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Abstract A simple general equilibrium production economy matches moments of the value premium and equity premium. Value firms have low productivity, but will eventually produce high cash flows. The present value of these temporally distant cash flows is especially sensitive to equity premium movements. The value premium is the reward for bearing this sensitivity. Capital adjustment costs are important. Without these costs, value firms would disinvest heavily, leading to high cash flows today, low cash-flow growth going forward, and little exposure to discount rate shocks. Empirical evidence verifies that value firms have higher cash-flow growth and supports other predictions. Received date July 17, 2017; Accepted date October 22, 2017 By Editor Raman Uppal The asset pricing literature typically takes a divide-and-conquer approach to the time-series and cross-section of stock returns. Papers that model the aggregate stock market and its time-varying risk premia rarely investigate the implications of their framework for the cross-section.1 This division is the natural result of the difficulties of solving heterogeneous firm models in a general equilibrium setting. The complex preferences required by asset prices typically render such models intractable. This division is unfortunate, however, since changes in the aggregate risk premium are important and undiversifiable sources of risk, which must have implications for the cross-section. This paper presents a general equilibrium model with heterogeneous firms that links the equity premium with the value premium. The model is essentially a real business-cycle model with idiosyncratic productivity and a variant of Campbell and Cochrane (1999) preferences. Though this setting is simple, it leads to an intractable and infinite dimensional state space. I apply the methods of Krusell and Smith (1998) to approximate and solve the model.2 The model replicates key features of the value premium. Both in the model and in the data, expected returns are linear in the log of book-to-market (B/M). Moreover, the model matches the slope of the relationship. The slope on log B/M is approximately six, indicating that a 20% higher B/M implies a 120-bp increase in expected returns over the next year. These cross-sectional predictions come with an equity premium and business cycle that are consistent with the data. The model matches the first two moments of aggregate excess returns, the risk-free rate, consumption, output, and investment, as well as excess return and dividend predictability regressions. Aggregate and cross-sectional dynamics interact to create the value premium. The story takes several steps, but each step is intuitive. The explanation begins with the nature of value firms (high B/M firms) in the model. Value firms are those that have suffered a series of bad productivity shocks and have low cash flow. Productivity recovers on average, however. Thus cash flows for value firms recover over time, leading to high cash-flow growth. High cash-flow growth then translates into high expected returns, since these temporally distant cash flows are more exposed to persistent discount rate shocks. To understand this, note that when the equity premium rises, all stock prices fall, since cash flows are discounted more aggressively. Stocks with temporally distant cash flows are hit harder, however. This is because the effects of a persistent rise in discount rates are compounded for distant cash flows. Since the equity premium rises in bad times, investors demand a premium in exchange holding these high cash-flow growth stocks. Cash-flow cyclicality has been implicated in other models of the value premium (Zhang 2005). I show that cash-flow cyclicality is not the main driver in my model. Indeed, value firms have less procyclical cash flows than growth firms. This is the natural result of optimal investment. In bad aggregate states, the household strongly values consumption, so value firms disinvest, leading to countercyclical cash flows. On the other hand, growth firms are so productive that it is efficient for them to give less consumption to the household in order to invest for the future. This story for the value premium hinges on two features of the model. The first is that idiosyncratic productivity mean reverts. This assumption is consistent with Fama and French’s (1995) finding that value firms have low but mean-reverting return on book equity. The second is that mean-reverting productivity has stronger effects on cash flow than optimal investment. Optimal investment implies that unproductive value firms should disinvest. Disinvestment results in higher cash flows today, and thus lower cash-flow growth going forward, reducing the exposure of value firms to discount rate shocks. The size of the value premium, then, depends on the strength of capital adjustment costs and the persistence of idiosyncratic productivity. Strong adjustment costs discourage investment and disinvestment, favoring the productivity channel and the value premium. Similarly, a low persistence of idiosyncratic productivity reduces the cross-firm differential in incentives for investment, thus reducing the investment channel and increasing the value premium. For parameter values that generate a data-like aggregate stock market and business cycles, the productivity channel dominates, and the value premium exists. This robustness is due to the strong adjustment costs that are required to generate a volatile Tobin’s q (and thus a volatile stock return). This story for the value premium appears to be at odds with the traditional view that value firms have low duration (Dechow, Sloan, and Soliman 2004, Zhang 2005, Da 2009) and that long duration stocks have lower expected returns (Weber Forthcoming). The duration of infinitely lived equities, however, is difficult to measure. I show that the duration measure used by Weber (Forthcoming) does a poor job of capturing cash-flow growth in the model. Indeed, half of model simulations produce a downward sloping measured term structure, implying that the model can be consistent with with Weber’s results. In contrast, measuring cash-flow growth is straightforward. I provide new empirical evidence demonstrating that value firms have higher cash-flow growth. The measures of cash flow I use are based on earnings. Earnings-based measures are important because the Miller and Modigliani (1961) imply that dividend policy is irrelevant to firm value in the model. I also provide empirical evidence that links value to discount rate shocks. To understand how I measure discount rate shocks, it helps to review the findings of Chen (2017a), as the model is an extension of that earlier paper. In Chen (2017a), I show that adding external habit into a representative firm real business-cycle model results in endogenous, time-varying consumption volatility. Precautionary savings motives lead consumption to be concave in wealth. Thus consumption is more sensitive to shocks in bad times. Physical investment, then, allows this countercyclical sensitivity to show up in aggregate consumption. The presence of heterogeneous firms does not affect these consumption volatility risk results. These results imply that variation in discount rates can be measured by consumption volatility, and, moreover, that value firms are more exposed to consumption volatility shocks. I present empirical evidence consistent with these predictions using Boguth and Kuehn’s (2013) measure of consumption volatility. The model is not consistent with the option-based evidence that the term structure of equity is downward sloping (Van Binsbergen, Brandt, and Koijen 2012). This problem is present in most consumption-based models without additional features such as rigid financial leverage (Belo, Collin-Dufresne, and Goldstein 2015), learning (Croce, Lettau, and Ludvigson 2014), or wage insurance (Marfè 2017). As these mechanisms work on the aggregate firm, they could potentially be introduced in my setting without disrupting the cross-sectional implications. Indeed, the empirical data show that disparate mechanisms are acting on the aggregate market and the cross-section: the aggregate term structure is downward sloping, but, in the cross-section, high growth stocks have higher returns (Chen 2012). The model is closely related to Zhang (2005), who shows that the value premium arises naturally in a Q-theoretical model with time-varying risk premia. The main innovation in this paper is general equilibrium. Zhang’s (2005) partial equilibrium model uses an exogenous SDF and leaves open the interpretation of aggregate predictability in his model. The use of general equilibrium shows that the value premium is consistent with a rational expectations model and provides additional predictions regarding consumption volatility. I also focus on the cash-flow growth channel, which is distinct from the cash-flow cyclicality channel emphasized by Zhang (2005) and other papers in the Q-theory literature (Carlson, Fisher, and Giammarino 2005; Cooper 2006). On the general equilibrium side, the model is closely related to Santos and Veronesi (2010), who study the cross-section of stock returns in an external habit model. This paper can be considered an extension of their model into a production economy. Adding production reverses the value-expected return relationship. Without production, value stocks are characterized by a high dividend price ratio. Mean reversion implies low dividend growth and low exposure to discount rate shocks. In contrast, a model with production characterizes value with book-to-market. Value firms are then low productivity firms, and mean reversion implies high cash-flow growth and high exposure to discount rate shocks. A handful of papers model the equity premium and the cross-section in a long-run risk setting. These papers find that the long-run risk framework is consistent with several facts about the cross-section. Avramov, Cederburg, and Hore (2011) find size, value, and momentum effects in an endowment economy. Favilukis and Lin (2016) and Ai and Kiku (2012) find value effects in production economies. While these papers successfully generate a large equity premium and volatile excess returns, they use the version of the long-run risk model which produces counterfactually high dividend predictability. This paper overcomes this issue by using the external habit framework, which drives aggregate asset prices with time-varying discount rates. Tsai and Wachter (2015) study the cross-section of returns in a rare disaster setting. They find that a model with rare booms and disasters can quantitatively match facts about both the equity premium and value premium. Unlike my model, theirs is not a risk-based explanation for the value premium. Thus, the two papers are complementary perspectives on these phenomena. 2. A General Equilibrium Model with Heterogeneous Firms The model is a real business-cycle model with external habit formation, capital adjustment costs, and idiosyncratic firm productivity. It is designed to have the minimal features for both an equity premium and an endogenous cross-section of firms. It differs from Chen (2017a) in that it (1) features heterogenous firms, (2) abstracts from labor, and (3) assumes zero steady-state growth. These deviations have little quantitative effect on the model's aggregate predictions. Markets are complete, and time is discrete and infinite. For the remainder of the paper, lowercase denotes logs, that is, ct≡ log Ct . 2.1 Representative household A unit measure of identical households j∈[0,1] chooses asset holdings to maximize lifetime utility E0{∑t=0∞βt(Cj,t−Ht)1−γ1−γ}, (1) where β is the time preference parameter, γ is the utility curvature, Cj,t is household j’s consumption, and Ht is the aggregate level of habit. Ht is determined by aggregate consumption and is taken as external by the household. I specify the evolution of habit using surplus consumption, rather than the level of habit itself. That is, let St≡Ct−HtCt, (2) be surplus consumption, where Ct is aggregate consumption. Then surplus consumption follows an AR1-process in logs st+1≡(1−ρs)s¯+ρsst+λ(ct+1−ct), (3) where s¯ is the log of steady-state surplus consumption, ρs is the persistence of surplus consumption, and λ is the conditional volatility of surplus consumption. This modeling approach leads to a simple stochastic discount factor and eases comparison with the existing literature (Campbell and Cochrane 1999, and Wachter 2006, among others). The habit process differs from the literature in that the conditional volatility λ is a constant. In most models, this conditional volatility is time-varying and countercyclical (e.g., Campbell and Cochrane 1999; Menzly, Santos, and Veronesi 2004). In Chen (2017a), I show that the introduction of production results in countercyclical consumption volatility that is quantitatively similar to the assumed countercyclical volatility of surplus consumption typical of endowment economy models. For comparability with Campbell and Cochrane (1999), I fix λ at their steady-state value λ=1S¯−1. (4) That markets are complete means that the household side of the model boils down to a simple stochastic discount factor Mt,t+1=β(Ct+1CtSt+1St)−γ. (5) 2.2 Heterogeneous firms A unit measure of heterogeneous firms is indexed by i∈[0,1] . The firms produce consumption goods according the the production function Π(Ki,t,Bi,t,At)=AtBi,tKi,tα, (6) where Ki,t is the firm’s capital, Bi,t is idiosyncratic productivity, At is aggregate productivity, and α is a production curvature parameter. Both productivity process are AR1 in logs: at+1=ρaat+σaϵa,t+1 (7) bi,t+1=ρbbi,t+σbϵb,i,t+1, (8) where ρa and ρb are the persistence of aggregate and idiosyncratic productivity (respectively), σa and σb are their respective conditional volatilities, and ϵa,t+1 and ϵb,i,t+1 are independent standard normal random variables. All heterogeneity in the models originates from the idiosyncratic productivity process (8). This approach is used for three reasons. The first is that it is a very simple way of introducing a cross-section of firms. The second is that a large literature documents substantial heterogeneity in productivity (Syverson (2011)). The third is that this approach is the standard way of modeling firm heterogeneity in both macroeconomics and finance (Hennessy and Whited 2005; Zhang 2005; Khan and Thomas 2008; Bloom 2009). We will see, however, that this approach has difficulties matching the tremendous heterogeneity in asset valuations that is seen in the data. Matching the heterogeneity in the data with additional sources of heterogeneity is an interesting question for future research; however, it is beyond the scope of this paper. Capital accumulates according to the usual capital accumulation rule, Ki,t+1=Ii,t+(1−δ)Ki,t, (9) where Ii,t is firm-level investment, and δ is the depreciation parameter. Firms face a convex capital adjustment cost Φ(Ii,t,Ki,t)=φ2(Ii,tKi,t−δ)2Ki,t. (10) where φ is the adjustment cost parameter. Adjustment costs are a pure loss. They do not represent payments to labor. Adjustment costs are included because production economies produce a counterfactually smooth Tobin’s q unless one includes an investment friction. Quadratic costs are chosen for simplicity. The firm’s objective is standard: max{Ii,t,Ki,t+1}E0{∑t=0∞M0,t[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]}. (11) It chooses investment and capital to maximize future dividends, discounted with the household’s SDF. 2.3 Recursive competitive equilibrium Equilibrium is defined recursively. Thus, in the remainder of this section I drop the time subscripts and represent the next period with primes. In this heterogeneous firm model, each firm’s capital stock Ki and idiosyncratic productivity Bi plays a part in the state of the economy. Let μ be the joint distribution of firm capital and idiosyncratic productivity. The aggregate state is the triple of μ, surplus consumption S, and aggregate productivity A. The recursive competitive equilibrium consists of a law of motion for the evolution of μ, Γ(μ,S,A) ; a law of motion for aggregate consumption C(μ,S,A) ; a capital policy for the firm G(Ki,Bi;μ,S,A) ; and a value function for the firm V(Ki,Bi;μ,S,A) such that Firm optimality holds: G(Ki,Bi;μ,S,A) and V(Ki,Bi;μ,S,A) solve V(Ki,Bi;μ,S,A)=max{I,Ki′}{ Π(Ki,A,Bi)−Φ(I,Ki)+∫−∞∞dF(ϵ′a)∫−∞∞dF(ϵ′b)M(A′;μ,S,A) ×V(Ki′,Bi′;μ′,S′,A′)}, (12) where the productivity processes are given by Equations (7) and (8), capital accumulation is given by Equation (9), the SDF is the household’s intertemporal marginal rate of substitution M(A′;μ,S,A)=β(C(μ′,S′,A′)S′C(μ,S,A)S)−γ, (13) S′ evolves according to Equation (3), μ′ is given by Γ(μ,S,A) , and F(ϵ′a) is the standard normal cumulative distribution function. 2. Firm decisions are consistent with the law of motion for consumption: C(μ,S,A)=∫dμ(Ki,Bi){Π(Ki,Bi,Z)−I(Ki,Bi;μ,S,A), (14) −Φ(I(Ki,Bi;μ,S,A),Ki)} (15) where I(Ki,Bi;μ,S,A)=G(Ki,Bi;μ,S,A)−(1−δ)Ki . 3. Firm decisions are consistent with the law of motion for the distribution of firms—that is, let B be the Borel algebra for ℝ+2 . Then μ′=Γ(μ,S,A) is given by ∀(K1,B1)∈B,μ′(K1,B1)=∫{(Ki,Bi)|G(Ki,Bi;μ,S,A)∈K1}dμ(Ki,Bi)∫{ϵ′b| exp (ρbb+σbϵ′b)∈B1}dF(ϵ′b). (16) 2.4 Krusell-Smith solution method I solve the model with a variant of the Krusell and Smith (1998) algorithm, similar to Khan and Thomas (2008). Fortran code can be found at https://sites.google.com/site/chenandrewy/. Like in Khan and Thomas (2008), I approximate the distribution capital and idiosyncratic productivity μ with the aggregate capital stock K. Thus, the approximate aggregate state is a triple of aggregate capital, surplus consumption, and aggregate productivity: (K, S, A). I discretize the aggregate and idiosyncratic productivity processes (7) and (8) using the Rouwenhorst (1995) method. I then conjecture that the laws of motion for aggregate consumption and capital follow the following log-linear forms: c= log C˜(K,S,Aj)=θ0,jC+θ1,jCk+θ2,jCsk′= log Γ˜(K,S,Aj)=θ0,jΓ+θ1,jΓk+θ2,jΓs, (17) where C˜(K,S,Aj) and Γ˜(K,S,Aj) are approximate laws of motion, j∈{1,…,NA} represents the discretized aggregate productivity state, NA is the number of discretized aggregate productivity states, and {θi,jC,θi,jΓ}i=0,j=1i=2,j=NA are coefficients that determine the laws of motion. Importantly, the use of aggregate productivity dependent coefficients allows for a nonlinear relationship between consumption and the aggregate state. The goal of the Krusell-Smith method is to find θi,jC,θi,jΓ such that Firms maximize value given the laws of motion θi,jC,θi,jΓ Estimates of (17) on simulated data using policies from step 1 produce coefficients close to θi,jC,θi,jΓ , and R2’s close to one. The most straightforward application of Krusell-Smith searches for this approximate equilibrium by performing a fixed-point iteration using the firm’s problem defined in Equation (12) and a simulation of a distribution of firms. However, no theorem suggests that this fixed-point iteration will converge, and indeed I find that it typically does not. To aid in finding equilibrium, I apply the “equilibrium-in-simulation” method (Krusell and Smith 1997); that is, I first solve solve the approximate equilibrium version of (12). I then plug the resultant value function into the following problem: G(Ki,Bi;K,S,A;C)=argmax{Ki′}{Π(Ki,A,Bi)−I−Φ(I,Ki)+∫−∞∞dF(ϵ′a)∫−∞∞dF(ϵ′b)M∗(A′;K,S,A;C)×V(Ki′,Bi′;K′,S′,A′) } (18) M∗(A′;K,S,A;C)=β(C˜(K′,S′,A′)S′CS)−γ. (19) This procedure introduces today’s aggregate consumption as an additional state variable and solves for a new investment policy which accounts for aggregate consumption. I then use this augmented investment policy G(Ki,Bi;K,S,A;C) in the simulation step. This allows me to find a “market-clearing” C at each date in the simulation. That is, at each date, I use a root finder to find the C that solves equation (14). Note that once the equilibrium is found, aggregate consumption from the simulation of the firms and that produced by the law of motion are equal, and so problem (18) with market clearing (14) and problem (12) produce identical choices. The presence of external habit significantly complicates the computationally demanding Krusell-Smith algorithm. External habit preferences introduce an additional aggregate state variable, surplus consumption, which is absent from the standard RBC economy. As a result, using the RBC equilibrium as an initial guess for the Krusell-Smith algorithm will cause the algorithm to fail. To address this problem, I apply a homotopy method. I solve a series of models with the following altered SDF M′=β(C′C(S′S)χ)−γ (20) where χ is an additional parameter that generates a series of models. I begin by solving a model with χ = 0. Here, the RBC model serves as a good initial guess. Once the program is fairly close to equilibrium, I increase χ by 0.1 and use the previous laws of motion as an initial guess for the new model. I repeat this process until χ=1.0 , which is equivalent to the model presented in (2). Surplus consumption also adds the difficulty that it is an endogenous state variable that is not predetermined. As a result, the habit process equation (3) must be solved at every date in the simulation step of the algorithm. Note that the simulation step involves simulating an entire distribution of firms, and so an entire distribution of decision rules must be accounted for in solving equation (3). This also significantly increases the computational burden of the algorithm. 2.5 Calibration to post-war U.S. data The model is calibrated to post-war (post-1947) U.S. data. This sample period is chosen because the World Wars introduce structural changes that may not be captured by the model. In particular, over the long sample (post-1929) HP-filtered output and investment are essentially uncorrelated. Aggregate quantities are taken from the BEA. Firm-level data are taken from CRSP/Compustat. B/M-sorted portfolios are taken from Ken French’s Web site. Aggregate asset-price moments are taken from Beeler and Campbell (2012). Table 1 shows the calibration. Preference parameters are chosen as much as possible to fit unconditional moments of asset prices. Since time preference β is reflected in risk-free assets, I choose it to fit the mean 30-day Treasury-bill return. The model and data Treasury-bill returns match nicely at about 1% per year. The persistence of surplus consumption ρs affects the persistence of asset prices. Thus I choose ρs to approximately match the annual persistence of the CRSP price/dividend ratio of 0.87. The two remaining preference parameters, the steady-state surplus consumption S¯ and utility curvature γ, jointly control risk aversion. Thus, it is difficult to identify these parameters separately. For ease of comparison with the literature on external habit, I choose γ = 2 to match Campbell and Cochrane’s (1999) value and then choose S¯ to fit the mean Sharpe ratio of the CRSP index. The model does a good job matching the data here: both Sharpe ratios are roughly 0.40. Table 1 Calibration Parameter Value Target Data Model Unconditional asset price moments β Time preference 0.89 Mean 30-day Treasury-bill return 0.89 0.96 ρs Persistence of surplusconsumption 0.86 Persistence of CRSP price/div 0.87 0.88 s¯ Steady-state surplus 0.06 Mean Sharpe ratio of 0.44 0.39 consumption CRSP index Long-run growth moments α Production curvature 0.35 Mean output/capital 0.41 0.42 δ Depreciation rate 0.08 Mean investment rate 0.07 0.08 Unconditional business cycle moments σa Volatility of TFP 0.03 Volatility of GDP (%) 1.61 1.70 ρa Persistence of TFP 0.92 Persistence of Solow residual 0.92 0.87 φ Adjustment cost 19 Volatility of cons. growth (%) 1.32 1.38 Firm level data ρb Persist of idio prod 0.65 Persistence of firm ROE 0.40 0.48 σb Vol of idio prod 0.80 Vol firm stock return (%) 0.35 0.34 Chosen outside of the model γ Utility curvature 2.00 For ease of comparison with Campbell-Cochrane (1999) Parameter Value Target Data Model Unconditional asset price moments β Time preference 0.89 Mean 30-day Treasury-bill return 0.89 0.96 ρs Persistence of surplusconsumption 0.86 Persistence of CRSP price/div 0.87 0.88 s¯ Steady-state surplus 0.06 Mean Sharpe ratio of 0.44 0.39 consumption CRSP index Long-run growth moments α Production curvature 0.35 Mean output/capital 0.41 0.42 δ Depreciation rate 0.08 Mean investment rate 0.07 0.08 Unconditional business cycle moments σa Volatility of TFP 0.03 Volatility of GDP (%) 1.61 1.70 ρa Persistence of TFP 0.92 Persistence of Solow residual 0.92 0.87 φ Adjustment cost 19 Volatility of cons. growth (%) 1.32 1.38 Firm level data ρb Persist of idio prod 0.65 Persistence of firm ROE 0.40 0.48 σb Vol of idio prod 0.80 Vol firm stock return (%) 0.35 0.34 Chosen outside of the model γ Utility curvature 2.00 For ease of comparison with Campbell-Cochrane (1999) The model, parameter values, and empirical moments are annual. Consumption is real nondurable goods and services consumption. Returns are adjusted for inflation using CPI. The volatility of GDP and relative volatility of consumption are logged and HP-filtered with a smoothing parameter of 6.25. Firm-level empirical moments use Compustat firms with market equity more than $600 million. Table 1 Calibration Parameter Value Target Data Model Unconditional asset price moments β Time preference 0.89 Mean 30-day Treasury-bill return 0.89 0.96 ρs Persistence of surplusconsumption 0.86 Persistence of CRSP price/div 0.87 0.88 s¯ Steady-state surplus 0.06 Mean Sharpe ratio of 0.44 0.39 consumption CRSP index Long-run growth moments α Production curvature 0.35 Mean output/capital 0.41 0.42 δ Depreciation rate 0.08 Mean investment rate 0.07 0.08 Unconditional business cycle moments σa Volatility of TFP 0.03 Volatility of GDP (%) 1.61 1.70 ρa Persistence of TFP 0.92 Persistence of Solow residual 0.92 0.87 φ Adjustment cost 19 Volatility of cons. growth (%) 1.32 1.38 Firm level data ρb Persist of idio prod 0.65 Persistence of firm ROE 0.40 0.48 σb Vol of idio prod 0.80 Vol firm stock return (%) 0.35 0.34 Chosen outside of the model γ Utility curvature 2.00 For ease of comparison with Campbell-Cochrane (1999) Parameter Value Target Data Model Unconditional asset price moments β Time preference 0.89 Mean 30-day Treasury-bill return 0.89 0.96 ρs Persistence of surplusconsumption 0.86 Persistence of CRSP price/div 0.87 0.88 s¯ Steady-state surplus 0.06 Mean Sharpe ratio of 0.44 0.39 consumption CRSP index Long-run growth moments α Production curvature 0.35 Mean output/capital 0.41 0.42 δ Depreciation rate 0.08 Mean investment rate 0.07 0.08 Unconditional business cycle moments σa Volatility of TFP 0.03 Volatility of GDP (%) 1.61 1.70 ρa Persistence of TFP 0.92 Persistence of Solow residual 0.92 0.87 φ Adjustment cost 19 Volatility of cons. growth (%) 1.32 1.38 Firm level data ρb Persist of idio prod 0.65 Persistence of firm ROE 0.40 0.48 σb Vol of idio prod 0.80 Vol firm stock return (%) 0.35 0.34 Chosen outside of the model γ Utility curvature 2.00 For ease of comparison with Campbell-Cochrane (1999) The model, parameter values, and empirical moments are annual. Consumption is real nondurable goods and services consumption. Returns are adjusted for inflation using CPI. The volatility of GDP and relative volatility of consumption are logged and HP-filtered with a smoothing parameter of 6.25. Firm-level empirical moments use Compustat firms with market equity more than $600 million. I choose aggregate technology parameters to fit moments of the real economy. The production curvature α and depreciation rate δ are chosen to fit the capital-output ratio and the mean growth-adjusted investment rate. These moments match nicely at 0.40 and 0.08 respectively. The volatility of aggregate productivity σa is chosen to fit the volatility of HP-filtered log gross domestic product (GDP) (I use a smoothing parameter of 6.25, as argued by Ravn and Uhlig (2002)). The data and the model match well in this dimension, producing a volatility of about 1.5%. The persistence of aggregate productivity is chosen to fit the persistence of the Solow residual with constant labor. A critical parameter of the production technology is the quadratic adjustment cost parameter φ . I choose this parameter value to hit the volatility of aggregate consumption growth (nondurables and services). The data and the model match well here, producing a volatility of about 1.5% per year. Firm-level technology parameters are chosen to fit the cross-sectional means of time-series moments. I target non-micro-cap firms (firms with market equity of more than 600 million). The persistence of idiosyncratic productivity ρb is chosen to fit the persistence of firm-level ROE. The volatility of idiosyncratic productivity σb is chosen to match the volatility of individual stock returns. 3. Aggregate Implications This section discusses the aggregate asset price and business-cycle performance of the model. The aggregate implications are similar to an earlier paper (Chen 2017a), and thus I keep the discussion brief. Some aggregate features, however, are helpful for understanding elements of the value premium. I review these features in some detail. The key aggregate feature of the model is endogenous countercyclical consumption volatility. This feature is illustrated in Figure 1, which shows scatterplots of consumption, consumption volatility, and stock prices from model simulations. Figure 1 View largeDownload slide Concave consumption and consumption volatility risk The figure plots data from 100 simulations the same length as the empirical sample. Each marker represents one year from a model simulation. x’s are quarters in the lowest quartile of surplus consumption. Squares are the top quartile. Consumption Ct and the aggregate stock price ∑i(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) are normalized to be 1 on average. Consumption is concave is the aggregate stock price, which leads to countercyclical consumption volatility. Figure 1 View largeDownload slide Concave consumption and consumption volatility risk The figure plots data from 100 simulations the same length as the empirical sample. Each marker represents one year from a model simulation. x’s are quarters in the lowest quartile of surplus consumption. Squares are the top quartile. Consumption Ct and the aggregate stock price ∑i(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) are normalized to be 1 on average. Consumption is concave is the aggregate stock price, which leads to countercyclical consumption volatility. The figure shows that consumption is concave in the aggregate stock prices (left panel). This shape implies that consumption is more sensitive to stock prices when prices are low, as the slope is steeper. This countercyclical sensitivity implies that consumption volatility is countercyclical (right panel), although the model contains only one homoscedastic aggregate shock (Equation (7)). These consumption dynamics originate from precautionary savings motives which make consumption concave in wealth (Carroll and Kimball 1996). External habit amplifies the concavity, leading to the visible cyclicality seen in Figure 1. Physical investment is also critical, as without it precautionary motives cannot affect aggregate quantities in equilibrium. Further details about this mechanism can be found in Chen (2017a). Countercyclical consumption volatility, then, leads to countercyclical risk premia. This is seen in the lognormal approximation of the maximum Sharpe ratio max{all assets}{Et(Rt+1−Rf,t+1)σt(Rt+1)}≈γ(λ+1)σt(Δct+1). (21) In bad times, σt(Δct+1) is high, and thus so are risk premia. Note that this mechanism is distinct from Campbell and Cochrane (1999), in which the habit sensitivity λ varies over time instead of consumption volatility. The model’s consumption volatility has similar quantitative implications with Campbell and Cochrane’s (1999) habit sensitivity. As a result, the model shares much of the quantitative performance of the Campbell-Cochrane model. Tables 2–4 illustrate this performance. Table 2 Unconditional aggregate asset price moments U.S. Data Model Mean 5 % 50 % 95 % Calibrated moments E(rf) (%) 0.89 0.96 −3.98 0.34 8.45 AC1(rf) 0.84 0.83 0.57 0.86 0.99 E(Rm−Rf)/σ(Rm) 0.44 0.46 0.26 0.46 0.69 Untargeted moments E(rm−rf) (%) 6.36 7.42 4.95 7.44 9.91 σ(rm−rf) (%) 16.52 18.20 12.25 17.78 25.19 AC1(rm−rf) 0.08 −0.06 −0.28 −0.06 0.14 σ(rf) (%) 1.82 3.87 0.55 2.25 12.20 E(pm−dm) 3.36 2.57 2.00 2.57 3.04 σ(pm−dm) 0.45 0.43 0.24 0.41 0.70 AC1(pm−dm) 0.87 0.88 0.72 0.90 0.95 U.S. Data Model Mean 5 % 50 % 95 % Calibrated moments E(rf) (%) 0.89 0.96 −3.98 0.34 8.45 AC1(rf) 0.84 0.83 0.57 0.86 0.99 E(Rm−Rf)/σ(Rm) 0.44 0.46 0.26 0.46 0.69 Untargeted moments E(rm−rf) (%) 6.36 7.42 4.95 7.44 9.91 σ(rm−rf) (%) 16.52 18.20 12.25 17.78 25.19 AC1(rm−rf) 0.08 −0.06 −0.28 −0.06 0.14 σ(rf) (%) 1.82 3.87 0.55 2.25 12.20 E(pm−dm) 3.36 2.57 2.00 2.57 3.04 σ(pm−dm) 0.45 0.43 0.24 0.41 0.70 AC1(pm−dm) 0.87 0.88 0.72 0.90 0.95 Figures are annual and real. Data moments are taken from Beeler and Campbell (2012) and correspond to 1947–2008. Model columns show means and percentiles from 100 simulations of the same length as the empirical sample. rm is the log return on the CRSP index, rf is the risk-free rate, pm−dm is the CRSP log price-dividend ratio. Δdm,t is CRSP dividend growth. Table 2 Unconditional aggregate asset price moments U.S. Data Model Mean 5 % 50 % 95 % Calibrated moments E(rf) (%) 0.89 0.96 −3.98 0.34 8.45 AC1(rf) 0.84 0.83 0.57 0.86 0.99 E(Rm−Rf)/σ(Rm) 0.44 0.46 0.26 0.46 0.69 Untargeted moments E(rm−rf) (%) 6.36 7.42 4.95 7.44 9.91 σ(rm−rf) (%) 16.52 18.20 12.25 17.78 25.19 AC1(rm−rf) 0.08 −0.06 −0.28 −0.06 0.14 σ(rf) (%) 1.82 3.87 0.55 2.25 12.20 E(pm−dm) 3.36 2.57 2.00 2.57 3.04 σ(pm−dm) 0.45 0.43 0.24 0.41 0.70 AC1(pm−dm) 0.87 0.88 0.72 0.90 0.95 U.S. Data Model Mean 5 % 50 % 95 % Calibrated moments E(rf) (%) 0.89 0.96 −3.98 0.34 8.45 AC1(rf) 0.84 0.83 0.57 0.86 0.99 E(Rm−Rf)/σ(Rm) 0.44 0.46 0.26 0.46 0.69 Untargeted moments E(rm−rf) (%) 6.36 7.42 4.95 7.44 9.91 σ(rm−rf) (%) 16.52 18.20 12.25 17.78 25.19 AC1(rm−rf) 0.08 −0.06 −0.28 −0.06 0.14 σ(rf) (%) 1.82 3.87 0.55 2.25 12.20 E(pm−dm) 3.36 2.57 2.00 2.57 3.04 σ(pm−dm) 0.45 0.43 0.24 0.41 0.70 AC1(pm−dm) 0.87 0.88 0.72 0.90 0.95 Figures are annual and real. Data moments are taken from Beeler and Campbell (2012) and correspond to 1947–2008. Model columns show means and percentiles from 100 simulations of the same length as the empirical sample. rm is the log return on the CRSP index, rf is the risk-free rate, pm−dm is the CRSP log price-dividend ratio. Δdm,t is CRSP dividend growth. Table 3 Predicting dividend growth and excess returns with the price-dividend ratio A: Predicting dividend growth ∑j=1LΔdm,t+j=α+β(pm,t−dm,t)+ϵt+L L U.S. data Model Mean 5 % 50 % 95 % β^ 1 0.003 0.009 −0.009 0.008 0.033 3 0.012 0.013 −0.015 0.013 0.048 5 0.044 0.016 −0.036 0.017 0.057 t-stat 1 0.112 0.492 −0.616 0.550 1.399 3 0.193 0.669 −0.828 0.754 2.055 5 0.482 0.678 −1.608 0.857 2.408 R2 1 0.000 0.010 0.000 0.006 0.032 3 0.001 0.017 0.000 0.012 0.049 5 0.011 0.026 0.000 0.015 0.080 A: Predicting dividend growth ∑j=1LΔdm,t+j=α+β(pm,t−dm,t)+ϵt+L L U.S. data Model Mean 5 % 50 % 95 % β^ 1 0.003 0.009 −0.009 0.008 0.033 3 0.012 0.013 −0.015 0.013 0.048 5 0.044 0.016 −0.036 0.017 0.057 t-stat 1 0.112 0.492 −0.616 0.550 1.399 3 0.193 0.669 −0.828 0.754 2.055 5 0.482 0.678 −1.608 0.857 2.408 R2 1 0.000 0.010 0.000 0.006 0.032 3 0.001 0.017 0.000 0.012 0.049 5 0.011 0.026 0.000 0.015 0.080 B: Predicting excess returns ∑j=1L(rm,t+j−rf,t+j)=α+β(pm,t−dm,t)+ϵt+L L U.S. Data Model Mean 5 % 50 % 95 % β^ 1 −0.12 −0.12 −0.26 −0.11 −0.01 3 −0.27 −0.23 −0.47 −0.22 −0.03 5 −0.42 −0.33 −0.64 −0.30 −0.02 t-stat 1 −2.63 −1.91 −3.30 −1.99 −0.24 3 −3.19 −2.50 −4.44 −2.54 −0.40 5 −3.37 −3.13 −6.72 −2.98 −0.31 R2 1 0.09 0.07 0.00 0.07 0.16 3 0.19 0.14 0.01 0.13 0.28 5 0.26 0.19 0.01 0.19 0.39 B: Predicting excess returns ∑j=1L(rm,t+j−rf,t+j)=α+β(pm,t−dm,t)+ϵt+L L U.S. Data Model Mean 5 % 50 % 95 % β^ 1 −0.12 −0.12 −0.26 −0.11 −0.01 3 −0.27 −0.23 −0.47 −0.22 −0.03 5 −0.42 −0.33 −0.64 −0.30 −0.02 t-stat 1 −2.63 −1.91 −3.30 −1.99 −0.24 3 −3.19 −2.50 −4.44 −2.54 −0.40 5 −3.37 −3.13 −6.72 −2.98 −0.31 R2 1 0.09 0.07 0.00 0.07 0.16 3 0.19 0.14 0.01 0.13 0.28 5 0.26 0.19 0.01 0.19 0.39 Figures are annual and real. Data moments are taken from Beeler and Campbell (2012) and correspond to 1947–2008. The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. rm,t is the log return on the CRSP index, rf,t is the risk-free rate, pm,t−dm,t is the CRSP log price-dividend ratio. t-statistics are Newey-West with 2(L−1) lags. Table 3 Predicting dividend growth and excess returns with the price-dividend ratio A: Predicting dividend growth ∑j=1LΔdm,t+j=α+β(pm,t−dm,t)+ϵt+L L U.S. data Model Mean 5 % 50 % 95 % β^ 1 0.003 0.009 −0.009 0.008 0.033 3 0.012 0.013 −0.015 0.013 0.048 5 0.044 0.016 −0.036 0.017 0.057 t-stat 1 0.112 0.492 −0.616 0.550 1.399 3 0.193 0.669 −0.828 0.754 2.055 5 0.482 0.678 −1.608 0.857 2.408 R2 1 0.000 0.010 0.000 0.006 0.032 3 0.001 0.017 0.000 0.012 0.049 5 0.011 0.026 0.000 0.015 0.080 A: Predicting dividend growth ∑j=1LΔdm,t+j=α+β(pm,t−dm,t)+ϵt+L L U.S. data Model Mean 5 % 50 % 95 % β^ 1 0.003 0.009 −0.009 0.008 0.033 3 0.012 0.013 −0.015 0.013 0.048 5 0.044 0.016 −0.036 0.017 0.057 t-stat 1 0.112 0.492 −0.616 0.550 1.399 3 0.193 0.669 −0.828 0.754 2.055 5 0.482 0.678 −1.608 0.857 2.408 R2 1 0.000 0.010 0.000 0.006 0.032 3 0.001 0.017 0.000 0.012 0.049 5 0.011 0.026 0.000 0.015 0.080 B: Predicting excess returns ∑j=1L(rm,t+j−rf,t+j)=α+β(pm,t−dm,t)+ϵt+L L U.S. Data Model Mean 5 % 50 % 95 % β^ 1 −0.12 −0.12 −0.26 −0.11 −0.01 3 −0.27 −0.23 −0.47 −0.22 −0.03 5 −0.42 −0.33 −0.64 −0.30 −0.02 t-stat 1 −2.63 −1.91 −3.30 −1.99 −0.24 3 −3.19 −2.50 −4.44 −2.54 −0.40 5 −3.37 −3.13 −6.72 −2.98 −0.31 R2 1 0.09 0.07 0.00 0.07 0.16 3 0.19 0.14 0.01 0.13 0.28 5 0.26 0.19 0.01 0.19 0.39 B: Predicting excess returns ∑j=1L(rm,t+j−rf,t+j)=α+β(pm,t−dm,t)+ϵt+L L U.S. Data Model Mean 5 % 50 % 95 % β^ 1 −0.12 −0.12 −0.26 −0.11 −0.01 3 −0.27 −0.23 −0.47 −0.22 −0.03 5 −0.42 −0.33 −0.64 −0.30 −0.02 t-stat 1 −2.63 −1.91 −3.30 −1.99 −0.24 3 −3.19 −2.50 −4.44 −2.54 −0.40 5 −3.37 −3.13 −6.72 −2.98 −0.31 R2 1 0.09 0.07 0.00 0.07 0.16 3 0.19 0.14 0.01 0.13 0.28 5 0.26 0.19 0.01 0.19 0.39 Figures are annual and real. Data moments are taken from Beeler and Campbell (2012) and correspond to 1947–2008. The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. rm,t is the log return on the CRSP index, rf,t is the risk-free rate, pm,t−dm,t is the CRSP log price-dividend ratio. t-statistics are Newey-West with 2(L−1) lags. Table 4 Business-cycle moments U.S. data Model Mean 5 % 50 % 95 % Calibrated moments σ(yhp) (%) 1.50 1.70 1.20 1.66 2.20 σ(Δc) (%) 1.32 1.38 0.91 1.35 1.95 Untargeted moments σ(chp)/σ(yhp) 0.49 0.47 0.39 0.47 0.56 σ(ihp)/σ(yhp) 2.68 3.45 3.04 3.44 3.90 ρ(yhp,chp) 0.84 0.99 0.98 1.00 1.00 ρ(yhp,ihp) 0.58 1.00 0.99 1.00 1.00 AC1(Δc) 0.52 0.04 −0.20 0.04 0.28 E(Adj cost/Y) (%) 1.01 0.63 0.96 1.64 E(Adj cost/I) (%) 5.92 2.89 5.48 10.61 U.S. data Model Mean 5 % 50 % 95 % Calibrated moments σ(yhp) (%) 1.50 1.70 1.20 1.66 2.20 σ(Δc) (%) 1.32 1.38 0.91 1.35 1.95 Untargeted moments σ(chp)/σ(yhp) 0.49 0.47 0.39 0.47 0.56 σ(ihp)/σ(yhp) 2.68 3.45 3.04 3.44 3.90 ρ(yhp,chp) 0.84 0.99 0.98 1.00 1.00 ρ(yhp,ihp) 0.58 1.00 0.99 1.00 1.00 AC1(Δc) 0.52 0.04 −0.20 0.04 0.28 E(Adj cost/Y) (%) 1.01 0.63 0.96 1.64 E(Adj cost/I) (%) 5.92 2.89 5.48 10.61 Figures are annual and real. Data moments correspond to 1947–2011. yhp, chp, and ihp are HP-filtered log GDP, log consumption, and log investment using a smoothing parameter of 6.25. Δc is consumption growth. Adj cost is the total adjustment cost paid in the year ∑iΦ(Ii,t,Ki,t ). The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. Table 4 Business-cycle moments U.S. data Model Mean 5 % 50 % 95 % Calibrated moments σ(yhp) (%) 1.50 1.70 1.20 1.66 2.20 σ(Δc) (%) 1.32 1.38 0.91 1.35 1.95 Untargeted moments σ(chp)/σ(yhp) 0.49 0.47 0.39 0.47 0.56 σ(ihp)/σ(yhp) 2.68 3.45 3.04 3.44 3.90 ρ(yhp,chp) 0.84 0.99 0.98 1.00 1.00 ρ(yhp,ihp) 0.58 1.00 0.99 1.00 1.00 AC1(Δc) 0.52 0.04 −0.20 0.04 0.28 E(Adj cost/Y) (%) 1.01 0.63 0.96 1.64 E(Adj cost/I) (%) 5.92 2.89 5.48 10.61 U.S. data Model Mean 5 % 50 % 95 % Calibrated moments σ(yhp) (%) 1.50 1.70 1.20 1.66 2.20 σ(Δc) (%) 1.32 1.38 0.91 1.35 1.95 Untargeted moments σ(chp)/σ(yhp) 0.49 0.47 0.39 0.47 0.56 σ(ihp)/σ(yhp) 2.68 3.45 3.04 3.44 3.90 ρ(yhp,chp) 0.84 0.99 0.98 1.00 1.00 ρ(yhp,ihp) 0.58 1.00 0.99 1.00 1.00 AC1(Δc) 0.52 0.04 −0.20 0.04 0.28 E(Adj cost/Y) (%) 1.01 0.63 0.96 1.64 E(Adj cost/I) (%) 5.92 2.89 5.48 10.61 Figures are annual and real. Data moments correspond to 1947–2011. yhp, chp, and ihp are HP-filtered log GDP, log consumption, and log investment using a smoothing parameter of 6.25. Δc is consumption growth. Adj cost is the total adjustment cost paid in the year ∑iΦ(Ii,t,Ki,t ). The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. Table 2 shows aggregate asset-price moments. The model produces a large and volatile equity premium, and a low and smooth risk-free rate. The large and volatile equity premium is typical of habit models (Jermann 1998; Campbell and Cochrane 1999). The smooth risk-free rate comes from countercyclical consumption volatility, which counterbalances the intertemporal substitution effects of habit models. Table 3 shows regressions of future dividend growth and excess returns on the price-dividend ratio. The table shows the model does a good job replicating facts about time-varying risk premia. Like in the data, the price dividend ratio has little predictive power for future dividend growth and has strong predictive power for future excess returns. Table 4 shows basic business-cycle moments. As intended by the calibration, the model produces low consumption volatility. Like in the data, investment is much more volatile than output and consumption is much less volatile. The model also replicates the comovement of consumption, investment, and GDP. 4. The Value Premium This section contains the main results. Section 4.1 shows that the model reproduces key features of the value premium. Sections 4.2–4.4 show how value is linked to low productivity, high cash-flow growth, and high expected returns. Section 4.5 finishes up by showing that cash-flow cyclicality does not explain the value premium. 4.1 Matching moments of the value premium We have seen that the model addresses the equity premium puzzles. External habit combined with production produces a large and volatile equity premium, a low and smooth risk-free rate, and asset price fluctuations that are linked to excess returns. This brings us to the main question of the paper. Is external habit consistent with the cross-section of stock returns? Table 5 examines this question. It shows regressions of next year’s returns on today’s log B/M. The regressions are firm-level and follow the Fama and MacBeth (1973) method. Table 5 Regressions of future returns on log B/M Dependent variable: Ri,t+1 U.S. data Model Mean 5 % 50 % 95 % Intercept 18.62 18.34 13.33 18.03 24.14 t-stat 5.74 3.88 2.99 3.88 4.80 log (B/M)i,t 5.70 5.84 3.58 5.84 7.86 t-stat 4.88 3.34 2.17 3.30 4.57 Dependent variable: Ri,t+1 U.S. data Model Mean 5 % 50 % 95 % Intercept 18.62 18.34 13.33 18.03 24.14 t-stat 5.74 3.88 2.99 3.88 4.80 log (B/M)i,t 5.70 5.84 3.58 5.84 7.86 t-stat 4.88 3.34 2.17 3.30 4.57 Figures are annual. The table shows Fama-Macbeth regressions of future returns on log (B/M). The model counterpart of B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Data moments correspond to 1947–2012. The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. Table 5 Regressions of future returns on log B/M Dependent variable: Ri,t+1 U.S. data Model Mean 5 % 50 % 95 % Intercept 18.62 18.34 13.33 18.03 24.14 t-stat 5.74 3.88 2.99 3.88 4.80 log (B/M)i,t 5.70 5.84 3.58 5.84 7.86 t-stat 4.88 3.34 2.17 3.30 4.57 Dependent variable: Ri,t+1 U.S. data Model Mean 5 % 50 % 95 % Intercept 18.62 18.34 13.33 18.03 24.14 t-stat 5.74 3.88 2.99 3.88 4.80 log (B/M)i,t 5.70 5.84 3.58 5.84 7.86 t-stat 4.88 3.34 2.17 3.30 4.57 Figures are annual. The table shows Fama-Macbeth regressions of future returns on log (B/M). The model counterpart of B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Data moments correspond to 1947–2012. The model columns show means and percentiles from 100 simulations of the same length as the empirical sample. In both the model and data, the log (B/M)i,t coefficient is positive and highly statistically significant. Moreover, the slopes are large and similar in magnitude, with a value of about six. This slope means that in both the model and the data, a 20% higher B/M implies a roughly 120-bp higher expected return. Though firm-level regressions provide the most statistical power and offer the simplest quantitative description of the value premium, the literature often examines value-weighted portfolio sorts. Table 6 shows summary statistics on decile B/M-sorted portfolios. Table 6 Summary statistics for 10 B/M-sorted portfolios E(Rport) σ(Rport) port U.S. data Model U.S. data Model Lo 7.7 9.6 20.9 19.9 2 8.0 10.3 17.1 21.4 3 8.1 10.6 16.7 22.5 4 8.6 10.9 17.6 22.9 5 9.5 11.1 18.3 23.4 6 9.7 11.3 17.6 23.9 7 9.8 11.5 19.3 24.5 8 11.6 11.7 21.1 24.9 9 11.9 12.0 20.4 25.6 Hi 13.3 12.3 25.7 26.7 E(Rport) σ(Rport) port U.S. data Model U.S. data Model Lo 7.7 9.6 20.9 19.9 2 8.0 10.3 17.1 21.4 3 8.1 10.6 16.7 22.5 4 8.6 10.9 17.6 22.9 5 9.5 11.1 18.3 23.4 6 9.7 11.3 17.6 23.9 7 9.8 11.5 19.3 24.5 8 11.6 11.7 21.1 24.9 9 11.9 12.0 20.4 25.6 Hi 13.3 12.3 25.7 26.7 Figures are annual. Returns are value weighted. Each year, firms are sorted into 10 value-weighted portfolios using B/M. Returns are calculated over the next year. The model counterpart of B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Data moments correspond to 1947–2012. Table 6 Summary statistics for 10 B/M-sorted portfolios E(Rport) σ(Rport) port U.S. data Model U.S. data Model Lo 7.7 9.6 20.9 19.9 2 8.0 10.3 17.1 21.4 3 8.1 10.6 16.7 22.5 4 8.6 10.9 17.6 22.9 5 9.5 11.1 18.3 23.4 6 9.7 11.3 17.6 23.9 7 9.8 11.5 19.3 24.5 8 11.6 11.7 21.1 24.9 9 11.9 12.0 20.4 25.6 Hi 13.3 12.3 25.7 26.7 E(Rport) σ(Rport) port U.S. data Model U.S. data Model Lo 7.7 9.6 20.9 19.9 2 8.0 10.3 17.1 21.4 3 8.1 10.6 16.7 22.5 4 8.6 10.9 17.6 22.9 5 9.5 11.1 18.3 23.4 6 9.7 11.3 17.6 23.9 7 9.8 11.5 19.3 24.5 8 11.6 11.7 21.1 24.9 9 11.9 12.0 20.4 25.6 Hi 13.3 12.3 25.7 26.7 Figures are annual. Returns are value weighted. Each year, firms are sorted into 10 value-weighted portfolios using B/M. Returns are calculated over the next year. The model counterpart of B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Data moments correspond to 1947–2012. The expected return columns show that, in both the model and data, expected returns are monotonically increasing in B/M. In the model, expected returns culminate to an economically significant decile 10-1 return of about 2.5% per year. The decile value premium is smaller than that of the data, but it comes with a much smaller dispersion in B/M. Recall that all heterogeneity in the model originates from a single AR1 idiosyncratic productivity process. For parsimony and to maintain clarity of the mechanism, the model abstracts from other sources of heterogeneity such as differences in financial frictions or life-cycle effects. As a result, the model generates a dispersion in B/M that is significantly less than the data. In the model, log B/M differs by 0.4 between the high and low deciles. In the data, this difference is 1.4. This difference in spreads means that the traditional high-low portfolio returns of the model are not comparable to the data. Reproducing the enormous B/M dispersion in the data is an interesting question, but is beyond the scope of this paper. A more effective way to illustrate the portfolio sort results is to interpret them as a nonparametric regression (Cochrane 2011). Figure 2 provides this interpretation. It plots the average returns of the 10 B/M-sorted portfolios against log B/M. Returns are equally weighted in this figure because the functional form in the data reflects equal weighting; that is, value-weighted returns do not result in a log linear pattern in the data. The figure shows that the model captures this log-linear form. The match is not only qualitative but quantitative too. The slope of the the relationship between expected returns and log B/M is similar in both model and data, consistent with the results of the firm-level regressions (Table 5). Figure 2 View largeDownload slide B/M decile returns as a function of B/M Each year, firms are sorted into 10 equal-weighted portfolios by B/M. Returns are calculated over the next year, and the expected return is the mean return for each portfolio. log (B/M) for each portfolio-year is the mean log (B/M), and log (B/M) in the figure averages over years. The data panel is computed from data from Ken French’s Web site. The model panel uses 100 simulations of the model of the same length as the empirical sample. As the model lacks leverage, B/M in the model is capital divided by the ex-dividend firm market value Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) . The model captures the functional form and the slope of the relationship between expected returns and B/M. Figure 2 View largeDownload slide B/M decile returns as a function of B/M Each year, firms are sorted into 10 equal-weighted portfolios by B/M. Returns are calculated over the next year, and the expected return is the mean return for each portfolio. log (B/M) for each portfolio-year is the mean log (B/M), and log (B/M) in the figure averages over years. The data panel is computed from data from Ken French’s Web site. The model panel uses 100 simulations of the model of the same length as the empirical sample. As the model lacks leverage, B/M in the model is capital divided by the ex-dividend firm market value Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) . The model captures the functional form and the slope of the relationship between expected returns and B/M. 4.2 Value and idiosyncratic productivity To understand where the value premium comes from, it helps to start with the meaning of value in the model. Figure 3 plots B/M and expected returns against the two firm state variables, idiosyncratic productivity and capital. The left panel shows that value firms are low productivity firms with high capital. Capital, however, is slow moving. As a result, value is primarily characterized by low productivity. The right panel shows how value is connected to expected returns. Expected returns decline strongly in idiosyncratic productivity. Figure 3 View largeDownload slide B/M and expected returns as a function of firm states Figures show laws of motion of the model. B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Idiosyncratic productivity is Bi,t (see Equations (6) and (8)). Low, medium, and high firm capital ( Ki,t ) are the 25th, 50th, and 75th percentiles of simulated data. Expected return is the expected return of the firm over the next year ( Et(Vi,t+1/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Aggregate states are fixed at their median values from simulated data. High B/M implies low idiosyncratic productivity and high expected returns. Figure 3 View largeDownload slide B/M and expected returns as a function of firm states Figures show laws of motion of the model. B/M is capital divided by the ex-dividend firm market value ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Idiosyncratic productivity is Bi,t (see Equations (6) and (8)). Low, medium, and high firm capital ( Ki,t ) are the 25th, 50th, and 75th percentiles of simulated data. Expected return is the expected return of the firm over the next year ( Et(Vi,t+1/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Aggregate states are fixed at their median values from simulated data. High B/M implies low idiosyncratic productivity and high expected returns. Overall, Figure 3 shows that value firms are low productivity firms that have high expected returns. The productivity relationship is consistent with the empirical facts about return on book equity (ROE) presented in Fama and French (1995). Fama and French find that value portfolios have low ROE, while growth portfolios have the opposite. Since ROE is net income divided by book equity, it is naturally mapped to the model’s concept of idiosyncratic productivity, which measures the amount of goods a firm can produce from a unit of capital (Equation (6)). Fama and French also find that these ROE differences revert over time. This dynamic is replicated by the model, as seen in Figure 4. The figure plots the ROE of B/M-sorted portfolios as a function of the time since portfolio formation. Value portfolios have low ROE at portfolio formation, but their ROE slowly increases over time. Growth portfolios follow the opposite pattern. Figure 4 View largeDownload slide Return on equity of B/M-sorted portfolios Return on equity for a portfolio-year is the total earnings of the portfolio ( ∑iΠi,t−δKi,t−Φ(Ii,t,Ki,t) ) divided by the total capital in the portfolio ( ∑iKi,t ). In each year of each simulation, firms are sorted into 10 portfolios based on the model counterpart to B/M: ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). The figure averages return on equity across 100 model simulations. Growth portfolios have high and mean reverting return on equity, and value portfolios have the opposite, consistent with the empirical results of Fama and French (1995). Figure 4 View largeDownload slide Return on equity of B/M-sorted portfolios Return on equity for a portfolio-year is the total earnings of the portfolio ( ∑iΠi,t−δKi,t−Φ(Ii,t,Ki,t) ) divided by the total capital in the portfolio ( ∑iKi,t ). In each year of each simulation, firms are sorted into 10 portfolios based on the model counterpart to B/M: ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). The figure averages return on equity across 100 model simulations. Growth portfolios have high and mean reverting return on equity, and value portfolios have the opposite, consistent with the empirical results of Fama and French (1995). The assumption that idiosyncratic productivity is slowly mean reverting is critical for these results. If productivity is close to a random walk, the model would counterfactually predict that differences in ROE are nearly permanent. Similarly, quickly mean reverting productivity would imply very transient differences in ROE between value and growth firms. As idiosyncratic productivity determines the cross-sectional distribution of cash flows, the mean reversion assumption is also critical for the value premium. Indeed, we will see that a few steps of logic lead from Figure 4 to the existence of a value premium. 4.3 Low productivity and cash-flow growth Mean-reverting productivity, then, implies that value firms have high cash-flow growth. This relationship is illustrated in Figure 5, which plots the cash-flow growth of portfolios sorted on B/M. Figure 5 View largeDownload slide Cash-flow growth of B/M-sorted portfolios Cash-flow growth is computed from 100 model simulations. In each year of each simulation, firms are sorted into 10 portfolios based on the model counterpart to B/M: ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Portfolio cash flows are constructed by value-weighting firm cash flows [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] . Average portfolio cash flow for each year after portfolio formation is then constructed by averaging across simulation years. Portfolio cash-flow growth is constructed by taking log first differences for the average cash-flow level. The value portfolio has negative cash flows in year one, so its year 2 growth rate is omitted from the plot. Value portfolios have, on average, higher cash-flow growth. Figure 5 View largeDownload slide Cash-flow growth of B/M-sorted portfolios Cash-flow growth is computed from 100 model simulations. In each year of each simulation, firms are sorted into 10 portfolios based on the model counterpart to B/M: ( Ki,t/(Vi,t−[AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)]) ). Portfolio cash flows are constructed by value-weighting firm cash flows [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] . Average portfolio cash flow for each year after portfolio formation is then constructed by averaging across simulation years. Portfolio cash-flow growth is constructed by taking log first differences for the average cash-flow level. The value portfolio has negative cash flows in year one, so its year 2 growth rate is omitted from the plot. Value portfolios have, on average, higher cash-flow growth. Soon after portfolio formation, value portfolios (darkest lines) have high dividend growth. Growth slows down quickly and eventually reaches the average growth rate of zero (the model abstracts from long run growth). Intuitively, value firms have low productivity, but this low productivity is temporary (Figure 4). Mean reversion implies that productivity grows, and so cash flow grows. This contrasts with growth portfolios (the lightest lines), which show the reverse pattern. When growth firms are declared as growth, they have very high productivity. Mean reversion then means that their productivity will fall, leading to low cash-flow growth. While mean reversion provides a simple story of the relation between value and cash-flow growth, the exact relationship is endogenous. Cash flow also depends on investment, which is not discussed in the simple story above. Indeed, low productivity tends to encourage disinvestment, increasing cash flow today and decreasing cash-flow growth. Thus, in principle, whether the productivity channel or the investment channel dominates depends on parameter choices. I come back to this issue when I discuss alternative parameter choices (Sections 5.1 and 5.2). 4.4 Cash-flow growth and expected returns So far, I have shown that value is characterized by low productivity and high cash-flow growth. To finish the story, I need to show that high cash-flow growth leads to high expected returns. This last link is due to the persistent discount rate shocks that drive the external habit model. High cash-flow growth means that cash flows are distributed far into the future, and thus are more exposed to persistent discount rate shocks. Investors, then, demand high returns in exchange for bearing this higher exposure. A number of previous papers discuss this link (Cornell 1999; Lettau and Wachter 2007; Santos and Veronesi 2010; Chen 2017b), but here I provide a new and simplified sketch of the intuition. This sketch is informal and steps outside of the general equilibrium model. Consider a growing perpetuity: P0=D1κ0−g, (22) where P0 is the price of the asset, D1 is the next period’s cash flow, κ0 is the discount rate and g is the growth rate of cash flows. Now suppose that the discount rate gets hit by an unexpected shock Δκ . The next-period price is then P1=D1(1+g)κ0+Δκ−g. (23) Taking a first-order Taylor approximation of the definition of the return gives us R1≡D1+P1P0≈(1+κ0)−(1+gκ0−g)Δκ. (24) If discount rates suddenly go up, the stock price takes a hit, and so we have a negative sign on the second term. Notice also that the second term is increasing in the cash-flow growth rate g, implying that discount rate shocks hit high cash-flow growth assets particularly hard. Intuitively, high cash-flow growth means that most of the cash flows will occur in the distant future, and these distant cash flows are hit multiple times by a persistent shock to discount rates. Informally, the law of one price implies that3 E0[R1−Rf]≈(1+gκ0−g)Covt(−Δκ,−M1)σ0(M1)E0(M1) (25) Provided that the discount rate shock is positively correlated with the SDF, this higher exposure to discount rate shocks commands a risk premium. In the model, this correlation is indeed positive (Chen 2017a). Like in most habit models, a negative shock both increases discount rates and increases the marginal utility of consumption. 4.5 Cash-flow cyclicality The seminal Zhang (2005) model of the value premium features a similar, q-theoretical model. Zhang’s analysis of the value premium focuses on cash-flow cyclicality, rather than the term structure. In this section, I demonstrate that cash-flow cyclicality is not driving the value premium. Figure 6 provides this demonstration. Figure 6 View largeDownload slide The cyclicality of value and growth cash flows Value firm plots are calculated using the median firm capital Ki,t and idiosyncratic productivity Bi,t of firms in the tenth decile of B/M-sorted portfolios. Growth firm plots use the respective capital and idiosyncratic productivity from the first decile. Cash flow is [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] and net investment is Ii,t−δKi,t . Aggregate capital Kt is fixed at its mean across 100 model simulations. Low, med, and high aggregate productivity At are the 25th, 50th, and 75th percentiles across simulations. Value firm cash flows are less procyclical than growth firms’, implying that cyclicality does not explain the model’s value premium. Figure 6 View largeDownload slide The cyclicality of value and growth cash flows Value firm plots are calculated using the median firm capital Ki,t and idiosyncratic productivity Bi,t of firms in the tenth decile of B/M-sorted portfolios. Growth firm plots use the respective capital and idiosyncratic productivity from the first decile. Cash flow is [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] and net investment is Ii,t−δKi,t . Aggregate capital Kt is fixed at its mean across 100 model simulations. Low, med, and high aggregate productivity At are the 25th, 50th, and 75th percentiles across simulations. Value firm cash flows are less procyclical than growth firms’, implying that cyclicality does not explain the model’s value premium. The left panels of Figure 6 plot the cash flows of value and growth firms against the two state variables that represent the business cycle in this model: surplus consumption and aggregate productivity. For both value and growth firms, cash flow declines in surplus consumption. Since high surplus consumption represents a good state, in this respect both value and growth firms have countercyclical cash flows. Regarding the magnitude of the countercyclicality, there is no apparent difference. On the other hand, growth firms are clearly more procyclical in terms of aggregate productivity. The top left panel shows that value firm cash flow decreases slightly in aggregate productivity (comparing solid to dotted lines). Growth firms (bottom left), on the other hand, show a clear positive relationship, as the dotted line is far above the solid one. Thus, the cyclicality of cash flows itself would then lead to a value discount, not a value premium. The right panels of Figure 6 show that these results are intuitive. These panels show net investment (investment net of depreciation) for value and growth firms. In bad times, that is, in states with low surplus consumption or low aggregate productivity, value firms are disinvesting. These are times when the household highly values consumption, and since value firms are unproductive, it is efficient for the value firms to discard their capital and provide cash flows to the household. This behavior leads to countercyclical cash flows for value firms. On the other hand, growth firms are investing in bad states. The household wants consumption, but since growth firms are so productive, it is efficient for the firm to give the household less consumption so that it can invest for the future. This behavior leads to procyclical and riskier cash flows for growth firms. Of course, the risk of holding a stock is not just the risk of its cash flow next period. Every cash flow into the infinite future affects the risk of the stock. Both the temporal distribution and the short-term cyclicality of a firm’s cash flows affect its risk and return. On net, the high cash-flow growth of value outweighs the lower cyclicality of its cash flows. Empirical evidence suggests that value firms have more cyclical cash flows (Cohen, Polk, and Vuolteenaho 2009; Santos and Veronesi 2010), and so this element of the model is somewhat at odds with the data. One can increase the cyclicality of value cash flows by including operating leverage or costly investment reversibility, but this model is complex enough without these features. Nevertheless, adding these features to the model and comparing the relative contribution of time horizon and cyclicality to the value premium is an interesting question for future research. 5. The Value Premium under Alternative Assumptions This section examines model predictions under alternative parameter choices and modeling assumptions. These examinations provide a deeper understanding of how optimal firm decisions lead to the investment and cash-flow dynamics discussed earlier. Specifically, I examine alternative assumptions for habit (Section 5.1) and the persistence of idiosyncratic productivity (Section 5.2). I also examine the role of general equilibrium (Section 5.3). 5.1 Habit and the value premium Habit plays several roles in the model’s results. Like in other models, habit adds to the amount of and variation in aggregate risk. These aggregate features work through the preferences of the representative agent, which have been studied extensively in previous papers. Habit plays a novel role in this heterogenous firm model, however. As habit affects the calibration of the capital adjustment cost parameter, the strength of habit indirectly affects the behavior of investment, and thus, the cross-sectional of cash flows and expected returns. These indirect effects are examined in Table 7. Each column examines model predictions using a different assumption for the steady-state surplus consumption S¯ . For each S¯ , I recalibrate the adjustment cost φ to roughly fit the relative volatility of consumption growth to GDP. Thus, the models in Table 7 should be interpreted as alternative fitted models rather than pure comparative statics. Table 7 Habit and the value premium Habit level Full Half Quarter Eighth Steady-state surplus consumption S¯ 0.06 0.12 0.24 0.48 Adjustment cost φ 19.0 10.0 3.0 0.1 σ(Δc)/σ(Δy) 0.50 0.55 0.56 0.57 σ(Rm) (%, annual) 21.2 10.9 5.0 2.1 CF growth value-growth (%, year 2) 29.4 24.5 12.6 −35.2 Slope of Ri,t+1 on log (B/M)t 5.98 0.99 −0.27 −0.09 Habit level Full Half Quarter Eighth Steady-state surplus consumption S¯ 0.06 0.12 0.24 0.48 Adjustment cost φ 19.0 10.0 3.0 0.1 σ(Δc)/σ(Δy) 0.50 0.55 0.56 0.57 σ(Rm) (%, annual) 21.2 10.9 5.0 2.1 CF growth value-growth (%, year 2) 29.4 24.5 12.6 −35.2 Slope of Ri,t+1 on log (B/M)t 5.98 0.99 −0.27 −0.09 Each column shows simulated moments from a different calibration of the model. All parameters are the same as the baseline (Table 1), except for S¯, φ , and σb. In all columns, I use σb=0.40 to make the numerical solution more stable. S¯ is chosen to show the effect of varying habit. Given S¯, φ is chosen so that the relative volatility of consumption to GDP ( σ(Δc)/σ(Δy) ) is approximately 0.50. σ(Rm) is the volatility of the return of the market. CF growth value-growth (%, year 2) is the difference in cash-flow growth between the eighth B/M-sorted portfolio and the third portfolio in year 2 (following Figure 5). Slope of Ri,t+1 on log (B/M)t comes from Fama-Macbeth regressions following Table 5. Table 7 Habit and the value premium Habit level Full Half Quarter Eighth Steady-state surplus consumption S¯ 0.06 0.12 0.24 0.48 Adjustment cost φ 19.0 10.0 3.0 0.1 σ(Δc)/σ(Δy) 0.50 0.55 0.56 0.57 σ(Rm) (%, annual) 21.2 10.9 5.0 2.1 CF growth value-growth (%, year 2) 29.4 24.5 12.6 −35.2 Slope of Ri,t+1 on log (B/M)t 5.98 0.99 −0.27 −0.09 Habit level Full Half Quarter Eighth Steady-state surplus consumption S¯ 0.06 0.12 0.24 0.48 Adjustment cost φ 19.0 10.0 3.0 0.1 σ(Δc)/σ(Δy) 0.50 0.55 0.56 0.57 σ(Rm) (%, annual) 21.2 10.9 5.0 2.1 CF growth value-growth (%, year 2) 29.4 24.5 12.6 −35.2 Slope of Ri,t+1 on log (B/M)t 5.98 0.99 −0.27 −0.09 Each column shows simulated moments from a different calibration of the model. All parameters are the same as the baseline (Table 1), except for S¯, φ , and σb. In all columns, I use σb=0.40 to make the numerical solution more stable. S¯ is chosen to show the effect of varying habit. Given S¯, φ is chosen so that the relative volatility of consumption to GDP ( σ(Δc)/σ(Δy) ) is approximately 0.50. σ(Rm) is the volatility of the return of the market. CF growth value-growth (%, year 2) is the difference in cash-flow growth between the eighth B/M-sorted portfolio and the third portfolio in year 2 (following Figure 5). Slope of Ri,t+1 on log (B/M)t comes from Fama-Macbeth regressions following Table 5. As the baseline model takes several days to solve, I set the volatility of idiosyncratic productivity σb to 0.40 rather than the baseline calibration of 0.80 for this exercise. This decreased idiosyncratic volatility means that the model doesn’t fit the volatility of firm stock returns as well as the baseline (Table 1), but it helps ensure that the solution algorithm converges, which is important when calibrating multiple models.4 Examining σb=0.40 is also interesting, as it helps examine the robustness of the value premium results. The “full” column of Table 7 shows that the value premium results are fairly robust. For σb=0.40 , the slope from regressing future returns on log (B/M)t is 5.98, close to the baseline model’s value of 5.84 and also close to the empirical value of 5.70 (Table 5). Moving across columns, we see that S¯ has mostly monotonic effects on the adjustment cost φ , the cash-flow growth differential, and the value premium slope. As S¯ increases, φ decreases, and the difference in cash-flow growth decreases (and ultimately reverses). The value premium slope generally decreases in S¯ , and ends up at approximately zero for S¯=0.24 and greater. To understand these relationships, it helps to recall that higher S¯ (weaker habit) implies a weaker consumption smoothing motive (Chen 2017a). Thus, starting from the full habit model, increasing S¯ would lead to an excessively volatile consumption growth, unless another parameter is adjusted. The intuitive adjustment is a decrease in φ . Decreasing φ encourages more volatile investment, and, through market clearing, smoother consumption. Thus, decreasing φ counteracts the effects of higher S¯ . Habit, then, affects cross-sectional cash-flow dynamics indirectly through the calibration of φ . At full habit, φ is large, and firms make relatively small capital adjustments in response to productivity shocks. As a result, the cash-flow dynamics are driven by the productivity process, and unproductive value firms experience high cash-flow growth as their productivity mean reverts. The dominance of the productivity channel is seen in the “full” column, which shows that value firms’ cash flows grow 29.4 percentage points more than growth firms’. Weaker levels of habit imply a low φ that results in a more prominent investment channel. The investment channel acts in exactly the opposite direction of productivity: low productivity encourages disinvestment, increasing cash flows today and decreasing cash-flow growth. At one quarter of the baseline level of habit (“quarter” column), the difference in cash-flow growth of 12.6 percentage points is less than half of the baseline. At one eighth habit, the investment channel dominates and the cash-flow growth pattern reverses, resulting in value firms having 35.2 percentage point lower cash-flow growth. These cash-flow dynamics result in a declining value premium. As we move from full habit to eighth habit, the slope on log (B/M)t decreases from about 6.0 to approximately zero. The value premium story is more complicated, however, as reducing habit also reduces aggregate risk, as seen in the market volatility σ(Rm) row. These results illustrate how investment affects cross-sectional cash-flow dynamics, and how the importance of this channel is determined by habit. Overall, the investment channel is limited in habit models. Even at one quarter of the baseline habit level, the productivity channel dominates and value firms have higher cash-flow growth. Moreover, for values of habit that result in aggregate return volatility of the same order of magnitude as the data ( S¯≤0.12 ), the value has higher cash-flow growth, and the value premium exists. 5.2 The persistence of idiosyncratic productivity and the value premium The cross-section of cash-flow dynamics is also affected by the persistence of idiosyncratic productivity ρb. Intuitively, investment is driven by expectations about productivity in the future, which, in turn is driven by ρb. These cash-flow dynamics, in turn, affect the value premium. The effect of the persistence of idiosyncratic productivity ρb is illustrated in Table 8. Each column examines a different choice of ρb. All other parameters are the same as those used in the baseline, except for σb, which is set to 0.40 to help with the stability of the solution algorithm.5 Table 8 The persistence of idiosyncratic productivity and the value premium Persistence of idio prod ρb 0.25 0.45 0.65 0.85 Persistence of firm ROE 0.25 0.41 0.58 0.73 CF growth value-growth (%, year 2) 50.8 39.6 29.4 14.5 Slope of Ri,t+1 on log (B/M)t 6.19 5.97 5.98 3.28 Persistence of idio prod ρb 0.25 0.45 0.65 0.85 Persistence of firm ROE 0.25 0.41 0.58 0.73 CF growth value-growth (%, year 2) 50.8 39.6 29.4 14.5 Slope of Ri,t+1 on log (B/M)t 6.19 5.97 5.98 3.28 Each column shows simulated moments from a different calibration of the model. Parameters are the same as the baseline (Table 1), except for σb and ρb. In all columns, I use σb=0.40 to make the numerical solution more stable. ρb is chosen to illustrate the effect of the persistence of idiosyncractic productivity. Firm ROE is net income divided by capital ( (Πi,t−δKi,t−Φ(Ii,t,Ki,t)/Ki,t ). CF growth Value-Growth (%, year 2) is the difference in cash-flow growth between the eighth B/M-sorted portfolio and the third portfolio in year 2 (following Figure 5). The slope of Ri,t+1 on log (B/M)t comes from Fama-Macbeth regressions (following Table 5). Table 8 The persistence of idiosyncratic productivity and the value premium Persistence of idio prod ρb 0.25 0.45 0.65 0.85 Persistence of firm ROE 0.25 0.41 0.58 0.73 CF growth value-growth (%, year 2) 50.8 39.6 29.4 14.5 Slope of Ri,t+1 on log (B/M)t 6.19 5.97 5.98 3.28 Persistence of idio prod ρb 0.25 0.45 0.65 0.85 Persistence of firm ROE 0.25 0.41 0.58 0.73 CF growth value-growth (%, year 2) 50.8 39.6 29.4 14.5 Slope of Ri,t+1 on log (B/M)t 6.19 5.97 5.98 3.28 Each column shows simulated moments from a different calibration of the model. Parameters are the same as the baseline (Table 1), except for σb and ρb. In all columns, I use σb=0.40 to make the numerical solution more stable. ρb is chosen to illustrate the effect of the persistence of idiosyncractic productivity. Firm ROE is net income divided by capital ( (Πi,t−δKi,t−Φ(Ii,t,Ki,t)/Ki,t ). CF growth Value-Growth (%, year 2) is the difference in cash-flow growth between the eighth B/M-sorted portfolio and the third portfolio in year 2 (following Figure 5). The slope of Ri,t+1 on log (B/M)t comes from Fama-Macbeth regressions (following Table 5). As ρb increases, the value-growth cash-flow growth differential decreases, as does the value premium. These effects are intuitive. High ρb implies that low productivity will persist well into the future, encouraging value firms to disinvest, and lowering the cash-flow growth of value firms. This decrease in cash-flow growth implies that there is less of a difference in exposure to discount rate shocks, and thus a smaller value premium. The value premium is fairly robust, however. For values of ρb between 0.25 and 0.85, the slope of future returns on log (B/M)t is the same magnitude as that of the data. ρb also has a direct effect on the persistence of firm ROE. Higher ρb results in higher ROE persistence. In the baseline calibration, ρb is set to 0.65 in order to match the empirical persistence of ROE of 0.48. The ρb=0.65 column of Table 8 shows a higher ROE persistence of 0.58, as a result of the lower σb assumed for this comparative static exercise. The robustness of the value premium slope, however, shows that the these calibration choices are not critical. 5.3 General equilibrium and the value premium The model is general equilibrium (GE), which plays an important role in pinning down difficult-to-observe investment frictions. As the previous section show, investment frictions have an important effect on the model’s cross-sectional asset pricing results. To explore the role of GE, I conduct a partial equilibrium (PE) experiment. First I take the laws of motion for consumption and aggregate capital (17) and apply parameters values from the calibration (Table 1). Note that these parameter values are calibrated using a GE model. I then plug these laws of motion into the firm’s problem (12) and solve for firm investment policies, but I change the adjustment costs for the firm’s problem to be 1/20th of their calibrated value. These lower adjustment costs are in line with partial equilibrium estimates which use a constant SDF (Whited 1992). Lastly, I simulate a panel of firms using these PE investment policies, updating aggregates using the GE laws of motion. This procedure follows the large literature on partial equilibrium dynamic firm models (Zhang 2005; Carlson, Fisher, and Giammarino 2005; Hennessy and Whited 2005). I am conjecturing an SDF, and then solving for optimal firm behavior given this SDF, but I do not go on to check that the SDF is consistent with firm behavior. As a result, markets will not clear, that is, equation (14) does not hold. Indeed, consumption is not clearly defined since I can calculate consumption either from the conjectured law of motion or by aggregating in the panel simulation. Table 12 shows that in this PE model, the value premium disappears. It shows Fama-Macbeth regressions of next year’s returns on today’s log B/M ratio. While the GE model matches the data quite nicely, in the GE model, the slope on log B/M becomes tiny and statistically insignificant. Figure 7 explains why the value premium goes away. It shows the cash-flow growth of value and growth firms, comparing the GE model to the PE model. In the GE model, there is a large spread in cash-flow growth, but in PE, the spread is tiny. Intuitively, a firms do not want to have high cash-flow growth because temporally distant cash flows raise its discount rate and lowers its value. The firm tries to reduce its discount rate by shifting its cash flows from the future to the present, that is, by disinvesting. The low adjustment costs of the PE model reduce the costs of this disinvestment, and thus result in a lower value premium. Figure 7 View largeDownload slide Partial equilibrium: Cash-flow growth of book-to-market sorted portfolios “GE” uses the baseline general equilibrium model. “PE” uses equilibrium aggregate laws of motion, but firm-level decision rules consistent with adjustment costs that closer to those from the partial equilibrium literature (1/20th of the value from Table 1). Cash-flow growth is the growth rate of average cash flows from buy-and-hold value-weighted B/M-sorted portfolios portfolios like in Figure 5. Value is the eighth highest B/M-sorted portfolio and growth is the third. The low adjustment costs implied by partial equilibrium calibrations lead to a small spread in cash-flow growth. Figure 7 View largeDownload slide Partial equilibrium: Cash-flow growth of book-to-market sorted portfolios “GE” uses the baseline general equilibrium model. “PE” uses equilibrium aggregate laws of motion, but firm-level decision rules consistent with adjustment costs that closer to those from the partial equilibrium literature (1/20th of the value from Table 1). Cash-flow growth is the growth rate of average cash flows from buy-and-hold value-weighted B/M-sorted portfolios portfolios like in Figure 5. Value is the eighth highest B/M-sorted portfolio and growth is the third. The low adjustment costs implied by partial equilibrium calibrations lead to a small spread in cash-flow growth. Note that the low elasticity of intertemporal substitution (EIS) implied by external habit preferences and the need to match aggregate consumption volatility are critical to the quantitative effects in this discussion. This low EIS means that the household has a strong desire to smooth consumption across time, and, through the SDF, the firm has a strong incentive to smooth cash flows. This strong smoothing motive combined with the volatility of consumption growth seen in U.S. data then imply large investment frictions. This stands in contrast to long-run risk and disaster models, which typically imply a large EIS, and therefore small investment frictions. 6. Evidence on the Value Premium Mechanism The previous sections describe a controversial story for the value premium. The story implies that value firms have high cash-flow growth, contradicting the traditional view that value firms have low duration (Dechow, Sloan, and Soliman 2004; Zhang 2005; Da 2009), and that long duration stocks have lower expected returns (Weber Forthcoming). Moreover, the story implies that value firms are more exposed to discount rate shocks, in contrast to Campbell and Vuolteenaho’s (2004) empirical results. This section provides evidence in favor of the model’s story for the value premium. Section 6.1 shows that, empirically, value firms have high cash-flow growth according to four different earnings-based definitions of cash flow. Section 6.2 shows that the measured term structure of equity implied by the model is consistent with empirical results of Weber (Forthcoming). Section 6.3 closes by providing empirical evidence linking value with discount rate shocks. All sections discuss how the results presented here can be reconciled with the literature. 6.1 Empirical evidence about value and cash-flow growth The traditional view is that value firms have low cash-flow growth. This notion is intuitively appealing. As explained in Bodie, Kane, and Marcus (2008), “growth stocks have high ratios, suggesting that investors in these firms must believe that the firm will experience rapid growth to justify the prices at which the stocks sell.” This natural idea is espoused in several theory papers, including Zhang’s (2005) theory of the value premium. While this view is intuitive, the empirical evidence is mixed at best. The bulk of the supporting evidence comes from two equity duration studies: Dechow, Sloan, and Soliman (2004) and Da (2009). Equity duration is extremely difficult to measure, however. Unlike bond duration, equity duration is extremely sensitive to the estimate of the discount rate and terminal value, both of which are notoriously difficult to pin down. The remaining evidence for the traditional view comes from a subset of the results in Chen (2017b). He examines numerous definitions of cash flow and cuts of the data, and finds support for the traditional view in the post-1963 sample using buy-and-hold portfolios and dividends as cash flow. Most of Chen’s other findings, however, strongly conflict with the traditional view. When the data features just one of the following: rebalanced portfolios, pre-1963 data, or earnings growth, he finds that value has higher cash-flow growth. Indeed, this evidence against the traditional view can be found in numerous other papers, going back as far as Lakonishok, Shleifer, and Vishny (1994). Other papers that find that value has higher cash-flow growth include Ang and Liu (2004), Bansal, Dittmar, and Lundblad (2005), Hansen, Heaton, and Li (2008), and Chen, Petkova, and Zhang (2008). These papers all focus on rebalanced portfolios, which matches the standard technique of rebalancing portfolios in evaluation of value premium returns (Fama and French 1993). In the remainder of this section, I present additional evidence against the traditional view. I focus on buy-and-hold portfolios because only this slice of the data shows supporting evidence for the traditional view. I also focus on earnings since the bulk of Chen’s (2017b) evidence regards dividend growth. Earnings are particularly of interest, because absent financial frictions, earnings (net of optimal investment) is the true determinant of firm value (Miller and Modigliani 1961). Indeed, dividends are irrelevant in most production-based asset pricing models, and one could include an arbitrary dividend policy in the model of this paper without altering the value premium. Earnings are also interesting because Skinner (2008) finds that since about 1980, firms repurchases have become the dominant source of payout, and that repurchases are determined by earnings. I look at four different variations of earnings. The first is the most common measure, earnings before extraordinary income (ib). This measure is stable and reflects the ongoing activities of the firm, but investors must face the consequences of extraordinary income, and thus the stock price should reflect these items. Thus, I also look at earnings (ni), which includes extraordinary income. Earnings, however, reflect depreciation charges, which are not represented in cash flows in the model. I thus also examine earnings plus depreciation (ni + dp). Lastly, cash flows in the model can also come from selling / buying capital. Thus I also examine earnings plus depreciation less net investment (ni + dp - capx + sppe). This last measure is closest in spirit to the cash flows of the model. I use tercile book-to-market sorted portfolios and CRSP and COMPUSTAT data from 1971–2011. The portfolios are buy-and-hold portfolios. Cash flows from delisted stocks are reinvested in the remaining stocks, following Chen’s (2017b) procedure. The choice of terciles is due to the use of net investment. Net investment is quite volatile, and the use of large portfolios averages out much of this volatility and paints a clearer picture of the typical cash-flow dynamics. The relatively short post-1971 sample is due to the limited availability of sales of plant, property, and equipment (sppe) data. Table 9 shows cash-flow levels. It shows the cash flow from a $1 investment in value or growth portfolios, averaged across portfolio formation years. The first thing that jumps out from the table is that value stocks and growth stocks have similar levels of cash flows per dollar of investment. In the first year after portfolio formation, value stocks pay 7 cents per dollar invested while growth stocks pay 6 cents, with respect to earnings before extraordinary income. Table 9 Compustat results: Mean cash flow of value and growth Mean cash flow per $ invested in year 0 E before extraordinary E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 1 0.063 0.084 0.073 0.029 0.029 0.005 2 0.067 0.093 0.098 0.031 0.034 0.030 3 0.072 0.102 0.113 0.035 0.037 0.034 Mean cash flow per $ invested in year 0 E before extraordinary E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 1 0.063 0.084 0.073 0.029 0.029 0.005 2 0.067 0.093 0.098 0.031 0.034 0.030 3 0.072 0.102 0.113 0.035 0.037 0.034 “Growth,” “neutral,” and “value” are buy-and-hold value-weighted tercile portfolios sorted on B/M. “E before extraordinary” is earnings before extraordinary income (ib). “E + Dep - Net Inv” is earnings plus depreciation less capital expenditures plus sales of plant, property, and equipment (ni + dp - capx + sppe). “Year” is year after portfolio formation. Cash flow is averaged across portfolio formation years. The sample is from 1971 to 2011. Table 9 Compustat results: Mean cash flow of value and growth Mean cash flow per $ invested in year 0 E before extraordinary E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 1 0.063 0.084 0.073 0.029 0.029 0.005 2 0.067 0.093 0.098 0.031 0.034 0.030 3 0.072 0.102 0.113 0.035 0.037 0.034 Mean cash flow per $ invested in year 0 E before extraordinary E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 1 0.063 0.084 0.073 0.029 0.029 0.005 2 0.067 0.093 0.098 0.031 0.034 0.030 3 0.072 0.102 0.113 0.035 0.037 0.034 “Growth,” “neutral,” and “value” are buy-and-hold value-weighted tercile portfolios sorted on B/M. “E before extraordinary” is earnings before extraordinary income (ib). “E + Dep - Net Inv” is earnings plus depreciation less capital expenditures plus sales of plant, property, and equipment (ni + dp - capx + sppe). “Year” is year after portfolio formation. Cash flow is averaged across portfolio formation years. The sample is from 1971 to 2011. While this result conflicts with the traditional view that growth stocks are young companies that have yet to pay much in cash flow, it is consistent with the empirical evidence that growth firms have higher return on book equity (Fama and French 1995). In fact, net of extraordinary income and investment, value firms pay much less. Using this definition, in the first year value pays half a cent per dollar while growth pays an order of magnitude more. The second pattern which emerges from the table is that value has higher cash-flow growth. There is little action in the cash flows of growth firms, but the value cash flows exhibit apparent growth. Table 10 shows growth rates of the cash flows from the previous figure. It also considers two additional definitions of cash flow: earnings (after extraordinary income) and earnings plus depreciation. By all definitions of cash flow, value portfolios have much higher cash-flow growth than growth portfolios in year two. Indeed, using the definition closest in spirit to the model (earnings plus depreciation less net investment), value experiences a huge 544% growth in cash flow between years one and two, while growth gets a meager 5% growth. Cash-flow growth is also monotonically increasing in B/M using all definitions. Cash-flow growth of value exceeds that of neutral which exceeds that of growth. An additional pattern which is seen in Table 10 is that the growth rates mean revert. Value begins with strikingly high cash-flow growth in year two, but growth slows down quickly. Growth portfolios follow the opposite pattern. Both the high cash-flow growth of value portfolios and its subsequent mean reversion will be seen in the model. Table 10 Compustat results: Growth rates of mean cash flow Growth of mean cash flow (% per year) E before extraordinary E Year Growth Neutral Value Growth Neutral Value 2 6.8 10.2 34.5 6.4 10.1 50.2 3 6.6 9.4 14.6 6.6 7.0 16.4 4 8.4 7.7 6.9 7.4 11.0 9.8 E + Dep E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 2 8.9 9.3 17.3 5.1 20.0 544.3 3 8.7 8.3 9.4 12.0 6.4 11.8 4 9.1 10.3 7.2 14.4 22.6 −1.7 Growth of mean cash flow (% per year) E before extraordinary E Year Growth Neutral Value Growth Neutral Value 2 6.8 10.2 34.5 6.4 10.1 50.2 3 6.6 9.4 14.6 6.6 7.0 16.4 4 8.4 7.7 6.9 7.4 11.0 9.8 E + Dep E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 2 8.9 9.3 17.3 5.1 20.0 544.3 3 8.7 8.3 9.4 12.0 6.4 11.8 4 9.1 10.3 7.2 14.4 22.6 −1.7 “Growth,” “neutral,” and “value” are buy-and-hold value-weighted tercile portfolios sorted on B/M. “E before extraordinary” is earnings before extraordinary income (ib). “E” is earnings (ni); “Dep” is depreciation (dp); and “Net inv” is capital expenditures less sales of plant, property, and equipment (capx-sppe). “Year” is year after portfolio formation. The growth rate in year t is the growth rate of mean cash flows between years t and t – 1. The sample is from 1971 to 2011. Table 10 Compustat results: Growth rates of mean cash flow Growth of mean cash flow (% per year) E before extraordinary E Year Growth Neutral Value Growth Neutral Value 2 6.8 10.2 34.5 6.4 10.1 50.2 3 6.6 9.4 14.6 6.6 7.0 16.4 4 8.4 7.7 6.9 7.4 11.0 9.8 E + Dep E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 2 8.9 9.3 17.3 5.1 20.0 544.3 3 8.7 8.3 9.4 12.0 6.4 11.8 4 9.1 10.3 7.2 14.4 22.6 −1.7 Growth of mean cash flow (% per year) E before extraordinary E Year Growth Neutral Value Growth Neutral Value 2 6.8 10.2 34.5 6.4 10.1 50.2 3 6.6 9.4 14.6 6.6 7.0 16.4 4 8.4 7.7 6.9 7.4 11.0 9.8 E + Dep E + Dep - Net inv Year Growth Neutral Value Growth Neutral Value 2 8.9 9.3 17.3 5.1 20.0 544.3 3 8.7 8.3 9.4 12.0 6.4 11.8 4 9.1 10.3 7.2 14.4 22.6 −1.7 “Growth,” “neutral,” and “value” are buy-and-hold value-weighted tercile portfolios sorted on B/M. “E before extraordinary” is earnings before extraordinary income (ib). “E” is earnings (ni); “Dep” is depreciation (dp); and “Net inv” is capital expenditures less sales of plant, property, and equipment (capx-sppe). “Year” is year after portfolio formation. The growth rate in year t is the growth rate of mean cash flows between years t and t – 1. The sample is from 1971 to 2011. Analyzing cash-flow growth for the longer term faces data limitations. Forty years of data provides only 10 nonoverlapping 4-year periods. Thus, it is probably best to focus on the year two and year three growth rates. Nevertheless, that the cash-flow patterns are common across multiple definitions of cash flow is reassuring. 6.2 The term structure of equity The notion that high cash-flow growth implies high expected returns conflicts with the traditional notion that the term structure of equity is downward sloping. This notion comes from the literature on dividend strips (Van Binsbergen, Brandt, and Koijen 2012), and more recently, from Weber’s (Forthcoming) work on duration. Dividend strips, however, are claims on the aggregate stock market, not on a market-neutral long-short portfolio. Thus, that aggregate market’s term structure is downward sloping does not contradict facts about the cross-section of stocks. In fact, the U.S. data show both that the aggregate term structure is downward sloping (Van Binsbergen, Brandt, and Koijen 2012), and that high cash-flow growth stocks have higher expected returns than low cash-flow growth stocks (see Section 6.1; Chen (2012)). Weber’s (Forthcoming) findings about duration do speak directly to the cross-section. The duration of infinitely lived equities, however, is a complicated object which may not line up with cash-flow growth. Indeed, Weber finds that duration is only mildly correlated with sales growth. Thus, the duration-implied term structure, may not be downward sloping in my model. To examine this issue, I measure duration following Weber. Duration is measured using Duri,t=∑s=1Ts×CFi,t+s/(1+κ)sPi,t+(T+1+κκ)Pi,t−∑s=1TCFi,t+s/(1+κ)sPi,t (26) CFi,t+s=Ki,t+s[Πi,t+s−δKi,t+s−Φ(Ii,t+s,Ki,t+s),Ki,t+s−Ki,t+s+1−Ki,t+sKi,t+s,] (27) where Pi,t is the ex-dividend price of the stock and CFi,t+1 is the expected cash flow at time t + 1. T = 15 years, and κ=0.113 , the mean annual return on a typical firm. Using T = 10, T = 20, κ=0.09 , or κ=0.13 does not significantly affect the results. The timing of Ki,t+s is adjusted because, in macro models, Ki,t+s is beginning of the period while in accounting, the book value is end of the period. Πi,t+s−δKi,t+s−Φ(Ii,t+s,Ki,t+s)Ki,t+s−1 and Ki,t+s+1−Ki,t+sKi,t+s−1 follow AR(1) processes. In each simulation, I estimate the AR(1) models using pooled OLS. Figure 8 shows the distribution of the slope of the term structure of equity implied by the duration. On average the slope is zero, but the slope is negative in half of simulations. Thus, it is quite possible for the model to be consistent with Weber’s finding that the slope is negative. Figure 8 View largeDownload slide Distribution of the measured slope of the term structure of equity Figure shows the distribution of slopes from 100 simulations the same size as the U.S. sample. For each firm-year in each simulation, duration is measured according to equations (26) and (27). Firms are sorted into 10 portfolios based on duration, and the returns over the next year are calculated. The slope for each simulation is calculated by regressing the average return on each portfolio on the average duration of each portfolio. The slope is noisy since measured duration is a poor measure of cash-flow growth (see Figure 9). Figure 8 View largeDownload slide Distribution of the measured slope of the term structure of equity Figure shows the distribution of slopes from 100 simulations the same size as the U.S. sample. For each firm-year in each simulation, duration is measured according to equations (26) and (27). Firms are sorted into 10 portfolios based on duration, and the returns over the next year are calculated. The slope for each simulation is calculated by regressing the average return on each portfolio on the average duration of each portfolio. The slope is noisy since measured duration is a poor measure of cash-flow growth (see Figure 9). Figure 9 shows why the slope is so often negative. The figure shows cash-flow growth for portfolios sorted on duration. Though duration generates some spread in cash-flow growth, the spread is muted compared to that generated by book-to-market (Figure 5). Figure 9 View largeDownload slide Measured duration and cash-flow growth Cash-flow growth is computed from 100 model simulations. In each year of each simulation, firms are sorted into 10 portfolios based on measured duration (equations (26)–(27)). Portfolio cash flows are constructed by value-weighting firm cash flows [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] . Average portfolio cash flow for each year after portfolio formation is then constructed by averaging across simulation years. Portfolio cash-flow growth is constructed by taking log first differences for the average cash-flow level. The long duration portfolio has negative cash flows in year one, so its year 2 growth rate is omitted from the plot. Measured duration generates some spread in cash-flow growth, but the spread is muted compared to that generated by B/M (Figure 5). Figure 9 View largeDownload slide Measured duration and cash-flow growth Cash-flow growth is computed from 100 model simulations. In each year of each simulation, firms are sorted into 10 portfolios based on measured duration (equations (26)–(27)). Portfolio cash flows are constructed by value-weighting firm cash flows [AtBi,tKi,tα−Ii,t−Φ(Ii,t,Ki,t)] . Average portfolio cash flow for each year after portfolio formation is then constructed by averaging across simulation years. Portfolio cash-flow growth is constructed by taking log first differences for the average cash-flow level. The long duration portfolio has negative cash flows in year one, so its year 2 growth rate is omitted from the plot. Measured duration generates some spread in cash-flow growth, but the spread is muted compared to that generated by B/M (Figure 5). Intuitively, duration comes from market prices and forecasts of cash flows. By including market prices, duration is implicitly a combination of discount rates and cash-flow growth. How discount rates affect duration is complicated, and depends a lot on the econometrician’s assumptions. By assuming that the terminal value is described by current market prices, Weber’s measure builds in a negative relationship between discount rates and duration. As a result, high cash-flow growth stocks, which have high discount rates, do not have especially high durations, leading to a flat (and often negative) term structure of equity. 6.3 Empirical evidence about value and discount rate shocks The model implies that value firms are more exposed to discount rate shocks. The existing literature is conflicted on this issue. Campbell and Vuolteenaho (2004) find that value stocks are less exposed to discount rate shocks. But Chen and Zhao (2009) show that alternative specifications of the Campbell and Vuolteenaho (2004) procedure lead to the opposite conclusion. Campbell et al. (2012) extend Campbell and Vuolteenaho (2004) to include stochastic volatility, and while they show the robustness of volatility betas to Chen and Zhao’s (2009) concerns, they don’t discuss the robustness of discount rate betas. This section adds the the debate by identifying discount rates through the lens of the model. The model’s discount rate shocks are due to time-varying consumption volatility (Section 3; Chen (2017a)). Thus, according to the model, value returns should have more negative reactions to consumption volatility shocks, or more simply, more negative consumption volatility betas. To show this I need an empirical measure of consumption volatility. I use Boguth and Kuehn’s (2013) measure, which comes from an estimation of a Markov chain model for the first and second moments of consumption growth. An advantage of Boguth and Kuehn’s measure is that they take advantage of the information in the components of consumption, which helps alleviate problems regarding identifying persistent volatility in the short post-war quarterly consumption data. Since the model is annual and lacks the consumption components used in Boguth and Kuehn (2013), I compute consumption volatility from the model’s laws of motion for the simulated results. Figure 10 shows consumption volatility betas for 10 book-to-market sorted portfolios. Consumption volatility betas are constructed by regressing excess returns on changes in consumption volatility. The left panel shows that, with a couple exceptions, consumption volatility betas decline monotonically in B/M. The right panel shows betas from the model. Here, consumption volatility is precisely measured using the laws of motion of the model and the betas are averaged over numerous simulations. As a result, we get a cleanly declining relationship between consumption volatility betas and B/M. Figure 10 View largeDownload slide Consumption volatility betas of book-to-market sorted portfolios Betas are the slopes form regressions of returns less the risk-free rate on contemporaneous changes in consumption volatility. Consumption volatility in the data is from Boguth and Kuehn (2013). Consumption volatility in the model comes from the model’s laws of motion. Changes in consumption volatility are normalized by their standard deviation. Returns come from value-weighted B/M-sorted portfolios with annual rebalancing. Portfolio 10 is high B/M (Value). Model figures are the average portfolio beta across 100 simulations of the same length as the empirical sample. Figure 10 View largeDownload slide Consumption volatility betas of book-to-market sorted portfolios Betas are the slopes form regressions of returns less the risk-free rate on contemporaneous changes in consumption volatility. Consumption volatility in the data is from Boguth and Kuehn (2013). Consumption volatility in the model comes from the model’s laws of motion. Changes in consumption volatility are normalized by their standard deviation. Returns come from value-weighted B/M-sorted portfolios with annual rebalancing. Portfolio 10 is high B/M (Value). Model figures are the average portfolio beta across 100 simulations of the same length as the empirical sample. The zig-zagging in the data panel is not surprising considering the short sample of quarterly post-war consumption and the high volatility of portfolio returns. Firm level results provide more statistical power, and are shown in Table 11. The table shows Fama-Macbeth regressions of consumption volatility betas on log B/M. Betas are constructed by regressing returns on changes in consumption volatility in rolling windows. The table shows that in both model and data, the relationship is negative, statistically significant, and similar in magnitude. The table uses forward-looking betas, that is, the windows for date t run from date t to 40 quarters after date t. I use forward-looking betas because theory predicts that it’s the future return covariance that matters. Using the more traditional backward-looking windows does not materially affect the data columns, but it does affect the model columns. This result is likely because firms in the model are characterized by stationary state variables and cannot display permanent differences like in the data. The window is long because the model only contains three aggregate technology states in order to maintain tractability. Shorter windows show a stronger relationship between consumption volatility betas and book-to-market in the data. Table 11 Regressions of consumption volatility betas on B/M Dependent var: Consumption vol beta U.S. data Model Mean 5% 50% 95% log (B/M)i,t −1.02 −2.68 −4.20 −2.64 −1.24 t-stat −2.57 −3.83 −6.08 −3.67 −1.56 Dependent var: Consumption vol beta U.S. data Model Mean 5% 50% 95% log (B/M)i,t −1.02 −2.68 −4.20 −2.64 −1.24 t-stat −2.57 −3.83 −6.08 −3.67 −1.56 Figures are annualized. Consumption volatility in the data is Boguth and Kuehn’s (2013) estimate. Consumption volatility in the model is computed from the model’s laws of motion. Standard errors are Newey-West with 12 lags. Regressions are firm-level Fama-Macbeth using weighted least squares where the weights are the inverse of the squared standard error of the consumption volatility beta estimate. Betas are constructed by regressing excess returns on changes in consumption volatility for 40 quarters into the future. Table 11 Regressions of consumption volatility betas on B/M Dependent var: Consumption vol beta U.S. data Model Mean 5% 50% 95% log (B/M)i,t −1.02 −2.68 −4.20 −2.64 −1.24 t-stat −2.57 −3.83 −6.08 −3.67 −1.56 Dependent var: Consumption vol beta U.S. data Model Mean 5% 50% 95% log (B/M)i,t −1.02 −2.68 −4.20 −2.64 −1.24 t-stat −2.57 −3.83 −6.08 −3.67 −1.56 Figures are annualized. Consumption volatility in the data is Boguth and Kuehn’s (2013) estimate. Consumption volatility in the model is computed from the model’s laws of motion. Standard errors are Newey-West with 12 lags. Regressions are firm-level Fama-Macbeth using weighted least squares where the weights are the inverse of the squared standard error of the consumption volatility beta estimate. Betas are constructed by regressing excess returns on changes in consumption volatility for 40 quarters into the future. Table 12 Partial Equilibrium Experiment: Regressions of Future Returns on B/M Dependent variable: Ri,t+1 US data GE model PE model Intercept 18.62 18.34 10.82 t-stat 5.74 3.88 3.60 log (B/M)i,t 5.70 5.84 0.53 t-stat 4.88 3.34 0.83 Dependent variable: Ri,t+1 US data GE model PE model Intercept 18.62 18.34 10.82 t-stat 5.74 3.88 3.60 log (B/M)i,t 5.70 5.84 0.53 t-stat 4.88 3.34 0.83 Figures are annual. The table shows Fama-Macbeth regressions of future returns on log (B/M). The GE model is the baseline model (Table 1). The PE model uses equilibrium aggregate laws of motion, but firm-level decision rules consistent with adjustment costs that closer to those from the partial equilibrium literature (1/20th of the value from Table 1). Table 12 Partial Equilibrium Experiment: Regressions of Future Returns on B/M Dependent variable: Ri,t+1 US data GE model PE model Intercept 18.62 18.34 10.82 t-stat 5.74 3.88 3.60 log (B/M)i,t 5.70 5.84 0.53 t-stat 4.88 3.34 0.83 Dependent variable: Ri,t+1 US data GE model PE model Intercept 18.62 18.34 10.82 t-stat 5.74 3.88 3.60 log (B/M)i,t 5.70 5.84 0.53 t-stat 4.88 3.34 0.83 Figures are annual. The table shows Fama-Macbeth regressions of future returns on log (B/M). The GE model is the baseline model (Table 1). The PE model uses equilibrium aggregate laws of motion, but firm-level decision rules consistent with adjustment costs that closer to those from the partial equilibrium literature (1/20th of the value from Table 1). A weakness of the results is that the overall level of the consumption volatility betas differs significantly between the model and data. This deviation is due to the precise measurement of consumption volatility in the model, as well as the single shock nature of the model. These two features mean that TFP, consumption volatility, and stock prices of portfolios move in lock-step, leading to highly negative consumption volatility betas. Softening the level of the betas would involve introducing additional sources of aggregate risk and is an interesting path for future research. 7. Conclusion I present a general equilibrium model with heterogeneous firms which links the time-varying equity premium with the value premium. The time-varying equity premium leads to a cross-sectional equity term premium. Value stocks endogenously have temporally distant cash flows, leading to the value premium. The value premium exists as long as the mean-reversion in productivity has stronger effects on cash-flow growth than optimal investment. For parameter values which generate data-like stock market volatility, I find that the productivity effects dominate, value stocks have temporally distant cash flows, and the value premium is positive. This explanation for the value premium is consistent with three empirical facts about value firms: (1) value firms have low but mean reverting ROE, (2) value firms have higher cash-flow growth, and (3) value firms have more negative consumption volatility betas. Empirical facts (2) and (3) are controversial, and I provide new empirical evidence showing their robustness. Moreover, the model illustrates how measured duration may be poorly correlated with cash-flow growth, and thus the measured term structure of equity can be downward sloping. This paper is a revised version of the second chapter from my PhD dissertation at the Ohio State University. It originated from conversations with Lu Zhang and would not have been possible without him. I would also like to thank Eric Engstrom, Aubhik Khan, John Pokorny, Valerio Poti, Steve Sharpe, René Stulz, and Julia Thomas; an anonymous referee; and seminar participants at the Federal Reserve Board for helpful comments. The views expressed herein are those of the author and do not necessarily reflect the position of the Board of Governors of the Federal Reserve or the Federal Reserve System. Footnotes 1 Aggregate time-series papers include Campbell and Cochrane (1999), Bansal and Yaron (2004), and Wachter (2013). Cross-sectional papers include Berk, Green, and Naik (2002), Zhang (2005), and Carlson, Fisher, and Giammarino (2005). 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The Review of Asset Pricing Studies – Oxford University Press
Published: Dec 1, 2018
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