economic growth, business cycles, and expected stock returns

7/28/2019 Economic Growth, Business Cycles, and Expected Stock Returns.

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Economic Growth, Business Cycles, and Expected

Stock ReturnsCaroline Mller

Job Market Paper

October, 2008

Abstract

This article presents a dynamic general equilibrium model which jointlyaccounts for main asset pricing phenomena of the time series and the cross sec-tion. Learning about changes in long-run output growth leads to time-variationin risk premia. Covariance with GDP growth through business cycles deter-mines a countercyclical equity premium. Countercyclical value (size) spreadsand value (size) premia arise in compensation for systematic dierences in thetime-varying exposure of rm fundamentals to business cycle uctuations. TheFama and French (1993) factors HML and SMB are shown to capture rms rel-ative exposure to recessionary periods of the economy. The model extends the

consumption-based CAPM (CCAPM) to an output-based CAPM (YCAPM).

Key words: asset pricing, general equilibrium, learning, macroeconomic risk,time-varying risk premia

HEC Paris, School of Management, 1, rue de la Libration, 78351 Jouy en Josas, Cedex, France.Correspondence to [email protected]. I would like to thank Fernando Alvarez, Lau-rent Calvet, Thierry Foucault, Francesco Franzoni, Evgenia Golubeva (discussant), Lars Hansen,Monique Jeanblanc, Stefano Lovo, Pierre Mella-Barral, Tomasz Michalski, Alan Moreira, JacquesOlivier, Pietro Veronesi, and seminar participants at HEC, and the University of Chicago, economicdynamics working group, for their comments, as well as participants at the 2008 FMA conference.

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1 Introduction

The "failure" of the CAPM, more precisely, its inability to account for the crosssection of average stock returns, has probably remained one of the main driversbehind research in asset pricing. Since Fama and French (1993), many empiricallysuccessful asset pricing models have been proposed. However, much less has been saidwhen it comes to providing economically founded explanations for the asset pricinganomalies detected.

What is the economic intuition behind the Fama and French (1993) factors? Whatrisk factors can possibly explain their existence? What would candidate explanatoryfactors imply for standard asset pricing models? Should we rethink common deni-

tions of risk?

While aggregate macroeconomic risk is generally accepted to be the source of riskpremia in asset markets, so far, theoretical asset pricing models have restricted itsdenition to the variability of consumption. However, consumption is just one ofmany macroeconomic time series. Moreover, it happens to be one of the smoothest,making it not the most favorable choice, given the size of risk premia that need to beexplained1.

Concerning the relation of risk premia to alternative macroeoconomic aggregates,Rangvid (2006), for example, provides evidence on the relation between expected

returns and output. He shows that the ratio of share price to GDP captures more ofthe variation over time in expected returns on the aggregate market, than do ratiosof price-to-earnings or price-to-dividends. Cooper and Priestley (2007) demonstratethat the output gap is a predictor of returns on stocks as well as bonds. Interestingly,there also exists a link between output and returns in the cross section: Liew andVassalou (2000) show that the Fama and French (1993) factors forecast GDP growth.

The present article develops an asset pricing model relating expected returns tovariations in output growth. The proposed dynamic general equilibrium model canrationalize risk premia in the time series, as well as the cross section, based on rmsexposure to variability in GDP growth. Risk premia obtain as a function of the co-

variation of rm fundamentals with changes in aggregate output. As fundamentalsof small and value rms are more adversely aected by recessionary periods of the

1 see Hansen and Singleton (1983), Mehra and Prescott (1985), or Grossman, Melino, and Shiller(1987), for example, for the literature on the "equity premium puzzle"

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economy2, this has implications for the cross section. Value and size premia arisein compensation for a comparatively higher exposure of rm fundamentals to nega-tive business cycle shocks. As the model operates in an environment of incompleteinformation, the agent continuously learns about exposures inferring from rm-levelproductivity observations. The dynamic structure of the model then allows generat-

ing countercyclical risk premia, reecting rms time-varying exposure to changes inGDP growth.

The model is constructed to combine a latent two-state Markov Switching Modelwith i.i.d. shocks to the underlying state variable. The two regimes are denedthrough GDP growth. The latter can be either high, should the economy be in anexpansion or low, in a contraction. The agent however, cannot observe the state of theeconomy and has to draw inferences from productivity observations at the rm level.Firm productivity is modeled to equate GDP growth3 supplemented by idiosyncraticshocks, characterized by a mean zero underlying distribution. However, consistentwith the above mentioned bias in the reaction of fundamentals of small and value

rms to recessions, the mean value of the underlying distribution exhibits a negativebias when dened cross-sectionally.

The agent evaluates rms in order to optimally allocate capital. Observing pro-ductivity4 at the rm level, she simultaneously learns about rms business cycleexposure, as well as the overall state of the economy, when considering her observa-tions in the aggregate. The agents belief about the time-varying state of the economythen leads to a countercyclical equity premium.

In the cross section, learning takes two eects. First, learning is at the source ofthe emergence of value (size) premia, as well as, value (size) spreads, a systematicdierence in the average market-to-book ratio of value and growth rms. The sizespread is commonly dened as the dierence in market-to-book ratios between smalland large rms. In the present model, it can also be understood as the dierence inaverage market values of these rms. Second, just as in the time series, time-varyingbeliefs about the state of the economy lead to countercyclicalities. Hence, the modelendogenizes the cyclical behavior of value and size premia, a stylized fact pointed outby Lettau and Ludvigson (2001), and Campbello, Chen, and Zhang (2006), respec-tively. It also accounts for cyclicalities in value (size) spreads, as observed by Cohen,Polk and Vuolteenaho (2003)5.

2 see Gertler and Gilchrist (1994), who provide evidence for small rms; see Xing and Zhang

(2005) for value versus growth rms3 for the purpose of simplication the model abstracts from labor markets and the economy is

closed4 In fact, the stucture of the model is such that productivity equals protability. The latter might

be a more plausible variable of observation for some readers.5 Cohen, Polk and Vuolteenaho (2003) provide evidence that the value premium is high when

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The emergence of empirical regularities in the cross section is driven by agentsoptimizing behavior with respect to time. Within the model, the agent disposes ofa limited amount of time to evaluate a continuum of rms. She therefore faces atrade o: learning more on one rm means learning less on the others. As learningreduces uncertainty surrounding rm-specic shocks to productivity, it positively af-

fects agents expected utility from investment. It can then be shown that due to thecross-sectional bias in the reaction of rm fundamentals to recessions, the marginalutility from learning is strictly higher for small and value rms at all times. Hence, itis optimal for the agent to always learn strictly more on these type of rms6. More-over, the optimal learning bias is increasing in recessions. As the resolution of uncer-tainty through learning negatively aects valuation ratios7, countercyclical value andsize spreads emerge. Concerning the emergence of risk premia in the cross section,the learning bias takes eect as soon as the agent becomes aware of an underlyingsystematic bias in fundamentals.

Combining countercyclical value (size) spreads and premia, the model can ac-

count for stock return predictability in the cross section8. Moreover, it provides anexplanation for the predictability of GDP growth by the Fama and French size andbook-to-market factors, as pointed out by Liew and Vassalou (2000).

Finally, the model extends the common consumption-based denition of risk(CCAPM) to an output-based denition (YCAPM). Within the model, variationin consumption obtains as a consequence of variation in output. The positive cor-relation of consumption with output makes it a valuable proxy for macroeconomicrisk, but not a risk factor by itself. Equity premia are exclusively generated by thecovariation of rms output with changes in GDP growth. A conditional CAPM

representation shows that systematic dierences in covariation lead to betas that di-verge in the cross section, and are countercyclical. Eventually, the model shows howthe conditional CAPM combines in one, what the "unconditional" Fama and French(1993) model splits into three dierent factor exposures.

Related Literature

the value spread is large and vice versa. As the value premium was found to be countercyclical(Lettau and Ludvigson (2001)), the value spread needs to be countercyclical, too. To the best ofmy knowledge, there is currently no empirical evidence on the time-series behavior of size spreads.

6 The current paper takes the empirically observed dierences in the reaction of rm fundamentalsto recessions as given. Mller (2008b) shows that this higher sensitivity is reected in systematicallyhigher leverage ratios. The latter might be thought of as causing the aforementioned bias.

7 see Pstor and Veronesi (2003), who show that the market-to-book ratio is increasing in uncer-tainty about protability

8 The model could also account for return predictability at frequencies higher than business cy-cles. Regime-switches at business-cycle frequencies, could be combined with i.i.d shocks at higherfrequencies, for example, where the cross-sectional bias in mean values of the underlying distributionis dened to be time-varying.

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This article is related to the literature on asset pricing and learning9. It draws onwork by Pstor and Veronesi (2003, 2005), but in comparison to these authors diersin motivation, its focus being on the role of learning in relation to expected returnsand return predictability. Revealing "learning" as a potential explanatory factor forthe emergence of time-varying risk premia, this article is related to recent work by

Hansen and Sargent (2007)10. While these authors obtain countercyclical uncertaintypremia from a combination of model and parameter uncertainty, the present papermakes use of the latter only.

As the model derives asset pricing implications from the real side of the economy,it is also related to the emerging literature on production-based asset pricing11. Berk,Green and Naik (1999) were the rst to establish a relation between investment deci-sions and expected returns. They show that as a consequence of optimal investmentdecisions rms assets and growth options change in predictable ways with marketvalue, which becomes a proxy for the state variable describing their relative impor-tance. More recent contributions include those of Gomes, Kogan, and Zhang (2003),

Carlson et al. (2004), or Gala (2006), for example. The present paper diers fromthis literature, as it does not explain empirical regularities from rms investmentdecisions. Instead, "equity return puzzles" are rationalized as reecting a higher ex-posure to systematic risk in form of changes in GDP growth. Therefore, the presentarticle introduces a link to macroeconomics and business cycle theory, which is absentfrom existing production-based asset pricing models.

Finally, in view of the obtained cyclicalities in risk premia, this paper is related toasset pricing models featuring habit formation. Santos and Veronesi (2005), for exam-ple, construct a "multiple endowments" economy, specifying cash ows exogenously.

They introduce habit persistence and obtain eects in the cross section through theinteraction of a time-varying aggregate risk premium with changes in the durationof an assets cash ow. However, the authors point to a "cash-ow risk puzzle", i.e.the cross-sectional dispersion in cash ow risk needed to match the cross-sectionalproperties of stock returns is found to be too large. Making the stochastic discountfactor exogenous, driven by investor sentiment, Lettau and Wachter (2005) circum-vent this problem. This approach, however, comes at the cost of permitting at best aweak correlation with macroeconomic aggregates and hence, with a general dicultyto provide fundamental explanations of cyclicalities in risk premia.

The paper is now organized as follows. Section 2 presents the production economy.

Section 3 establishes its link to nancial markets and results for stock valuation andstock return predictability. Section 4 calibrates the model and section 5 concludes.

9 see Detemple (1986), David (1997), Veronesi (2000) and Brennan and Xia (2001), for example10 see also Hansen (2007)11 see Cochrane (2006) for a detailed review

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2 The Economy

I consider a representative agent economy. While the economy can be thought of asbeing of innite horizon, the representative agents investment horizon is nite anddenoted by T. The agent is endowed with initial capital B0 that needs to be allocated

for the time of the investment period.

The economy is populated with a continuum of heterogeneous rms i 2 I, wherethe set of rms I is assumed exogeneously xed. There is a linear technology, pro-ducing output, Yit, from capital, Bit, for each rm i 2 I, such that

Yit = tBit. (1)

Firm productivity is assumed to follow a mean-reverting Ornstein Uhlenbeckprocess12 of the form

dt = ( t) dt + dZt (2)

with standard Brownian motion Zt, constant speed of mean reversion 2 R++ andconstant volatility 2 R++. Productivity, t, depends on the state of the economy. The variable creating state dependency is long-run output growth, 2 R++.It can be either high, H, should the economy be in an expansion, or low, L, in acontraction.

At some point in time, t, where t 2 [0; T), rm-specic shocks to productivity,i , are assumed to spread over all rms in the economy. For the purpose of sim-

plication, t

is assumed to be singular. However, shocks could occur at arbitraryfrequencies within the investment period T. Technology shocks are idiosyncratic, thatis they are i.i.d and drawn from a single underlying distribution whose mean value is zero, whatever the actual state of the economy . Hence,

i i:i:d: N

0;b2t (3)for all rms i 2 I. The time t rate of variance, b2t 2 R++, is assumed known by

the agent.

Once technology shocks have occurred, the rm-specic process for productivitydenotes

12 Empirical evidence on the mean reversion of rm protability was provided by Penman (1991),Fama and French (2000), and Pakos (2001), among others. Evidence on the non-stationarity of realoutput was provided by Nelson and Plosser (1982), Cheung and Chinn (1996), Rapach (2002), andDavid, Lumsdaine, and Papell (2003), for example.

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dit = ( + i it) dt + dZt + 1dZi ;t t t T (4)

with constant variances ; 1 2 R++. The process now contains two standardBrownian motions Zt and Zi ;t, for all i

2I13. The rst is common to all rms and

relates to systematic risk. The second relates to idiosyncratic variations driven byrm-specic technology shocks. As the number of shocks is spanned by a correspond-ing number of Brownian motions, markets are complete.

Although technology shocks are idiosyncratic, they are assumed to display a sys-tematic bias in the cross section. This bias relates exclusively to the bad state ofthe economy. Empirical evidence shows that economic fundamentals of small andvalue rms are more negatively aected by recessionary periods of the economy thanthose of large and growth rms. Xing and Zhang (2005) provide evidence comparingfundamentals of value and growth rms, while similar ndings for small and large

rms are provided in Gertler and Gilchrist (1994).Hence, while rms are assumed observationally equivalent to the agent14, there is

a latent cross section dened through systematic dierences in the reaction of rmfundamentals to changes in GDP growth. Consequently, rms can be of type "small"or of type "large", they can be of type "value" or of type "growth". Each rm i 2 Ican be thought of as having been randomly assigned to one of two categories alongeach dimension at time 0.

In order to reect the above described bias from rm fundamentals, let me denethe unconditional cross-sectional mean, 2 R. It obtains as the aggregate (average)value of the under (3) dened technology shocks of rms i 2 , that is

Ri2i di,where = ;;s;l for = H; L. As there are no systematic dierences in thereaction of rm fundamentals during expansions, it follows that H =

H 0. Inrecessions, however, the mean values in the cross-section of technology shocks displaythe following characteristics

L < L (5)

denotes the higher exposure to negative business cycle shocks of value rms, ,compared to growth rms, ,

sL < lL (6)

13 Note that consequently, it replaces t in (1) for all t t T.

14 One could think of assuming the agent to distinguish rms by size. However, as long as sheremains oblivious to associated dierences in exposures of rm fundamentals to recessions, such anassumption would not bear on subsequent results.

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denotes the higher exposure of fundamentals of small rms, s, compared to thoseof large rms, l.

I take the empirically observed dierence in reactions of rm fundamentals asgiven. Introducing bond markets, Mller (2008b) shows that this dierence is re-

ected in a systematic cross-sectional divergence in leverage ratios. One might inturn consider the dierence in leverage ratios to be at the source of the bias in re-actions of rm fundamentals. It would seem rather plausible that a rm with littlenancial slack lacks a buer in recessionary periods of the economy. Consequently,its earnings, ouput, and dividends should react more adversely to negative businesscycle shocks.

I will now turn to the implications of the features of the constructed productioneconomy for rm valuation.

3 Financial Markets

Firms are all-equity nanced. Capital Bit of rms i 2 I is then equal to rms bookvalue of equity, and rm protability is given by

it =YitBit

, (7)

which is nothing but the instantaneous accounting return on equity, with Yitdenoting rms earnings at time t. Consistent with the denition of a regime-switching

production economy as in (4), the level of protability can be either high, H

, shouldthe economy be in an expansion, or low, L, in a contraction.

Firms i 2 I pay out dividends Dit for all t 2 [0; T]. In order to smooth dividendsover time, they are payed out as a constant fraction c, c 2 R++, of book equity

Dit = cBit. (8)

Capital of rm i then follows the process

dBit = (it

c)Bitdt. (9)

As the economys consumption good is immediately perishable and non-storable,the assumed dividend policy leads to the following equilibrium restriction on con-sumption for all t 2 [0; T]

Ct =Ri2I

Ditdi. (10)

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For the purpose of simplication, I will assume in the remaining analysis thatc = 0. Note however, that all results go through for c 2 R++, where c < it 8i, 8t,with marginal quantitative impacts15.

The representative agent with risk aversion > 1, has preferences dened by

power utility. She aims to allocate capital across rms such as to maximize expectedutility from terminal wealth16,

maxBit

Et

W1T1

8i 2 I. (11)

The market-clearing condition is given by WT = BT Ri2I

BiTdi. As rms wereassumed observationally equivalent to the agent, her priors on rm fundamentals areunbiased. Consequently, at time 0, it is optimal to equally allocate initial capital B0across rms i 2 I. Capital can be reallocated instantaneously and at no cost.

There is a risk less bond in zero net supply, whose yield is normalized to zero forsimplicity. Standard arguments then imply that the state-price density obtains fromthe pricing equation

t = 1Et

WT

, (12)

where denotes the Lagrange multiplier from the utility maximization problemof the representative agent.

Stocks are dened as contingent claims to be liquidated at the end of the agents

investment period, with a market clearing under WT = BT. The agent expectsmarkets to be perfectly competitive, thus leaving no abnormal earnings in equilibrium.She therefore assumes MiT = BiT for all i 2 I. The current market value of rmsstock is then given by the pricing equation

Mit = Et

TBiT

t

. (13)

Recalling the latent Markov structure of the model, I will now turn to the devel-opment of agents beliefs about the actual regime of the economy, , when inferringfrom protability observations it across rms i

2I. These time-varying beliefs will

be at the source of cyclicalities in risk premia of the time series and the cross section.

15 The assumption of c < it 8i, 8t, ensures that the economy does not stagnate.16 Note that even in the presence of intermediate consumption, i.e. c 6= 0, the investment problem

reduces to the maximization of terminal wealth as c was assumed exogeneous.

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3.1 The State Price Density

The state variable , dening whether the economy is in an expansion or contraction,is assumed to follow a two state, continuous-time Markov switching process withtransition probability matrix between time t and t + 4 given by

P (4) = 1 4 44 1 4

(14)

where is the probability that during an innitesimal time interval , meanproductivity shifts from the high state, H, to the low state, L, while denotesthe inverse.

The agent observes productivity, it, across rms i 2 I. As the rm-specicshocks to productivity, i , disappear at the market level for = H; L, the process ofinference when forming beliefs about the overall state of the economy is as dened in

(2), for all t 2 [0; T].Lemma 1 The posterior probability of the good state, t = Pr

= Hjzt

, where

the ltration is generated by zt = ft : 0 t Tg, follows the law of motion

dt = ( + ) (s t) dt + h(t)dZ0;t (15)

where

dZ0;t =1

(dt Et (dtjzt)) (16)

h (t) = H L

t (1 t) (17)s =

+ (18)

and Z0;t is a Wiener process with respect to zt, and s the probability of H

under the Markov chain stationary distribution.

Proof: See David (1997) Theorem 1, Theorem 9.1 in Liptser and Shiryayev (1977)

By means of the posterior belief distribution, the eects of the Poisson processesimplicit in the continuous-time Markov chain are smoothed out. Therefore, althoughchanges of the state variable through time occur in jumps, learning leads to continuous

processes.

Lemma 2 The values 0 and 1 are entrance boundaries for , i.e., there isa probability zero that equals either of these two values in any nite time.

Proof: see David (1997)

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Given the posterior belief of the good state, t, the dynamics of aggregate pro-ductivity as dened in (2) can be reexpressed as follows

Lemma 3 The process for aggregate productivity, which accountsfor the agents beliefs about the actual state of the economy obtains as

dt = tH + (1 t) L t dt + dZ0;t. (19)Proof: see Appendix

This process serves as the basis for stock valuation. It allows importing intothe agents valuation of rms equity the fact that the same rm-level protabilityobservation can lead to very dierent conclusions on underlying rm value, whenmade at dierent stages of the business cycle. Low protability is usually a badsign for rm productivity and valuation, but it becomes much less so when observedduring a recession, a period when the protability of the overall economy is low.

Given the agents belief about the state of the economy, the state-price densityobtains from pricing equation (12) where WT = BT = MT. The rst equality is themarket-clearing condition. The second, where BT

Ri2I

BiTdi, is a consequence ofperfectly competitive capital markets, leaving no abnormal earnings in equilibriumfor all i 2 I.

Proposition 1 For t 2 [0; t), the state-price density is given by

t = 1Bt

24

Et[T] expne

AH0 (T t) A1 (T t) to

+ (1 Et[T]) expn eAL0 (T t) A1 (T t) to35 (20)

whereEt[T] =

s + (t s) exp f ( + ) (T t)g . (21)eA0 (), for = H; L, and A1 () are given in the Appendix.Proof: see Appendix

Note that the state-price density obtains as a weighted average of the relevantstochastic discount factor for each regime of the economy. Note also, that the weightsare given by Et[T], the conditional expectation of the probability of the good state

of the economy for time T. This is a consequence of the agents investment problem.Taking dividend policy c as given, she allocates initial capital B0, as to maximizeexpected wealth at the end of her investment period. Hence, the only probabilitydistribution relevant to her investment problem, is that expected for time T. Thelatter obtains from the forward equation of t.

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3.2 Optimal Learning Allocations

The Separation Theorem17 provides the possibility of separating the ltering problemfrom the study of portfolio allocation within Markovian structures of incompleteinformation. While this result is not novel, so far, the estimation itself was not

associated with considerations of optimization. This changes with the present setup.The agents investment period being nite, her time to learn is limited. Combinedwith a continuum of rms to observe, this leads to a trade-o for the agent: learningmore on one rm means learning less on the others. I will show in the following,how empirical regularities in the cross section can arise from the optimizing behaviorinduced by such a trade-o, as soon as there is a bias in underlying rm fundamentals.

I assume that at some point in time t, where t 2 [0; T), the under (3) denedproductivity shocks spread over all rms i 2 I. This immediately bears on ltration.Observing protability, it, across rms i 2 I, the agent now simultaneously developsbeliefs about the state of the economy, as well as rms exposure to it, when trying

to infer the value of underlying rm-specic technology shocks.

The posterior belief about rm-specic shocks to productivity, bit, obtains for alli 2 I from the ltration generated by t and

it . The process followed by t is as

dened in Lemma 3, the dynamics of

it are as follows

d

it = (

tH + (1 t) L

+ i it)dt + 0dZ0;t + 1dZi ;t t t T (22)

with constant variances 0; 1 2 R++. The process contains two standard Brown-

ian motions. The rst, Z0;t, relates to systematic risk. The second, Z

i ;t, diersacross rms i 2 I, driven by rm-specic technology shocks.

Accounting for the above dened ltration, the posterior mean and variance ofrm-specic shocks to output growth obtains as follows

Lemma 4 Suppose that at t = t and for = H; L the prior distribution of theidiosyncratic shock to productivity of rm i 2 I for = ;;s;l is normal,i N

0;b2t. Then the posterior distribution of i at time t, t t T,

conditional onzt =n

t ;

it

: t t T

ois also normal, i jzt N

hb

it;

b2it

i,

where the posterior mean b

it follows the process

dbit = b2it 1 01

d eZi ;t (23)17 see Dothan and Feldman (1986) or Gennotte (1986), for example. More recently, Feldman (2005)

establishes a more general version in form of a state space representation theorem.

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where eZi ;t is a standard Brownian motion given in the Appendix.The posterior variance b2it is given by

b2it = "b2t + 1 012

(it t

)#1

(24)

where it 2 [0; t] denotes the aggregate time spent observing rm i between 0 andt.

Proof: see Appendix

The posterior variance, b2it, of rm i 2 I is a direct function of it, the aggregatetime having been spent learning about rm i between 0 and t. The higher the valueassociated with it, i.e. the more time has been spent learning about rm i, thelower the uncertainty surrounding expectations about the value of the underlying

shock to productivity. As the value function VBit; t;bit;b2it; T t of rms i 2I18 is decreasing in uncertainty of the posterior belief distribution, the reduction ofuncertainty positively aects agents expected utility from investment.

Due to the nite horizon of the investment problem, the agent will have to decidehow to optimally allocate the time at her disposition across rms i 2 I of the economy.Hence, ltration hides a maximization problem with respect to time, which can bedenoted as follows

maxit2[0;Tt]

Et B

1T

1 8 i 2 I, (25)where it is the aggregate time to be optimally allocated towards rm i 2 I,

between time t and T. The maximization problem is subject to the constraintRi2I

RTt

isdids T t for all t 2 [0; T], i.e. aggregate time spent learning can-not exceed the time being at the agents disposal.

At every instant t, the agent therefore decides on the optimal amount of time tobe allocated towards each rm i 2 I, given that she has time T t at her disposition.The solution to this optimization problem implies an optimal instantaneous learningratio between any two rms of the economy, which I denote by

it

jt

, where i6= j,

with i; j 2 I. This learning ratio obtains as a consequence of the fact that marginalutilities have to equate for an equilibrium to obtain. The agent therefore allocates

18 see equation (84) in the appendix

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her time such that expected marginal utilities from learning, i.e. @V()=@it, equalacross all rms i 2 I, at every instant t 2 [0; T] 19.

As long as no technology shocks have occurred, i.e. t 2 [0; t), the problem istrivial. As capital allocation is unbiased, the exact same amount of time is opti-

mally spent learning across rms i 2 I of the economy. For t 2 [t

; T], however,rm fundamentals exhibit a cross-sectional bias, as observed by Gertler and Gilchrist(1994), and Xing and Zhang (2005), respectively. As a consequence, and in imme-diate response, learning allocations will reect this bias. In fact, as soon as t = t,marginal utilities do no longer equal when time is allocated equally across rms i 2 I.Instead, marginal utilities from learning are now strictly higher for small and valuerms, ceteris paribus. This result would seem intuitive. In fact, small and valuerms are riskier as they are aected more strongly by negative business cycle shocks.Learning about such rms should therefore be more benecial, as the same amountof time invested will protect the agent from more negative "surprises" on average,than when allocated towards large and growth rms. In equilibrium, the risk-averse

agent therefore learns strictly more on small and value rms20.

For the sake of space, subsequent results are presented for value and growth rmsonly. They are, however, equally valid comparing small and large rms. Note that theoptimal learning ratio in the cross section obtains by aggregating

itjt

across i 2 ; sand j 2 ; l, respectively.

Proposition 2 The optimal learning ratio between value and growth rms8 t 2 [t; T], is given by

tt = Et [T] Ht

H

t

! + (1 Et [T]) Lt

L

t

! (26)= Et [T] + (1 Et [T]) t

L L

19 Example: The agent disposes of 4 days in an economy of 4 rms. Assuming that these rms

exhibit the same protability, it is optimal for the agent to allocate the same amount of time towardseach rm. This means 1 day in total for each rm, and implies the allocation of the same innitesimalunit of time towards each rm at very instant t. The latter is, of course, a theoretical implication,based on the presumption that time is inifnitely divisible. In practice, the agent could be thoughtof as starting by studying for half an hour each rms company report, before revisiting the decisionproblem again.

20 Generally speaking, the result that risk aversion would drive us towards learning more about

comparatively riskier "underlyings", seems like a rather close reection of reality. Taking a look ata typical daily newspaper or TV journal, for example, it would seem that the "worse" the event, themore we get informed about it.

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wheret

L L

> 1 (27)

and @t()@(LL)

> 0.

t

Ri2

itdi is the time t aggregate time to be

optimally spent learning about rms i

2, = ; between 0 and T

t.

Proof: see Appendix

Corollary 1 The optimal learning ratio

t

t

is strictly larger than one, and

increasing (decreasing) in contractions (expansions) 8 t 2 [t; T].

Accounting for Lemma 2, and the fact that t () > 1, the learning ratio is strictlylarger than one for all t 2 [t; T], i.e. conditional on a bias from fundamentals, theagent learns strictly more on value rms. Moreover, t ()


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are supportive for the above results, as small and value stocks will turn out to be thosewhose returns exhibit higher variability, as well as a higher correlation with marketreturns. On the other hand, the number of analysts following a company is found togrow in its size. This would seem to contradict above predictions. However, takinginto account that the size eect has disappeared, the contradiction is less pronounced.

Finally, Bhushan provides indicative evidence that industries with lower market-to-book ratios are followed by a larger number of analysts, which is what one wouldexpect given above results.

3.2.1 Learning and Valuation Ratios

Having established a systematic divergence in learning allocations in the cross section,I will now turn to determine its eects on rm valuation. The reduction of uncertaintythrough learning bears on valuation ratios.

Proposition 3 The market-to-book ratio of rm i is decreasing in it,the time spent learning about it, 8i 2 I, t 2 [0; T].

Proof: see Appendix

The above result was established by Pstor and Veronesi (2003), who derive anegative relation between the resolution of rm-specic uncertainty and the level ofvaluation ratios. The latter is a result of the convex nature of compounding. Higheruncertainty is equivalent to an increase in both the probability that an individualrms future growth rate of book equity will be persistently low, or persistently high.Because of risk aversion, the impact of the latter on valuations will be strictly higher,

which in turn increases expected future book values and ultimately todays market-to-book ratios. As learning reduces idiosyncratic variations surrounding expectationson rm fundamentals, it will lead to a decrease in valuation ratios.

Consequently, the equilibrium learning bias towards value rms translates into asystematic dierence in market-to-book ratios between value and growth rms: thevalue spread. The latter is expressed as the dierence in aggregate valuation ratios,where the aggregate (average) market-to-book ratio of rms i 2 I of type , where = ; , is denoted by

Mt

Bt R

i2MitBit

di.

Proposition 4 For t

2[t; t), the value spread obtains as

MtBt

Mt

Bt= (1 s (t s) exp f ( + ) (T t)g)

exp

AL0 (T t) + (1 ) A1 (T t) t

Et[T] Ht + (1 Et[T]) Lt t (28)

16


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where

t = exp

1

2(1 )2 A2 (T t)2b Lt 2

exp

1

2

(1

)2 A2 (T

t)2b L

t

2

, (29)t = exp

n eA0 (T t) A1 (T t) to for = H; L (30)and

MtBt

Mt

Bt> 0. (31)

A0 (), eA0 (), A1 (), A2 () are given in the Appendix, and Lt Ri2Lit di.Proof: see Appendix

Similarly, the learning bias towards small rms will translate into a size spread.For the sake of space, I will restrict myself to the value spread in the subsequentanalysis.

As can be seen from Proposition 4, the value spread is a direct function of the

equilibrium learning bias through

L

t

, the aggregate optimal time allocated to-

wards rms i 2 in the low state, L. As a consequence, the cross-sectional dierencein mean values of rm-specic productivity shocks causes a divergence in market-to-book ratios to the extent that it is reected in a systematic divergence in learning

allocations in equilibrium. Note also, that @M

t

Bt

M

t

Bt =@t > 0, for all , , . AsGDP growth, t, is increasing in troughs and decreasing at peaks, the value spreadis negatively correlated with business cycles and exhibits countercyclical dynamics, astylized fact pointed out by Cohen, Polk and Vuolteenaho (2003) 21.

The agent thus learns about output growth, but, so far, is still assumed unawareof its systematic bias in the cross section. I will now turn to study the eects of anawareness of systematic dierences in the reaction of rm fundamentals to recession-ary periods of the economy.

3.3 Learning about Systematic Risks in the Cross-SectionI assume that at some point in time t, where t 2 (t; T), the agent becomes awarethat output and earnings of small and value rms are systematically more adverselyaected by recessions. Her believed means are bL and bL, where bL > bL. For

21 see Footnote 5

17


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simplicity, I will assume that believed means equal actual ones L and L. Notehowever, that this does not imply that the agent eliminated uncertainty, which wasshown to be impossible. Again, for the sake of space, results are presented for valueand growth rms only.

For subsequent results to hold, I will have to make an assumption on the relationbetween the dierential in cross-sectional means of productivity shocks in state L,

and the corresponding dierence in optimal learning times of value rms,

L

t

,

and growth rms,

L

t

.

Assumption 1: The dierence in mean values of the underlying distribution oftechnology shocks between value and growth rms in state L, fullls the

following condition 8t 2 [t; T]

L

L 0. (36)

R ;;t, S ;;t and R ;;t, S ;;t are given in the Appendix.

Proof: see Appendix

The value premium obtains as a function of both, the cross-sectional means ofproductivity shocks in a recession, L and L, as well as the corresponding optimal

learning times Lt and Lt . It is important to note, however, that only givenAssumption 1 will there be a value premium. This has implications for the roleof learning in explaining the cross section of stock returns. Assuming the bias in

rm fundamentals to be directly observable, for example, b Lt 2 equals zerofor = ; , as there is no uncertainty left to be reduced. At the same time, asL L > 0, complete information violates Assumption 1. Such a situation wouldthen lead to a growth premium.

Hence, in the present model setup, learning proofs necessary for a value premiumto obtain. Moreover, calibrations show that the size of the value premium is highly

sensitive to the dierence in learning times, L

t and L

t , while the actual divergence inmean values, L and L, is of minor importance. This is interesting when comparedto ndings by Santos and Veronesi (2006), who point to growth premia arising insetups featuring habit formation. The authors show that cash ow risk alone isinsucient to counteract such premia. It would therefore seem that learning mightplay a potentially important role for the explanation of asset pricing anomalies in thecross section.

Corollary 2 The value (size) premium is countercyclical 8t 2 [t; T).

Proof: see Appendix

The countercyclicality of value and size premia was pointed out by Lettau andLudvigson (2001), and Campello, Chen, and Zhang (2006), respectively. Because ofits countercyclical nature, the value (size) premium then preserves individual assetsreturn dynamics through business cycles and consequently, the time series propertiesof their aggregate, the market portfolio. In the present model, the time-series behavior

19


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of the cross-section is driven by two sources. First, agents time-varying beliefs aboutthe state of economy lead to cyclicalities in risk premia. Second, these movements areamplied by a time-varying countercyclical learning ratio. Hence, one would expectthe magnitude of induced cyclicalities to be larger for the cross-section than for thetime-series, which is conrmed when calibrating the model.

Changes in return expectations for t 2 [t; T], will bear on portfolio allocations.Priors being initially unbiased, capital B0 was equally allocated across rms i 2 I.As the agent is updating her beliefs, she optimally reallocates capital across rms.However, as long as t < t these reallocations will not be in a systematic fashion,so that

Ri2

Bitdi Bt = B for t 2 [0; t). Once, however, that return expectationsdiverge in the cross section, misvaluations appear. In fact, (small) value rms becomeovervalued, as the market value of their equity does not account for their compara-tively higher downside risk from fundamentals. (Large) growth rms, on the otherhand, are undervalued. Consequently, at t = t, a cross-sectional bias in capital allo-cation emerges, such that Bt < B

t and B

st < B

lt for t

2[t; T]. The bias in capital

allocation will however only counteract part of the cross-sectional divergence in mar-ket values. In fact, in equilibrium, it exactly osets the dierence in cross-sectionalmarket values arising from expected return dierentials, such that the magnitude ofvalue (size) spreads remains unchanged.

Looking at results in conjunction, the return dynamics of the market portfolio -the equity premium - obtain as a function of changes in aggregate rm productivity,or GDP growth. Cross-sectional dierences in expected returns, as well as theirdynamics over time, are determined by time-varying systematic dierences in thecovariation of rms output with GDP growth.

The model therefore explains the empirical success of the Fama-French three fac-tor model (1993) as a consequence of its ability to have depicted factor-mimickingportfolios that capture rms time-varying exposure to business cycle uctuations.This exposure is split into variations with overall GDP growth, as captured by themarket portfolio, and additional, rm-specic exposure to recessionary periods of theeconomy. The latter is captured by the factors HML and SMB, as the output of smalland value rms displays a systematically stronger covariation with GDP growth forthese periods of the business cycle.

Additionally, results are obtained from a framework demonstrating that book-to-

market and size are associated with persistent dierences in protability, as found byFama and French (1995). The relation was established to exist only conditional onbook-to-market ratios being high or low, a nding supported by above results. Also,within the model, stock prices forecast the reversion of earnings growth, after rankingrms on size and book-to-market, just as observed by Fama and French, who ndmarket, size and book-to-market factors in earnings like those in returns.

20


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I will now turn to the implications of the obtained results for standard assetpricing models.

3.4 Conditional CAPM

The single risk factor of the economy are variations in GDP growth. These dene thebusiness cycle by means of regime switches in the overall mean value of rm produc-tivity. Moreover, changes in productivity during recessions dier systematically inthe cross section through dierences in mean values of the underlying distribution oftechnology shocks. Hence, returns arise in compensation for rms relative exposureto macroeconomic risk. The conditional beta representation captures this sensitivity.

Proposition 6 For t 2 [t; T], the cross sectional beta for rms i 2 denotes

t

covt

dRt ; dRmt

vart (dR

m

t )

=R ;;t

m

R;t

=

1e(Tt)

+ SR ;;t S ;;t

1e(Tt)

+ Sm

R;t Sm

;t

(37)

where SR ;;t, S ;;t and SmR;t, S

m;t are given in the Appendix.

Proof: see Appendix

Concerning the above expression, two things should be mentioned. First, as soonas variations in output of rms i 2 are stronger than those of overall GDP, i.e.R ;;t >

mR;t, the resulting beta is strictly larger than one. As higher variability is

induced by stronger downside risk from fundamentals, higher betas duly induce higherexpected returns. Second, the higher the risk from fundamentals of rms i 2 , thehigher their beta ceteris paribus. Consequently, as R;;t > R ;;t, a systematic

cross sectional divergence in betas between (small) value, i.e. high beta, and (large)growth, i.e. low beta, rms arises.

Corollary 3 The cross sectional dierential in betas, t t , is countercyclical8t 2 [t; T].

Proof: see Appendix.

Because of these dynamics in betas over time, a CAPM in unconditional form must"fail". Note however, that time-variation in the cross-sectional risk exposure accountsfor only part of the overall time-variation in the value premium. The remainingportion is due to a time-varying price of risk22. This nding is in agreement with

empirical evidence presented by Daniel and Titman (1997) who show that the rmcharacteristic "size" or "book-to-market" is still related to expected returns aftercontrolling for the stocks loading with respect to the mimicking portfolio.

22 As can be seen in section 4, only a very small portion of the value premium will be due todierences in risk exposures.

21


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3.5 YCAPM

The economys single risk factor are variations in output (Y) growth. Expectedreturns over and above the risk-free rate are determined by the covariation of rmoutput with changes in GDP growth. I introduce a YCAPM which diers from

the consumption CAPM (CCAPM) in that the latter denes risk as a function ofcovariation with consumption growth. The introduction of consumption into thepresent setup would immediately give way to consumption risk premia23. However,these would reect only part of the overall risk in the economy. Moreover, as c wasassumed exogeneous and constant, variability in consumption would obtain as a directfunction of variability in output. The latter then remains the single risk factor of theeconomy.

Proposition 7 For t 2 [0; T], the market (equity) premium obtains as

Et[Rmt ]

covt

dt

t; dRmt =

m;t

mR;t (38)

where

m;t =

1 e(Tt)

Sm;t

(39)

and

mR;t =

1 e(Tt)

+ SmR;t Sm;t

. (40)

Sm;t and SmR;t are as denoted in the Appendix.

Proof: see Appendix

Note, that as risk premia are determined by covariation with output growth, thecorresponding volatility that is found to be priced at the market level is . Theobtained denition of risk thus allows for the introduction of higher variability andtherefore the potential for higher levels of risk premia, than those commonly obtain-able from consumption-based frameworks. Moreover, the speed of mean reversionof productivity, , becomes a priced risk factor. Generally speaking, the lower thespeed of mean reversion, the higher the required risk premium, ceteris paribus. Cal-ibrations show that scaling risks from changes in output by , allows matching theequity premium with low levels of risk aversion.

Corollary 4 The equity premium is countercyclical 8t 2 [0; T].Proof: see Appendix

23 Recall that it suces to assume c 6= 0 in (9), for example.

22


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Again, agents time-varying beliefs about the state of the economy, lead to cycli-calities in risk premia. Countercyclicality in the time-series behavior of the marketpremium was pointed out by Campbell and Cochrane (1999), for example.

4 Calibration

The model is calibrated using the following parameter values for the mean-revertingprocess followed by productivity24 25: H = 0:011643, L = 0:003577, = 0:0951, = 0:245, and = 0:0328611. The rst four parameter values are maximum likeli-hood estimates of a two-state Markov-Switching Model applied to US real GDP datafor the period of 1952 to 1984. The annual volatility of real GDP growth, , obtainsfrom quarterly NIPA data for the period of 1967 to 2002. Volatility parameters as-sociated with idiosyncratic shocks to rm productivity,

bt, 0, and 1, are assumed

to equal 0:07, following Pstor and Veronesi (2005).

Data for the rst and second moments of asset returns in the time series and crosssection, as well as valuation ratios, obtain from the merged CRSP-COMPUSTATdatabase for the period of 1948 to 2001. CAPM s (Table IV) obtain running time-series regressions of excess returns on respective M/B-decile portfolios. The latter areconstructed following the standard procedure of Fama and French (1992). Returns onportfolios are from July of year t to June of year t+1. Data for the "growth" portfolioare based on averages of small and large high-M/B portfolios, data for the "value"portfolio are based on averages of small and large low-M/B portfolios. Results forthe zero-investment portfolio HML obtain from their dierential.

The remaining parameters of the model are the coecient of relative risk aversion, the speed of mean reversion of productivity (or GDP growth) , and the averageinvestment period . The latter is chosen to take values between 4 and 7, as Atkinsand Dyl (1997) report an average holding period of rms common stock of 4.01 yearsfor NYSE quoted rms over the period of 1975 to 1989, and of 6.99 years for Nasdaqquoted rms over the period of 1983 to 1991. The speed of mean reversion ischosen to vary between 0:1 and 0:4. Risk aversion adjusts in order to match thedata. Results in Table II to Table V are obtained assuming to equal 0:5, i.e. theagent is unbiased in her beliefs about the actual state of the economy.

As can be seen from Table II, concerning the time series, a coecient of rela-

tive risk aversion of 4 (combined with = 6 and = 0:1) allows obtaining a highequity premium of 8:8 percent, and annual equity return volatility of 14:83 percent.

24 see equation (4)25 Note that all results obtain in closed form, except for the derivative @t

@t

, which is approximated

via simulation.

23


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Consequently, the Sharpe ratio displays a reasonable 0:59. The model is thus ableto replicate main moments of the time series, maintaining a low coecient of riskaversion. While the expected return on the market portfolio is slightly higher thanobserved in the data, volatility is slightly lower. Varying parameters and showsthat volatility decreases in the speed of mean reversion of productivity (GDP growth).

Table III displays results for ratios from fundamentals of the time-series. Again,a low coecient of relative risk aversion of 4 (combined with = 7 and = 0:15),allows matching observed data.

Turning to the cross section, given that mean productivity of the bad state, L,obtained as 0:003577 from the Markov-Switching regression, cross-sectional meanvalues of rm-specic productivity shocks for state L, were chosen to equal 0:004 and0:003 for L and L, respectively. The corresponding cross-sectional divergencein optimal learning times is chosen as 10=1 for value rms in bad times. Concerningresults from calibration as displayed in Table V, two things should be mentioned.

First, compared to the time series, the coecient of relative risk aversion has to beincreased in order to match observed empirical moments of the cross section. Setting to 10, for example, leads to a value premium of 3:43 percent, with an average expectedreturn on value rms of 9:4 percent, and an average expected return on growth rmsof 5:97 percent. Second, the coecient of relative risk aversion needed to replicatethe value spread is strictly lower than the one necessary for the value premium to bematched. Hence, the model comes with a tradeo for the cross section: value spreadsand premia cannot be matched simultaneously. However, independently, the modelreplicates their size, as well as time variation in both empirical regularities.

Note also, that the magnitude of the value premium proofs sensitive to the choice

of learning ratio, while the actual divergence in cross-sectional means of the distribu-tion of technology shocks has much less of an impact ceteris paribus. Figure 5 showsthat for given parameter values, increasing the optimal learning ratio in state L, i.e.

L

t =L

t

, from 0 to 10, for example, induces a change in the size of the value

premium of approximately 3:4 percent. On the other hand, changing the dierentialin cross-sectional means from 0 to 0:02 percent, while keeping the learning ratio at aconstant 10, leads to variations of approximately 0:075 percent, only.

It is important to realize that in the present model, dierences in equity premiaacross rms arise as a function of both dierences in risk exposures, as well as dif-

ferences in risk prices. Going back to the above mentioned value premium of 3:43percent, for example, it can be seen from Table V, that it is associated with an un-conditional beta26 of1:01 for value rms, and 0:99 for growth rms. The market price

26 Note that I assume an unbiased agent, i.e. = 0:5, for all measures that are "unconditional".Another possibility would have been to take an average of calibrated parameter values for the period1948-2001, for example.

24


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of risk implied by parameter values = 10, = 0:35, and = 6 is 6:8 percent. It isthen straightforward to verify that only 2:62 percent of the dierential in expectedreturns of value rms and the market return is compensation for risk exposure, whilethe remaining part of the dierence is explained by a higher price of risk for theserms. For growth rms, the expected return is lower than the market premium, and

8:19 percent of this dierence is due to risk exposure27.

The model endogenizes time variation in risk premia, as well as valuation ratios.Comparing Figure 1 and Figure 4, it is apparent that time variation in the equitypremium is relatively small, while the one generated for the value premium is compar-atively important. Recall that for both premia time variation is induced by learning.However, while learning aects the time series through time-varying beliefs about thestate of the economy only, in the cross section, it takes a second eect through timevariation in the learning ratio. In the present model context, learning therefore notonly accounts for most of the absolute size of the value premium, it also gives rise toimportant time variation across business cycles. The same holds true for the value

spread. One is therefore inclined to conclude that learning might be a particularlyimportant explanatory factor for regularities in the cross section.

Note that the induced movements in empirical regularities in the time series andthe cross section coincide with NBER recessions. While on average, displayed vari-ations are at slightly higher frequencies than business cycles, premia peak in allrecessions28.

Comparing the present model to models generating time-variation in risk premiathrough habit formation, it seems noteworthy that in the present context relativerisk aversion is comparatively low and constant. Campbell and Cochrane (1999),for example, report levels of risk aversion of 60 and higher to match the equitypremium. Moreover, relative risk aversion implied by models of habit formationis countercyclical. This in turn presumes that economic conditions impact agentsattitude towards risk.

Finally, in view of the interest rate risk puzzle, it is worth studying the modelsimplications for the risk-free rate, which so far had been normalized to zero for sim-plicity. Application of Its lemma to the state-price density of Proposition 1 lead toan implied risk-free rate that denotes

27 I would like to thank Lars Hansen for suggesting to decouple risk exposure and risk price.

28 note also that the value spread as well as the Sharpe Ratio are countercyclical

25


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rf;t =

1 e(Tt)

S;;t

tH + (1 t) L

t (41)1 e

(Tt)

2 1 e(Tt)

S;;t 2 S;t1;t S;t2;twhere S;;t, S;t1;t, and S;t2;t, are as denoted in the Appendix

29. For the periodof 1948-2001, calibrations yield a long-run risk-free rate of 1:06%, and an annualvolatility of 4:79%, in comparison to an actual average rate of the 3-month TreasuryBill of 1:44%, and an average standard deviation of 3:08%. This result leads to twoconclusions with respect to the interest rate risk puzzle. First, the model comes closeto inducing observed interest rate volatility, while matching the equity premium.Second, in terms of the level of the risk-free rate, the puzzle seems slightly reversed.While consumption-based models are known to imply far too high risk-free rates when

matching the equity premium, the present model instead implies a negative long-runrate.

5 Conclusion

The proposed general equilibrium model relates variations in GDP growth to riskpremia in equity markets. It establishes variability in output as a priced risk fac-tor, and reveals the potential signicance of learning for the understanding of assetpricing anomalies. The model justies equity premia of the time series and the cross

section qualitatively, as well as quantitatively. Risk premia rationalize as they arisein compensation for exposure to non-diversiable macroeconomic risk, their cyclicalbehavior obtains endogenously thanks to the inference structure of the model. TheFama and French (1993) factors are given economic content in representing the factor-mimicking portfolios of rms time-varying relative exposure to recessionary periodsof the economy. Finally, the model shows how the conditional CAPM combines inone, what the "unconditional" Fama and French model splits into three dierentfactor exposures.

29 see the second part to the Proof of Proposition 1.

26


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References

[1] Atkins, A. B., and Dyl, E. A., 1997, Transaction costs and holding periods forcommon stocks, Journal of Finance, Vol. LII, No.1

[2] Berk, J., Green, R., and Naik, V., 1999, Optimal Investment, growth options,and security returns, Journal of Finance 54, 1153-1607

[3] Bhushan, R., 1989, Firm characteristics and analyst following, Journal of Ac-counting and Economics, vol. 11, Issues 2-3, 255-274

[4] Brennan, M. J. and Xia, Y., 2001, Stock price volatility and equity premium,Journal of Monetary Economics, 47, 249-283

[5] Campbell, J.Y., and Cochrane, J.H., 1999, By Force of Habit: A consumption-based explanation of aggregate stock market behavior, Journal of Political Econ-omy, vol. 107, no. 2

[6] Campello M., Chen L., and Zhang L., 2006, Expected Returns, Yield Spreads,and Asset Pricing Tests, working paper, University of Michigan

[7] Carlson, M., Fisher, A., and Giammarino, R., 2004, Corporate Investment andasset price dynamics: Implications for the cross-section of returns, forthcoming,Journal of Finance

[8] Cheung, Y.-W., and Chinn, M. D., 1996, Deterministic, Stochastic, and Seg-mented Trends in Aggregate Output: A Cross-Country Analysis, Oxford Eco-nomic Papers 48: 134-162

[9] Cochrane, J., 2006, Financial Markets and the real economy, working paper,University of Chicago

[10] Cochrane, J., 1991, Production-based asset pricing and the link between stockreturns and economic uctuations, Journal of Finance 46, 207-234

[11] Cohen, R., Polk, C. and Vuolteenaho, T., 2003, The Value Spread, The Journalof Finance, Vol. LVIII, No. 2

[12] Cooper, I., and Priestley, R., 2007, Time-varying risk premia and the outputgap, working paper, BI Norwegian School of Management

[13] Daniel, K., and Titman, S., 1997, Evidence on the characteristics of cross-sectional variation in stock returns, Journal of Finance, 52, 1-33

[14] David, A., 1997, Fluctuating Condence in Stock Markets: Implications forReturns and Volatility, The Journal of Financial and Quantitative Analysis, Vol.32, No. 4, pp. 427-462

27


28/57

[15] Detemple, J., 1986, Asset pricing in a production economy with incompleteinformation, Journal of Finance, 41, 383-390

[16] Dothan, M. U. and Feldman, D., 1986, Equilibrium Interest Rates and Multi-period Bonds in a Partially Observable Economy, The Journal of Finance, Vol.

XLI, No. 2

[17] Fama, E. and French F., 1992, The cross-section of expected stock returns, Jour-nal of Finance 48, 1927-1942

[18] Fama, E. F., and French, K. R., 1993, Common risk factors in the returns onstocks and bonds, Journal of Financial Economics 33, 3-56

[19] Fama, E. F., and French, K. R., 1995, Size and Book-to-Market Factors inEarnings and Returns, Journal of Finance, 50, 131-155

[20] Fama, E. F., and French, K. R., 2000, Forecasting protability and earnings,

Journal of Business 73, 161-175

[21] Feldman, D., 2005, Incomplete Information Equilibria: Separation TheoremsAnd Other Myths, Annals of Operations Research, forthcoming

[22] Gala, V. D., 2006, Investment and Returns, working paper, University of Chicago

[23] Gennotte, G., 1986, Optimal Portfolio Choice Under Incomplete Information,Journal of Finance, 41, 3

[24] Gertler, M. and Gilchrist S., 1994, Monetary Policy, Business Cycles, and theBehavior of Small Manufacturing Firms, The Quarterly Journal of Economics,

Vol. 109, No. 2, pp. 309-340

[25] Gomes, J., Kogan, L., and Zhang, L., 2003, Equilibrium cross section of returns,Journal of Political Economy 111 (4), 693-732

[26] Grossman, S., Melino, A., and Shiller R. J., 1987, Estimating the Continuous-time Consumption-based Asset-Pricing Model, Journal of Business and Eco-nomic Statistics 5, 315-328

[27] Hamilton, J. D., 1989, A new approach to the economic analysis of nonstationarytime series and the business cycle, Econometrica, Vol. 57, No. 2, pp. 357-384

[28] Hansen, L. P., 2007, Beliefs, Doubts and Learning: Valuing Macroeconomic Risk,American Economic Review, Vol. 97, No. 2

[29] Hansen, L. P., and Sargent T., 2007, Fragile Beliefs and the Price of ModelUncertainty, working paper, NYU

28


29/57

[30] Hansen, L. P., and Singleton, K. J., 1983, Stochastic Consumption, Risk Aver-sion, and the Temporal Behavior of Asset Returns, Journal of Political Economy91, 249-268

[31] Judd, Kenneth L., 1985, The Law of Large Numbers with a Continuum of IID

Random Variables, Journal of Economic Theory, 35

[32] Karlin, S., and Taylor, H. M., 1975, A First Course on Stochastic Processes,Academic Press, New York

[33] Kogan, L., 2004, Asset prices and real investment, forthcoming, Journal of Fi-nancial Economics

[34] Lettau M. and Ludvigson S., 2001, Resurrecting the (C)CAPM: A cross-sectionaltest when risk premia are time-varying, Journal of Political Economy, vol. 109(6), 1238-1287

[35] Lettau, M., and Wachter, J., 2005, Why is long-horizon equity less risky? Aduration-based explanation of the value premium, working paper, NYU, SternSchool of Business

[36] Liew, J., and Vassalou, M., 2000, Can book-to-market, size and momentum berisk factors that predict economic growth?, Journal of Financial Economics 57,221-45

[37] Liptser, R.S and Shiryayev, A.N., 1977, Statistics of Random Processes I, II,Springer-Verlag, N.Y.

[38] Mehra, R., and Prescott, E., 1985, The Equity Premium: A Puzzle, Journal ofMonetary Economics 15, 145-161

[39] Mller, C., 2008a, Dynamic Asset Allocation under Macroeconomic Uncertaintyand Learning, working paper, HEC, School of Management

[40] Mller, C., 2008b, Macroeconomic Fluctuations, Comovements, and Return Pre-dictability in Asset Markets, working paper, HEC, School of Management

[41] Nelson, C. R., and Plosser, C. I., Trends and random walks in macroeconomictime series, Journal of Monetary Economics 10, 1982, 139-162

[42] Pakos, M., 2001, Mean reversion in protability measures, working paper, Uni-versity of Chicago

[43] Pstor L. and Veronesi P., 2003, Stock Valuation and Learning about Protabil-ity, Journal of Finance 58, 5

29


30/57

[44] Pstor, L., and Veronesi, P., 2005, Technological Revolutions and Stock Prices,working paper, University of Chicago

[45] Penman, S. H., 1991, An evaluation of accounting rate-of-return, Journal ofAccounting, Auditing, and Finance 6, 233-256

[46] Petkova, R., and Zhang, L., 2004, Is Value Riskier Than Growth?, forthcoming,Journal of Financial Economics

[47] Rangvid, J., 2006, Output and expected returns, Journal of Financial Eco-nomics, 81, 595-624

[48] Rapach, D. E., 2002, Are real GDP levels nonstationary? Evidence from paneldata tests, Southern Economic Journal 68(3), 473-495

[49] Santos, T., and Veronesi, P., 2006, Habit formation, the cross-section of stockreturns and the cash-ow risk puzzle, working paper, University of Chicago

[50] Veronesi, P., 1999, Stock market overreaction to bad news in good times: arational expectations equilibrium model, Review of Financial Studies, Vol. 12,No. 5, pp. 975-1007

[51] Xing Y. and Zhang L., 2005, Value versus Growth: Movements in EconomicFundamentals, working paper, University of Rochester

[52] Zhang L., 2005, The value premium, Journal of Finance 60 (1), 67-103

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6 Appendix

6.1 Proof of Lemma 3

Aggregate productivity follows the process

dt = ( t) dt + dZt. (42)Replacing into dZ0;t from Lemma 1, I obtain

dZ0;t =1

( ( E[jzt])dt) + dZt

which - when plugged back into the initial expression - yields

dt dt = (E[jzt] t)dt + dZ0;t (43)Q.E.D.

6.2 Proof of Lemma 4

The process followed by the agents posterior belief about rm-specic shock bit,8 i 2 I, where = ;;s;l for = H; L, is conditional on the ltration zt =n

t ;

it

: t t T

owith processes

dt = (

tH + (1 t) L

t)dt + dZ0;t (44)and

d

it = (

tH + (1 t) L+ i it)dt + 0dZ0;t + 1dZi ;t. (45)

Applying results from Liptser and Shiryaev (1977), the process for bit = Et [i ]can be expressed as

dbit = b2itC0 (P 0)1 deZt (46)where C = (0; )0,

eZt =

eZ0;t;

eZi ;t

0

and

P = 00 1 ,where the process followed by eZt is given by

deZt = P 1 dtd

it

Et

dtd

it

.

31


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The process followed by the posterior variance denotes

db2itdt

= b

2it2C0 (PP

0)1C. (47)

Simple algebra then yields

C0 (P

0)1

=

0;

1 0

1

(48)

which when replaced into (46) yields

dbit = b2it 1 01

d eZi ;t. (49)Next, I obtain

g

C0 (PP 0)

1C =

1 0

12

(50)

which when replaced into (47) yields

db2itdt

= b2it2 g, (51)a Riccati dierential equation with solution

b2it = b2t + g (it t)1 (52)where it 2 [0; t] denotes the aggregate time spent observing rm i between 0 and

t.

Q.E.D.

6.3 Lemma A1

For t 2 [t; T], conditional on regime = H; L, the original processes can be writtenas follows,

dbit = t dt (53)

d

t = +bit t dt + d eZ0;t (54)dbit = b2it 1 01

d eZi ;t (55)

db2it = b2it2 g dt (56)32


33/57

where bit ln[Bit] 8 i 2 I, and = ;;s;l.

The conditional value function obtains as

VBit; t ;bit;b2it; T t = Et "(BiT)11 j = H; L# (57)=

B1it1 e

A0(Tt)+(1)A1(Tt)t+(1)A2(Tt)bit+ 12 (1)2A2(Tt)2b2it

where

A0 (T t) = (1 ) A2 (T t)+2

2

(1 )22

(T t) + 1 e

2(Tt)

2 2A1(T t)

(58)

A1 (T t) = 1 e(Tt)

A2 (T t) = (T t) A1 (T t) . (59)

6.3.1 Proof of Lemma A1

Given that

V

bit;

t ;

b

it;

b2it; t

=

1

1

E

he(1)

RTt

bisds j zt

ione is looking for a function ft ;bit;b2it; t, such thatI) f

t ;bit;b2it; t e(1) Rt0 bisds = Xit

is a martingale, i.e. E[XiTjzt] = Xit,andII) f

T;biT;b2iT; T = 1 8T;biT;b2iT.

Hence, XiT = e(1)

RT0

bisds and Xit = Eh

e(1)RTt

bisdsjzti

.

Applying Its Lemma, assuming eZ0;t to be uncorrelated with eZi ;t8i 2 I for eachregime = H; L, one obtains that

df

t ;bit;b2it; t =

(f

+bit t f2 b2it2 g+ft +

12

f2 + 1

2fb2it2 g

)dt

33


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+fd eZ0;t + fb2it 1 01

d eZi ;twhere fx and fxx denote the respective rst and second order partial derivatives,

where x = ; ;2; t.

Knowing that E[XiTjzt] = Xit with Xit as dened under I), one can write that

f

+bit t f2 b2it2 g + ft + 12f2 + 12f b2it2 g = 0.I then conjecture a solution of the form

f

t ;bit;b2it; t = e(1)[a(t)+c(t)t+d(t)bit;+h(t)b2it] 8t ;bit;b2it.

Applying the method of undetermined coecients and rearranging terms, yieldsthe solution as denoted in (57), which in turn can be veried to fulll the above PDE.

Q.E.D.

6.4 Proof of Proposition 1

As the Proof of Lemma A1, replacing 1 by , as t = 1Et

BT

. As pro-ductivity shocks i are idiosyncratic, their market-level aggregate equals zero, i.e.Ri2I

i di = 0 for = H; L. The economy as a whole therefore grows at rate t, suchas dened in (2). Hence, I obtain that

t Et "BT

Bt

j = H; L# = e eA0(Tt)A1(Tt)t (60)where

eA0 (T t) = A2 (T t) + 22 22

+1 e2(Tt)

2 2A1 (T t)

.

The state-price density is given by

t = 1Et

BT

= 1

nEt [T] Et

hBHT

i+ (1 Et [T]) Et

hBLT

io(61)In order to obtain E[T j zt], I integrate dt from Lemma 1, obtaining

T = t +

ZTt

( + ) (s u) du +ZTt

h (u) dZ0;u. (62)

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35/57

It follows that

E[T j zt] = t +ZTt

( + ) (s E[u j zt]) du. (63)

Assuming that t is xed, and denoting T = E[T j zt], where t = t, then

T = t +ZTt

( + ) (s u) du, (64)

@T@T

= ( + ) (s T) , (65)and it follows that

T = E[T j zt] = s + (t s) e(+)(Tt). (66)The state-price density then obtains as

t = 1Bt s + (t s) e(+)(Tt) Ht + 1 s + (t s) e(+)(Tt) Lt (67)

Q.E.D.

The risk-free rate rf obtains from applying Its lemma to t, and is given by

rf;t =

1 e(Tt)

S;;t

tH + (1 t) L

t (68)S;t1;t S;t2;t

1 e(Tt)

2

1 e(Tt)

S;;t

2

where

S;;t =

@Et[T]@t

Ht Lt

Et [T] Ht + (1 Et [T]) Lt

, (69)

S;t1;t =@Et[T]

@t

Ht Lt


, (70)

and

S;t2;t =Et [T]

@Ht@t

+ (1 Et [T]) @Lt

@t

Et [T] Ht + (1 Et [T]) Lt. (71)

@Et[T]@t = @t

@te(+)(Tt), as can be veried from (66), @Et[T]@t , and @

t@t for = H; L,

obtain as easily, and are neglected for the sake of space. Note, that as rf was assumednormalized to 0, there will be no drift term when applying Its lemma to t insubsequent Proofs.

Q.E.D.

35


36/57


For t 2 [0; t), rm productivity does not dier across rms i 2 I. Consequently, itis optimal for the agent to equally allocate her time across the continuum of rms.Subsequent derivations are then for t 2 [t; T].

Let me denote

V

Bit; t ;bit;b2it; T t = B1it1 Ft ;bit;b2it; T t . (72)

The time t optimal amount of time (it) 2 [0; T t], to be spent observing rm

i in regime , needs to satisfy the following optimality condition for = H; L

@

@it F

t ;

b

it;

b (it)

2 ; T t

= 0. (73)

Recalling the expression forb2it from (52), the marginal utility from learning aboutrm i 2 I during regime obtains as

@Fi ()@it

= 12

(1 )2 A2 (T t)2 g

(bt)2 + g (it t)2 Fi () (74)where

Fi () F

t ;bit;b (it)2 ; T t 8i 2 I. (75)

The equilibrium condition for optimal learning allocations is then given by

@Fi ()@it

= @Fj ()@jt

8i 6= j with i; j 2 I, (76)

which simplies to

F

t ;bit;b ((it))2 ; T t

F

t ;bjt;b jt2 ; T t =

(bt)2 + g (it)2(bt)2 + g jt2 8i 6= j with i; j 2 I

(77)where (it)

2 [0; T t] denotes the time t optimal time to learn about rm i 2 Ifor t 2 [t

; T] for = H; L.

For the sake of space, I will restrict myself to the comparison of value rms, i 2 ,and growth rms, i 2 . In order to obtain an expression for the aggregate (average)ratio of optimal learning times of both types of rms, I need to aggregate both sides of(77) across rms of each type. For the RHS it suces to replace it by

t Ri2

itdi

36


37/57

and jt by

t Rj2

jtdj. The LHS is slightly more complex. I will apply a law

of large numbers in strong form30 by means of the Glivenko-Cantelli Theorem. Itfollows from the latter that the cross-sectional distribution of the i.i.d. shocks to rmproductivity equals its stationary distribution. Accounting for (3), (5), and (6), thestationary distribution for and = H; L, is given by

f

; ;b t = 1b t p2 exp0B@ 2

2b t 21CA . (78)

Denoting F () Ri2

F () di, I can then write that

F () =Zi2

eA0(Tt)+(1)A1(Tt)t+(1)A2(Tt)

bit+ 12 (1)2A2(Tt)2b((it))2di

= eA0(Tt)+(1)A1(Tt)t

Zi2

e(1)A2(Tt)bit+ 12 (1)2A2(Tt)2b((it))2di, (79)

which can be reexpressed as

F () = eA0(Tt)+(1)A1(Tt)tZi2

Eth

e(1)A2(Tt)biti di. (80)

Applying the Glivenko-Cantelli Theorem, I can write that

F () = eA0(Tt)+(1)A1(Tt)t Z10

Et he(1)A2(Tt)bt i f; ;b t dt= eA

0 (Tt)+(1)A1(Tt)t E

hEt

he(1)A2(Tt)

bt ii (81)which by the law of iterated expectations yields

F() = eA0(Tt)+(1)A1(Tt)t+(1)A2(Tt)+ 12 (1)2A2(Tt)2b

t

2. (82)

As L < L, it follows that a priori, that is when

t =

t , for = H; L, as

> 1 and A2 (T t) > 0,F

L

()FL () = e

(1)A2(Tt)(LL) < 1. (83)

30 see Judd (1985) for a proof of existence for a continuum of i.i.d. random variables

37


38/57

As@[b2t +g(itt)]1

@it< 0, it follows that

@Ft ;bit;b(it)2;Tt

@it< 0, 8i 2 I, = H; L.

Condition (77) then implies that

L

t

>

L

t

8 t 2 [t; T] for an equilibrium toobtain. As H = H, F

H () =FH () = 1 and hence

H

t

=

H

t

8t 2 [t; T].

Given the conditional value function of (57), recalling that initial capital allocationis unbiased for = H; L, the unconditional value function can be expressed as

V

Bit; t;bit;b2it; T t (84)=

B1it1

hEt [T] F

Ht ;bHit ;b2it; T t+ (1 Et [T]) FLt ;bLit;b2it; T ti .

I can then express the optimal learning ratio between value and growth rms as

tt

= Et [T]

Ht

H

t

!+ (1 Et [T]) Lt

L

t

!(85)

= Et [T] + (1 Et [T]) t

L L

where t () > 1 8 t 2 [t; T]. Note also that @t()@(LL) > 0.Q.E.D.


The market value of rm i 2 Ican be expressed as Mit = Et [BiTT] =t =

1Et B1iT =t,where Et

B1iT

= Et [T] Et

hBHiT1i

+ (1 Et [T]) Eth

BLiT1i

. The market-

to-book ratio of rms i 2 I then obtains as

MitBit

= Et[T] F

Ht ;bHit ;b Hit 2 ; T t

Et[T] Ht + (1 Et[T]) Lt

+ (1 Et[T]) F

Lt ;bLit;b

Lit2

; T t

Et[T] Ht + (1 Et[T]) Lt, (86)

where

F

t ;bit;b ((it))2 ; T t

= eA0(Tt)+(1)A1(Tt)t+(1)A2(Tt)

bit+ 12 (1)2A2(Tt)2b((it))2. (87)38


39/57

I then aggregate market-to-book ratios across rms i of type , deningM

t

BtR

i2MitBit

di, whereRi2

Bitdi = Bt = Bt, given that capital allocation is unbiased. As

in the Proof of Proposition 2, I will make use of the Glivenko-Cantelli Theorem. Notehowever, that the stationary distribution in the present application diers slightlyfrom (78). The reason is that true mean values are replaced by the agents unbiasedprior expectations on idiosyncratic technology shocks. Hence, = 0 for = H; L.The stationary distribution is then given by

f

; 0;b t = 1b t p2 exp0B@ ()2

2b t 21CA . (88)

I can therefore now write that

Mt

Bt = Et[T] Et[T] Ht + (1 Et[T]) Lt 1Zi2

eAH0 (Tt)+(1)A1(Tt)t+(1)A2(Tt)

bHit+ 12 (1)2A2(Tt)2b((Hit))2di+ (1 Et[T])


1

Zi2

eAL0 (Tt)+(1)A1(Tt)t+(1)A2(Tt)

bLit+ 12 (1)2A2(Tt)2b((Lit))2di (89)

which I reexpress as

M

t

Bt= Et[T] Ht + (1 Et[T]) Lt 1 (90)24 Et[T] eAH0 (Tt)+(1)A1(Tt)t Ri2Et heA2(Tt)bHit i di

+ (1 Et[T]) eAL0 (Tt)+(1)A1(Tt)t

t

Ri2

Et

heA2(Tt)

bLiti di35 .

Applying the Glivenko-Cantelli Theorem, I obtain

Mt

Bt= Et[T]

Ht + (1

Et[T])

Lt

1

266664Et[T] eAH0 (Tt)+(1)A1(Tt)t

R10

Et

heA2(Tt)

bHt i fH; 0;b Ht dH+ (1 Et[T]) eAL0 (Tt)+(1)A1(Tt)t

R10 Et

heA2(Tt)

bLt i fL; 0;b Lt dL

37777539


40/57

=

Et[T] Ht + (1 Et[T]) Lt1

24Et [T] eAH0 (Tt)+(1)A1(Tt)t E

hEt

heA2(Tt)

bHtii+ (1 Et [T]) eAL0 (Tt)+(1)A1(Tt)t EhEt heA2(Tt)bLt ii 35 . (91)

Applying the law of iterated expectations, accounting for the under (88) dened

stationary distribution f

; 0;b Ht and simplifying, I obtain thatMtBt

= Et [T] eAH0 (Tt)+(1)A1(Tt)t+

12(1)2A2(Tt)

2bHt2


+ (1 Et [T]) eAL0 (Tt)+(1)A1(Tt)t+

12(1)2A2(Tt)

2

bL

t

2

Et [T] Ht + (1 Et [T]) Lt(92)

As optimal learning allocations between value and growth rms only dier during

bad times, that is

L

t

>

L

t

8t 2 [t; T], the value spread obtains as

MtBt

Mt

Bt=

(1 Et[T])Et[T] Ht + (1 Et[T]) Lt

eAL0 (Tt)+(1)A1(Tt)t (93)

e12 (1)2A2(Tt)2bLt 2 e 12 (1)2A2(Tt)2bLt 2! .It follows that as Ht Lt < 0, @M

t

Bt=@t@M

t

Bt=@t > 0, 8;; . As GDP growth,

t, is decreasing at peaks and increasing in troughs, the value spread is negativelycorrelated with business cycles.

Q.E.D.


As@b(it)2

@it= < 0 8 i 2 I ; = H; L, the result follows given that @

MitBit

=@b (it)2 > 0

8i 2 I, = H; L.Q.E.D.

40


41/57


At t = t, where t 2 (t; T), the agent is assumed to become aware of the dierencein cross-sectional means of the underlying distribution of technology shocks duringrecessions. Consequently, for = H, the stationary distribution remains as denoted

in (88), whereas for = L, it is the one of (78).In order to obtain an expression for the value premium, I now turn to derive

return expectations for rms i 2 . These obtain from the standard pricing equationt covt

d;t;t

; dRt

, where Rt denotes the return on rms i 2 , and ;t their

state price density. Given the agents knowledge of , the latter, is given by

;t = 1B;t

Et[T] H;t + (1 Et[T]) L;t

(94)

where

;t = e eA0(Tt)A1(Tt)tA2(Tt)

for = H; L (95)and B;t

Ri2

Bitdi.

Applying Its Lemma, the following dynamics for ;t obtain

d;t;t

= ;;td eZ0;t ;;td eZ;t (96)where

;;t =

A1 (T t) S ;;t

(97)

with

S ;;t =

@Et[T]@t

H;t L;tEt[T] H;t + (1 Et[T]) L;t

, (98)

and

;;t = S ;;tb t2 1 01

(99)

where

S ;;t =(1 Et[T]) A2 (T t) L;t

Et[T] H;t + (1 Et[T]) L;t. (100)

Applying now Its Lemma to Mt = EthBT;Ti =;t, the return process for

rms i 2 obtains as follows

dRt dMt

Mt= t dt + R ;;td eZ0;t + R ;;td eZ;t (101)

41


42/57

wheret = ;;t R ;;t + ;;t R ;;t (102)

withR ;;t =

A1 (T t) + SR ;;t S ;;t

, (103)

where

SR ;;t (104)

=

@Et[T]@t

F

Ht ; 0;b Ht 2 ; T t FLt ; L;b Lt 2 ; T t

Et[T] F

Ht ; 0;b Ht 2 ; T t+ (1 Et[T]) FLt ; L;b Lt 2 ; T tand

R

;;t = SR ;;t + S ;;tb t2 1 01 (105)

where

SR ;;t (106)

=

(1 ) (1 Et[T]) A2 (T t) F

Lt ; L;b Lt 2 ; T tEt[T] F

Ht ; 0;

b

H

t

2

; T t

+ (1 Et[T]) F

Lt ; L;

b

L

t

2

; T t .

where

FH

() F

Ht ; 0;b Ht 2 ; T t (107)= e

AH0 (Tt)+(1)A1(Tt)t+12(1)2A2(Tt)

2bHt2

and

FL

() F

Lt ; L;b

L

t

2

; T t

(108)

= eAL0 (Tt)+(1)A1(Tt)t+(1)A2(Tt)L+

12(1)2A2(Tt)

2bLt 2.

For t 2 [t; T], the value premium, i.e. the dierence between expected returnsof value and growth rms, is then given by

42


43/57

Et[Rt ] Et[Rt ] = t t

=

;;t R;;t ;;t R;;t

+ ;;t R;;t ;;t R ;;t . (109)For the value premium to be strictly positive, I make the following assumption,

8t 2 [t; T],

L L 0, t is strictly increasing in t.Q.E.D

43


44/57

6.9 Proof of Corollary 2

The value premium is given by

t t

;;t R;;t ;;t R;;t

(113)

+ ;;t R;;t ;;t R;;t .We have that ;;t > ;;t and given that

t t > 0 (see assumption in

the Proof of Proposition 5), it follows that R;;t > R ;;t. Functional analysis

shows that given estimated parameter values (Table I),@;;t

@t> 0, and

@;;t

@t> 0,

8;; . Moreover, for values of ;; , that give rise to a value (size) premium, i.e.t t > 0, st lt > 0, it follows that for = ; s,

@R;;t

@t> 0, and

@R;;t

@t< 0,

and for = ; l,@

R;;t

@t< 0, and

@R;;t

@t> 0. It then follows that for values of

;; , that give rise to a value (size) premium,@

t

@t

< 0, for = ; l, and@

t

@t

> 0,

for = ; s. Hence, @

t t =@t > 0 for values of ;; , where t t > 0. AsGDP growth, t, is increasing in troughs and decreasing at peaks, the value premiumis negatively correlated with business cycles. The same holds for the size premium.

Q.E.D


As technology shocks are idiosyncratic at the market level, the state price density,mt , for all t

2[0; T], denotes

mt = 1Bt


(114)

where

t = Et

"BTBt

j = H; L#

= eeA0(Tt)A1(Tt)t for = H; L. (115)

Hence, mt = t, and applying Its Lemma, the dynamics for t obtain as follows

dt

t = m

;td eZ0;t, (116)where

m;t =

A1 (T t) Sm;t

(117)

with

44


45/57

Sm;t =

@Et[T]@t

Ht Lt


. (118)

Applying Its Lemma to Mmt

= Et [BTT] =t, where Mm

t Ri2I Mitdi is theaggregate market value, the return process of the total wealth portfolio obtains asdRmt

dMmtMmt

= mt dt + mR;tdeZ0;t (119)

where

mt = m;t mR;t, (120)

and

mR;t = A1 (T t) + SmR;t Sm;t (121)where

SmR;t =

@Et[T]@t

F

Ht ; 0; 0; T t FLt ; 0; 0; T t

Et[T] F(Ht ; 0; 0; T t) + (1 Et[T]) F (Lt ; 0; 0; T t)(122)

with

F(t ; 0; 0; T t) = eA0(Tt)+(1)A1(Tt)t for = H; L. (123)

Recall that the return process for rms i 2 is given by

dRt = t dt + R ;;td eZ0;t + R ;;td eZ;t. (124)

The cross-sectional beta then obtains as

t covt

dRt ; dR

mt

vart (dRmt )

=mR;t R ;;t

mR;t

2 (125)

= R

;;t

mR;t= A1(T t) + SR

;;t S ;;tA

economic growth, business cycles, and expected stock returns

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