risk, return, and dividends

ARTICLE IN PRESS

Journal of Financial Economics 85 (2007) 1–38

0304-405X/$

doi:10.1016/j

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Risk, return, and dividends$

Andrew Anga,�, Jun Liub

aColumbia University, 3022 Broadway 805 Uris, New York, NY 10027, USAbRady School of Management, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA

Received 14 December 2004; received in revised form 30 November 2005; accepted 25 January 2007

Available online 7 April 2007

Abstract

Using only the definition of returns, together with a transversality assumption, we demonstrate

that given a dividend process, any one of three variables—expected return, return volatility, and the

price–dividend ratio—completely determines the other two. By parameterizing only one of these

processes, common empirical specifications place strong, and sometimes counter-factual, restrictions

on the dynamics of the other variables. Our findings lend insight into the nature of the risk–return

relation and the predictability of stock returns.

r 2007 Elsevier B.V. All rights reserved.

JEL classification: G12

Keywords: Risk–return tradeoff; Risk premium; Stochastic volatility; Return predictability

- see front matter r 2007 Elsevier B.V. All rights reserved.

.jfineco.2007.01.001

cially thank John Cochrane, as portions of this manuscript originated from conversations between

authors. We are grateful to an anonymous referee for comments that greatly improved the article.

nk John Campbell, Joe Chen, Bob Dittmar, Chris Jones, Erik Luders, Sydney Ludvigson, Jiang

Willard, and seminar participants at an NBER Asset Pricing meeting, the Financial Econometrics

t the University of Waterloo, the Western Finance Association, Columbia University, Copenhagen

ool, ISCTE Business School (Lisbon), Laval University, London School of Economics, Melbourne

ool, Norwegian School of Management (BI), Vanderbilt University, UCLA, UC Riverside, Rice

ilburg University, University of Arizona, University of Maryland, University of Michigan,

f North Carolina, University of Queensland, University of Southern California, and Vanderbilt

r helpful comments. Andrew Ang acknowledges support from the NSF.

nding author. Tel.: +1212 854 9154; fax: +1 212 662 8474.

dresses: [email protected] (A. Ang), [email protected] (J. Liu).

p://www.columbia.edu/�aa610 (A. Ang).

www.elsevier.com/locate/jfec

dx.doi.org/10.1016/j.jfineco.2007.01.001

mailto:[email protected]

http://www.columbia.edu/~aa610

http://www.columbia.edu/~aa610

ARTICLE IN PRESSA. Ang, J. Liu / Journal of Financial Economics 85 (2007) 1–382

1. Introduction

Using the dividend process of a stock, we fully characterize the relations among expectedreturns, stock volatility, and price–dividend ratios, and derive over-identifying restrictionson the dynamics of these variables. We show that given the dividend process, any one ofthree variables—the expected return, stock return volatility, or the price–dividend ratio—completely determines the other two. These relations are not merely technical restrictionsbut lend insight into the nature of the risk–return relation and the predictability of stockreturns.Our method of using the dividend process to characterize the risk–return relation

requires no economic assumptions other than transversality, which ensures that theprice–dividend ratio exists and is well defined. (We use the terms dividend and cashflowsinterchangeably and define them to be the total payout received by a holder of an equitysecurity.) In deriving our relations, we do not rely on agent preferences, equilibriumconcepts, or a pricing kernel. This is in contrast to previous work that requires equilibriumconditions, particularly the utility function of a representative agent, to pin down therisk–return relation. For example, in a standard capital asset pricing model (CAPM) orMerton (1973) model, the expected return of the market is a product of the relative risk-aversion coefficient of the representative agent and the variance of the market return.The intuition behind our risk–return relations is a simple observation that, by definition,

returns equal the sum of capital gain and dividend yield components. Hence, the return isdetermined by price–dividend ratios and dividend growth rates. In particular, if we specifythe expected return process, we can compute price–dividend ratios given the dividendprocess. Going the other way, the price–dividend ratio, together with cashflow growthrates, can be used to infer the process for expected returns. Given the dividend process,these relations between expected returns and price–dividend ratios arise from a dynamicversion of the Gordon model.Less standard is that, given cashflows, the return volatility also determines

price–dividend ratios and vice versa. The second moment of the return is a function ofprice–dividend ratios and dividend growth rates. Thus, using dividends and price–dividendratios, we can compute the volatility process of the stock. Going in the opposite direction,if dividends are given and we specify a process for stochastic volatility, we can back out theprice–dividend ratio because the second moment of returns is determined by price–divi-dend ratios and dividend growth. In continuous time, we show that expected returns, stockvolatility, and price–dividend ratios are linked by a series of differential equations.Our risk–return relations are empirically relevant because our conditions impose

stringent restrictions on asset pricing models. Many common empirical applications oftendirectly specify only one variable among expected returns, risk, or the price–dividend ratio.Often, this is done without considering the dynamics of the other two variables. Our resultsshow that once the cashflow process is determined, specifying the expected returnautomatically pins down the diffusion term of returns and vice versa. Hence, specifyingone variable among the expected return, return volatility, or the price–dividend ratiomakes implicit assumptions about the dynamics of the other variables. Our relations canbe used as checks of internal consistency for empirical specifications that usuallyconcentrate on only one variable among predictable expected returns, stochastic volatility,or price–dividend ratio dynamics. More fundamentally, the over-identifying restrictionsamong expected returns, volatility, and prices provide additional restrictions, even before

ARTICLE IN PRESSA. Ang, J. Liu / Journal of Financial Economics 85 (2007) 1–38 3

equilibrium conditions are imposed, on stock return predictability and the risk–returntradeoff. Thus, our relations allow us to explore the implications for the joint dynamics ofcashflows, expected returns, return volatility, and prices.

We illustrate several applications of our risk–return conditions with popular empiricalspecifications from the literature on the predictability of expected returns, time-varyingvolatility, and the risk–return tradeoff. For example, Poterba and Summers (1986) andFama and French (1988b) estimate slow, mean-reverting components of returns. Often,empirical researchers regress returns on persistent instruments that vary over the businesscycle, such as dividend yields or risk-free rates to capture these predictable components.We show that with IID dividend growth, the stochastic volatility generated by thesemodels of mean-reverting expected returns is several orders too small in magnitude tomatch the time-varying volatility present in data. A second example is that many empiricalstudies model dividend yields, or log dividend yields, as a slow, mean-reverting process. Ifdividend growth is IID, an AR(1) process for dividend yields surprisingly implies that therisk–return tradeoff is negative. This result does not change if we allow dividend growth tobe predictable and heteroskedastic, where both the conditional mean and conditionalvolatility are functions of the dividend yield.

Third, it is well known that volatility is more precisely estimated than first moments (seeMerton, 1980). Since Engle (1982), a wide variety of ARCH or stochastic volatility modelshave been used to successfully capture time-varying second moments in asset prices.1 If wespecify the diffusion of the stock return, then stock prices and expected returns are fullydetermined assuming a dividend process. Hence, assuming a process for the stock returnvolatility provides an alternative way to characterize the risk–return tradeoff, rather thandirectly estimating conditional means as a function of return volatility as is commonlydone in the literature (see, e.g., Glosten, Jagannathan, and Runkle, 1993).

The idea of using the dividend process to characterize the relation between risk andreturn goes back to at least Grossman and Shiller (1981) and Shiller (1981), who argue thatthe volatility of stock returns is too high compared to the volatility of dividend growth.Campbell and Shiller (1988a,b) linearize the definition of returns and then iterate to derivean approximate relation for the log price–dividend ratio. They use this relation to measurethe role of cashflow and discount rates in the variation of price–dividend ratios byassuming that the joint dynamics are homoskedastic. Our approach is similar in that weuse the definition of returns to derive relations among risk, return, and prices. However,our relations link expected returns, stochastic volatility, and price–dividend ratios moretightly than the log-linearized price–dividend ratio formula of Campbell and Shiller.Furthermore, we are able to provide exact characterizations between the conditionalsecond moments of returns and prices (the stochastic volatility of returns and theconditional volatility of expected returns, dividend growth, and price–dividend ratios) thatCampbell and Shiller’s framework cannot easily handle.

1Most of the stochastic volatility literature does not consider implications of time-varying conditional volatility

for expected returns. Exceptions to this are the GARCH-in-mean models that parameterize time-varying

variances of an intertemporal asset pricing model. Bollerslev, Engle, and Wooldridge (1988), Harvey (1989),

Ferson and Harvey (1991), Scruggs (1998), and Brandt and Kang (2004), among others, estimate models of this

type. In contrast, most stochastic volatility models are used for derivative pricing, which only characterize the

dynamics of the risk–neutral measure rather than deriving the implied expected returns of a stock under the real

measure.


Our risk–return conditions are related to a series of papers that characterize therisk–return tradeoff in terms of the properties of a representative agent’s utility function orthe properties of the pricing kernel (see, among others, Bick, 1990; Stapleton andSubramanyam, 1990; Pham and Touzi, 1996; Decamps and Lazrak, 2000; Luders andFranke, 2004; Mele, 2005). In particular, He and Leland (1993) show that the risk–returnrelation is a direct function of the curvature of the representative agent’s utility and derivea partial differential equation that the drift and diffusion term of the price process mustsatisfy. In contrast to these papers, we use dividends, rather than preferences, to pin downthe risk–return relation. This has the advantage that dividends are observable, whichallows a stochastic dividend process to be directly estimated. Indeed, a convenientassumption made by many theoretical and empirical asset pricing models is that dividendgrowth is IID. In contrast, there is still no consensus on the precise form that arepresentative agent’s utility should take.The remainder of the paper is organized as follows. Section 2 derives the risk–return and

pricing relations for an economy with an underlying variable that captures the time-varying investment opportunity set. In Section 3, we apply these conditions to variousempirical specifications in the literature. Section 4 concludes. We relegate most proofs tothe Appendix and some proofs are available upon request.

2. The model

Suppose that the state of the economy is described by a single state variable xt, whichfollows the diffusion process

dxt ¼ mxðxtÞdtþ sxðxtÞdBxt , (1)

where the drift mxð�Þ and diffusion sxð�Þ are functions of xt. For now, we treat xt as a scalarand discuss the extension to a multivariate xt below. We assume that there is a risky assetthat pays the dividend stream Dt, which follows the process

dDt

Dt

¼ mdðxtÞ þ1

2ðs2dxðxtÞ þ s2dðxtÞÞ

� �dtþ sdxðxtÞdBx

t þ sdðxtÞdBdt , (2)

or equivalently,

Dt

D0¼ exp

Z t

0

mdðxsÞdsþ sdxðxtÞdBxt þ sdðxsÞdBd

s

� �.

Without loss of generality, we assume that dBxt is uncorrelated with dBd

t . Note that thedividend growth process is potentially correlated with x through the sdx dBx

t term.By definition, the price of the asset Pt is related to dividends, Dt, and expected returns,

mr, by

Et½dPt� þDt dt

Pt

¼ mr dt. (3)

By iterating Eq. (3), we can write the price as

Pt ¼ Et

Z T

t

e�ðR s

tmr duÞ

Ds dsþ e�ðR T

tmr duÞ

PT

� �. (4)


We show how to determine the drift mrð�Þ and diffusion srð�Þ of the return process dRt

from prices and dividends,

dRt ¼ mrðxtÞdtþ srðxtÞdBrt , (5)

under a transversality condition.

Assumption 2.1. The transversality condition

limT!1

Et½e�ðR T

tmr duÞ

PT � ¼ 0 (6)

holds almost surely.

Assumption 2.1 rules out specifications like the Black and Scholes (1973) and Merton(1973) models, which specify that the stock does not pay dividends. Equivalently, Black,Scholes, and Merton assume that the capital gain represents the entire stock return becausethere are no intermediate cashflows in these economies. By assuming transversality, we canexpress the stock price in Eq. (4) as the value of discounted cashflows:

Pt ¼ Et

Z 1t

e�ðR s

tmr duÞ

Ds ds

� �. (7)

An alternative way to compute the stock price is to iterate the definition of returnsdRt ¼ ðdPt þDt dtÞ=Pt forward under the transversality condition

limT!1

exp �

Z T

t

dRu �1

2s2r du

� �� PT ¼ 0

to obtain

Pt ¼

Z 1t

e�ðR s

tdRu�

12s

2r duÞ

Ds ds.

This equation holds path by path. As Campbell (1993) notes, we can take conditionalexpectations of both the left- and right-hand sides to obtain

Pt ¼ Et

Z 1t

e�ðR s

tdRu�

12s

2r duÞ

Ds ds

� �,

which can be shown to be equivalent to Eq. (7).The following proposition characterizes the relation among dividend growth, the drift

and diffusion of the return process dRt, and price–dividend ratios, subject to thetransversality condition.

Proposition 2.1. Suppose the state of the economy is described by xt, which follows Eq. (1),and a stock is a claim to the dividends Dt, which follow the process in Eq. (2). If the

price– dividend ratio Pt=Dt is a function f ð�Þ of xt, then the cumulative stock return process,dRt, satisfies the following equation:

dRt ¼ðmx þ sdxsxÞf

0þ 1

2s2xf 00 þ 1

fþ md þ

1

2ðs2dx þ s2dÞ

� �dt

þ sxðln f Þ0 dBxt þ sdx dBx

t þ sd dBdt . ð8Þ


Conversely, if the return Rt satisfies the diffusion equation2

dRt ¼ mrðxtÞdtþ srxðxtÞdBxt þ srd ðxtÞdBd

t , (9)

and the stock dividend process is given by Eq. (1), then the price– dividend ratio Pt=Dt ¼ f ðxtÞ

satisfies the following relation:

ðmx þ sdxsdÞf0þ 1

2s2xf 00 � mr � md �

12ðs2dx þ s2d Þ

� �f ¼ �1, (10)

and the diffusion of the stock return is determined from the relations

srxðxÞ ¼ sxðln f Þ0, (11)

srdðxÞ ¼ sd . (12)

The most important economic implication of the relations in Eqs. (8)–(12) is that giventhe dividend process, specifying one variable among the price–dividend ratio, the expectedstock return, and stock return volatility determines the other two. In other words, supposethat the dividend cashflows are given. If we denote j ) k as meaning that the process j

implies the process k, then we can write

f )mr

srx

(both from Eq. ð8Þ.

Thus, parameterizing prices, f, determines expected returns, mr, and stock return volatility,srx. The expected stock return alone determines both the stock price and the volatility ofthe return:

mr )f from Eq. ð10Þ;

srx from Eq. ð11Þ;

(

where we solve for srx after solving for f. Finally, given the dividend dynamics (or that srd

as a function of x is known), specifying a process for time-varying stock volatility, srx,determines the price of the stock and the expected return of the stock:

srx )f from Eq. ð11Þ;

mr from Eq. ð10Þ;

(

where the last implication for srx ) mr follows after noting that srx determines f and f

determines mr from Eq. (8).Thus, with dividends specified, there is only one degree of freedom between expected

returns, return volatility, and price–dividend ratios. More generally, if the dividend processcan also be specified, then we can choose two variables out of the dividend, expectedreturn, stochastic volatility, and price–dividend ratio processes, with our two choicescompletely determining the dynamics of the other two variables.In Proposition 2.1, expected returns, stochastic volatility, and dividend yields are linked

to each other by a series of differential equations. Thus, by fixing a dividend process andassuming a process for one variable among expected returns, return volatility, or dividendyields, we might be able to derive analytic solutions for the dynamics of the variables

2Since dBxt and dBd

t are independent, the diffusion term srðxtÞ of the return process in Eq. (5) is given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2rxðxtÞ þ s2rd ðxtÞ

q.


not explicitly modeled by working in continuous time. However, the relations inProposition 2.1 are fundamental, and the same intuition can be obtained in discrete time,which we discuss next.

2.1. Discrete-time intuition

We now provide some intuition on the relations among dividends, expected returns,price–dividend ratios, and return volatility in Proposition 2.1 using a discrete-time model.From the definition of returns, we can write

Rtþ1 �Ptþ1 þDtþ1

Pt

¼ mr;t þ sr;t�tþ1, (13)

where mr;t is the one-period expected return, sr;t is the conditional volatility, and �tþ1 isan IID shock with unit standard deviation. To determine prices from expected returns, orvice versa, we take conditional expectations of both sides of Eq. (13):

Pt ¼Et½Ptþ1 þDtþ1�

mr;t

.

We can iterate this forward to obtain a telescoping sum. Assuming transversality allows usto express the stock price as the stream of discounted cashflows:

Pt ¼X1j¼1

Et

Yj�1k¼0

1

mr;tþk

!Dtþj

" #. (14)

Thus, knowing the cashflow series provides a mapping between Pt and the mr;t process.This is just a dynamic version of a standard Gordon dividend discount model. Hence, thebasic Gordon model intuition allows us to infer prices from the expected return process, orvice versa, if dividends are given.

What is more surprising is that the volatility process determines prices, and vice versa,given the dividend series. To demonstrate this equivalence between volatility and prices indiscrete time, we multiply the definition of the return by �tþ1 and take conditionalexpectations:

Et

�tþ1ðPtþ1 þDtþ1Þ

Pt

� �¼ Et½�tþ1ðmt;r þ sr;t�tþ1Þ� ¼ sr;t.

We can rearrange this expression to write the stock price in terms of conditional volatilityand return innovations:

Pt ¼Et½�tþ1ðPtþ1 þDtþ1Þ�

sr;t.

Iterating forward and assuming appropriate transversality conditions, we obtain

Pt ¼X1j¼1

Et

Yj�1k¼0

�tþjþ1

sr;tþj

!Dtþj

" #. (15)

Thus, if the dividend stream is fixed, we can invert out Pt from the sr;t process andvice versa, in a similar fashion to inverting out prices from expected returns from theGordon model.


We can infer expected returns, mr;t, and stochastic volatility, sr;t, from each other byequating the price process. If expected returns are specified, then Eq. (14) allows us toinvert a price process. Then, with a price process, we can extract the sr;t process fromEq. (15). Going from sr;t to mr;t is simply the reverse procedure. Thus, with dividendsspecified, expected returns, prices, and return volatility are all linked, and knowingone process automatically pins down the other two. Thus, we obtain the same intuition inProposition 2.1 in discrete time. In the rest of our analysis, we use continuous time, whichallows us to obtain closed-form solutions.

2.2. Further comments on the proposition

The relations among prices, expected returns, and volatility outlined by Proposition 2.1arise only through the definition of returns and by imposing transversality. We have notused an equilibrium model, nor do we specify a pricing kernel, to derive the relationsbetween risk and return. Conditions (8)–(12) can be easily applied to various empiricalapplications because empirical models often assume a process for one or more of mr, srx,and f. Proposition 2.1 characterizes the functional form of the expected return, stochasticvolatility, or stock price after choosing a parameterization of only one of these variables.The relations among prices, expected returns, and volatility in Proposition 2.1 must hold

in any equilibrium model. In an equilibrium model with a (potentially endogenous)dividend process where transversality holds, prices, returns, and volatility are simulta-neously determined after specifying a complete joint distribution of state variables, agentpreferences, and technologies. Similarly, if a pricing kernel is specified, together with thecomplete dynamics of the state variables in the economy, the relations in Proposition 2.1must also hold. Hence, the relations in Eqs. (8)–(12) can be viewed as necessary but notsufficient conditions for equilibrium asset pricing models.The major advantage of the setup of Proposition 2.1 over an equilibrium framework is

that many empirical specifications in finance parameterize the conditional mean orvariance of returns (e.g., predictability regressions that specify the conditional mean orstochastic volatility models) without specifying a full underlying equilibrium model. Inthese situations, Proposition 2.1 implicitly pins down the other characteristics of returnsand prices that are not explicitly assumed. In a proof available upon request, we show thatan empirical specification of a particular conditional mean, variance, or price process doesnot necessarily uniquely determine a pricing kernel. This is especially useful for anempirical researcher who can write down a particular expected return or volatility processknowing that there exists at least one (and potentially an infinite number of) pricingkernels that can support the researcher’s choice of the expected return or volatility process.In Proposition 2.1, there are two effects if we relax the assumption of transversality.

First, the transversality assumption (Assumption 2.1) ensures that the price–dividend ratiois a function of x by Feynman-Kacs. The requirement that Pt=Dt ¼ f ðxtÞ is not satisfied ineconomies that only assume geometric Brownian motion processes for the stock processand do not specify cashflow components (like Black and Scholes, 1973; Merton, 1973). Inthese economies, there is also no state variable describing time-varying investmentopportunities as the mean and variance are constant. Second, if we relax the transversalitycondition, the ordinary differential equation (ODE) defining the price–dividend ratio inEq. (10) could have additional terms with derivatives with respect to time t, and an


additional boundary condition. This is due to the fact that when transversality does nothold, the price–dividend ratio is also potentially a function of time t.

Proposition 2.1 also applies to total returns, rather than excess returns. While someempirical studies focus on matching the predictability of total returns (see, e.g., Fama andFrench, 1988a,b; Campbell and Shiller, 1988a) and the volatility of total returns (see, e.g.,Lo and MacKinlay, 1988), we often build economic models to explain time-varying excessreturns, rather than total returns. Time-varying total returns can be partially driven bystochastic risk-free rates. Short rates could be included as a state variable in xt, especiallysince Ang and Bekaert (2006) and Campbell and Yogo (2006), among others, find thatrisk-free rates have predictive power for forecasting excess returns. In Section 3, weexplicitly investigate the implications of a system where risk-free rates linearly predictexcess returns. An alternative way to handle excess returns is to adjust Proposition 2.1 tosolve for conditional excess returns, since the nominal risk-free rate is known at time t overvarious horizons. Note that with daily or weekly returns, there is negligible differencebetween total and excess returns.

Finally, although Proposition 2.1 is stated for a univariate state variable xt, theequations generalize to the case where xt is a vector of state variables. In the multivariateextension, the ordinary differential equation in Eq. (10) becomes a partial differentialequation, where mx, sx, md , sd , srx, and srd represent matrix functions of x. An integrabilitycondition is required to ensure that the pricing function is well defined. This allows thevector of diffusion terms of the return process to imply a price–dividend ratio and anexpected return process that are unique up to an integration constant. A proof of themultivariate case is available upon request.

3. Empirical applications

Proposition 2.1 can be used to characterize the joint dynamics of expected returns ðmrÞ,return volatility ðsrÞ, dividend yields ðD=PÞ or price–dividend ratios, and dividend growthðdD=DÞ. In Table 1, we provide a brief summary of various model specifications in thefinance literature. We list some possible model specifications between mr, sr, D=P, anddD=D in each row. If a particular model specifies the dynamics of one of these fourvariables, we denote which variable is specified by bold font in the first column. The checkmarks in the second column denote which of these four variables are specified, while thequestion marks denote the variables whose dynamics are implied by parameterizingthe other two variables. The third column lists selected papers that assume a model for thevariable in bold font.

For example, in the first row of Table 1, Fama and French (1988a), Hodrick (1992),Poterba and Summers (1986), and Cochrane (2001) are examples of studies thatparameterize the expected return process. Fama and French (1988a) and Hodrick (1992)assume that expected returns are a linear function of dividend yields, whereas Poterba andSummers (1986) assume a slow mean-reverting process for expected returns. If a dividendprocess is also assumed together with a model for expected returns, then the dynamics ofstock volatility ðsrÞ and dividend yields ðD=PÞ are completely determined by the expectedreturn and dividend growth processes. Another example is the fourth row, where a largeliterature assumes a process for stochastic volatility (see, among others, Stein and Stein,1991; Heston, 1993). Combined with an assumption about dividends, Proposition 2.1completely determines the risk–return tradeoff and prices.

ARTICLE IN PRESS

Table 1

Possible model specifications (p¼ specified; ? ¼ implied)

Models mr sr D=P dD=D Selected literature

Expected returnsp

? ?p

Poterba and Summers (1986)

and dividends Fama and French (1988a,b)

Hodrick (1992)

Cochrane (2001)

Dividend yields ? ?p p

Campbell and Shiller (1988a,b)

and dividends

Dividend yieldsp

?p

? Stambaugh (1999)

and expected returns Lewellen (2004)

Campbell and Yogo (2005)

Stochastic volatility ?p

?p

Stein and Stein (1991)

and dividends or Heston (1993)

dividend yields

Risk–return relation

and

p p? ? Ferson and Harvey (1991)

stochastic volatility Glosten, Jagannathan, and Runkle

(1993), Chacko and Viceira (2005),

Liu (2007)

The table reports various possible model specifications between expected returns, mr, the volatility of returns, sr,

dividend yields, D=P (or, equivalently, price–dividend ratios), and dividend growth, dD=D. If a model specifies

one of these four variables, the specified variable is highlighted in bold in the first column. The ‘‘p’’ marks in the

second column indicate which of these four variables are specified, while the ‘‘?’’ marks indicate the variables

whose dynamics must be implied by the dynamics of the other two variables. The third column lists selected

papers that parameterize the variable denoted in bold. For example, in the first row, expected returns are specified

by, among others, Fama and French (1988a,b) and if a dividend process is also assumed, then the dynamics of

stock volatility ðsrÞ and dividend yields ðD=PÞ are completely determined by the expected return and dividend

growth processes.

A. Ang, J. Liu / Journal of Financial Economics 85 (2007) 1–3810

Our goal in this section is to illustrate how Proposition 2.1 can be applied to variousempirical models that have been specified in the literature. Investigating the joint dynamicsof expected returns, volatility, prices, and dividends produces sharper predictions ofrisk–return tradeoffs and expected return predictability. We work mainly with theassumption that dividends are IID, which is made in many exchange-based economicmodels. Many economic frameworks advocate IID dividend growth, including thetextbook expositions by Campbell, Lo, and MacKinlay (1997) and Cochrane (2001).Following this literature, many of our examples make the assumption of IID dividendgrowth for illustrative purposes. This also highlights the nonlinearities induced by thepresent value relation without specifying additional nonlinear dynamics in the cashflowprocess. Nevertheless, we also examine a system where dividend growth is predictable andheteroskedastic. We also examine features of the dividend growth process implied bycommon specifications of the expected return and stochastic volatility processes.In Section 3.1, we briefly confirm that Proposition 2.1 nests the special Shiller (1981) case

of constant expected returns, IID dividend growth, and constant price–dividend ratios.Section 3.2 analyzes the case of specifying expected returns and dividend growth byfocusing on a system where the risk-free rate can predict excess returns. In Section 3.3, weconsider a common mean-reverting specification for dividend yields combined with IIDdividend growth or dividend growth that is predictable and heteroskedastic. Section 3.4


investigates the implications of the Stambaugh (1999) model for dividend growth and therisk–return tradeoff, while Section 3.5 examines the implications for expected returns fromvarious models of stochastic volatility. Finally, we parameterize the risk–return tradeoffand stochastic volatility in Section 3.6.

3.1. IID dividend growth

If dividend growth is IID, then time-varying price–dividend ratios can result only fromtime-varying expected returns. The following corollary shows that under IID dividendgrowth, time-varying expected returns, price–dividend ratios, and time-varying volatilityare different ways of viewing a predictable state variable driving the set of investmentopportunities in the economy.

Corollary 3.1. Suppose that dividend growth is IID, so that md ¼ md and sd ¼ sd are

constant in Eq. (2). If the state variable describing the economy satisfies Eq. (1) and stock

returns are described by the diffusion process in Eq. (9), where srd ¼ srd is a constant, then

the following statements are equivalent:

1.

3

(19

The price– dividend ratio f ¼ f is constant.
2. The expected return mr ¼ mr is constant. 3. The volatility of stock returns is the same as the volatility of dividend growth, or srx ¼ 0 in
Eq. (9).

We can interpret the term srx in Eq. (9) as the excess volatility of returns that is not dueto fundamental cashflow risk. Shiller (1981) argues that the volatility of stock returns is toohigh compared to the volatility of dividend growth in an environment with constantexpected returns. Cochrane (2001) provides a pedagogical discussion of this issue andclaims that excess volatility is equivalent to price–dividend variability, if cashflows are notpredictable. Corollary 3.1 is the mathematical statement of this claim.

3.2. Specifying expected returns and dividends

In an environment where the price–dividend ratio is stationary, time-varyingprice–dividend ratios must reflect variation in either discount rates or cashflows, or both.If dividend growth is IID, then the only source of time variation for price–dividend ratiosis discount rates. We investigate two parameterizations of the expected return processwhile assuming that dividend growth is IID. First, we assume that expected returns arelinear functions of dividend yields. Second, we assume that the expected stock returnis a mean-reverting function of a predictable state variable, which we specify to be therisk-free rate.

3.2.1. Dividend yields linearly predicting returns

A large number of empirical researchers find that stock returns can be predicted byprice–dividend ratios or dividend yields in linear regressions.3 The following corollary

Papers examining the predictability of aggregate stock returns by dividend yields include Fama and French

88a), Campbell and Shiller (1988a,b), Hodrick (1992), Goetzmann and Jorion (1993), Stambaugh (1999),


investigates the effect of linear predictability of returns by log dividend yields on the priceprocess.

Corollary 3.2. Assume that dividend growth is IID, so md ¼ md and sd ¼ sd are constant in

Eq. (2) and sdx ¼ 0. Suppose that the log dividend yield lnðD=PÞ linearly predicts returns in

the predictive regression:

dRt ¼ ðaþ bxtÞdtþ srxðxtÞdBxt þ sd dBd

t , (16)

where the predictive instrument x ¼ � ln f is the log dividend yield and srx is a constant. Then

the dividend yield x follows the diffusion:

dxt ¼ mxðxtÞdtþ sxðxtÞdBxt , (17)

where the drift mx and diffusion sx are given by

mxðxÞ ¼ md þ12s2d þ

12s2rx � a� bxþ ex,

sxðxÞ ¼ �srx. ð18Þ

Corollary 3.2 implies the sign of sx is negative, indicating that shocks to returns and logdividend yields are conditionally negatively correlated. Since the relative volatility of logdividend shocks ðsd Þ is small compared to the total variance of returns, the negativeconditional correlation of returns and log dividend yields is large in magnitude. This is truein the data: Stambaugh (1999) reports that the conditional correlation between leveldividend yield innovations and innovations in returns is around �0:9 for US returns, andAng (2002) reports a similar number for the correlation between shocks to log dividendyields and returns. Note that log dividend yields predicting returns makes the strong(counterfactual) prediction that returns are homoskedastic.We calibrate the resulting log dividend yield process by estimating the regression implied

from the predictive relation (16). We use aggregate S&P500 market data at a quarterlyfrequency from 1935 to 2001. In Panel A of Table 2, we report summary statistics of logstock returns, both total stock returns and stock returns in excess of the risk-free rate(three-month T-bills), together with dividend growth and dividend yields. From Panel A,we set the mean of dividend growth at md ¼ 0:05 and dividend growth volatility atsd ¼ 0:07. The volatility of dividend growth is much smaller than the volatility of totalreturns and excess returns, which are very similar, at approximately 18% per annum. Thisallows us to set s2rx ¼ ð0:18Þ

2� ð0:07Þ2, or srx ¼ 0:15. Empirically, the correlation between

dividend growth and total or excess returns is close to zero (both correlations being around�0.08). This justifies our assumption in setting sdx ¼ 0.In Panel B of Table 2, we report linear predictability regressions of continuously

compounded returns over the next year on a constant and log dividend yields. Since thedata are at a quarterly frequency, but the regression is run with a one-year horizon on theleft-hand side, the regression entails the use of overlapping observations, which inducesmoving average error terms in the residuals. We report Hodrick (1992) standard errors inparentheses, which Ang and Bekaert (2006) show to have good small-sample propertieswith the correct empirical size. Goyal and Welch (2003), among others, document that

(footnote continued)

Engstrom (2003), Goyal and Welch (2003), Valkanov (2003), Lewellen (2004), Ang and Bekaert (2006), and

Campbell and Yogo (2006).

ARTICLE IN PRESS

Table 2

Summary statistics and predictive regressions

Panel A: Summary statistics

Total returns Excess returns Dividend growth Dividend yields

Mean Stdev Mean Stdev Mean Stdev Mean Stdev

1935:Q1–2001:Q4 0.125 0.169 0.070 0.173 0.053 0.066 0.040 0.015

1935:Q1–1990:Q4 0.121 0.173 0.066 0.178 0.059 0.071 0.044 0.013

Panel B: log dividend yield predictive regressions

Total returns Excess returns

Const Log Const Log

div yield div yield

1935:Q1–2001:Q4 0:452 0:100 0:439 0:113½2:41� ½1:76� ½2:40� ½2:03�

1935:Q1–1990:Q4 0:812 0:219 0:819 0:238½3:25� ½2:80� ½3:34� ½3:10�

Panel C: Risk-free rate predictive regressions

k ¼ 1 Quarter k ¼ 4 Quarters

Const Risk-free Const Risk-free

rate rate

1952:Q1–2001:Q4 0:153 �1:720 0:118 �1:056½3:65� ½2:25� ½2:85� ½1:38�

1952:Q1–1990:Q4 0:152 �1:793 0:112 �1:061½3:40� ½2:31� ½2:54� ½1:35�

Panel A reports means and standard deviations of total returns, returns in excess of the risk-free rate (three-month

T-bills), and dividend growth. All returns and growth rates are continuously compounded and means and

standard deviations for quarterly returns or growth rates are annualized. Panel B reports predictive regressions of

gross (or excess returns) on a constant and the log dividend yield. The regressions are run with continuously

compounded returns at an annual horizon on the left-hand side of the regression, but at a quarterly frequency. In

Panel C, we report predictive regressions of annualized continuously compounded excess returns over a k ¼ one-

quarter and k ¼ four-quarter horizon on a constant and annualized continuously compounded three-month T-bill

rates. These regressions are also run at a quarterly frequency. In Panels B and C, robust Hodrick (1992) t-statistics

are reported in parentheses. The stock data are S&P500 returns including reinvested dividends from Standard &

Poors and the frequency is quarterly.

A. Ang, J. Liu / Journal of Financial Economics 85 (2007) 1–38 13

dividend yield predictability declined substantially during the 1990s, so we also reportresults for a data sample that ends in 1990.

The coefficients in the total return regressions are similar to the regressions using excessreturns. For example, over the whole sample, the coefficient for the log dividend yield is0.10 using total returns, compared to 0.11 using excess returns. Hence, although weperform our calibrations for total returns, similar conditional relations also hold for excessreturns. The second line of Panel B shows that when the 1990s are removed from thesample, the magnitude of the predictive coefficients increases by a factor of approximatelytwo. To emphasize the linear predictive relation in Eq. (16), we focus on calibrations usingthe sample without the 1990s. Nevertheless, we obtain similar qualitative patterns for the


implied functional form for the drift of the price process when we calibrate parametervalues using data over the whole sample.Since the predictive regressions are run at an annual frequency, the estimated coefficients

in Panel B allow us to directly match a and b, since we can discretize the drift in Eq. (16) asapproximately ðaþ bxÞDt. Hence, we set a ¼ 0:81 and b ¼ 0:22. Together with thecalibrated values for md ¼ 0:05, sd ¼ 0:07, and srx ¼ 0:15, we compute the implied drift ofthe log dividend yield using Eq. (18). Fig. 1 plots the drift of the log dividend yield, whichshows it to be almost linear. Hence, if log dividend yields predict returns and dividendgrowth is IID, then linear approximations for log dividend yields will be very accurate.This implies that log-linearized systems like Campbell and Shiller (1988a,b) contain littleapproximation error for the dynamics of the log dividend yield. However, if we model thelevel dividend yield as predicting returns in Eq. (16) similar to Fama and French (1988a),then the implied drift of the level dividend yield is highly nonlinear, becoming stronglymean-reverting at high levels of the dividend yield, but behaves like a random walk at lowdividend yield levels.

3.2.2. Predictable mean-reverting components of returns

As a second example of specifying an expected return process, we assume that excessreturns are predictable by risk-free rates (see, e.g., Fama and Schwert, 1977; Campbell,1987; Lee, 1992; Ang and Bekaert, 2006; Campbell and Yogo, 2006). Ang and Bekaert(2006) find that the predictive power of risk-free rates to predict excess returns is muchstronger at short horizons than dividend yields, which is confirmed by Campbell and Yogo(2006). Denoting the risk-free rate as x, we consider the following system where the

–4.5 –4 –3.5 –3 –2.5

–0.1

–0.05

0

0.05

0.1

0.15

0.2

0.25

Log Dividend Yield

Drift

Fig. 1. Implications of returns predicted by log dividend yields. Note: We plot the implied drift of the log dividend

yield (Eq. (18)) using the calibrated parameter values a ¼ 0:81, b ¼ 0:22, md ¼ 0:05, sd ¼ 0:07, sdx ¼ 0, and

srx ¼ 0:15. The calibration is done using quarterly S&P500 data from 1935 to 1990.


risk-free rate predicts excess returns:

dRt ¼ ðxt þ aþ bxtÞdtþ srxðxtÞdBxt þ sd dBd

t , (19)

where the short rate x follows the Ornstein-Uhlenbeck process,

dxt ¼ kðy� xtÞdtþ sx dBt þ sxd dBdt . (20)

These equations imply that the term structure is a Vasicek (1977) model and that the excessstock return is predicted by the short rate. The setup also allows dividend growth and risk-free rates to be correlated through the sxd parameter.

In Panel C of Table 2, we report coefficients of predictive regressions for excessreturns over a one-quarter and a four-quarter horizon. We use annualized, conti-nuously compounded three-month T-bill rates as the predictive variable over the post-1952sample because interest rates were pegged by the Federal Reserve prior to the 1951Treasury Accord. The results confirm Ang and Bekaert’s (2006) findings that the pre-dictive power of the risk-free rate is best visible at short horizons, where the coefficienton the risk-free rate is �1:72 with a robust t-statistic of 2.25. Risk-free rate predict-ability is slightly stronger in the sample ending in 1990, where the predictive co-efficient is �1:79 with a t-statistic of 2.31. At a four-quarter horizon, the risk-freecoefficients drop to around �1:06 for both samples and are no longer significant at the5% level.

For our calibrations, we use the regression coefficients from the 1952–2001 sample at aquarterly horizon, giving us values of a ¼ 0:15 and b ¼ �1:72. The unconditional mean ofshort rates over this sample is y ¼ 0:053. We also match the annual risk-free rateautocorrelation of 0:787 ¼ expð�kÞ, the unconditional risk-free rate volatility of 0.0275,and the correlation of risk-free rates and dividend growth of 0.214 in the data. Thus, sx

and sxd satisfy

ð0:0275Þ2 ¼s2x þ s2xd

2kand 0:214 ¼

sxd

sxsd

.

We also assume that the mean of dividend growth and the volatility of dividend growth areconstant at md ¼ 0:05 and sd ¼ 0:07, respectively.

Our goal is to characterize the behavior of price–dividend ratios and the impliedstochastic volatility induced by the predictability of the excess return by risk-free rates. Wecan solve for price–dividend ratios exactly using Eq. (7) to obtain

Pt

Dt

¼ Et

Z 1t

exp �

Z s

t

ðxu þ aþ bxuÞdu

� �exp md ðs� tÞ þ sdðB

ds � Bd

t Þ� ��

ds

¼

Z 1t

exp � a� md �1

2s2d

� �ðs� tÞ

� �

�Et exp �

Z s

t

ð1þ bÞxu du

� �exp �

1

2s2dðs� tÞ þ sdðB

ds � Bd

t Þ

� �� ds

¼

Z 1t

exp � a� md �1

2s2d

� �ðs� tÞ

� �E

Qt exp �

Z s

t

ð1þ bÞxu du

� �� ds,


where the measure Q is determined by its Radon-Nikodym derivative with respect to theoriginal measure

exp

Zsd dBd

t �1

2s2d dt

� �.

By Girsanov’s theorem, the dynamics of the short rate x under Q is

dxt ¼ �kðxt � y� sd sxd=kÞdtþ sx dBt þ sxd dBQdt .

Hence, we can write the price–dividend ratio as

Pt

Dt

¼

Z 1t

exp � ð1þ bÞ yþsd sxd

k

� �þ a� md �

1

2sd

� �s

� ð1þ bÞ xt � y�sd sxd

k

� �1� e�ks

k

þðs2x þ s2xd Þð1þ bÞ2

2k2s�

2ð1� e�ksÞ

kþ

1� e�2ks

2k

� �!ds. ð21Þ

In the top panel of Fig. 2, we graph the risk premium, aþ bx, and the total expectedreturn, xþ ðaþ bxÞ, as a function of the dividend yield. The top panel of Fig. 2 shows thatthe expected return is a strictly increasing, convex function of the dividend yield. Thus,high dividend yields forecast high expected total and excess returns. The bottom panel ofFig. 2 plots the implied risk–return tradeoff of the excess return predictability system. We

plot the risk premium and total expected returns against sr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2rxðxÞ þ s2d

q. There are two

notable features of this bottom plot.First, the range of the implied volatility of returns is surprisingly small, not showing

much variation around 0.084, which is not much different from the volatility ofdividend growth at 0.07. The implied volatility is also much smaller than the standarddeviation of returns in the data, which is around 0.18. The intuition behind thisresult is that large changes in the price–dividend ratio, f, are required to produce a largeamount of stochastic volatility through the relation srx ¼ sxðln f Þ0 in Eq. (11) ofProposition 2.1. When expected returns are mean-reverting, only the terms in the sum(7) close to t change dramatically when x changes. One way for small changes in x toinduce large changes in f is for the predictive coefficient to be extremely large in magnitude,but this causes total expected returns to be unconditionally negative. We can also generatelarger amounts of heteroskedasticity if the mean reversion coefficient in the predictivevariable, k, is close to zero, which corresponds to the case of permanent changes inexpected returns.Second, the risk–return relation in Fig. 2 is downward sloping, so that high volatility

coincides with low risk premia. This is due to the convexity imbedded in the present valuerelation. When expected returns are low, price–dividend ratios are high. A standardduration argument implies that there are relatively large price movements resulting fromsmall changes in expected returns at high price levels and relatively small price movementsresulting from small changes in expected returns at low price levels. Hence, the risk–returnrelation is downward sloping. This result suggests that we might need an additionalvolatility factor to explain the amount of heteroskedasticity present in stock returns.

ARTICLE IN PRESS

Expected Return versus Dividend Yield

0.04 0.045 0.05 0.055 0.06 0.065

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

Dividend Yield

Expecte

d R

etu

rn

Risk Premium

Total Expected Return

Risk-Return Relation

0.0836 0.0838 0.084 0.0842 0.0844 0.0846 0.0848

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

Volatility of Total Returns

Expecte

d R

etu

rn

Risk Premium

Total Expected Return

Fig. 2. Implications of excess returns predicted by risk-free rates. Note: In the top panel, we graph the conditional

expected excess return, aþ bx, and the total expected excess return, xþ aþ bx, where x is the risk-free rate as a

function of the dividend yield, which we obtain from inverting Eq. (21). We use the parameter values k ¼ 0:240,sx ¼ 0:019, sxd ¼ 2:85� 10�4, y ¼ 0:053, md ¼ 0:05, sd ¼ 0:07, a ¼ 0:15, and b ¼ �1:72. In the bottom panel, we

graph the risk–return tradeoff for expected excess and total returns as a function of sr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2rxðxÞ þ s2d

q. To

produce the plots, we use quadrature to solve the price–dividend ratio in Eq. (21), and then numerically take

derivatives of the log price–dividend ratio to compute srxð�Þ from Eq. (11). The calibrations are done using

quarterly S&P500 data from 1952 to 2001.



Alternatively, heteroskedastic dividend growth could change the shape of the risk–returntradeoff, which we examine in the next section.

3.3. Specifying dividend yields and dividends

Many studies, like Stambaugh (1999), Lewellen (2004), and Campbell and Yogo (2006),specify the dividend yield to be a mean-reverting process. We now investigate the implieddynamics of expected returns and the risk–return tradeoff implied by mean-revertingdividend yields. Our first case uses IID dividend growth, while our second exampleconsiders predictable and heteroskedastic dividend growth.

3.3.1. IID dividend growth

Corollary 3.3. Assume that dividend growth is IID, so md ¼ md and sd ¼ sd are constant in

Eq. (2), and that sdx ¼ 0. Suppose that the level dividend yield x ¼ 1=f , where f ¼ P=D,follows the constant elasticity of variance (CEV) process

dxt ¼ kðy� xtÞdtþ sxgt dBx

t . (22)

Then the drift mr and diffusion srx of the return process dRt in Eq. (9) satisfy

mrðxÞ ¼ kþ md þ1

2s2d �

kyxþ s2x2ðg�1Þ þ x,

srxðxÞ ¼ �sxg�1. ð23Þ

If dividend yields are mean-reverting, Corollary 3.3 shows that returns areheteroskedastic, as srx ¼ �sxg�1. For the special case of a Cox, Ingersoll, and Ross(1987) (CIR) process where g ¼ 0:5, high dividend yields x tend to coincide with low returnvolatility, since in this special case srx ¼ �sx=

ffiffiffixp

. This is opposite to the behavior of thesevariables in data because during recessions or periods of market distress, dividend yieldstend to be high and stock returns tend to be volatile. For a CEV process with g ¼ 1,Corollary 3.3 states that the return volatility must be constant, even though expectedreturns are time-varying.We calibrate the parameters k, y, and s in Eq. (22) to match the moments of the

dividend yield. We match the quarterly autocorrelation, 0:96 ¼ expð�k=4Þ; the uncondi-tional mean y ¼ 0:044; and the unconditional variance ð0:0132Þ2 ¼ s2y=ð2kÞ for a CIRprocess. For a CEV process with g ¼ 1, we also calibrate s by matching the unconditionalvariance of dividend yields, using the relation ð0:0132Þ2 ¼ s2y2=ð2k� s2Þ. Dividendgrowth has a low correlation with dividend yields, at 0.05 in the data, so the assumptionthat sdx ¼ 0 is realistic.We characterize the behavior of expected returns and the risk–return tradeoff implied by

mean-reverting dividend yields in Fig. 3. The top panel graphs the drift of returns as amonotonic, increasing function of dividend yields. Both the cases where dividend yieldsfollow a CIR process or a CEV process with g ¼ 1 produce very similar drift functions.However, Corollary 3.3 shows that expected returns might not always be monotonicallyincreasing functions of the dividend yield. For example, if dividend yields follow a CIR

ARTICLE IN PRESS

Drift of Returns

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

Level Dividend Yield

Drift o

f R

etu

rns

CIR Process

CEV Process

Risk-Return Tradeoff

0.1 0.15 0.2 0.25 0.3 0.35–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

Return Volatility

Drift o

f R

etu

rns

CIR Process

CEV Process

Fig. 3. Implications of mean-reverting dividend yields and IID dividend growth. Note: In the top panel, we graph

the drift of the stock return as a function of the level dividend yield given by Eq. (22) in the solid line when the

level dividend yield follows a CIR process ðg ¼ 0:5Þ or in the dashed line when the level dividend yield follows a

CEV process ðg ¼ 1Þ. To produce the plot, we use the calibrated parameter values md ¼ 0:05, sd ¼ 0:07, andsdx ¼ 0. We match the quarterly autocorrelation, 0:96 ¼ expð�k=4Þ, the long-term mean y ¼ 0:044, and the

unconditional variance of the level dividend yield. For the CIR process, s is given by ð0:0132Þ2 ¼ s2y=ð2kÞ, whilefor the CEV process with g ¼ 1, ð0:0132Þ2 ¼ s2y2=ð2k� s2Þ. In the bottom panel, we plot the implied risk–return

tradeoff. The calibrations are done using quarterly S&P500 data from 1935 to 1990.



process, then mr is given by

mrðxÞ ¼ kþ md þ1

2s2d �

ky� s2

xþ x,

which could increase steeply as dividend yields x approach zero if ky4s2.While low dividend yields do not coincide with high expected returns for the parameter

values calibrated to data, Corollary 3.3 shows that low dividend yields can forecast highreturns in well-defined dynamic economies. To provide some intuition behind this result,we use the definition of a discrete-time expected return:

mr;t ¼Et Ptþ1½ �

Pt

þEt Dtþ1½ �

Pt

.

In a one-period model (or in a setting where Ptþ1 ¼ 0Þ, mr;t ¼ Et½Dtþ1�=Pt, so low pricesimply high expected returns. However, given Et½Dtþ1� in a multiperiod setting, low Pt canimply low mr;t if (i) low prices today imply low conditional prices next period or (ii) lowprices imply a large positive Jensen’s term. The former does not occur if dividend yields aremean-reverting, but large Jensen’s terms can arise in practice (see, e.g., Pastor andVeronesi, 2006).The bottom panel of Fig. 3 plots the risk–return relation implied by mean-reverting

dividend yields and IID dividend growth. The risk–return relation is strongly downwardsloping if dividend yields follow a CIR process. For a CEV process with g ¼ 1, therisk–return relation is degenerate because the implied return volatility is constant.4

Reasonable economic models usually imply that the risk premium is a weakly, or strictly,increasing function of volatility, so downward-sloping risk and total expected returnrelations could arise if the risk-free rate decreases faster than the risk premium increaseswhen volatility rises. Without this effect, a much less restrictive conditional mean mxð�Þ isrequired in Eq. (22), rather than the standard AR(1) kðy� xÞ formulation, in order for therisk–return tradeoff to be positive when dividend growth is IID.

3.3.2. Predictable and heteroskedastic dividend growth

A number of recent studies suggest that dividend growth is predictable (see Bansal andYaron, 2004; Hansen, Heaton, and Li, 2005; Lettau and Ludvigson, 2005; Ang andBekaert, 2006) and that dividend growth exhibits significant heteroskedasticity (see Calvetand Fisher, 2005). In this section, we examine the implied risk–return tradeoff for a systemwhere dividend yields are mean-reverting but dividend growth exhibits predictable andheteroskedastic components which are functions of the dividend yield.

Corollary 3.4. Assume that the level dividend yield x ¼ 1=f , where f ¼ P=D, follows the CIR

process

dxt ¼ kðy� xtÞdtþ sffiffiffiffiffixt

pdBx

t , (24)

and that the log dividend level follows the process

d lnDt ¼ ðaþ bxtÞdtþ bffiffiffiffiffixt

pdBd

t , (25)

4In the case where the log dividend yield x ¼ � ln f follows an AR(1) process and dividend growth is IID (see

Corollary 3.2), the risk–return relation is also degenerate because the return volatility is constant while expected

returns vary over time.


where the correlation between dBxt and dBd

t is zero. Then the drift mr and diffusion srx of the

return process dRt in Eq. (9) satisfy

mrðxÞ ¼ kþ aþðs2 � kyÞ

xþ 1þ bþ

1

2b2

� �x,

srxðxÞ ¼ �sffiffiffixp . ð26Þ

In Corollary 3.4, the level dividend yield is mean-reverting but is constrained to bepositive through the square-root process. In Eq. (25), dividend growth is predictable by thedividend yield, which is what Ang and Bekaert (2006) find. The conditional volatility ofdividend growth increases as the dividend yield increases. This is economically reasonable,as during periods of market distress, dividend yields are high because prices are low, andthere is larger uncertainty about future cashflows.

To match the dynamics of the dividend yield, we set k ¼ 0:16, y ¼ 0:044, and s ¼ 0:0365to match the autocorrelation, mean, and variance of the dividend yield. To calibrate theconditional mean of log dividend processes in Corollary 3.4, we regress annualizedquarterly dividend growth onto the dividend yield:

4gtþ1 ¼ aþ bdyt þ �tþ1,

where gtþ1 ¼ lnðDtþ1=DtÞ is quarterly dividend growth and dyt is the level dividend yield.Over the 1952–2001 sample, a ¼ 0:026 and b ¼ 0:415, with robust t-statistics of 2.63 and3.70, respectively. This result is consistent with the positive OLS coefficients for thedividend yield predicting dividend growth reported by Ang and Bekaert (2006).

It can be shown that the unconditional variance of dividend growth for s4t is given by

E½ðlnDs � lnDt � E½lnDs � lnDt�Þ2� ¼ b2

þbsk

� �2 !

ðs� tÞy�b2s2

k3ð1� e�kðs�tÞÞy.

This formula for s� t ¼ 1 allows us to match the unconditional variance of annualdividend growth. The volatility of annualized dividend growth, gt;tþ4 ¼ gtþ1þ

gtþ2 þ gtþ3 þ gtþ4, is 0.0932 in the data, which is matched by a value of b ¼ 0:444. Asexpected from Eqs. (24) and (25), the unconditional correlation between dividend growthand dividend yields is controlled by the parameter b. In the data, the correlation betweenannual dividend growth and dividend yields is 0.16, which is only slightly larger than thecorrelation implied by the model parameters at 0.06.

In the top panel of Fig. 4, we plot the drift and volatility of returns implied bypredictable and heteroskedastic dividend growth per Eq. (26). In the solid line, theexpected return assumes a concave shape that increases with the dividend yield. For the

return volatility in the dashed line, we plotffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2rx þ s2rd

qas a function of the dividend yield,

x, where srd ¼ bffiffiffixp

. The volatility of returns is highest when dividend yields are low (orprices are high). This implication seems to be counterfactual as stock return volatility increasesduring periods of market distress when prices are low and dividend yields are high. However, theconditional volatility curve is slightly U-shaped and increases also when dividend yields are high.

The bottom panel plots the implied risk–return tradeoff. First, the risk–return tradeoff doesnot have a unique one-to-one correspondence. This is due to the U-shape pattern of returnvolatility increasing at high dividend yields. Thus, according to this specification for dividend

ARTICLE IN PRESS

Drift and Volatility of Returns

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11−0.4

−0.3

−0.2

−0.1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


Drift a

nd V

ola

tilit

y o

f R

etu

rns

Drift

Volatility


0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36

0

0.1

0.2

0.3

Return Volatility

Drift o

f R

etu

rns

Fig. 4. Implications of mean-reverting dividend yields and non-IID dividend growth. Note: The top panel plots

the conditional drift and volatility of returns implied by predictable and heteroskedastic dividend growth in

Eq. (26). In the bottom panel, we plot the implied risk–return tradeoff. We use the parameters k ¼ 0:16,y ¼ 0:044, s ¼ 0:0365, a ¼ 0:026, b ¼ 0:415, and b ¼ 0:444. The calibrations are done using S&P500 data from

1952 to 2001.



cashflows, the risk–return tradeoff will be particularly difficult to pin down for low tomoderate return volatility levels. However, the general shape of the risk–return tradeoff isdownward sloping, similar to the IID dividend growth case in Fig. 3. Thus, either considerablymore heteroskedasticity in dividends is needed, or a richer nonlinear specification for thedividend yield is required to generate an upward-sloping risk–return relation.

3.4. Specifying dividend yields and expected returns

We take the Stambaugh (1999) model as a well-known example of a system that specifiesthe joint dynamics of dividend yields and expected returns. Stambaugh assumes that thedividend yield x ¼ D=P follows an AR(1) process and that the stock return is a linearfunction of the dividend yield. We modify the Stambaugh system slightly to use a CIRprocess or a CEV process to ensure that prices are always positive. Hence, the Stambaughmodel specifies

dRt ¼ ðaþ bxtÞdtþ srxðxtÞdBxt þ sdðxtÞdBd

t ,

dxt ¼ kðy� xtÞdtþ sxgt dBx

t , ð27Þ

where x is the dividend yield, x ¼ D=P, with g ¼ 0:5 ðg ¼ 1:0Þ for a CIR (CEV) dividendyield process. Stambaugh uses this system to assess the small-sample bias in a predictiveregression where the dividend yield is an endogenous regressor. By jointly specifyingdividend yields and expected returns, Stambaugh implicitly implies the dynamics ofdividend growth and the risk–return tradeoff.

A further application of Proposition 2.1 implies that the drift of dDt=Dt in Eq. (2) can bewritten as a function of the dividend yield x:

mdðxÞ þ1

2s2dðxÞ ¼ a� kþ ðb� 1Þxþ

kyx� s2xx2ðg�1Þ, (28)

assuming that sxd ¼ 0, which is true empirically. Hence, by assuming that dividend yieldsare mean-reverting and that dividend yields monotonically predict expected returns,Proposition 2.1 implies that dividend yields must predict dividend growth.

We graph Eq. (28) in the top panel of Fig. 5, which shows that dividend growth is ahighly nonmonotonic function of dividend yields. For very low dividend yields, dividendgrowth is a decreasing function of dividend yields. However, for dividend yields above 3%,dividend growth is an increasing function of dividend yields. Since dividend yields haveonly been below 2% for a short episode during the late 1990s, we should expect that, onaverage, dividend yields positively predict dividend growth. This result is the opposite tothe intuition of Campbell and Shiller (1988a,b) who claim that high dividend yields mustforecast either high future returns or low future dividends.5

To provide some discrete-time intuition on this result, we use the definition of expectedreturns to write

mr;t ¼ Et

Ptþ1 þDtþ1

Pt

� �¼

Dt

Pt

Et

1

Dtþ1=Ptþ1

� �þ 1

� �Et

Dtþ1

Dt

� �.

5If we model dividend yields, D=P ¼ x, in Eq. (27) to be an AR(1) process, then the drift of dividend growth

takes on a concave shape as a function of the dividend yield, which decreases to �1 as the level dividend yield

approaches zero from the right-hand side.


Given the expected return, mr;t, in a multiperiod model, high Dt=Pt implies a highEt½Dtþ1=Dt� if high dividend yields cause a large Jensen’s term or high dividend yieldsforecast high dividend yields next period. The latter result cannot occur if dividend yieldsare mean-reverting. In contrast, in a one-period setting (or where Ptþ1 ¼ 0Þ, high dividendyields forecast low dividend growth for a given expected return, mr;t:

mr;t ¼Dt

Pt

Et

Dtþ1

Dt

� �.

Hence, the result that dividend yields positively forecast future dividend growth can occuronly in a dynamic model.The middle panel of Fig. 5 plots the return–risk tradeoff implied by the Stambaugh

model. If we assume that dividend growth is homoskedastic and set sd ¼ 0:07, we caninvestigate the risk–return tradeoff implied by the ðaþ bxÞ expected return assumption in

the drift of the stock return and the volatility of the stock return, sr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2rx þ s2d

q. Since the

dividend yield is mean-reverting according to Eq. (27) in the Stambaugh system, thediffusion term of the return process takes the form srxðxÞ ¼ �sxg�1, similar to Eq. (23).Fig. 5 shows that the risk–return tradeoff is monotonically downward sloping for a CIRdividend yield process and the return volatility is constant if dividend yields follow a CEVprocess. We can induce a positive risk–return tradeoff only by relaxing the assumption thatdividend yields nonmonotonically predict expected returns, rather than the linear ðaþ bxÞ

drift term in Eq. (27), or by assuming a richer conditional mean specification for thedynamics of dividend yields.Finally, we consider the implied behavior of dividend growth heteroskedasticity from

the Stambaugh model. Following Calvet and Fisher (2005), we set the conditional mean ofdividend growth to be constant, at md ¼ 0:05, but solve endogenously for dividend growthheteroskedasticity. Using Eq. (28), we plot the conditional volatility of dividend growth,jsd j, as a function of the level dividend yield in the bottom panel of Fig. 5. Interestingly,the implied volatility of dividend growth is a nonmonotonic function of the dividend yieldand increases in periods of both low and high dividend yields, which would roughlycorrespond to the peaks and troughs of business cycle variation. A multifrequency modelof dividend growth heteroskedasticity, like Calvet and Fisher, where shocks to dividendgrowth occur jointly over different frequencies, could potentially match this pattern.

3.5. Specifying stochastic volatility and dividends

The dynamics of time-varying variances of stock returns have been successfully capturedby a number of models of stochastic volatility. If the dividend process is specified,

Fig. 5. Implications of the Stambaugh (1999) model. Note: In the top panel, we graph the drift of dividend growth

dDt=Dt from the Stambaugh (1999) system given in Eq. (28), where the level dividend yield x is mean-reverting

and dividend yields linearly predict stock returns in Eq. (27). We use the parameter values k ¼ 0:16, y ¼ 0:044,a ¼ �0:08, and b ¼ 4:59. If dividend yields follow a CIR process with g ¼ 0:5, s is given by ð0:0132Þ2 ¼ s2y=ð2kÞ,while if dividend yields follow a CEV process with g ¼ 1, ð0:0132Þ2 ¼ s2y2=ð2k� s2Þ. The middle panel plots the

risk–return tradeoff from the Stambaugh system assuming that dividend growth is homoskedastic, with sd ¼ 0:07.In the bottom panel, we follow Calvet and Fisher (2005) and assume that dividend growth is heteroskedastic with

a constant mean of md ¼ 0:05 and plot the implied conditional volatility of dividend growth. All calibrations are

done using quarterly S&P500 data from 1935 to 1990.

ARTICLE IN PRESS

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Drift

of D

ivid

en

d G

row

th

CIR Process

CEV Process

0.1 0.15 0.2 0.25 0.3 0.35−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Return Volatility

Expecte

d R

etu

rn

CIR Process

CEV Process

0 0.02 0.04 0.06 0.08 0.1

0

0.2

0.4

0.6

0.8

1

1.2


Div

idend G

row

th H

ete

roskedasticity

CIR Process

CEV Process

Drift of Dividend Growth


Conditional Volatility of Dividend Growth



Proposition 2.1 shows that the presence of stochastic volatility implies that stock returnsmust be predictable. We now use Proposition 2.1 to characterize stock return predictabilityby parameterizing the variance process. Thus, Proposition 2.1 can be used to shed light onthe nature of the aggregate risk–return relation, on which there is no theoretical orempirical consensus. This is an entirely different approach from the current approach inthe literature of estimating the risk–return tradeoff, which uses different measures ofconditional volatility in predictive regressions to estimate the conditional mean of stockreturns (see, e.g., Glosten, Jagannathan, and Runkle, 1993; Scruggs, 1998; Ghysels, Santa-Clara, and Valkanov, 2005).We look at two well-known stochastic volatility models, the Gaussian model of Stein

and Stein (1991) in Section 3.5.1 and the square root model of Heston (1993) in Section3.5.2.6 In both cases, we assume that dividend growth is IID, i.e., md ¼ md and sd ¼ sd areconstant in Eq. (2), and set sdx ¼ 0 to focus on the relations between risk and returninduced by the nonlinear present value relation.

3.5.1. The Stein– Stein (1991) model

In the Stein and Stein (1991) model, time-varying stock volatility is parameterized tobe an Ornstein-Uhlenbeck process. The Stein–Stein model in our setup can be writtenas

dRt ¼ mrðxtÞdtþ xt dBxt þ sd dBd

t ,

dxt ¼ kðy� xtÞdtþ sx dBxt . ð29Þ

The variance of the stock return is x2 þ s2d , so the stock return variance comprises aconstant component s2d , from dividend growth, and a mean-reverting component x2.Empirically, shocks to returns and shocks to volatility dynamics are strongly negativelycorrelated, which is termed the leverage effect, so sx is negative. The correlation ofdividend growth with squared returns is very small, at �0:07, which justifies ourassumption of setting sdx ¼ 0.The following corollary details the implicit restrictions on the expected return of the

stock mrð�Þ by assuming that stochastic volatility follows the Stein–Stein model.

Corollary 3.5. Suppose that dividend growth is IID, so md ¼ md and sd ¼ sd are constant in

Eq. (2). If the stock variance is determined by srxðxÞ ¼ x in Eq. (9), and x follows the mean-

reverting process (29) according to the Stein and Stein (1991) model, then the expected stock

return mrðxÞ as a function of x is given by

mrðxÞ ¼ md þ1

2s2d þ

1

2sx þ

kysx

xþ1

2�

ksx

� �x2 þ C�1 exp �

1

2

x2

sx

� �, (30)

where C is an integration constant C ¼ f ð0Þ, where f ð0Þ is the price– dividend ratio at time

t ¼ 0.

The expected return in Eq. (30) is a combination of several functional forms. First, theexpected return has a constant term, md þ

12s2d þ

12sx, which is the case in a standard

exchange equilibrium model with IID consumption growth and CRRA utility. Second, theexpected return contains a term proportional to volatility, kysx

x. This specification is implied

6It can be shown that for a log volatility model with IID dividend growth, the price–dividend ratio is not well

defined because the unconditional dividend yield cannot be computed.


by models of first-order risk aversion, developed by Yaari (1987) and parameterized byEpstein and Zin (1990). Third, the expected stock return is proportional to the variance,ð12� k

sxÞx2. A term proportional to variance would result in a CAPM-type equilibrium like

the standard Merton (1973) model. Finally, the last term, C�1 expð� 12 x2=sxÞ, can be

shown to be the dividend yield in this economy. Since the price–dividend ratio is only onecomponent of Eq. (30), the Stein–Stein model predicts that dividend yields are not asufficient statistic to capture the time-varying components of expected returns. Weemphasize that the risk–return tradeoff in Eq. (30) is not derived using an equilibriumapproach. The only economic assumptions behind the risk–return tradeoff is the IIDdividend growth process, the transversality condition necessary to derive Proposition 2.1,and the volatility dynamics of the Stein–Stein model.

To calibrate the parameters in Eq. (29), we set md ¼ 0:05, sd ¼ 0:07, and

y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0:18Þ2 � s2d

q. We set the parameters k ¼ 4 and sx ¼ �0:3. These parameter values

are meant to be illustrative, and are consistent with stochastic volatility models estimatedby Chernov and Ghysels (2002), among others. These parameter values imply that theunconditional standard deviation of volatility is 11%. We set C ¼ 26:1, which matches theaverage price–dividend ratio in the data of 24.5. In Fig. 6, we plot points corresponding to arange of plus and minus three unconditional standard deviations of x for these parameter values.

The top panel of Fig. 6 plots the expected return as a function of the dividend yieldimplied by the Stein–Stein model. Interestingly, because the Stein–Stein modelparameterizes volatility, jxj, rather than variance, there is no one-to-one correspondencebetween expected returns and dividend yields. We show two branches corresponding tonegative and positive x. The negative x branch produces a much steeper relation betweenexpected returns and dividend yields than the positive x branch. For positive x below theaverage dividend yield (4.4%), there is a nonmonotonic hook-shaped relation betweenexpected returns and dividend yields. However, one failure of the Stein–Stein model is thatit cannot account for the variation of dividend yields observed in data. In the top plot ofFig. 6, dividend yields range only from approximately 3.8% to 5.6% for a plus and minusthree standard deviation bound of x around its mean, which is substantially smaller thanthe approximately 1–10% range of dividend yields in the data.

In the bottom plot of Fig. 6, we graph the implied risk–return tradeoff. Again, becausethe Stein–Stein model assumes an AR(1) process for x, there are multiple risk–returntradeoff curves. The risk–return tradeoff for negative x is always sharply increasing,whereas the risk–return tradeoff for positive x has a pronounced nonmonotonic U-shapepattern for levels of volatility less than 20%. For volatility values higher than 15%, theexpected stock return becomes a sharply increasing function of volatility. According to theStein–Stein model, the risk–return relation will be very hard to pin down empiricallybecause of the nonmonotonic relation and multiple correspondence between risk andreturn. Studies like French, Schwert, and Stambaugh (1987) and Bollerslev, Engle, andWooldridge (1988) find only weak support for a positive risk–return tradeoff, whileGhysels, Santa-Clara, and Valkanov (2005) find a significant and positive relation. On theother hand, Campbell (1987) and Nelson (1991) find significantly negative relations.Glosten, Jagannathan, and Runkle (1993) and Scruggs (1998) report that the risk–returntradeoff is negative, positive, or close to zero, depending on the specification employed.Brandt and Kang (2004) find a conditional negative, but unconditionally positive, relationbetween the aggregate market mean and volatility. From Fig. 6, it is easy to see that

ARTICLE IN PRESS

Implied Drift of Returns as a Function of the Dividend Yield

0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058−0.5

0

0.5

1

1.5

2

2.5

Dividend Yield

Drift o

f R

etu

rns

Positive x

Negative x

Implied Drift of Returns as a Function of Volatility

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

–0.2

0

0.2

0.4

0.6

0.8

1

Volatility

Drift o

f R

etu

rns

Positive x

Negative x

Fig. 6. Implications of the Stein and Stein (1991) model. Note: In the top panel, we graph the implied stock return

as a function of the dividend yield implied by the Stein–Stein (1991) model, which is the system in Eq. (29). In the

bottom panel we plot the implied risk–return tradeoff in Eq. (30). To produce the plots, we use the parameters

y ¼ 0:17, k ¼ 4, sx ¼ �0:3, C ¼ 26:1, which matches the average price–dividend ratio, md ¼ 0:05, sd ¼ 0:07, andsdx ¼ 0. The calibration is done using quarterly S&P500 data from 1935 to 1990.



depending on the sample period of low, average, or high volatility, the expected returnrelation could be flat, upward sloping, or downward sloping.

To understand why the risk–return relation in the top panel of Fig. 6 generally slopesupwards for large absolute values of x, consider the following intuition. Theprice–dividend ratio f in the Stein–Stein economy is given by f ¼ C�1 expð� 1

2x2=sxÞ,

which is a decreasing function of volatility x because sx is negative (due to the leverageeffect). Note that for an infinite amount of volatility, the price of the stock is intuitivelyzero. If x is high (and f is low), x is likely to be lower (and f is likely to be higher) in the nextperiod because of mean reversion. The return comprises a capital gain and a dividendcomponent. Since the dividend is IID, for high enough values of x, the higher f next periodcauses the expected capital gain component to be large and hence the expected total returnto be large. Thus, very high volatility levels correspond to high expected returns.Mathematically, it is the quadratic term that dominates in Eq. (30) which is responsible forthe nonmonotonicity of the risk–return tradeoff.

3.5.2. The Heston (1993) model

In the Heston (1993) model, the variance follows a square-root process similar to CIR,which restricts the variance to be always positive. This modest change in the stochasticvolatility process produces a large change in the behavior of the risk premium, as thefollowing corollary shows.

Corollary 3.6. Suppose that dividend growth is IID, so md ¼ md and sd ¼ sd are constant in

Eq. (2) and that sdx ¼ 0. Suppose that returns are described by the Heston (1993) model:

dRt ¼ mrðxtÞdtþffiffiffiffiffixt

pdBx

t þ sd dBdt ,


pdBx

t . ð31Þ

Then the expected return mrðf Þ as a function of the price– dividend ratio f ¼ P=D is given by

mrðf Þ ¼ md þ1

2s2d þ

kysþ

s2� k

ln

f

C

� �þ

1

f, (32)

where C is an integration constant C ¼ f ð0Þ, where f ð0Þ is the price– dividend ratio at time

t ¼ 0. The expected stock return mrðxÞ as a function of the return variance, x, is given by

mrðxÞ ¼ md þ1

2s2d þ

kysþ

1

2�

ks

� �xþ C�1 exp �

x

s

. (33)

The top panel of Fig. 7 shows that the expected return is a monotonically increasingfunction of dividend yields (Eq. (32)). Unlike the Stein–Stein model, the Heston modelparameterizes the stock variance, so there is a unique one-to-one mapping betweendividend yields and expected returns. To produce the plot, we use the parameter valuesmd ¼ 0:05, sd ¼ 0:07, sdx ¼ 0, y ¼ ð0:18Þ2 � s2d , and k ¼ 4. We set s ¼ �0:2 to reflect theleverage effect. These parameter values for y, k, and s are very close to the valuesadvocated by Heston (1993). To match the average price–dividend yield in data, we setC ¼ 28:06.

In the bottom panel of Fig. 7, we plot the risk–return tradeoff implied by the Hestonmodel. We can interpret the Heston risk–return tradeoff in Eq. (33) to have threecomponents: a constant term, a term linear in the variance x, and a third term f ¼

C�1 expð�x=sÞ that can be shown to be the dividend yield. Unlike the Stein–Stein model,

ARTICLE IN PRESS

Implied Drift of Returns as a Function of the Dividend Yield

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11−6

−4

−2

0

2

4

6

Dividend Yield

Drift

of

Re

turn

s

Implied Drift of Returns as a Function of Volatility

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−0.5

0

0.5

1

1.5

2

2.5

Volatility

Drift

of

Re

turn

s

Fig. 7. Implications of the Heston (1993) model. Note: In the top panel, we graph the implied drift of the stock

return as a function of the dividend yield given by Eq. (32) implied by the Heston (1993) model, which is described

in Eq. (31). In the bottom panel, we plot the implied risk–return tradeoff given in Eq. (33). To produce the plots,

we use the parameters y ¼ 0:0275, k ¼ 4, s ¼ �0:2, and C ¼ 28:06, which matches the average price–dividend

ratio, md ¼ 0:05, sd ¼ 0:07, and sdx ¼ 0.



the risk–return relation implied by the Heston model is always positive. Mechanically, thisis because the expected return in the Heston economy in Eq. (33) does not have a negativeterm proportional to volatility that enters the risk–return tradeoff in the Stein–Stein model(see Eq. (30)). The term proportional to volatility allows the expected return in theStein–Stein solution to initially decrease before increasing. In the Heston model, no suchinitial decrease can occur and the expected stock return in Eq. (33) is dominated by thelinear term ð1

2� k

sÞx. Since empirical estimates of the mean-reversion of the variance, k, arelarge and s is small and negative due to the leverage effect, the risk–return tradeoff isupward sloping.

3.6. Specifying the risk– return tradeoff and stochastic volatility

Our last application parameterizes the risk–return tradeoff and stochastic volatility.There are various assumptions made about the risk–return tradeoff in the literature. Forexample, in two recent asset allocation applications involving stochastic volatility, Liu(2007) assumes that the Sharpe ratio is increasing in variance, following Merton (1973),while Chacko and Viceira (2005) assume that the Sharpe ratio is a decreasing function ofvolatility. Cochrane and Saa-Requejo (2000) assume that the Sharpe ratio is constant. Inour analysis, we work with the Heston (1993) model of stochastic volatility and analyzetwo cases of the risk–return tradeoff: (i) we assume that expected returns are proportionalto volatility and (ii) we assume that expected returns are proportional to variance. We nowcharacterize the dynamics of cashflows implied by these two assumptions about therisk–return tradeoff.

We assume that the return and stochastic variance process follow the Heston model,

dRt ¼ mrðxtÞ þffiffiffiffiffixt

pdBx

t þ sd ðxtÞdBdt ,


pdBx

t ,

and the risk–return tradeoff is characterized by

mrðxÞ ¼ Axd,

with d ¼ 0:5 or 1. Using Proposition 2.1, the drift of dividend growth is given by

mdðxÞ þ1

2sdðxÞ ¼ Axd �

kys�

1

2�

ks

� �x� C�1 exp �

x

s

. (34)

Eq. (34) shows that the cashflow drift directly inherits the risk–return tradeoff, along withother terms reflecting the dynamics of the Heston volatility process.

We characterize the drift of dividend growth in Eq. (34) in Fig. 8 using a value of A ¼ 5for the two cases d ¼ 0:5 and 1. In the top panel, we graph the drift of dividend growth inEq. (34) as a function of return volatility. To fix total return volatility, we assume thatdividend growth is homoskedastic, with sd ¼ 0:07, and graph the drift in Eq. (34) against

srðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffixþ s2d

q. In both the cases for d ¼ 0:5 and 1, very high volatility levels correspond

to low expected dividend growth. However, when expected returns are proportional tovolatility, the drift of dividend growth is nonmonotonic and both low and high volatilityforecast low future growth in dividends. In the middle panel of Fig. 8, we plot the drift ofdividend growth as a function of the dividend yield. High dividend yields correspond tolow future dividend growth, but the drift function can also be nonmonotonic for the

ARTICLE IN PRESS

Implied Drift of Dividend Growth as a Function of Total

Return Volatility

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4–2

–1.5

–1

–0.5

0

0.5

1

Total Return Volatility

Drift

of D

ivid

en

d G

row

th

Proportional to Variance

Proportional to Volatility

Implied Drift of Dividend Growth as a Function of the

Dividend Yield

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11–3.5

–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1


Drift

of

Div

idend G

row

th

Proportional to Variance


Conditional Volatility of Dividend Growth

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Stochastic Volatility x

Div

idend G

row

th H

ete

roskedasticity Proportional to Variance




d ¼ 0:5 case. In particular, the drift of dividend growth increases as dividend yieldsincrease for low dividend yield levels (around 4%).

Finally, we can gauge the implied heteroskedasticity of dividend growth from these twocommon specifications for the risk–return tradeoff and stochastic volatility by followingCalvet and Fisher (2005) and setting the conditional mean of dividend growth to be aconstant, at md ¼ 0:05. From Eq. (34), we can invert for the conditional volatility ofdividend growth, jsdðxÞj, as a function of the stochastic Heston component of the totalreturn. We plot this in the bottom panel of Fig. 8. Clearly, the implied dividend growthheteroskedasticity is a highly nonmonotonic function of return volatility, increasing whenreturn volatility is both very high and very low.

4. Conclusions

We derive conditions on expected returns, stock volatility, and price–dividend ratiosthat asset pricing models must satisfy. In particular, given a dividend process, specifyingonly one variable among the expected return process, the stochastic volatility process, orthe price–dividend ratio process, completely determines the other two. For example,specifying the dividend stream allows the volatility of stock returns to pin down theexpected return, and thus the risk–return tradeoff. We do not need to specify a completeequilibrium model to characterize these risk–return relations, but instead derive theseconditions using only the definition of returns, together with a transversality assumption.

Our conditions between risk and return are empirically relevant because many popularempirical specifications assume dynamics for only one or a combination of the expectedreturn, volatility, and price–dividend variables, without considering the implicit restric-tions on the dynamics of the other variables. Our relations allow us to investigate the jointdynamics of expected returns, return volatility, prices, and cashflows. We show that someof the implied restrictions made by empirical models that specify only one or two of thesevariables can result in implied dynamics of the other variables that are eithercounterfactual, or hard to match in equilibrium models.

One important implication of our examples is that future asset pricing models shouldtake into account predictability and heteroskedasticity of the dividend growth process.Even common specifications of expected return or volatility processes imply rich patternsof dividend growth predictability and heteroskedasticity. In this regard, important stridesin recognizing complex dividend dynamics have recently been made by Bansal and Yaron(2004), Calvet and Fisher (2005), and Hansen, Heaton, and Li (2005), who emphasize therole of non-IID dividend dynamics in equilibrium economies. Our results also point theway forward to developing an empirical methodology that can exploit our over-identifyingconditions to create more powerful tests to investigate the risk–return tradeoff, the

Fig. 8. Implications of the risk–return tradeoff using the Heston (1993) model. Note: In the top panel, we graph

the drift of dividend growth as a function of volatility implied by an assumption of the risk–return tradeoff and

the Heston (1993) model, given in Eq. (34). We plot Eq. (34) against total return volatility, srðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffixþ s2d

q,

where x is the time-varying Heston variance component of the return, assuming that sd ¼ 0:05. In the middle

panel, we plot the implied drift of dividend growth as a function of the dividend yield. The bottom panel plots the

conditional volatility of dividend growth, jsd ðxÞj as a function of the Heston variance, x, assuming that the

conditional mean of dividend growth is constant at md ¼ 0:05. In all three panels, we assume that the risk–return

tradeoff is proportional to volatility, mrðxÞ ¼ Affiffiffixp

, or proportional to variance, mrðxÞ ¼ Ax. We produce the plots

using the calibrated parameters y ¼ 0:0275, k ¼ 4, s ¼ �0:2, C ¼ 28:06, sdx ¼ 0, and A ¼ 5.


predictability of expected returns, the dynamics of stochastic volatility, and present valuerelations in a unifying framework.

Appendix A. Proof of Proposition 2.1

Eq. (8) follows from a straightforward application of Ito’s lemma to the definition of thereturn:

dRt ¼dPt þDt dt

Pt

, (A1)

which we rewrite as dRt ¼ df t=f t þ dDt=Dt þ 1=f t dt. Note that we assume that dBdt and

dBxt are uncorrelated by assumption.The definition of returns in Eq. (A1) allows us to match the drift and diffusion terms in

Eq. (8) for Rt. Hence, the price–dividend ratio f, the expected return mr, and the volatilityterms srx and srd are determined by rearranging the drift, and the dBx

t and dBdt diffusion

terms, respectively. If the expected return mrð�Þ is determined, Eq. (10) defines a differentialequation for f, which determines f. Once f is determined, we can solve for srx from Eq. (11).If the return volatility srx is specified, we can solve for f from Eq. (11) up to a multiplicativeconstant and this determines the expected return mr in Eq. (10). &

Appendix B. Proof of Corollary 3.1

Statements (2) and (3) are equivalent from Eq. (11) of Proposition 2.1. Assume thatf ¼ f is a constant. Then, using Eq. (10), we can show that mr ¼ f

�1þ md þ

12s2d , which is a

constant. Hence (2) follows from (1). Finally, to show that (1) follows from (2), supposethat mr ¼ mr is a constant. From Eq. (10), f satisfies the following ODE:

mxf 0 þ 12s2xf 00 � ðmr � md �

12s2dÞf ¼ �1. (B1)

Since the term on f is constant, it follows that the price–dividend ratio P=D ¼ f ¼

ðmr � md �12s2dÞ�1 is the solution. Note that this is just the Gordon formula, expressed in

continuous time. Hence, the price–dividend ratio is constant. &

Appendix C. Proof of Corollary 3.2

Using Eq. (11) of Proposition 2.1, we have srx ¼ sxðxÞðln f Þ0 ¼ �sx, since x ¼ � ln f .From Eq. (10), we have

aþ bx ¼mxðxÞf

0þ 1

2s2rxf 0

0þ 1

fþ md þ

1

2s2d . (C1)

Substituting f 0=f ¼ �1 and f 00=f ¼ 1 and re-arranging this expression for mxðxÞ yields Eq.

(18). &

Appendix D. Proof of Corollary 3.3

This is a straightforward application of Eq. (8) of Proposition 2.1, using f ¼ 1=x for thelevel dividend yield and f ¼ expð�xÞ for the log dividend yield. &


Appendix E. Proof of Corollary 3.5

Using Eq. (11) of Proposition 2.1, we have x ¼ sxðln f Þ0, from which we can solve theprice–dividend ratio f to be

f ¼ C exp1

2

x2

sx

� �, (E1)

where C is the integration constant C ¼ f ð0Þ.We set C to match the unconditional price–dividend ratio. This entails computing

E0½expð�lx2t Þ� for a diffusion process dxt ¼ �kðxt � yÞdtþ sx dBt for l ¼ �1=ð2sxÞ, since

sx is negative. We know that xt is a normal random variable with mean xt and variance s2tgiven by

xt ¼ ðx0 � yÞe�kt þ y,

s2t ¼ s2x

Z t

0

e�2ks ds ¼s2x2kð1� expð�2ktÞÞ. ðE2Þ

Thus, we have

E0½expð�lx2t Þ� ¼ E0½expð�lðxt � xtÞ

2� 2lðxt � xtÞxt � lx2

t Þ�

¼ expð�lx2t �

12 lnð1þ 2ls2t Þ þ

12l

2x2t ðs�2t þ 2lÞ�1Þ. ðE3Þ

We compute the last equality using Lemma A.1 of Ang and Liu (2004). Letting t!1, weobtain

E½expð�lx2Þ� ¼ expð�ly2 � 12lnð1þ ls2x=kÞ þ

12l2y2ð2k=s2x þ 2lÞ�1Þ, (E4)

as the unconditional mean of xt is y and the variance of the Ornstein-Uhlenbeck process iss2x=ð2kÞ. &

Appendix F. Proof of Corollary 3.6

The proof is similar to Corollary 3.5, except now the price–dividend ratio f is given by

f ¼ C expx

s

, (F1)

where C is the integration constant C ¼ f ð0Þ.We set C to match the average price–dividend ratio in the data. Cox, Ingersoll, and Ross

(1987) show that the process

dxt ¼ �kðxt � yÞdtþ sffiffiffiffiffixt

pdBt

has the steady-state density function

f ðxÞ ¼on

GðnÞxn�1e�ox,


where o ¼ 2k=s2 and n ¼ ð2kyÞ=s2. Hence, we can compute the average price–dividendratio using

E½expð�lxÞ� ¼

Z 10

on

GðnÞxn�1e�ðoþlÞx ¼

on

ðoþ lÞn,

with l ¼ �1=s. &

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