Business-Cycle Pattern of Asset Returns: A General Equilibrium Explanation

Transcription

1 Business-Cycle Pattern of Asset Returns: A General Equilibrium Explanation Abstract I develop an analytical general-equilibrium model to explain economic sources of business-cycle pattern of aggregate stock market returns. With concave production functions and capital accumulation, a technology shock has a pro-cyclical direct effect and a counter-cyclical indirect effect on expected returns. The indirect effect, reflecting the feedback effect of consumers behavior on asset returns, dominates the direct effect and causes counter-cyclical variations of expected returns. I show that the conditional mean, volatility, and Sharpe ratios of asset returns all vary counter-cyclically and they are persistent and predictable, and that stock market behavior has forecasting power for real economic activity. JEL Classification: D51, E30, G11, G12 Keywords: Counter-cyclical variation, capital accumulation, decreasing returns to capital, overlapping-generation model

2 1 Introduction Numerous papers have empirically documented that aggregate stock market returns display a counter-cyclical behavior. For example, Fama and French (1989) find that expected excess returns on bonds and stocks are typically high during recessions and low at business peaks. Schwert (1989) shows clear evidence of a major increase in volatility of equity returns during recessions. Whitelaw (1997) and Lettau and Ludvigson (2004) document that conditional Sharpe ratios of aggregate stock market returns are low at the peak of the business cycle and high at the trough. Brandt and Kang (2004) report that the expected return, conditional volatility, and conditional Sharpe ratios of aggregate market index returns all vary counter-cyclically, and that the conditional volatility of asset returns seems to lead economic recessions. The business-cycle pattern of asset returns is identified via linking the expected return and conditional volatility to variables which forecast business-cycles. 1 The literature has a few equilibrium models that establish the business-cycle pattern of asset returns. For example, Rouwenhorst (1995) uses a representative agent framework to study asset pricing implications of an equilibrium real-business-cycle model. He numerically shows that expected asset returns vary counter-cyclically, and the economic source appears to remain elusive in the numerical exercise. Campbell and Cochrane (1999) use a habit formation model to illustrate numerically that both the expected return and stock market volatility are decreasing functions of surplus consumption ratios, a proxy for economic conditions. Because the dynamics of the surplus consumption ratios is exogenously specified, the fundamental source of the counter-cyclical behavior is still unclear. This paper analytically illustrates the role of capital accumulation, coupled with decreasing returns to capital, in causing the counter-cyclical fluctuations of asset returns. The intuition underlying the theoretic model stems from capital accumulations in response to technology shocks. In a production economy with a concave technology, a positive and persistent technology shock has two offsetting effects on expected returns. As firms become more productive, future dividends 1 For example, Fama and French (1989) use the term spread, default spread, and dividend yield; Schwert (1989) uses short-term interest rates, yields on corporate bonds, and growth rates of industrial production; Whitelaw (1997) uses dividend yield, default spread, commercial paper-treasury spread, and one-year Treasury yield; Lettau and Ludvigson (2004) use consumption-wealth ratio and some of these conditioning variables; and Brandt and Kang (2004) adopt a latent VAR system without relying on predictors. 1

3 increase, and holding capital constant, expected returns increase. The positive shock also leads to capital accumulation and raises the level of future capital stocks, which in turn lowers the future marginal product of capitals and expected returns for a given level of technology. 2 The former effect (or the direct effect) reflects the wealth effect of a technology shock on asset returns and constitutes a pro-cyclical response of asset returns to the shock, and the latter effect (or the indirect effect) characterizes the substitution effect of the technology shock and renders a counter-cyclical response. The overall response of asset returns to the shock depends on which effect dominates in equilibrium. Based on the intuition, I build a tractable general-equilibrium model on a two-period-lived overlapping-generations (OLG) framework with a production technology that has fixed labor input and diminishing returns to capital. The productivity shock follows a first-order autoregressive process. There are no investment adjustment costs so that the relative price of capital is always one. Ríos-Rull (1994, 1996) argue that a reasonably calibrated OLG model has essentially identical empirical implications for asset prices and business cycle properties as does a representative-agent model. Using the OLG framework, De Long et al. (1990) and Spiegel (1998) examine the issue of stock market volatility relative to volatility of fundamentals, Constantinides, Donaldson and Mehra (2002) and Storesletten, Telmer and Yaron (2001) study the equity premium puzzle. With the two-period OLG model I derive analytical solutions, decompose the equilibrium response of asset returns to a technology shock into the two offsetting effects, and quantify their relative magnitudes. 3 Specifically, if the capital income share of the production function α = 1 2 or α = 1 3, then 1) both the share price and value of investment (in either real or financial assets) vary pro-cyclically; 2) both the expected return and conditional volatility of asset returns vary countercyclically and increase with the long-term level of productivity; 3) a productivity shock yields both a pro-cyclical direct effect (or wealth effect) and a counter-cyclical indirect effect (or substitution effect) on asset returns; and 4) the indirect effect, or the feedback effect of consumer behavior on asset prices, dominates the direct effect and constitutes the main source of the counter-cyclical behavior of asset returns. I also show that both the expected return and conditional volatility of 2 Lettau (2003) uses a similar decomposition of asset price responses to technology shocks to study why the premium of equity is small over the risk-free rate and over a real long-term bond. 3 Analytical results would be difficult to obtain in an infinitely-lived-agent setting. There is also a caveat. The two-period OLG model suits the low-frequency phenomenon well and the business cycle is a relatively high-frequency phenomenon, but the economic intuition illustrated in this article carries over to a representative-agent model or a more realistic OLG model with many periods of life for agents. 2

4 asset returns are persistent and predictable, and that asset market behavior has forecasting power for real economic activity. The indirect effect or the capital accumulation channel, which characterizes the feedback effect of investors portfolio allocation decisions on asset returns, is crucial to the counter-cyclical variations. Mainstream asset pricing models are developed in an exchange economy or a production economy with a linear technology (see surveys in Campbell (2000) and Cochrane (2001)). An exchange economy implies a perfectly inelastic supply of capitals, and a production economy without investment irreversibility or capital adjustment costs assumes a perfectly elastic supply of capitals. In either economy, the indirect effect is absent. Dividends per unit of capital are exogenous and the amount of capitals has no impact on asset returns. Asset return processes affect agents optimal portfolio decisions but the optimal decision rules do not affect the return dynamics even at the aggregate level. In a general equilibrium framework, however, both the return process and the allocation decisions are endogenous. Under perfect competition, an agent makes an optimal allocation decision taking as given all prices including the return process. At the aggregate level, the return process is endogenously determined such that the markets clear. Under imperfect competition, agents have market power and their behavior affects directly market prices and the return process. Agents form a rational expectation about the feedback from their choices to the return process and take the feedback into account when making portfolio decisions. 4 Above all, there exists a strategic relation between asset return process and optimal asset allocation at either the aggregate or disaggregate level. A caveat is in order. This paper studies the qualitative but not the quantitative feature of business-cycle pattern of asset returns. 5 In particular, I focus on examining the economic sources that qualitatively generate the counter-cyclical variations, and I do not quantitatively match to 4 Cuoco and Cvitanic (1998) examine an optimal consumption and investment problem for a large investor whose portfolio choices affect the instantaneous expected returns on the traded assets. Basak (1997) studies in an exchange economy a consumption-portfolio problem of an agent who acts as a price-leader in all markets and the implications of his behavior on equilibrium security prices. 5 The literature has shown that standard RBC models have counter-factual quantitative asset pricing implications (see., e.g., Jermann 1998, and Boldrin, Christiano and Fisher 2001). Generally, two additional features are needed to solve the quantitative failure: frictions at the household level like habit formation preferences and borrowing constraints that prevent inter-temporal consumption smoothing, and frictions at the firm level like capital adjustment costs, investment irreversibility, and multi-production sectors with limited inter-sectoral factor mobility. 3

5 the real-world data the magnitude of counter-cyclical variations predicted by this model. The literature has focused on and made significant progress in understanding the question whether a calibrated equilibrium business cycle model can generate realistic moments of asset returns, which is a core component in such analysis (e.g., Rouwenhorst 1995, Jermann 1998, Boldrin, Christiano and Fisher 2001, and Lettau 2003). By abstracting from the quantitative analysis and, hence, shying away from other features that have been key to the quantitative success, I am able to stress the importance of one particular feature, i.e., the diminishing return technology, for asset returns even in a simplified framework as a distinctive feature of this model. As a result, the qualitative analysis in the simple framework complements the quantitative success achieved in the literature; also, the analytic exercise furthers our understanding of the mechanism giving rise to counter-cyclical expected returns and, more generally, of the business-cycle-model implications for asset returns. The remainder of the paper proceeds as follows. Section 2 describes the environment, sets up firms and consumers problems, and characterizes the competitive equilibrium. Section 3 conducts partial equilibrium analyses of asset pricing and portfolio allocation, respectively. Section 4 studies general equilibrium properties of asset pricing and portfolio allocation. Section 5 concludes with a summary of the paper s main findings. 2 Model 2.1 Environment Consider an infinite-time-horizon economy consisting of overlapping generations of two-period-lived agents. The economy operates in discrete time, starting at time t = 1. At each date t 1, a [0, 1] continuum of identical agents of generation t are born, who are young in period t, old in period (t + 1), and dead in period (t + 2) and beyond. Each generation of agents are homogeneously endowed with one unit of time at date t and nothing at date (t + 1). Each agent has access to a risk-free interest-bearing storage technology which she can use to store goods for one period with a constant rate of return r f. There are a [0, 1] continuum of infinitely-lived representative firms who are endowed with k 1 units of capital goods at time 1. All production in the economy takes place in firms that own stocks of physical capital in the economy. Firms have no access to the storage technology. There is one 4

6 single physical good in each period. The physical good can be used for either consumption or investment and is perfectly reversible from capital good to consumption good or vice versa. The state of the economy at time t, denoted by s t, can be thought of as a history of the economy between dates 1 and t. To complete the setup, a generation of a [0, 1] continuum of identical agents, called the initial old, are present at time t = 1 and live for only one period. The initial old are endowed with one share of assets to claim dividends paid out by the firms. 2.2 Firms Each period, the representative firm hires labor to produce a single good based on a constantreturn-to-scale production function y = F (k, h) = zbk α h 1 α, 0 < α 1, where k and h represent the amount of capital stock (available at the beginning of each period) and labor service used in the production, α and 1 α are the income shares of capital and labor, respectively, and B is the long-term level of productivity. The output of the economy is uncertain because of a random shock to the total factor productivity z. Output in history s t is written as y (s t ) = z (s t ) B (k (s t 1 )) α (h (s t )) 1 α. (1) All variables are state-dependent except for constant terms, but I suppress s t from all variables for notational convenience. The technology shock to the production process, z t, is the source of uncertainty in the economy. At the beginning of each period t, a realization of the technology shock z t is observed by both firms and agents in this economy. I assume z t to have a first-order autoregressive (AR(1)) dynamics: ln z t+1 = ρ ln z t + ε t+1 with ε t+1 i.i.d. N [ 0, σ 2 ε], (2) where 0 ρ 1. If ρ = 1, the technology shock follows a random walk process and it has an ever-lasting impact on the economy. If ρ = 0, the shock is i.i.d and its impact on the economy is transitory. If 0 < ρ < 1, the shock is positively serially correlated and covariance-stationary, and its impact on the economy dies away over time. The higher ρ is, the more persistent is the impact of the shock to the economy. 5

7 I assume capital stock to depreciate at a constant rate 0 δ 1. The capital stock evolves as k t+1 = (1 δ) k t + i t, (3) where i t denotes the gross investment made by the firm at time t. Taking as given prices and wages, the representative firm maximizes his value to shareholders which is equal to the present discounted value of all current and future expected cash flows: [F P ] max {h t+j,i t+j } t 1 E t j=0 p t+j ( z t+j Bk α t+jh 1 α ) t+j w t+jh t+j i t+j subject to k t+1 = (1 δ) k t + i t. Here, p t is the price of one unit of date t goods denominated in units of date 1 goods, w t is the (real) wage rate denominated in units of date t goods, and E t [ ] stands for the expectation conditional on the information set Φ t available at the beginning of period t. The history of realized shocks to technology is s t {z τ } t τ=1 Φ t. 6 The first-order conditions for the efficient allocation of labor and investment are respectively given by and w t = (1 α) z t Bk α t h α t (4) p t = E t [ pt+1 ( αzt+1 Bk α 1 t+1 h1 α t δ)]. (5) The dividends to the shareholders are the residual value of the output produced after the factor payment to labor has been made and investment has been financed. Then the dividend at date t is d t = z t Bk α t h 1 α t w t h t i t. (6) 6 The firm faces a dynamic problem. If the state of the economy follows a Markovian process then, denoting by V (k t, s t) the firm value at time t, the Bellman equation for the firm s problem is: V (k t, s t) = max {h t,i t } pt ( ztbk α t h 1 α t w th t i t ) + Et [V (k t+1, s t+1)]. 6

8 2.3 Consumers The preference of an individual of generation t is described by an exponential utility function u (c t,2 ), where c t,2 stands for her consumption at her second period of life (i.e. at time t + 1). No interim consumption c t,1 is counted in her utility. I assume that there is no inter-generational altruism in the economy, that is, all old people consume everything before they are gone, and no bequests are made. 7 Each period young agents work for the firms and get the wage payment. Since no leisure enters the utility function, I set h t = 1. With the labor income, young people decide to split their spending between investment and storage. Each individual has free access to the perfectly competitive asset market to sell and/or purchase the shares. Since the individual consumes everything when old, no investments or storages are made in the second period of their lives. There are no intra-generational or inter-generational trades on loans in equilibrium. Agents of the same generation are homogenous, so no intra-generational trades occur. Since the old people will not be around next period, no young agents are willing to trade loans with the current old. Given the prices, a young agent of generation t 1 solves the following problem: [HP ] subject to max E t [u (c t,2 )] {c t,2,x t,θ t} P t x t + θ t w t h t c t,2 x t (P t+1 + d t+1 ) + θ t ( 1 + r f ) c t,2 0. Here, P t is the ex-dividend share price denominated in units of date t goods, x t is the number of 7 A caveat is in order. A typical general-equilibrium model endogeneizes the risk-free rate. In my model, because there are no inter-generational transfers and each agent lives for two periods and only cares about period 2 consumption, the risk-free rate is constant. Given the availability of a risk-free storage technology delivering a constant rate of return r f, the no-arbitrage condition requires the risk-free rate to equal the return on the storage technology that is exogenously given. As a tradeoff, this result greatly simplifies the analytical exercise of this paper. Moreover, as empirical studies show that the fluctuation in the risk-free rate is unlikely to be a main source of the business-cycle pattern of asset returns, the exogenously given risk-free rate in my model is an innocuous modeling feature. 7

9 shares purchased, and θ t is the amount of storage. For the initial old, the solution to her problem is just the autarchy : she consumes all her dividend payments and capital gains at time t = 1. Since u ( ) is a strictly increasing function, the budget constraints are binding in equilibrium. Denoting by u j ( ), j = 1, 2, the first- and second-order differentials of u ( ), the first-order necessary condition for x t is given by ( E t [u 1 (c t,2 ) ((P ))] t+1 + d t+1 ) P t 1 + r f = 0. (7) Define r t+1 = P t+1+d t+1 P t 1 as the net asset return, and equation (7) becomes ( E t [u )] 1 (c t,2 ) r t+1 r f = 0. (8) 2.4 Market Clearing In this economy, there are three markets operating at each point of time: goods market, asset market, and labor market. The labor market clears at h t = 1, and I can ignore the labor market. The two remaining market-clearing conditions are: Goods market : c t 1,2 + i t + θ t = y t, and (9) Asset market : x t = 1. (10) Using the Walras Law, I choose to clear the asset market and the goods market clears automatically. 2.5 Competitive Equilibrium The competitive equilibrium in this economy is defined as a sequence of allocations {c t,2, x t, θ t, y t, k t, i t, d t } t=1 and a sequence of prices {p t, P t, w t } t=1 satisfying: 1. Given(p t, w t ), {y t, k t, i t, d t } solves [F P ], i.e., the value maximization problem of the representative firm; 2. Given (p t, P t, w t ), {c t,2, x t, θ t } solves [HP ], i.e., the utility maximization problem facing each young agent of generation t 1, and the initial old consume all their wealth obtained from dividend payments and share sales; 3. Market clears: x t = 1; and 8

10 4. k 1, z 1, and x 0 = 1 are given. 2.6 Returns of Real and Financial Assets Proposition 1 In this economy, the share price is equal to the capital stock, i.e., P t = k t+1 for any t 1. Proof. This proof is based on Rouwenhorst (1995). I can rewrite the agent s period-by-period budget constraints into a lifetime budget constraint as p t (P t x t + θ t ) + p t+1 c t,2 = p t w t h t + p t+1 ( x t (P t+1 + d t+1 ) + θ t (1 + r f )), (11) where the left-hand and the right-hand sides of the inequality stand for the lifetime uses and sources of income, respectively. Applying the Lagrange multiplier method to the consumer s utility-maximization problem, I obtain the following first-order necessary conditions for c t,2 and x t, respectively: c t,2 : u 1 (c t,2 ) Λp t+1 = 0 (12) x t : E t [p t+1 (P t+1 + d t+1 ) p t P t ] = 0, (13) where Λ is the Lagrange multiplier associated with the lifetime budget constraint. The dividend paid by firms at time t + 1 (equation (6)) is d t+1 = y t+1 w t+1 h t+1 i t+1 = z t+1 Bk α t+1h 1 α t+1 (1 α) z t+1bk α t+1h α t+1 h t+1 k t+2 + (1 δ) k t+1 = αz t+1 Bk α t+1h 1 α t+1 k t+2 + (1 δ) k t+1. So, d t+1 + k t+2 k t+1 = αz t+1 Bk α 1 t+1 h1 α t δ, which is substituted into equation (5) to obtain [ ( )] dt+1 + k t+2 p t = E t p t+1. k t+1 Thus, I obtain equation (13) for P t = k t+1. Q.E.D. 9

11 Since the total number of shares x t = 1, Proposition 1 implies that the firm s value at date t is equal to the capital stock, i.e., V t = P t = k t+1. Therefore, Tobin s q = 1 as the price of capital, measured in units of current-period output, is normalized to be one. Proposition 1 also implies the following no-arbitrage condition in this economy: Corollary 1 Define the net investment return as rt+1 I = αz t+1bkt+1 α 1 h1 α t+1 δ, then r t+1 = rt+1 I. That is, the net asset return is equal to the net investment return. Using a more sophisticated production function, Cochrane (1991) proves that the investment return equals the asset return if the firm has access to a complete financial market. Corollary 1 is a simplified version of the result with a zero capital adjustment cost. Corollary 2 Define φ t P t x t as the amount of income invested in the risky asset by a young agent of generation t. In equilibrium, φ t = k t+1, i.e., the total amount of investment in financial assets is equal to the total amount of investment in real assets (or physical capitals). Proof. Trivial given Proposition 1 and x t = 1 in equilibrium. Q.E.D. 3 Asset Returns and Portfolio Allocations: Partial Equilibrium In this section, I study both the asset returns given (aggregate) portfolio allocations and the portfolio allocations given asset returns. The analysis on asset returns does not depend on the form of the utility function. 3.1 Time-varying Asset Returns This part of analysis resulting in various propositions basically follows the derivations in Rouwenhorst (1995). Corollary 1 allows me to express asset returns in real terms r t+1 = αz t+1 Bkt+1 α 1 h1 α t+1 δ, (14) which, using Corollary 2 and h t+1 = 1, can be further rewritten as r t+1 = αbφ α 1 t z t+1 δ = b t z t+1 δ, with (15) b t αbφ α 1 t. (16) 10

12 The expected return and standard deviation of the risky asset, conditional on the information set Φ t, are µ r,t E t [r t+1 ] = b t E t [z t+1 ] δ, (17) σ r,t Std t [r t+1 ] = b t Std t [z t+1 ]. (18) Equation (17) implies that the expected return in excess of the risk free return is µ e r,t E t [r t+1 r f ] = b t E t [z t+1 ] A, (19) where A = r f + δ. The realized return can be decomposed into its expected and unexpected components: r t+1 = E t [r t+1 ] + b t ɛ t+1, (20) where ɛ t+1 = z t+1 E t [z t+1 ]. The return on the risky assets follows a one-factor representation, with the technology shock z as the factor. The term b measures the sensitivity of the asset returns to this factor, and the term ɛ is the unexpected component of the factor realization with a conditional mean of zero and a conditional variance of V ar t [z t+1 ], respectively. (The unconditional mean and variance of ɛ are zero and E [V ar t [z t+1 ]], respectively.) Combining equation (18) with equation (20), I obtain r t+1 = E t [r t+1 ] + Std t [r t+1 ] ɛ t+1 Std t [z t+1 ] µ r,t + σ r,t ɛ t+1, (21) where ɛ t+1 ɛ t+1 Std t[z t+1 ] is the standardized unexpected component of the factor realization. Equation (17) implies that E t [z t+1 ] can be loosely interpreted as the factor risk premium, so the factor sensitivity term b is the asset beta. The asset beta is a function of the time-varying marginal product of capital of the risky production process and is itself state-dependent and timevarying. Equation (18) shows that the conditional volatility of the risky asset is directly affected by the state-dependent and time-varying asset beta as well. Proposition 2 Given the evolution of technology shocks as specified in equation (2), if ρ < 1, then 11

13 1) { µ z,t E t [z t+1 ] = exp ρ ln z t + 1 } 2 σ2 ε, σ z,t Std t [z t+1 ] = µ z,t exp {σ 2 ε } 1, ln µ z,t = 1 2 (1 ρ) σ2 ε + ρ ln µ z,t 1 + ρε t, and ln σ z,t = 1 ( )] [σ 2 (1 ρ) ε 2 + ln e σ2 ε 1 + ρ ln σ z,t 1 + ρε t. 2) Moreover, given today s realizations of the technology shock z > 1 (< 1), the higher the transition coefficient ρ, the higher (lower) the expected return and volatility of the factor for the coming period. 3) Both the expected return and volatility of the factor are non-decreasing with respect to the technology shock z. Proposition 2 suggests that both the conditional mean and volatility of the factor are persistent, state-dependent and time-varying. The two conditional moments of the factor exhibit the same level of persistence determined by the AR(1) transition coefficient ρ. The conditional mean of the factor risk is proportionate to the conditional volatility of the factor risk at a positive constant 1 e σ2 ε 1 over time, indicating that the two conditional moments of the factor co-vary in lockstep and in the same direction. If ρ = 0, the technology shock is i.i.d., and both conditional moments of the factor remain constant over time and across histories. Proposition 3 If α < 1, the factor sensitivity b, E t [r t+1 ], and Std t [r t+1 ] all decline as the aggregate portfolio allocation φ increases. If α = 1, the factor sensitivity b = B is a constant, and both conditional moments of the risky asset return are independent of the aggregate portfolio allocation. Proposition 3 implies that when α < 1 both the conditional mean and volatility of the risky asset respond negatively to the amount invested in the asset. The more market demand for the asset is, the higher market price, and thus the lower expected return, the risky asset has (if the future cash flow does not change). We derive this result in a production economy; the result differs from the widely-made assumption in the optimal portfolio allocation literature, which is typically developed in an exchange economy, that the allocation decision does not affect the asset return process (see, e.g., Kandel and Stambaugh 1996, and Barberis 2000). The latter is valid in the production economy only when the income share of capital α equals one: with the production function linear in capital inputs, the factor sensitivity is a constant, and the return process is independent of the (aggregate) 12

14 allocation decision. However, empirical evidence from macroeconomics and asset pricing literatures soundly rejects the case of α = 1 or a constant factor sensitivity (Prescott 1986, and Harvey 1989). Proposition 3 further gives an important implication of decreasing returns to capital for expected returns and their conditional volatility. On the one hand, when the income share of capital α is set to one, the production technology has constant returns to capital, capital accumulation has no role in affecting the asset return process, and the returns are independent from the allocation decision. On the other hand, in equilibrium the household s allocation decision realizes capital accumulation; for capital accumulation to impact the asset returns, the technology with decreasing returns to capital is indispensable. 3.2 Portfolio Allocations Equation (8) delivers the first-order condition characterizing the agent s optimal decision. It can be rewritten as E t [u 1 (c t,2 )] E t [ r t+1 r f ] + Cov t [u 1 (c t,2 ), r t+1 r f ] = 0. (22) Using the generalized Stein s lemma as in Gron, Jorgensen and Polson (2004), I rewrite the covariance term in equation (22) as Cov t [ u 1 (c t,2 ), r t+1 r f ] = E Q t [u 2 (c t,2 )] Cov t [c t,2, r t+1 r f ], (23) where E Q is the expectation taken under the measure Q induced by size-biasing the volatility distribution. 8 Define v t φt w t as the proportion of income invested in the risky asset by a young agent of generation t. The optimal proportion v t is then given by v t = 1 [ E t rt+1 r f ], (24) γ V ar t [r t+1 ] where γ EQ t [u 2(c t,2 )] E t[u 1 (c t,2 )] is the volatility-adjusted risk aversion coefficient. With the exponential utility function, γ is a constant and is interpreted as the modified (absolute) risk aversion 8 Let X be a random variable with a stochastic volatility distribution so that X V is distributed N(µ, σ 2 V ) and V has density p(v ). The size-biased volatility-adjusted distribution Q is given by q(v ) = V p(v )/E(V ). 13

15 coefficient (Gron, Jorgensen and Polson, 2004). Equation (24) suggests that the optimal portfolio allocation decision follows a conditional meanvariance rule, where the degree of risk aversion is adjusted by taking into account that asset returns are generated from a fat-tailed stochastic volatility distribution. Proposition 4 The optimal proportion of wealth invested in the risky asset increases 1) as the expected return increases, or 2) as the conditional variance or the modified risk aversion decreases. 4 Asset Returns and Portfolio Allocations: General Equilibrium The above two partial equilibrium analyses fail to recognize the fact that both the asset return and portfolio allocation are simultaneously endogenous and should be jointly determined in the entire system. 4.1 Strategic Portfolio Allocations and Asset Returns Using Corollary 2 and substituting equation (14) into equation (24), I obtain which, for α < 1, becomes φ t y t (1 α) v t = 1 γ E t [ αzt+1 Bφ α 1 t A ] V ar t [αz t+1 Bφ t α 1 ], (25) = 1 [ E t αzt+1 Bφ α 1 t A ] γ V ar t [αz t+1 Bφ α 1 t ] = 1 [ E t [z t+1 ] γ αbφ α 1 t V ar t [z t+1 ] By rearranging terms in equation (26), I have φ t 2α 1 = y t (1 α) γ [ φ α 1 t µ z,t αbσz,t 2 A (αb) 2 φ t 2α 2 V ar t [z t+1 ] A (αb) 2 σ 2 z,t ] ]. (26). (27) Equation (27) characterizes the general equilibrium outcome of the portfolio allocation. In principle, with one unknown φ in one equation, I can solve for φ. Unfortunately, I cannot without specifying a value for the parameter α since α enters the equation as an exponent of the unknown φ. I study below several special cases given different values of α. 14

16 4.1.1 Case 1: α = 1 With α = 1, the production function becomes linear in capital inputs and does not employ labor services as the input: y t = z t Bk t. To keep the economy going, I further assume for this case that the firm s output is allocated to the young agents who then make investment decisions to finance their consumption of their second-period lives. Then, I obtain b t = B and r t = Bz t δ. (28) The expected (excess) return and conditional volatility of assets are µ e r,t = Bµ z,t A and σ r,t = Bσ z,t. (29) The portfolio allocation rule is v t = 1 γ µ e r,t σ 2 r,t [ ] = 1 1 γ Bσ z,t e σε 2 1 A B 2 σz,t 2. (30) Neither the expected (excess) return nor the allocation rule depends on the level of aggregate allocation. The factor sensitivity is constant over time and is equal to the long-term level of productivity B. With the advent of a positive technology shock, current output increases (equation (28)), and the factor risk premium increases as well (Proposition 2). Both the expected (excess) return and conditional volatility of asset returns rise (equation (29)) and vary pro-cyclically, which is inconsistent with empirical findings [see, e.g., Fama and French (1989) and Schwert (1989)] Case 2: α = 1 2 Since a neat solution for a t is obtained with α = 1 2, I focus my analysis on this case, though such chosen value for α is not empirically justified. This case clearly illustrates 1) how a nonlinear technology differs from a linear technology in terms of its impact on asset pricing, and 2) how the portfolio allocation decision at the aggregate level feeds back to affect asset pricing. Overall, this case shows the difference between a general-equilibrium approach and a partial-equilibrium analysis 15

17 which is the pervasive approach used in the optimal portfolio allocation literature. When α = 1 2, the solution to equation (27) is k t+1 = P t = φ t = B 2 y 2 t µ 2 z,t ( 2Ayt + γb 2 σ 2 z,t) 2, (31) which implies that v t = 2B 2 y t µ 2 z,t ( 2Ayt + γb 2 σ 2 z,t) 2. (32) Lemma 1 With the income share of capital α = 1 2, i) the proportion of wealth invested into the risky asset increases 1) as the risk premium of the factor, µ z, increases, or 2) as the conditional variance of the factor, σz, 2 or the level of modified risk aversion, γ, or the risk-free rate, r f hence A, decreases. ii) If 2Ay t γb 2 σz,t 2 > 0 (< 0), the allocation into the risky asset increases (decreases) with respect to the long-term level of productivity, B, but decreases (increases) with respect to the current output, y. iii) The value of investment in the risky asset increases with the current output. Lemma 1 implies that, although the proportion of wealth invested in the risky asset may not necessarily increase as current output increases, the value of investment does. The share price, the real investment, and the capital stock accumulation also increase as current output increases. In other words, both the share price and the value of investment (in both real and financial assets) vary pro-cyclically. The factor sensitivity or the asset beta is b t = Ay t γb2 σ 2 z,t y t µ z,t = A µ z,t + γb2 σ 2 z,t 2y t µ z,t. (33) Lemma 2 With the income share of capital α = 1 2, the factor sensitivity or asset beta increases 1) as the risk-free rate or the modified risk-aversion or the long-term productivity of the technology or the conditional variance of the factor increases, or 2) as the factor premium or current output decreases. Lemma 2 implies that: a) the factor sensitivity or asset beta co-varies in the same direction as the long-term level of productivity; and b) the asset beta varies counter-cyclically. Substituting equation (33) into equation (19) and equation (18), I respectively calculate the 16

18 expected (excess) return and the conditional volatility of the risky asset as follows: µ e r,t E t [ r t+1 r f ] = γb2 σ 2 z,t 2y t, (34) and σ e r,t Std t [ r t+1 r f ] = σ z,t = exp {σ 2 ε} 1 ( µ z,t ( A + γb2 σ 2 z,t 2y t A + γb2 σz,t 2 2y t ) ), (35) where the last equality in equation (35) is obtained by using Proposition 2. Moreover, equation (34) and equation (35) imply that the conditional Sharpe ratio of the risky asset is SR r,t µe r,t σ e r,t Given Lemma 2, we easily obtain the following proposition: = γb 2 σ 2 z,t exp {σ 2 ε } 1 ( 2Ay t + γb 2 σ 2 z,t). (36) Proposition 5 (Long-term and Cyclical Behavior of Asset Returns: α = 1 2 ) Income share of capital 1) The expected (excess) asset returns increase as the modified risk aversion or the long-term productivity of the technology or the volatility of technology innovations increases, and decreases as the current output increases. 2) The conditional volatility of asset returns increase as the risk-free rate or the modified risk aversion or the long-term productivity of the technology or the volatility of technology innovations increases, and decreases as the current output increases. 3) The conditional Sharpe ratio of asset returns increases as the modified risk aversion or the long-term productivity of the technology or the conditional factor variance increases, and decreases as the volatility of technology innovations or the current output increases. Proposition 5 links the expected (excess) return and volatility of financial assets to the fundamentals of the economy via the factor sensitivity or asset beta. In particular, both the expected (excess) return and the conditional volatility of asset returns vary counter-cyclically, consistent with empirical evidences (see, e.g., Fama and French 1989, Schwert 1989, and Brandt and Kang 2004). Proposition 5 also shows that the conditional Sharpe ratios, i.e., the prices of risk, vary counter-cyclically, consistent with the empirical findings of Whitelaw (1997), Brandt and Kang (2004), and Lettau and Ludvigson (2004). 17

19 4.1.3 Case 3: α = 1 3 Based on Prescott (1986), I set α = 1 3, i.e., the capital share of income is one third. When α = 1 3, there are two solutions to equation (27). After dropping the solution which is negative for every t and not economically sensible, I obtain the solution to equation (27) as φ t = B 432A 3 yt 3 Q t, (37) with Then, I have Q t = 36γB2 Ayt 2 µ z,t σz,t Ayt 2 µ z,t B ( γ 2 B 3 σz,t Ay2 t µ ) z,t γ 3 B 5 σz,t 6 + γ 2 B 3 σz,t 4 B ( γ 2 B 3 σz,t Ay2 t µ ). z,t v t = 3 φ t = 2 y t B 288A 3 yt 4 Q t, (38) and b t = B 1 3 A 2 y 2 t Q 2 3 t. (39) With b t as given in equation (39), I obtain the expected (excess) return and the conditional volatility from equation (19) and equation (18), respectively. Similar to Case 2, I show analytically that, when α = 1 3, both the share price and the value of investment (in both real and financial assets) vary pro-cyclically. 9 Then I have Proposition 6 (Long-term and Cyclical Behavior of Asset Returns: α = 1 3 ) Income share of capital The factor sensitivity, the expected (excess) return, the conditional volatility, and the conditional Sharpe ratio of asset returns increase 1) as the modified risk aversion or the long-term productivity of the technology increases, or 2) as the current output decreases. 4.2 Source of Cyclical Pattern of Asset Returns With a non-linear production technology (α = 1 2 or α = 1 3 ), both the expected return and the conditional volatility of asset returns are analytically shown to vary counter-cyclically. It is natural to examine what causes the business-cycle pattern of asset returns. 9 Note that φ t y t = B3 A 3 y 4 t γσz,tq 2 t/ B ( ) γ 2 B 3 σz,t Ayt 2 µ z,t > 0. 18

20 4.2.1 Decomposition of Equilibrium Responses In general equilibrium a technology shock has two effects on expected returns. On the one hand, holding constant the aggregate allocation, as a positive productivity shock hits the economy the factor premium and, hence, the expected return increase immediately (Proposition 2). This direct effect reflects the wealth effect of a technology shock. On the other hand, as the expected return increases, the aggregate amount of investment in the risky assets increases too (Proposition 4), and all else equal, the expected return declines due to the diminishing return to capital for a non-linear production technology (Proposition 3). This indirect effect essentially captures the substitution effect of a technology shock. From equation (17), the impact of technology shocks on expected asset returns can be decomposed as follows: dµ r,t dz t = dµe r,t dz t = b t µ z,t z t + µ z,t b t z t. (40) Denote π 1 b t µ z,t z t and π 2 µ z,t b t z t. Then, π 1 and π 2 characterize the direct and indirect effects associated with a technology shock, respectively Direct Effect Using Proposition 2, I obtain the direct effect as π 1 b t µ z,t z t = b t µ z,t ρ z t 0. (41) The direct effect or the wealth effect hinges critically on the persistence level of the technology shock. The more persistent the technology shock is, the larger is the direct effect. If ρ = 0, then π 1 = 0. That is, no direct effect exists at all if the technology shock is i.i.d. If ρ > 0, then π 1 > 0. If ρ = 1, the technology shock follows a random walk process and has a permanent impact on asset returns, yielding the strongest wealth effect. In the case of a linear technology, b t = B is a constant, π 1 = Bµ z,t ρ z t 0. Moreover, the indirect effect does not exist, and the direct effect is equivalent to the general equilibrium effect. The direct effect posits that the expected return varies pro-cyclically. With a positive productivity shock, the output increases, and there is a positive correlation between the output 19

21 and the expected return Indirect Effect In addition to a direct effect, a technology shock leads to capital accumulation resulting in an indirect effect or a substitution effect. Using equation (16), I obtain the indirect effect as Since α 1 and φt z t negative. π 2 µ z,t b t z t = µ z,t b t z t = µ z,t (α 1) b t φ t φ t z t 0. (42) 0, i.e., the aggregate investment varies pro-cyclically, 10 the indirect effect is The indirect effect depends on whether or not the production technology is linear. With a linear technology (α = 1), the expected return does not depend on the total amount invested, the factor sensitivity b = B remains constant, and the indirect effect does not exist. That is, bt z t = 0 and π 2 = 0. With a non-linear production technology (α < 1), the factor sensitivity b = αbφ α 1 declines with respect to the aggregate amount of investment φ, and so does the expected return. Opposite to the direct effect, the indirect effect posits that the expected return varies countercyclically to the extent that the output increases but the expected return decreases in response to a positive technology shock General-equilibrium Effects As shown in equation (40), a general-equilibrium response of the expected return to a technology shock is a combination of the pro-cyclical direct effect and the counter-cyclical indirect effect. If the direct effect dominates, the expected return varies pro-cyclically. If the indirect effect dominates, the expected return vary counter-cyclically. Therefore, the counter-cyclical variation of expected returns shown in Section 4.1, where α = 1 2 or α = 1 3, implies that the counter-cyclical indirect effect (or substitution effect) dominates the pro-cyclical direct effect (or wealth effect). 11 For example, when ρ = 0, i.e., the technology shock is i.i.d, the direct effect is zero, and the indirect effect 10 In principle, φ t z t to get its sign. When α = 1 2, φ t z t = 11 When α = 1 2, dµ r,t dz t can be obtained by applying the implicit function theorem to equation (24), but it is not trivial [ ] 2ρAyt + γb 2 σz,t 2 > 0. 2φ t z t(2ay t +γb 2 σ 2 z,t) = γb2 σ 2 z,t 2z t y t (ρ 1) 0 with equality if ρ = 1. 20

22 dominates the direct effect (for α < 1), resulting in the counter-cyclical variation of the expected return. Interestingly, as the direct effect strengthens with respect to the persistence of technology shocks, when ρ = 1, i.e., the technology shock is a random walk, the direct effect and the indirect effect cancels against each other, yielding no general-equilibrium effect on the expected return. Theorem 1 In case of a concave production technology with the income share of capital α = 1 2 or α = 1 3, the counter-cyclical indirect effect (or the feedback effect of the portfolio allocation decision on asset returns) dominates the pro-cyclical direct effect, and the expected (excess) return, conditional volatility, and conditional Sharpe ratios of asset returns vary counter-cyclically. 4.3 Persistence and Predictability of Conditional Mean and Volatility As noted earlier, the equilibrium properties of this economy remain the same across different values of α < 1. For simplicity, I focus below on the case of α = 1 2. Equation (34) and equation (35) imply that µ e r,t = γb2 σ 2 z,t 2y t or ln µ e r,t = ln ( γb 2 2 ) + 2 ln σ z,t ln y t, (43) and σ r,t = ( ) e σ2 ε 1 A + γb2 σz,t 2 = A e 2y σ2 ε 1 + e σ2 ε 1µ e r,t. (44) t With Proposition 2, equation (43) implies that ln µ e r,t = (1 ρ) C + ρ ln µ e r,t 1 (ln y t ln y t 1 ) (1 ρ) ln y t 1 + 2ρε t, (45) where C is given by ( ) γb 2 ( ) C = ln + σε 2 + ln e σ2 ε 1. 2 Equation (45) shows that the persistence of the expected return is determined by the persistence level of the technology evolution process, ρ. The more persistent the technology evolution is, the more persistent is the expected return. Equation (45) also shows that the economic growth rate, measured by ln y t ln y t 1, is negatively related to the expected (excess) return. A higher growth rate foreshadows a lower expected return, suggesting that the expected return varies counter-cyclically. 21

23 The relation between the expected return and the economic activity sheds light on the economic source of return predicability. We can forecast asset returns using various economic variables such as dividend yield, term premium, default premium, short-term interest rates, GDP growth rate, investment-to-capital ratio, consumption-to-wealth ratio, and so on, which are either related to or characteristics of real economic activity like business cycles (see a survey in Campbell 2000). Mirroring the return predictability by business-cycle-related variables, asset returns are also associated with subsequent economic activity. A re-arrangement of equation (45) gives ln y t = (1 ρ) C + ρ ln y t 1 ( ln µ e r,t ln µ e r,t 1) (1 ρ) ln µ e r,t 1 + 2ρε t. (46) Equation (46) implies that 1) a change in expectation about asset returns, measured by ln µ e r,t ln µ e r,t 1, is negatively associated with the output level; and 2) a higher (lower) expected return signals a higher (lower) subsequent output level. Given the current output level, a higher (lower) expected return predicts a higher (lower) future economic growth rate. This finding provides a rationale to use financial prices and yields as leading indicators for the real economy (Stock and Watson 1989). This also justifies the conventional wisdom that stock market serves as a barometer of the state of the economy. Along this line, researchers have documented that financial market behavior has forecasting power for real economic activity (see, e.g., Barro 1990, Fama 1990, and Cochrane 1991). Theorem 2 With the income share of capital of a production function α = 1 2 or α = 1 3, both the expected return and the conditional volatility of asset returns are persistent, time-varying, and predictable. activity. Moreover, the behavior of financial market has forecasting power for real economic 5 Conclusions In this paper, I develop an analytical general-equilibrium model to qualitatively establish and explain economic sources of the business-cycle pattern of stock market returns documented in Fama and French (1989), Schwert (1989), Whitelaw (1997), and Brandt and Kang (2004), among others. With a concave production function, I analytically show that the expected return, conditional volatility and Sharpe ratios of asset returns all vary counter-cyclically and co-vary positively with the long-term level of productivity. A productivity shock yields a pro-cyclical direct effect (or 22

24 wealth effect) and a counter-cyclical indirect effect (or substitution effect) on asset returns. The indirect effect, which characterizes the feedback effect of consumers behavior on asset returns, dominates the direct effect and constitutes the main source of the counter-cyclical variations of asset returns. I also analytically show that in general equilibrium with a non-linear production function: 1) both the share price and the value of investment (in either real or financial assets) vary procyclically; and 2) both the conditional mean and volatility of asset returns are persistent and predictable, and the asset market behavior has forecasting power for real economic activity. This simple model sheds light on the economic source of the predictability of asset returns. 23

25 Appendix: Proofs Proof of Corollary 1: Using Proposition 1, the net asset return r t+1 P t+1+d t+1 P t 1 = k t+2+d t+1 k t+1 1 = αz t+1 Bk α 1 t+1 h1 α t+1 δ ri t+1. Q.E.D. Proof of Proposition 2: 1) If ρ < 1, then equation (2) implies that ln z t+1 ln z t N ( ρ ln z t, σ 2 ε). Then use the formula on the first two moments of a lognormal distribution to obtain the first two equations. The remaining two equations are obtained using the first two equations and equation (2). 2) µz,t ρ = µ z,t ln z t > 0 (< 0) and σz,t ρ = σ z,t ln z t > 0 (< 0) if z t > 1 (< 1). 3) µz,t z t = µ z,t ρ z t 0 and σz,t z t = σ z,t ρ z t 0. Q.E.D. Proof of Proposition 3: The proof is trivial since b = αbφ α 1 is a decreasing function of φ as α < 1 (the law of diminishing returns to capital). If α = 1, b = B is a constant. Q.E.D. Proof of Lemma 1: The first part is trivial given equation (32) and equation (31). For the second part, realize that vt B = 4Bytµ2 z,t(2ay t γb 2 σz,t) 2 > 0 (< 0) and (2Ay t+γb 2 σz,t) 2 3 v t y t = 2B2 µ 2 z,t(2ay t γb 2 σ 2 z,t) (2Ay t+γb 2 σ 2 z,t) 3 < 0 (> 0) if 2Ay t γb 2 σ 2 z,t > 0 (< 0). For the last part, note that φ t = 1 2 v ty t, so φt y t = 1 2 Proof of Proposition 6: Note that b t B = B 4 3 A 2 yt 2 γσz,tq t / B ( γ 2 B 3 σz,t Ay2 t µ z,t) > 0, b t γ = B 7 3 A 2 yt 2 σz,tq t / B ( γ 2 B 3 σz,t Ay2 t µ z,t) > 0, and b t y t = B 7 3 A 2 y t γσz,tq t / B ( γ 2 B 3 σz,t Ay2 t µ z,t) < 0. ( vt y t y t + v t ) = 2γB4 y tµ 2 z,t σ2 z,t (2Ay t+γb 2 σ 2 z,t) 3 > 0. Q.E.D. µ e r,t X SR r,t X Because µ e r,t = b t µ z,t A, σ r,t = b t σ z,t, and SR r,t = btµz,t A b tσ z,t, we have = bt X µ z,t, σr,t X = bt X = bt X σ z,t, and A b 2 t σz,t, for X=B, γ, and y t, respectively. Q.E.D. 24

26 References [1] Abel, Andrew B., 1999, Risk premia and term premia in general equilibrium, Journal of Monetary Economics 43, [2] Barberis, Nicholas, 2000, Investing for the long run when returns are predictable, Journal of Finance 55, [3] Barro, Robert J., 1990, The stock market and investment, Review of Financial Studies 3, [4] Basak, Suleyman, 1997, Consumption choice and asset pricing with a non-price-taking agent, Economic Theory 10, [5] Boldrin, Michele, Lawrence J. Christiano, and Jonas D. Fisher, 2001, Habit persistence, asset returns, and the business cycle. American Economic Review 91, [6] Brandt, Michael W., and Qiang Kang, 2004, On the relationship between the conditional mean and volatility of stock returns: A latent VAR approach, Journal of Financial Economics 72, [7] Campbell, John Y., 2000, Asset pricing at the millennium, Journal of Finance 55, [8] Campbell, John Y., and John H. Cochrane, 1999, By force of habit: A consumption-based explanation of aggregate stock market behavior, Journal of Political Economy 107, [9] Cochrane, John H., 1991, Production-based asset pricing and the link between stock returns and economic fluctuations, Journal of Finance 46, [10] Cochrane, John H., 2001, Asset Pricing, Princeton University Press, Princeton, NJ. [11] Constantinides, George M., John B. Donaldson, and Rajnish Mehra, 2002, Junior can t borrow: A new perspective on the equity premium puzzle, Quarterly Journal of Economics 117, [12] Cuoco, Domenico, and Jakša Cvitanic, 1998, Optimal consumption choices for a large investor, Journal of Economic Dynamics and Control 22,