paper the value premium

counter-By linking risk and expected return to economic primitives my model vides a unified framework to rationalize many empirical regularities in thecross-section of returns in relatio

The Value Premium

WHY DO VALUE STOCKS EARN HIGHER EXPECTED RETURNSthan growth stocks? This pears to be a troublesome anomaly for rational expectations, because according

ap-to conventional wisdom, growth options hinge upon future economic conditionsand must be riskier than assets in place In a widely used corporate financetextbook, Grinblatt and Titman (2001, p 392) contend that “Growth opportuni-ties are usually the source of high betas, , because growth options tend to be

most valuable in good times and have implicit leverage, which tends to increasebeta, they contain a great deal of systematic risk.” Gomes, Kogan, and Zhang(2003) also predict that growth options are always riskier than assets in place,

as these options are “leveraged” on existing assets Growth stocks, which derivemarket values more from growth options, must therefore be riskier than valuestocks, which derive market values more from assets in place Yet, historically,growth stocks earn lower average returns than value stocks

I investigate how risk and expected return are determined by economic itives, such as tastes and technology, in the neoclassical framework with ratio-nal expectations and competitive equilibrium (e.g., Kydland and Prescott (1982)and Long and Plosser (1983)) A workhorse of many fields of economics, thisframework has been under strenuous attack in finance (e.g., Shleifer (2000))

prim-∗William E Simon Graduate School of Business Administration, University of Rochester Thispaper is based on chapter three of my doctoral dissertation at the Wharton School of the University

of Pennsylvania I thank my advisors Andrew Abel, Craig MacKinlay, Amir Yaron, and especially Joao Gomes for their training and inspiration I also acknowledge helpful comments from Michael Brandt, Domenico Cuoco, Kent Daniel, Gary Gorton, Rick Green (the editor), Skander Van den Heuvel, Ming Huang, Donald Keim, Leonid Kogan, Martin Lettau, Ralitsa Petkova, Nick Souleles, Robert Stambaugh, Yunguang Yang, and participants at numerous workshops I am especially indebted to an anonymous referee for many constructive criticisms Naiping Liu taught me how

to build Fortran 90 MEX routines in Matlab Financial support from the Dean’s Fellowship for Distinguished Merits at the Wharton School is gratefully acknowledged All remaining errors are

my own.

67

Yet, despite frequent claims of inefficient markets, what is missing, it seems,

is a clear delineation of what the neoclassical world implies about risk andexpected return Filling this gap seems extremely important

I demonstrate that, contrary to the conventional wisdom, assets in place aremuch riskier than growth options, especially in bad times when the price ofrisk is high This mechanism can potentially explain the value anomaly, a highspread in expected return between value and growth strategies even thoughtheir spread in unconditional market beta is low

My explanation relies on two salient features of the model, costly reversibilityand countercyclical price of risk Costly reversibility implies that firms facehigher costs in cutting than in expanding capital.1 Through optimal capitalinvestment, this asymmetry gives rise to cyclical behavior of value and growthbetas

In bad times, value firms are burdened with more unproductive capital,finding it more difficult to reduce their capital stocks than growth firms do.The dividends and returns of value stocks will hence covary more with eco-nomic downturns In good times, growth firms invest more and face higheradjustment costs to take advantage of favorable economic conditions Expand-ing capital is less urgent for value firms since their previously unproductivecapital now becomes productive As expanding capital is relatively easy, thedividends and returns of growth firms do not covary much with economicbooms The net effect is a high dispersion of risk between value and growthstrategies in bad times and a low or even negative dispersion of risk in goodtimes

Costly reversibility is also consistent with a low unconditional dispersion

of risk between value and growth Bad times characterized by disinvestmentoccur less often and last for shorter periods than good times A low unconditionaldispersion of risk arises, as high positive dispersion of risk in bad times is offset

by low or even negative dispersion in good times

With rational expectations, the value premium equals the risk dispersionbetween value and growth times the price of risk When the price of risk isconstant, the average value premium must be accounted for entirely by theunconditional beta dispersion This seems at odds with the empirical evidence

in Fama and French (1992).2 It is well known that time-varying price of risk

1 Abel and Eberly (1994, 1996) study firms’ optimal investment with costly reversibility in a partial equilibrium setting Ramey and Shapiro (2001) provide direct empirical evidence for costly reversibility A large portion of the literature on capital investment is devoted to examining the implications of a special case of costly reversibility, that is, irreversible investment, which says that the cost of cutting capital is infinite so that investment can never be negative Dixit and Pindyck (1994) survey the literature on irreversible investment and Kogan (2000, 2001) examines the implications of irreversibility on investment and time-varying return volatility in a two-sector general equilibrium model.

2 However, Petkova and Zhang (2003) show, using the longer sample from 1927 to 2001 than the short sample from 1963 to 1991 used by Fama and French (1992), that the unconditional market beta spread between value and growth is 0.41, much higher than the effective zero reported by Fama and French.

improves the performance of the conditional asset pricing models; my tion is to analyze the impact of this time-variation on capital investment andexpected return within the neoclassical framework.

contribu-I find that because discount rates are higher in bad times with the cyclical price of risk, firms’ expected continuation values are on average lowerthan those with constant price of risk Value firms want to disinvest even more

counter-in bad times The time-varycounter-ing price of risk thus counter-interacts with and propagatesthe effect of asymmetry, resulting in a high average value premium, more thanthe amount attributable to the unconditional dispersion of risk alone

By linking risk and expected return to economic primitives my model vides a unified framework to rationalize many empirical regularities in thecross-section of returns in relation to the value premium: (i) Value is riskierthan growth, especially in bad times when the price of risk is high (Lettauand Ludvigson (2001), Petkova and Zhang (2003)); (ii) high book-to-market sig-nals persistently low profitability and low book-to-market signals persistentlyhigh profitability (Fama and French (1995)); (iii) the expected value premium

pro-is atypically high at times when the value spread (in book-to-market) pro-is wide(Cohen, Polk, and Vuolteenaho (2003)); and (iv) the earnings growth spreadbetween value and growth is a positive predictor of the value-minus-growthreturn (Asness et al (2000)).3In contrast, it is not clear how these patterns can

be explained by the behavioral overreaction hypothesis advocated by DeBondtand Thaler (1985) and by Lakonishok, Shleifer, and Vishny (1994), since it isrelatively detached from economic fundamentals

Finally, the model also yields a rich array of new refutable hypotheses viding fresh directions for future empirical research:4(i) Value firms disinvestmore than growth firms in bad times, and growth firms invest more than valuefirms in good times; (ii) the expected value premium and the value spread areboth countercyclical; (iii) the degree of asymmetry correlates positively withthe expected value premium across industries; (iv) the industry cost of capi-tal increases with the industry book-to-market and the cross-sectional disper-sion of individual stock returns within the industry; and finally, (v) the de-gree of asymmetry correlates positively with the industry cost of capital acrossindustries

pro-My work is related to that of Berk, Green, and Naik (1999), who construct adynamic real options model in which assets in place and growth options change

in predictable ways This pattern in turn imparts predictability in risk andexpected returns The real options model in Berk et al features exogenousproject-level cash f low and systematic risk My neoclassical model differs in thatall firm-level variables, except for the exogenous idiosyncratic productivity, aredetermined endogenously in competitive equilibrium My model can thereforeshed light on more fundamental determinants of firm-level cash f low, risk, andexpected return

3 The model in Gomes et al (2003) can also generate patterns (ii) and (iii) but through different economic mechanisms See Section II.C for more discussion on that paper.

4 Schwert (2003) highlights the importance for structural models to derive new testable ses and go beyond the stage of explaining the existing stylized facts.

hypothe-Gomes et al (2003) represent another theoretical attempt to link risk and pected returns to size and book-to-market in a dynamic equilibrium model Mywork differs primarily in its explanation of the value premium Gomes et al.assume that all firms have equal growth options, implying that investmentplans are independent of current productivity Since more profitable growthfirms cannot invest more, by construction, they have to pay out more divi-dends Growth firms have shorter cash-f low duration than value firms This iscounterfactual.5 Gomes et al then rely on this pattern to generate a positiveexpected value premium, based on equity duration risk (e.g., Cornell (1999)).

ex-By relaxing the equal-growth assumption, my model allows firms to conditioninvestment plans optimally on their current productivity A new mechanismfor the value premium arises, as asymmetry and the countercyclical price ofrisk cause assets in place to be harder to reduce, and hence to be riskier thangrowth options especially in bad times when the price of risk is high

The outline for the rest of the paper is as follows The equilibrium investmentmodel is constructed in Section I I present the main findings concerning thevalue premium in Section II and explore other model predictions in Section III.Section IV brief ly discusses the related literature Finally, Section V concludes

I The Model

I construct a neoclassical industry equilibrium model (e.g., Lucas and Prescott(1971)) augmented with aggregate uncertainty.6Section I.A describes the eco-nomic environment Section I.B presents the value-maximizing behavior offirms I then discuss aggregation in Section I.C and define the competitiveequilibrium in Section I.D Appendix A contains the proofs and Appendix Boutlines the solution methods

A Environment

The industry is composed of a continuum of competitive firms that produce

a homogeneous product Firms behave competitively, taking the price in theproduct market as given

A.1 Technology

Production requires one input, capital, k, and is subject to both an aggregate shock, x, and an idiosyncratic shock, z The next two assumptions concern the

nature of the productivity shocks:

5 Smith and Watts (1992) document that high book-to-market firms are more likely to pay out dividends Dechow, Sloan, and Soliman (2002) report that equity duration is strongly negatively correlated with book-to-market.

6 Most of the extant industry equilibrium models abstract from aggregate uncertainty Examples include Hopenhayn (1992), Cooley and Quadrini (2001), and Gomes (2001).

ASSUMPTION1: The aggregate productivity shock has a stationary and monotone Markov transition function, denoted by Q x (x t+1| x t ), as follows:

x t+1= x(1 − ρ x)+ ρ x x t + σ x ε x

ASSUMPTION2: The idiosyncratic productivity shocks, denoted by z jt , are

uncor-related across firms, indexed by j, and have a common stationary and monotone Markov transition function, denoted by Q z (z jt+1| z jt ), as follows:

nontriv-The production function is given by

where 0< α < 1, and y jt and k jt are the output and capital stock of firm j at period t, respectively The production technology exhibits decreasing-return-to-

scale

A.2 Stochastic Discount Factor

I follow Berk et al (1999) and parameterize directly the pricing kernel out explicitly modeling the consumer’s problem Since my focus is on the pro-duction side, this strategy seems reasonable I assume the pricing kernel tobe

eral equilibrium, I can link c to the aggregate state variable in a reduced-form

way by letting c t = a + bx t with b > 0.7 Equation (4) now follows immediately

by definingγ t to be Ab.

It is well known that power utility has many limitations, one of which isthat it implies a constant price of risk, given an exogenous, homoscedastic con-sumption growth process I thus assume in (5) thatγ tis time-varying and de-

creasing with the demeaned aggregate productivity x t − x, where γ1< 0 I

re-main agnostic about the precise economic sources of the countercyclical price ofrisk.8

A.3 Industry Demand

The inverse industry demand function is denoted by P(Y t ), where P t is the

output price and Y t is the total output in the industry at time t I follow Caballero and Pindyck (1996) and parameterize P(·) as

where 0< η < 1 and 1/η can be interpreted as the price elasticity of demand.

B Firms

I now summarize the decisions of firms The timing of events is standard

Upon observing the shocks at the beginning of period t, firms make optimal

investment decisions

B.1 Value Maximization

I suppress the firm index j for notational simplicity The profit function for

an individual firm with capital stock k t and idiosyncratic productivity z t, facing

aggregate shock x t and log output price p t ≡ log P t, is

π(k t , z t ; x t , p t)= e x t +z t +p t k α t − f , (7)

where f denotes the nonnegative fixed cost of production, which must be paid

every period by all the firms in production A positive fixed cost captures theexistence of fixed outside opportunity costs for some scarce resources such asmanagerial labor used by the firms

7 Since there exists a large number of firms, the law of large numbers implies that firm-specific shocks do not affect the aggregate consumption Moreover, the stationarity of the economy implies that the level of aggregate capital stock affects consumption only indirectly through aggregate shock, given the initial level of aggregate capital.

8 The specific functional form of the kernel relates naturally to the time-varying risk aversion in Campbell and Cochrane (1999) Other possibilities include loss aversion in Barberis, Huang, and Santos (2001) and limited market participation in Guvenen (2002).

Let v(k t , z t ; x t , p t) denote the market value of the firm I can use Bellman’sprinciple of optimality to state the firm’s dynamic problem as

v(k t , z t ; x t , p t)= max

k t+1,i t



π(k t , z t ; x t , p t)− i t − h(i t , k t)+

 

M t+1v(k t+1, z t+1; x t+1, p t+1)Q z (d z t+1| z t )Q x (d x t+1| x t)

,(8)subject to the capital accumulation rule

where i t denotes investment at time t and δ is the constant depreciation rate.

The first three terms in the right-hand side of (8) ref lect current dividend,

denoted by d t, i.e., profitπ minus investment, i, minus adjustment cost, h.

Following Lucas (1967), I model adjustment cost directly as a deduction from

the profit function The functional form of h is asymmetric and quadratic:

andχ{·} is the indicator function that equals one if the event described in{·}

is true and zero otherwise Figure 1 provides a graphical illustration of the

specification of h.

The quadratic adjustment cost is standard in the Q-theoretical literature of

investment I model the adjustment cost to be asymmetric also, that is,θ−>

1996) Firms face higher costs per unit of adjustment in contracting than inexpanding their capital stocks.9

B.2 Beta and Expected Return

PROPOSITION1: The risk and expected return of firm j satisfy

Figure 1 Asymmetric adjustment cost This figure illustrates the specification of capital

ad-justment cost, equations (10) and (11) The investment rate, i /k, is on the x-axis and the amount

of adjustment cost, h(i, k), is on the y-axis The adjustment cost is assumed to be

θ t ≡ θ+· χ {i t≥0}+ θ−· χ {i t <0}

andχ{·} is an indicator function that equals one if the event described in{·} is true and zero

otherwise Moreover,θ−> θ+> 0, implying that firms face higher costs in adjusting capital stocks

downward than upward.

and d jt is the dividend at time t, d jt ≡ π jt − i jt − h(i jt , k jt).10The quantity of risk

is given by

β j t ≡ −Covt [R j t+1, M t+1]/Var t [M t+1] (14)

and the price of risk is given by

λ mt≡ Vart [M t+1]/E t [M t+1]. (15)

10Note that v(k jt , z jt , x t , p t ) is the cum dividend firm value, in that it is measured before dividend

is paid out Define v e

j t ≡ v j t − d j t to be the ex dividend firm value, then R jt+1 reduces to the usual

definition R+1= (v e + d+1 )/v e.

Proof: See Appendix A.

C Aggregation

Having described the optimization behavior of firms, I am now ready to acterize the aggregate behavior of the industry The output price will be deter-mined in the competitive equilibrium to equate industry demand and supply

char-in the product market It is immediate that the char-industry output, and hence theprice, will depend on the cross-sectional distribution of firms

Letµ tdenote the measure over the capital stocks and idiosyncratic shocks

for all the firms in the industry at time t Let i(k t , z t ; x t , p t ) and y(k t , z t ; x t , p t)denote, respectively, the optimal investment decision and output for the firm

with capital k t and idiosyncratic productivity z t facing log price p tand aggregate

productivity x t Define to be any measurable set in the product space of k and

z, and let (µ t , x t , x t+1) be the law of motion for the firm distributionµ t Then

(·, ·, ·) can be stated formally as

Although the exact condition is somewhat technical, the underlying intuition

is quite straightforward Equation (16) says that the law of motion for theindividual states for the firms is obtained simply by combining their optimaldecision rules concerning capital accumulation, as formalized in (17) The totalindustry output can be now written as

as a value function v∗(k t , z t ; x t , p∗t ) for each firm; and (iii) a law of motion of firm

t , z t ; x t , p∗t ) and v∗(k t , z t ; x t , p∗t ) solve the

value-maxi-mization problem (8) for each firm;

firms in the industry, that is, (18) holds The law of motion of firm

(17) hold.

r Product market clearing

PROPOSITION2: There exists a unique value function v(k, z, x, p) that satisfies (8) and is continuous, increasing, and differentiable in k, z, x, and p, and concave

in k In addition, a unique industry equilibrium exists.

II Main Findings

In this section, I first calibrate the model in Section II.A Section II.Bpresents the main quantitative results and Section II.C investigates the eco-nomic sources of the value premium within the model

A Calibration

Calibration of an economic model involves restricting some parameter valuesexogenously and setting others to replicate a benchmark data set as a modelsolution (e.g., Dawkins, Srinivasan, and Whalley (2001)) Once calibrated, themodel can be used to assess the effects of an unobservable change in exogenousparameter values The model solution provides predictions of the way in whichthe economy is likely to respond to the change, while the pre-change solutionserves as the reference point

Table I summarizes the key parameter values in the model All model rameters are calibrated at the monthly frequency to be consistent with theempirical literature I break down all the parameters into three groups Thefirst group includes parameters that can be restricted by prior empirical or

pa-Table I

Benchmark Parameter Values

This table lists the benchmark parameter values used to solve and simulate the model I break all the parameters into three groups Group I includes parameters whose values are restricted by prior empirical or quantitative studies: capital share,α; depreciation, δ; persistence of aggregate

productivity,ρ x; conditional volatility of aggregate productivity,σ x; and inverse price elasticity of demand,η Group II includes parameters in the pricing kernel, β, γ0 , andγ1 , which are tied down

by matching the average Sharpe ratio and the mean and volatility of real interest rate The final group of parameters is calibrated with only limited guidance from prior empirical studies I start with a reasonable set of parameter values and conduct extensive sensitivity analysis in Tables III and IV.

0.30 0.01 0.951/3 0.007/3 0.50 0.994 50 −1000 10 15 0.97 0.10 0.0365

quantitative studies: capital share α; depreciation δ; persistence ρ x and ditional volatility σ x of aggregate productivity; and inverse price elasticity ofdemandη Because of the general consensus concerning their numerical values,

con-these parameters provide essentially no degrees of freedom for generating thequantitative results

The capital shareα is set to be 30%, similar to that in Kydland and Prescott

(1982) and in Gomes (2001) The monthly rate of depreciation, δ, is set to be

0.01, which implies an annual rate of 12%, the empirical estimate of Cooper andHaltiwanger (2000) The persistence of the aggregate productivity process,ρ x,

is set to be 0.951/3 = 0.983, and its conditional volatility, σ x, 0.007/3 = 0.0023.

With the AR(1) specification for x tin (1), these monthly values correspond to0.95 and 0.007 at the quarterly frequency, respectively, consistent with Cooleyand Prescott (1995) Finally, I follow Caballero and Pindyck (1996) and set theinverse price elasticity of demandη to be 0.50.

The second group of parameters includes those in the pricing kernel:β, γ0, and

γ1 These parameters can be tied down by aggregate return moments implied

by the pricing kernel The log pricing kernel in (4) and (5) implies that the

real interest rate R ft and the maximum Sharpe ratio S t can be written as,respectively,

11The long-run average aggregate productivity, ¯x, determines the long-run average scale of the

economy, but does not affect stock returns directly Equations (22) and (23) imply that returns are

not directly affected by the level of ¯x, but by business cycle f luctuation, that is, x t − ¯x The degree

of this f luctuation is already pinned down byσ x , the conditional volatility of the x t process Thus ¯x

is purely a scaling constant, and I set ¯x such that the long-term average capital stock is normalized

to be 1 This is done by solving the firm’s problem without uncertainty in closed form and then

imposing the steady-state condition This implies that ¯x = −5.70 Other normalization schemes yield quantitatively similar results Normalizing ¯x is standard in the literature (e.g., Cooley and

Prescott (1995), Boldrin et al (2001)).

close to those in the data reported by Campbell and Cochrane (1999) and byCampbell, Lo, and MacKinlay (1997) As these parameters are pinned downtightly by the aggregate return moments, they provide no degrees of freedom

in matching cross-sectional moments of returns, which are my focus here.Importantly, a γ0 of 50 does not necessarily imply extreme risk aversion,nor does aγ1of−1,000 Because the pricing kernel is exogenously specified inthe model, the criterion of judging whether its parameters are representative

of reality should be whether the aggregate return moments implied by thepricing kernel mimic those in the data After all, I do not claim any credits inexplaining time series predictability; my contribution is to endogenize cross-sectional predictability of returns, given time series predictability

The calibration for the third group of parameters has only limited guidancefrom prior studies and I have certain degrees of freedom in choosing their val-ues There are five parameters in this group: (i) the adjustment cost coefficient,

θ+; (ii) the degree of asymmetry,θ−/θ+; (iii) the conditional volatility of cratic productivity,σ z; (iv) the persistence of idiosyncratic productivity,ρ z; and

idiosyn-(v) the fixed cost of production, f I first choose their benchmark values by

us-ing available studies and by matchus-ing key moments in the data I then conductextensive sensitivity analysis

First,θ+can be interpreted as the adjustment time of the capital stock given

one unit change in marginal q (e.g., Shapiro (1986) and Hall (2001)) The

first-order condition with respect to investment for the value-maximization problemsays thatθ+· (i/k) = q − 1, where q is the shadow price of additional unit of capital If q rises by one unit, the investment-capital ratio (i /k) will rise by

1/θ+ To cumulate to a unit increase, the f low must continue at this level for

θ+periods.

The empirical investment literature has reported a certain range for this justment time parameter Whited (1992) reports this parameter to be between0.5 and 2 in annual frequency, depending on different empirical specifications.This range corresponds to an adjustment period lasting from 6 to 24 months.Another example is Shapiro (1986), who finds the adjustment time to be abouteight calendar quarters or 24 months I thus set the benchmark value of θ+

ad-to be 15, which corresponds ad-to the average empirical estimates, and conductsensitivity analysis by varyingθ+from 5 to 25.

The empirical evidence on the degree of asymmetry, θ−/θ+, seems scarce.Here I simply follow Hall (2001) and set its benchmark value to be 10 (Table IIIcontains comparative static experiments on this parameter)

To calibrate parameters ρ z andσ z, I follow Gomes (2001) and Gomes et al.(2003) and restrict these two parameters using their implications on the degree

of dispersion in the cross-sectional distribution of firms One direct measure

of the dispersion is the cross-sectional volatility of individual stock returns.Moreover, since disinvestment in recessions is intimately linked to the valuepremium, as argued in Section II.C below, it is important for the model to matchthe average rate of disinvestment as well

These goals are accomplished by settingρ z = 0.97 and σ z = 0.10 These values

imply an average annual cross-sectional volatility of individual stock returns

of 28.6%, approximately the average of 25% reported by Campbell et al (2001)and 32% reported by Vuolteenaho (2001) Furthermore, the average annual rate

of disinvestment is 0.014, close to 0.02 in the data reported by Abel and Eberly(2001)

The value of σ z is also in line with the limited empirical evidence P ´astorand Veronesi (2003) show that the average volatility of firm-level profitabil-ity has risen from 10% per year in the early 1960s to about 45% in the late1990s.12The calibrated conditional volatility of firm-level productivity is 10%per month, corresponding to 35% per year, which seems reasonable given therange estimated by P ´astor and Veronesi

The unconditional volatility of idiosyncratic productivity is about 32 timesthat of aggregate productivity Such a high idiosyncratic shock is necessary togenerate a reasonable amount of dispersion in firm characteristics within themodel However, even with such a high firm-level shock, firm value and in-

vestment rate are much more sensitive to changes in aggregate productivity x t than to changes in idiosyncratic productivity z t.13The reason is that x taffects

the stochastic discount factor, while z t does not; shocks at the firm-level aremainly cash f low shocks that can be integrated out, while shocks at the aggre-gate level consist primarily of discount rate shocks, consistent with Vuolteenaho(2001)

Finally, I am left with the fixed cost of production, f Since f deducts the

firm’s profit given in (7), it has a direct impact on the market value of the firm

I thus calibrate f to be 0.0365 such that the average book-to-market ratio in the

economy is 0.54, which matches approximately that in the data, 0.67, reported

by Pontiff and Schall (1999)

Table II reports the set of key moments generated using the benchmark rameters I simulate 100 artificial panels each with 5,000 firms and 900 months

pa-I then compute the return and quantity moments for each sample and reportthe cross-sample averages in Table II The corresponding moments in the dataare also reported for comparison

Table II suggests that the model does a reasonable job of matching thesereturn and quantity moments Importantly, the fit seems reasonable not onlyfor the moments that serve as immediate targets of calibration, but also forother moments The mean and volatility of industry return are comparable

to those computed using the industry portfolios of Fama and French (1997).The volatility of aggregate book-to-market ratio is 0.24, close to that of 0.23reported by Pontiff and Schall (1999) The average rate of investment is 0.135

in the model, close to 0.15 in the data reported by Abel and Eberly (2001) Insum, the calibrated parameter values seem reasonably representative of thereality

12 It is a topical area to explain this upward trend in firm-level profitability associated with the trend in idiosyncratic volatility documented in Campbell et al (2001) But this is outside the scope

of this paper.

13 These results were reported in a previous version of the paper, but not in the current version

to save space They are available upon request.

Table II

Key Moments under the Benchmark Parametrization

This table reports a set of key moments generated under the benchmark parameters reported

in Table I The data source for the average Sharpe ratio is the postwar sample of Campbell and Cochrane (1999) The moments for the real interest rate are from Campbell et al (1997) The data moments for the industry returns are computed using the 5-, 10-, 30-, and 48-industry portfolios in Fama and French (1997), available from Kenneth French’s web site The numbers of the average volatility of individual stock return in the data are from Campbell et al (2001) and Vuolteenaho (2001) The data source for the moments of book-to-market is Pontiff and Schall (1999), and the annual average rates of investment and disinvestment are from Abel and Eberly (2001).

Average annual value-weighted industry return 0.13 0.12–0.14 Annual volatility of value-weighted industry return 0.27 0.23–0.28 Average volatility of individual stock return 0.286 0.25–0.32

B Empirical Predictions

I now investigate the empirical predictions of the model concerning the crosssection of returns I show that (i) the benchmark model with asymmetry and acountercyclical price of risk is capable of generating a value premium similar tothat in the data And (ii) without these two features, an alternative parameterset does not exist that can produce the correct magnitude of the value premium.Therefore, at least in the model, asymmetry and countercyclical price of riskare necessary driving forces of the value premium

Table III reports summary statistics, including means, volatilities, and conditional betas for portfolio HML and for 10 portfolios sorted on book-to-market, using both the historical data and 10 artificial data simulated in themodel.14The book value of a firm in the model is identified as its capital stock,

un-and the market value is defined as the ex dividend stock price, as in footnote

10 I follow Fama and French (1992, 1993) in constructing HML and 10 to-market portfolios for each simulated panel I repeat the entire simulation

book-100 times and report the cross-simulation averages of the summary statistics

in Table III From Panel A, the benchmark model is able to generate a positiverelation between book-to-market and average returns The benchmark modelgenerates a reliable value premium, measured as the average HML return,which is quantitatively similar to that in the data

14 The historical data on 10 book-to-market portfolios are those used in Davis, Fama, and French (2000) and are available from Kenneth French’s web site The sample ranges from July 1927 to December 2001.

Table III

Properties of Portfolios Sorted on Book-to-Market

This table reports summary statistics for HML and 10 book-to-market portfolios, including mean,

m, volatility, σ, and market beta, β Both the mean and the volatility are annualized The average

HML return (the value premium) is in annualized percent Panel A reports results from historical data and benchmark model with asymmetry and countercyclical price of risk (θ−/θ+= 10 and γ1 =

−1000) Panel B reports results from two comparative static experiments Model 1 has symmetric adjustment cost and constant price of risk (θ−/θ+= 1 and γ1 = 0), and Model 2 has asymmetry and constant price of risk (θ−/θ+= 10 and γ1 = 0) All the model moments are averaged across

100 artificial samples All returns are simple returns.

Panel A: Data and Benchmark Panel B: Comparative Statics

HML 4.68 0.14 0.12 4.87 0.43 0.12 2.19 0.09 0.04 2.54 0.11 0.04 Low 0.11 1.01 0.20 0.09 0.85 0.23 0.08 0.95 0.30 0.08 0.94 0.30

To evaluate the role of asymmetry and the countercyclical price of risk, I duct comparative static experiments in Panel B of Table III by varying two keyparameters governing the degree of asymmetry,θ−/θ+, and the time-variation

con-of the log pricing kernel,γ1 Two cases are considered: Model 1 has ric adjustment cost and the constant price of risk (θ−/θ+= 1 and γ1= 0) andModel 2 has asymmetry and constant price of risk (θ−/θ+= 10 and γ1= 0) Allother parameters remain the same as in the benchmark model

symmet-Panel B of Table III shows that, without asymmetry or time-varying price

of risk, Model 1 displays a small amount of the value premium Introducingasymmetry in Model 2 increases the amount somewhat, but it is still lower thanthat in the benchmark model In short, asymmetry and the time-varying price

of risk seem indispensable for generating the value premium in the benchmarkmodel

However, the importance of these features established in Table III is tional on the benchmark calibration of Model 1 It is possible that even withoutthese two features, an alternative parameter set may exist in Model 1 that willproduce the correct magnitude for the value premium I thus conduct exten-sive sensitivity analysis on Model 1 by varying its parameter values from thebenchmark calibration

condi-Panels A–H of Table IV report the results from the following eight ative static experiments on Model 1: Low Volatility (σ = 0.08, Panel A); High

compar-Table IV

The Performance of Model 1 (θ − /θ+ = 1 and γ1 = 0) under Alternative

Parameter Values

This table reports summary statistics for HML and 10 book-to-market portfolios, including

annu-alized mean, m, and volatility, σ, and market beta, β, generated from Model 1 without asymmetry

and countercyclical price of risk The average HML returns are in annualized percent Nine native parameter values are considered: Low Volatility (σ z = 0.08); High Volatility (σ z = 0.12); Fast

alter-Adjustment (θ+= 5); Slow Adjustment (θ+= 25); Low Fixed Cost (f = 0.0345); High Fixed Cost ( f = 0.0385); Low Persistence (ρ z = 0.95); High Persistence (ρ z = 0.98); and High Volatility, Slow

Adjustment, High Fixed Cost, and High Persistence (Panel I) All moments are averaged across

100 artificial samples All returns are simple returns.

Low Volatility High Volatility Fast Adjustment Slow Adjustment

HML 1.78 0.07 0.03 2.28 0.10 0.04 1.57 0.07 0.04 2.31 0.08 0.03 Low 0.08 0.95 0.30 0.08 0.94 0.29 0.09 0.96 0.30 0.07 0.95 0.29

Low Fixed Cost High Fixed Cost Low Persistence High Persistence

Volatility (σ z = 0.12, Panel B); Fast Adjustment (θ+= 5, Panel C); Slow ment (θ+= 25, Panel D); Low Fixed Cost (f = 0.0345, Panel E); High Fixed Cost (f = 0.0385, Panel F); Low Persistence (ρ z = 0.95, Panel G); and High Persis-

Adjust-tence (ρ z = 0.98, Panel H) These experiments cover a wide range of empirically

plausible parameter values A conditional volatility of 12% per month for the

idiosyncratic productivity corresponds to 42% per year, close to the upper bound

of 45% estimated by P ´astor and Veronesi (2003) As argued in Section II.A, thetwo alternative values ofθ+cover the range of its empirical estimates The twovalues of fixed cost of production imply a wide range of industry book-to-market,from 0.29 to 9.58 Finally, a persistence level of 0.98 for the idiosyncratic pro-ductivity is close to that of the aggregate productivity, and is likely to be anupper bound.15

Importantly, Table IV shows that the amount of value premium generatedfrom the eight alternative parameter sets of Model 1 is uniformly much lowerthan that in the data and that in the benchmark model The table also indi-cates that the magnitude of the value premium increases with the persistenceand conditional volatility of idiosyncratic productivity, the adjustment time pa-rameter, and the fixed cost of production.16A natural question is then whetherModel 1 can generate the correct magnitude of the value premium by combin-ing all the extreme parameter values used in Panels B, D, F, and H Panel I inTable IV reports that this is not true The value premium generated from thisparameter set is still lower than that in the data by 1.5% per annum

In sum, the simulation results indicate that (i) an alternative parameterset does not exist that will produce the correct magnitude for the value pre-mium in Model 1 without asymmetry and the countercyclical price of risk And(ii) once these two ingredients are incorporated, the benchmark model is able

to generate a value premium consistent with the data I conclude that, at least

in the model, asymmetry and the countercyclical price of risk are necessarydriving forces of the value premium

C Causality

I now focus on the causal relation of asymmetry and the countercyclical price

of risk to the value premium I first demonstrate that productivity difference

is what determines the value or growth characteristics of firms to begin with

I then investigate how productivity difference transforms to difference in riskand expected return through optimal investment Finally, I examine how thestructural link between productivity and expected return is affected by the deepparameters governing the degree of asymmetry and time-variation in the price

16 The prediction that the value premium increases with the fixed cost of production is consistent with Carlson et al (2004).

Panel A: Return on Equity (ROE) Panel B: Time-Series of ROE

Time Series

Value

Figure 2 The value factor in profitability (ROE) Following Fama and French (1995), I

mea-sure profitability by return on equity, that is [k t + d t]/k t−1, where k tdenotes the book value of

equity and d tis the dividend payout Thus profitability equals the ratio of common equity income for

the fiscal year ending in calender year t and the book value of equity for year t − 1 The profitability

of a portfolio is defined as the sum of [k jt + d jt ] for all firms j in the portfolio divided by the sum

of k jt−1; thus it is the return on book equity by merging all firms in the portfolio For each portfolio

formation year t, the ratios of [ k t +i + d t +i]/k t +i−1 are calculated for year t + i, where i = −5, , 5 The ratio for year t + i is then averaged across portfolio formation years Panel A shows the

11-year evolution of profitability for value and growth portfolios Time 0 on the horizontal axis

is the portfolio formation year Panel B shows the time series of profitability for value and growth portfolios Value portfolio contains firms in the top 30% of the book-to-market ratios and growth portfolio contains firms in the bottom 30% of the book-to-market ratios The figure is generated under the benchmark model, and varyingθ−/θ+andγ1 yields similar results.

time series for each simulated panel with 5,000 firms and 900 months I thenrepeat the same analysis on 100 simulated panels and report the cross-sampleaverage results in Figure 2.17

Figure 2 demonstrates that, consistent with Fama and French (1995), to-market is associated with persistent differences in profitability In the model,growth firms are on average more profitable than value firms for 5 years be-fore and 5 years after portfolio formation The profitability of growth firmsimproves prior to portfolio formation and deteriorates thereafter, and the oppo-site is true for value firms This pattern is driven by the mean-reverting behav-

book-ior of the idiosyncratic productivity, z t The difference in profitability betweenvalue and growth is also confirmed in Panel B, where profitability is examinedchronologically In sum, idiosyncratic productivity corresponding empirically

to firm-level profitability is what determines value or growth characteristicfor a specific firm, given that it is the only source of firm heterogeneity in themodel

17 The figure is generated under the benchmark model The results from varying the two etersθ−/θ+ andγ1 are qualitatively similar, and are hence omitted.