he equiy marke porfolio and he 5 FF porfolios. We ran predicive regressions in which he explanaory variables are he one-period lagged values of he 3- monh Treasury bill rae ( T-bill ), he erm spread ( Term ), he defaul spread ( Def ), he level of real capial ( k-level ) and he innovaion of he capial o wealh raio ( ω innov ). The laer innovaion is compued in view of he high correlaion beween he levels of capial and his raio, and is obained from he regression of he capial o wealh raio on real capial, boh expressed in levels. The corresponding - saisics are correced for auocorrelaion and heeroskedasiciy using he Newey-Wes esimaor wih four lags. Resuls are repored in Table II. 3

25 Inser Table II abou here Generally speaking, he level of R s for he porfolios is very low, which vindicaes he claim in he lieraure ha excess or oal reurns on hese are very hard o predic. Resuls for he aggregae dividend yield however are surprisingly encouraging, for a variable also deemed o be difficul o predic, wih all variables bu he defaul spread highly significan. Noe ha he nominal rae has a posiive impac on he dividend yield and a negaive one on he equiy marke premium. One possible explanaion for he firs relaionship is ha he nominal rae increases wih inflaion as do dividend yields. As o he second relaionship, when he level of ineres raes increases because of inflaion, he risk premium (which is expressed in real erms) decreases ex pos as he inflaion ax reduces he real profiabiliy of firms 13. The level of real capial has a negaive impac on he dividend yield. The likely explanaion is ha dividend policy, as implied by Modigliani-Miller s irrelevance proposiion, is a residual decision so ha, oher hings being equal, a firm ha faces profiable invesmen projecs funds hem (a leas parly) by paring wih cash ha would oherwise be devoed o paying dividends. The impac of our key variable, he capial o wealh raio, is posiive on boh he dividend yield and he marke premium. Firs, when ha par of a echnological skock ha is capured by ω is posiive, expeced profis and dividends increase. This obains for he 5 FF porfolios as well as he marke as a whole. Second, he firs order expansion of ω is (1-(m/k)). When is innovaion is posiive, he relaive weigh of real balances decreases, which implies less expeced opporuniy coss and hen more expeced reurns and dividends Uncondiional CAPMs 13 Inflaion, in addiion o being a ax on real money holdings (paricularly imporan for firms), increases he real ax burden on firms as, for insance, he laer canno correcly depreciae heir exising physical capial as replacemen cos accouning is fiscally forbidden. 4

26 We repor firs ess of alernaive uncondiional CAPMs [Eq. (6)]. Table III repors (second pass) cross-secional regressions using he excess reurns on he 5 FF porfolios sored by size and book-o-marke. This able presens cross-secional regressions using he excess reurns (over he hreemonh T-bill rae) on 5 Fama-French porfolios sored by size and book-o-marke. The full-sample facor loadings, which are he independen variables in he cross-secional regressions and are no shown o save space, have been compued in ime-series simple regressions (for each of he 5 porfolios) in which he dependen variable is he excess reurn on a given porfolio [Eq. (5)]. The cross-secion regression (Fama-Macbeh) coefficiens are obained by Ordinary Leas Squares (OLS) and displayed on he firs rows of Table III. The -saisics ( (NW) ) are correced for auocorrelaion and heeroskedasiciy using he Newey-Wes esimaor wih four lags o assess he saisical significance of he independen variables. Since he laer are esimaes from a ime series regression, we have also repored he -saisics ( (S) ) adjused for errors- in-variables according o he procedure esablished by Shanken (199), a more difficul es o pass. Noe however ha when he homoskedasiciy assumpion made by Shanken (199) is relaxed, Jagannahan and Wang (1996) and more recenly Shanken and Wang (007) have shown ha he bias may be relaively small if i exiss a all. This is why we repor boh -saisics. To assess he overall fi of each compeing model, we have compued he adjused R used by Jagannahan and Wang (1996), ( R (JW) ), which measures he proporion of cross-secional variaion in expeced reurns explained by he model. Panel A of Table III repors he cross-secional resuls for he sandard CAPM and he Fama-French 3-facor model (FF3). Panel B refers o a non-moneary economy and shows resuls for he C-CAPM, he Y-CAPM, and he K-CAPM derived from Eqs (1), () and (3), respecively. Panel C exhibis resuls for a pure exchange moneary economy where he independen variables are various combinaions of credi goods, oupu, oal consumpion, real money holdings and 3-monh T-bill rae, all expressed in growh raes. Panel D repors resuls for a producion moneary economy where he independen variables are various combinaions of he growh raes of real capial, 3- monh T-bill rae, and capial o wealh raio. When all hree variables are presen, we 5

27 have our model [Eq. (7)]. Panel E is he same as Panel D wih he excess reurn on he marke porfolio subsiued for he growh rae of real capial. Inser Table III abou here Panel A confirms he plain failure of he sandard uncondiional CAPM, where he R (JW) is only 0.31 and he boh he consan and he excess reurn on he marke porfolio are srongly significan, wih he wrong negaive sign on he laer variable. By conras, he 3-facor FF model is much more successful, like in he lieraure where he repored adjused R lies roughly in he range depending on he sudies 14. Here he R (JW) is 0.81 and he reurn on he HML is posiive and very significan, and he reurn on he SMB porfolio is posiive and border-line significan. Overall, his model performs raher well and remains he hard-o-bea reference model, bu sill has he puzzling wrong sign (hardly significan however) on he marke porfolio. Panel B shows he complee failure of boh he Y-CAPM and he K-CAPM for which he independen variable has no explanaory power. This was expeced since hese models have no heoreical grounding. The C-CAPM fares beer, wih a high R (JW) of However, he srongly significan and negaive sign obained for aggregae consumpion is less expeced bu confirms he well esablished failure of he uncondiional C-CAPM o explain he reurns on he 5 FF porfolios 15. Resuls on more complex versions of a pure exchange moneary economy where money is presen are repored in Panel C. The inroducion of money, eiher hrough real balances or he nominal ineres rae, ha show up everywhere as very significan, slighly improves he C-CAPM alhough consumpion reains he wrong sign. In accordance wih Yogo s (006) resul ha disaggregaing consumpion beween durable and non-durable goods, he former being significan, we find ha when credi goods c is subsiued for oal consumpion c, resuls improve since he sign of c is posiive and almos significan when he nominal rae is used. The models based upon oupu 14 See for insance Fama and French (199), Jagannahan and Wang (1996), Leau and Ludvigson (001) or Pekova (006). These sudies differ by he period considered and/or he frequency of daa. 15 For a recen discussion on his failure, see Yogo (006). 6