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

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 dividends1For 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 andLudvigson (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.

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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. Rıos-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 = 12 or

α = 13 , 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 counter-

cyclically 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 of2Lettau (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.3Analytical 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-frequencyphenomenon, but the economic intuition illustrated in this article carries over to a representative-agent model or amore realistic OLG model with many periods of life for agents.

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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 dis-

aggregate 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 to4Cuoco 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 exchangeeconomy a consumption-portfolio problem of an agent who acts as a price-leader in all markets and the implicationsof his behavior on equilibrium security prices.

5The 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 neededto solve the quantitative failure: frictions at the household level like habit formation preferences and borrowingconstraints that prevent inter-temporal consumption smoothing, and frictions at the firm level like capital adjustmentcosts, investment irreversibility, and multi-production sectors with limited inter-sectoral factor mobility.

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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 rf .

There are a [0, 1] continuum of infinitely-lived representative firms who are endowed with k1

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

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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 st, 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 constant-

return-to-scale production function y = F (k, h) = zBkαh1−α, 0 < α 6 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 st is written as

y (st) = z (st)B (k (st−1))α (h (st))1−α . (1)

All variables are state-dependent except for constant terms, but I suppress st from all variables for

notational convenience.

The technology shock to the production process, zt, is the source of uncertainty in the economy.

At the beginning of each period t, a realization of the technology shock zt is observed by both firms

and agents in this economy. I assume zt to have a first-order autoregressive (AR(1)) dynamics:

ln zt+1 = ρ ln zt + εt+1 with εt+1 ∼ i.i.d. N[0 , σ2

ε

], (2)

where 0 6 ρ 6 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.

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I assume capital stock to depreciate at a constant rate 0 6 δ 6 1. The capital stock evolves as

kt+1 = (1− δ) kt + it, (3)

where it 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:

[FP ]

max{ht+j ,it+j}t>1

Et

∞∑j=0

pt+j

(zt+jBk

αt+jh

1−αt+j − wt+jht+j − it+j

)subject to

kt+1 = (1− δ) kt + it.

Here, pt is the price of one unit of date t goods denominated in units of date 1 goods, wt is

the (real) wage rate denominated in units of date t goods, and Et [·] stands for the expectation

conditional on the information set Φt available at the beginning of period t. The history of realized

shocks to technology is st ≡ {zτ}tτ=1 ⊆ Φt.6

The first-order conditions for the efficient allocation of labor and investment are respectively

given by

wt = (1− α) ztBkαt h−αt (4)

and

pt = Et[pt+1

(αzt+1Bk

α−1t+1 h

1−αt+1 + 1− 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

dt = ztBkαt h

1−αt − wtht − it. (6)

6The firm faces a dynamic problem. If the state of the economy follows a Markovian process then, denoting byV (kt, st) the firm value at time t, the Bellman equation for the firm’s problem is:

V (kt, st) = max{ht,it}

pt(ztBk

αt h

1−αt − wtht − it

)+ Et [V (kt+1, st+1)] .

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2.3 Consumers

The preference of an individual of generation t is described by an exponential utility function

u (ct,2), where ct,2 stands for her consumption at her second period of life (i.e. at time t + 1).

No interim consumption ct,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 ht = 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 ]

max{ct,2,xt,θt}

Et [u (ct,2)]

subject to

Ptxt + θt 6 wtht

ct,2 6 xt (Pt+1 + dt+1) + θt

(1 + rf

)ct,2 > 0.

Here, Pt is the ex-dividend share price denominated in units of date t goods, xt is the number of7A 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 period2 consumption, the risk-free rate is constant. Given the availability of a risk-free storage technology delivering aconstant rate of return rf , the no-arbitrage condition requires the risk-free rate to equal the return on the storagetechnology 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 thebusiness-cycle pattern of asset returns, the exogenously given risk-free rate in my model is an innocuous modelingfeature.

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shares purchased, and θt 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 uj (·) , j = 1, 2, the first- and second-order differentials of u (·), the first-order necessary

condition for xt is given by

Et

[u1 (ct,2)

((Pt+1 + dt+1)− Pt

(1 + rf

))]= 0. (7)

Define rt+1 = Pt+1+dt+1

Pt− 1 as the net asset return, and equation (7) becomes

Et

[u1 (ct,2)

(rt+1 − rf

)]= 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 ht = 1, and I can ignore the labor market.

The two remaining market-clearing conditions are:

Goods market : ct−1,2 + it + θt = yt, and (9)

Asset market : xt = 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

{ct,2, xt, θt, yt, kt, it, dt}∞t=1 and a sequence of prices {pt, Pt, wt}∞t=1 satisfying:

1. Given(pt, wt) , {yt, kt, it, dt} solves [FP ], i.e., the value maximization problem of the

representative firm;

2. Given (pt, Pt, wt), {ct,2, xt, θt} 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: xt = 1; and

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4. k1, z1, and x0 = 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., Pt = kt+1 forany 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

pt (Ptxt + θt) + pt+1ct,2 = ptwtht + pt+1

(xt (Pt+1 + dt+1) + θt

(1 + rf

)), (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 ct,2 and

xt, respectively:

ct,2 : u1 (ct,2)− Λpt+1 = 0 (12)

xt : Et [pt+1 (Pt+1 + dt+1)− ptPt] = 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

dt+1 = yt+1 − wt+1ht+1 − it+1

= zt+1Bkαt+1h

1−αt+1 − (1− α) zt+1Bk

αt+1h

−αt+1ht+1 − kt+2 + (1− δ) kt+1

= αzt+1Bkαt+1h

1−αt+1 − kt+2 + (1− δ) kt+1.

So,dt+1 + kt+2

kt+1= αzt+1Bk

α−1t+1 h

1−αt+1 + 1− δ,

which is substituted into equation (5) to obtain

pt = Et

[pt+1

(dt+1 + kt+2

kt+1

)].

Thus, I obtain equation (13) for Pt = kt+1. Q.E.D.

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Since the total number of shares xt = 1, Proposition 1 implies that the firm’s value at date t

is equal to the capital stock, i.e., Vt = Pt = kt+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 rIt+1 = αzt+1Bkα−1t+1 h

1−αt+1 − δ, then

rt+1 = rIt+1. 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 φt ≡ Ptxt as the amount of income invested in the risky asset by a youngagent of generation t. In equilibrium, φt = kt+1, i.e., the total amount of investment in financialassets is equal to the total amount of investment in real assets (or physical capitals).

Proof. Trivial given Proposition 1 and xt = 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

rt+1 = αzt+1Bkα−1t+1 h

1−αt+1 − δ, (14)

which, using Corollary 2 and ht+1 = 1, can be further rewritten as

rt+1 = αBφα−1t zt+1 − δ = btzt+1 − δ, with (15)

bt ≡ αBφα−1t . (16)

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The expected return and standard deviation of the risky asset, conditional on the information

set Φt, are

µr,t ≡ Et [rt+1] = btEt [zt+1]− δ, (17)

σr,t ≡ Stdt [rt+1] = btStdt [zt+1] . (18)

Equation (17) implies that the expected return in excess of the risk free return is

µer,t ≡ Et[rt+1 − rf

]= btEt [zt+1]−A, (19)

where A = rf + δ.

The realized return can be decomposed into its expected and unexpected components:

rt+1 = Et [rt+1] + btεt+1, (20)

where εt+1 = zt+1 − Et [zt+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 ε is the unexpected component of the factor realization with a conditional

mean of zero and a conditional variance of V art [zt+1], respectively. (The unconditional mean and

variance of ε are zero and E [V art [zt+1]], respectively.) Combining equation (18) with equation (20),

I obtain

rt+1 = Et [rt+1] + Stdt [rt+1]εt+1

Stdt [zt+1]≡ µr,t + σr,tεt+1, (21)

where εt+1 ≡ εt+1

Stdt[zt+1] is the standardized unexpected component of the factor realization.

Equation (17) implies that Et [zt+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 time-

varying. 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

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1)

µz,t ≡ Et [zt+1] = exp{ρ ln zt +

12σ2ε

},

σz,t ≡ Stdt [zt+1] = µz,t√

exp {σ2ε} − 1,

lnµz,t =12

(1− ρ)σ2ε + ρ lnµz,t−1 + ρεt, and

lnσz,t =12

(1− ρ)[σ2ε + ln

(eσ

2ε − 1

)]+ ρ lnσz,t−1 + ρεt.

2) Moreover, given today’s realizations of the technology shock z > 1 (< 1) , the higher the transitioncoefficient 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 thetechnology 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√eσ

2ε−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, Et [rt+1], and Stdt [rt+1] all decline as theaggregate 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 portfolioallocation.

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)

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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

Et [u1 (ct,2)]Et[rt+1 − rf

]+ Covt

[u1 (ct,2) , rt+1 − rf

]= 0. (22)

Using the generalized Stein’s lemma as in Gron, Jorgensen and Polson (2004), I rewrite the

covariance term in equation (22) as

Covt

[u1 (ct,2) , rt+1 − rf

]= EQt [u2 (ct,2)]Covt

[ct,2, rt+1 − rf

], (23)

where EQ is the expectation taken under the measure Q induced by size-biasing the volatility

distribution.8

Define vt ≡ φtwt

as the proportion of income invested in the risky asset by a young agent of

generation t. The optimal proportion vt is then given by

vt =1γ

Et[rt+1 − rf

]V art [rt+1]

, (24)

where γ ≡ −EQt [u2(ct,2)]Et[u1(ct,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 aversion8Let X be a random variable with a stochastic volatility distribution so that X|V is distributed N(µ, σ2V ) and

V has density p(V ). The size-biased volatility-adjusted distribution Q is given by q(V ) = Vp(V )/E(V ).

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coefficient (Gron, Jorgensen and Polson, 2004).

Equation (24) suggests that the optimal portfolio allocation decision follows a conditional mean-

variance 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 theexpected 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

vt =1γ

Et[αzt+1Bφ

α−1t −A

]V art [αzt+1Bφtα−1]

, (25)

which, for α < 1, becomes

φtyt (1− α)

=1γ

Et[αzt+1Bφ

α−1t −A

]V art [αzt+1Bφtα−1]

=1γ

[Et [zt+1]

αBφtα−1V art [zt+1]− A

(αB)2 φt2α−2V art [zt+1]

]. (26)

By rearranging terms in equation (26), I have

φt2α−1 =

yt (1− α)γ

[φtα−1µz,tαBσ2

z,t

− A

(αB)2 σ2z,t

]. (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

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:

yt = ztBkt.

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

bt = B and rt = Bzt − δ. (28)

The expected (excess) return and conditional volatility of assets are

µer,t = Bµz,t −A and σr,t = Bσz,t. (29)

The portfolio allocation rule is

vt =1γ

µer,tσ2r,t

=1γ

[1

Bσz,t√eσ2

ε − 1− A

B2σ2z,t

]. (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)].

4.1.2 Case 2: α = 12

Since a neat solution for at is obtained with α = 12 , 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

which is the pervasive approach used in the optimal portfolio allocation literature.

When α = 12 , the solution to equation (27) is

kt+1 = Pt = φt =B2y2

t µ2z,t(

2Ayt + γB2σ2z,t

)2 , (31)

which implies that

vt =2B2ytµ

2z,t(

2Ayt + γB2σ2z,t

)2 . (32)

Lemma 1 With the income share of capital α = 12 ,

i) the proportion of wealth invested into the risky asset increases 1) as the risk premium of thefactor, µz, increases, or 2) as the conditional variance of the factor, σ2

z , or the level of “modified”risk aversion, γ, or the risk-free rate, rf hence A, decreases.ii) If 2Ayt−γB2σ2

z,t > 0 (< 0), the allocation into the risky asset increases (decreases) with respectto 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

bt =Ayt + 1

2γB2σ2z,t

ytµz,t=

A

µz,t+γB2σ2

z,t

2ytµz,t. (33)

Lemma 2 With the income share of capital α = 12 , 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 technologyor the conditional variance of the factor increases, or 2) as the factor premium or current outputdecreases.

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

expected (excess) return and the conditional volatility of the risky asset as follows:

µer,t ≡ Et[rt+1 − rf

]=γB2σ2

z,t

2yt, (34)

and

σer,t ≡ Stdt

[rt+1 − rf

]=σz,tµz,t

(A+

γB2σ2z,t

2yt

)

=√

exp {σ2ε} − 1

(A+

γB2σ2z,t

2yt

), (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

SRr,t ≡µer,tσer,t

=γB2σ2

z,t√exp {σ2

ε} − 1(2Ayt + γB2σ2

z,t

) . (36)

Given Lemma 2, we easily obtain the following proposition:

Proposition 5 (Long-term and Cyclical Behavior of Asset Returns: Income share of capitalα = 1

2)1) The expected (excess) asset returns increase as the “modified” risk aversion or the long-termproductivity of the technology or the volatility of technology innovations increases, and decreases asthe current output increases.2) The conditional volatility of asset returns increase as the risk-free rate or the “modified” riskaversion or the long-term productivity of the technology or the volatility of technology innovationsincreases, and decreases as the current output increases.3) The conditional Sharpe ratio of asset returns increases as the “modified” risk aversion or thelong-term productivity of the technology or the conditional factor variance increases, and decreasesas 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

4.1.3 Case 3: α = 13

Based on Prescott (1986), I set α = 13 , i.e., the capital share of income is one third.

When α = 13 , 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

φt =B

432A3y3t

Qt, (37)

with

Qt =

−36γB2Ay2t µz,tσ

2z,t + 12Ay2

t µz,t

√B(γ2B3σ4

z,t + 48Ay2t µz,t

)−γ3B5σ6

z,t + γ2B3σ4z,t

√B(γ2B3σ4

z,t + 48Ay2t µz,t

) .

Then, I have

vt =32φtyt

=B

288A3y4t

Qt, (38)

and

bt = 12 · 223B

13A2y2

tQ− 2

3t . (39)

With bt 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 = 13 , 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: Income share of capitalα = 1

3)The factor sensitivity, the expected (excess) return, the conditional volatility, and the conditionalSharpe ratio of asset returns increase 1) as the “modified” risk aversion or the long-term productivityof 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 = 12 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.

9Note that ∂φt∂yt

= 1144

B3A−3y−4t γσ2

z,tQt/√B(γ2B3σ4

z,t + 48Ay2tµz,t

)> 0.

18

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:dµr,tdzt

=dµer,tdzt

= bt∂µz,t∂zt

+ µz,t∂bt∂zt

. (40)

Denote π1 ≡ bt∂µz,t∂zt

and π2 ≡ µz,t∂bt∂zt

. Then, π1 and π2 characterize the direct and indirect effects

associated with a technology shock, respectively.

4.2.2 Direct Effect

Using Proposition 2, I obtain the direct effect as

π1 ≡ bt∂µz,t∂zt

= btµz,tρ

zt> 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

π1 = 0. That is, no direct effect exists at all if the technology shock is i.i.d. If ρ > 0, then π1 > 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, bt = B is a constant, π1 = Bµz,tρzt

> 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

and the expected return.

4.2.3 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

π2 ≡ µz,t∂bt∂zt

= µz,t∂bt∂zt

= µz,t (α− 1)btφt

∂φt∂zt

6 0. (42)

Since α 6 1 and ∂φt∂zt

> 0, i.e., the aggregate investment varies pro-cyclically,10 the indirect effect is

negative.

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∂zt

= 0

and π2 = 0. With a non-linear production technology (α < 1), the factor sensitivity b = αBφα−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 counter-

cyclically to the extent that the output increases but the expected return decreases in response to

a positive technology shock.

4.2.4 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 = 12 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 effect10In principle, ∂φt

∂ztcan be obtained by applying the implicit function theorem to equation (24), but it is not trivial

to get its sign. When α = 12, ∂φt∂zt

= 2φt

zt(2Ayt+γB2σ2z,t)

[2ρAyt + γB2σ2

z,t

]> 0.

11When α = 12,dµr,t

dzt=

γB2σ2z,t

2ztyt(ρ− 1) 6 0 with equality if ρ = 1.

20

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 = 12

or α = 13 , 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 = 12 .

Equation (34) and equation (35) imply that

µer,t =γB2σ2

z,t

2ytor lnµer,t = ln

(γB2

2

)+ 2 lnσz,t − ln yt, (43)

and

σr,t =√eσ2

ε − 1

(A+

γB2σ2z,t

2yt

)= A

√eσ2

ε − 1 +√eσ2

ε − 1µer,t. (44)

With Proposition 2, equation (43) implies that

lnµer,t = (1− ρ)C + ρ lnµer,t−1 − (ln yt − ln yt−1)− (1− ρ) ln yt−1 + 2ρεt, (45)

where C is given by

C = ln(γB2

2

)+ σ2

ε + ln(eσ

2ε − 1

).

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 yt−ln yt−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

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 yt = (1− ρ)C + ρ ln yt−1 −(lnµer,t − lnµer,t−1

)− (1− ρ) lnµer,t−1 + 2ρεt. (46)

Equation (46) implies that 1) a change in expectation about asset returns, measured by lnµer,t −

lnµer,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 = 12 or α = 1

3 , boththe expected return and the conditional volatility of asset returns are persistent, time-varying, andpredictable. Moreover, the behavior of financial market has forecasting power for real economicactivity.

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

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 pro-

cyclically; 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

Appendix: Proofs

Proof of Corollary 1:

Using Proposition 1, the net asset return rt+1 ≡ Pt+1+dt+1

Pt− 1 = kt+2+dt+1

kt+1− 1 =

αzt+1Bkα−1t+1 h

1−αt+1 − δ ≡ rIt+1. Q.E.D.

Proof of Proposition 2:

1) If ρ < 1, then equation (2) implies that ln zt+1| ln zt ∼ N(ρ ln zt, σ2

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 = µz,t ln zt > 0 (< 0) and ∂σz,t

∂ρ = σz,t ln zt > 0 (< 0) if zt > 1 (< 1) .

3) ∂µz,t∂zt

= µz,tρzt

> 0 and ∂σz,t∂zt

= σz,tρzt

> 0. Q.E.D.

Proof of Proposition 3:

The proof is trivial since b = αBφα−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 =

4Bytµ2z,t(2Ayt−γB2σ2

z,t)(2Ayt+γB2σ2

z,t)3 > 0 (< 0) and

∂vt∂yt

= −2B2µ2z,t(2Ayt−γB2σ2

z,t)(2Ayt+γB2σ2

z,t)3 < 0 (> 0) if 2Ayt − γB2σ2

z,t > 0 (< 0) .

For the last part, note that φt = 12vtyt, so ∂φt

∂yt= 1

2

(∂vt∂ytyt + vt

)=

2γB4ytµ2z,tσ

2z,t

(2Ayt+γB2σ2z,t)

3 > 0. Q.E.D.

Proof of Proposition 6: Note that

∂bt∂B = 36 · 2

23B

43A2y2

t γσ2z,tQ

− 23

t /√B(γ2B3σ4

z,t + 48Ay2t µz,t

)> 0,

∂bt∂γ = 24 · 2

23B

73A2y2

t σ2z,tQ

− 23

t /√B(γ2B3σ4

z,t + 48Ay2t µz,t

)> 0, and

∂bt∂yt

= −24 · 223B

73A2ytγσ

2z,tQ

− 23

t /√B(γ2B3σ4

z,t + 48Ay2t µz,t

)< 0.

Because µer,t = btµz,t −A, σr,t = btσz,t, and SRr,t = btµz,t−Abtσz,t

, we have∂µer,t∂X = ∂bt

∂Xµz,t,∂σr,t∂X = ∂bt

∂Xσz,t, and∂SRr,t∂X = ∂bt

∂XA

b2tσz,t, for X=B, γ, and yt, respectively. Q.E.D.

24

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