Correlations ∗

Paul Ehling†

BIChristian Heyerdahl-Larsen‡

LBS

Current Draft: July 2012

∗We would like to thank Knut Kristian Aase, Suleyman Basak, João Cocco, Ilan Cooper, StephanDieckmann, Giulia Di Nunno, Mike Gallmeyer, Francisco Gomes, João Gomes, Trond Stølen Gus-tavsen, Burton Hollifield, Philipp Illeditsch, Tom Lindstrøm, Iñaki Rodríguez Longarela (EFA dis-cussant), Anna Pavlova, Richard Priestley, Skander Van den Heuvel (FIRS discussant), Amir Yaron(AFA discussant), Bernt Øksendal, and participants at a BI Brown Bag, the Workshop on RiskMeasures and Stochastic Games with Applications to Finance and Economics at the Department ofMathematics of the University of Oslo, the Arne Ryde Workshop in Financial Economics at LundUniversity, the NHH-UiO Macro Workshop, Nordic Finance Network (NFN) Workshop, SIFR - In-stitute for Financial Research, Texas A&M, Fourth Annual Empirical Asset Pricing Retreat at theUniversity of Amsterdam, Symposium on Stochastic Dynamic Models in Finance and Economics atthe University of Southern Denmark, EFA 2007 meetings, European Meeting of the EconometricSociety 2008, FIRS Conference 2009 Prague, a EIEF Brown Bag, AFA 2011 meetings, 4th FinancialRisk International Forum in Paris 2011, a Banco de España internal seminar, a LBS Brown Bag,Bank of Canada and Johannes Gutenberg-Universität Mainz for helpful comments and suggestions.We are grateful to Nam Huong Dau for excellent research assistance. Part of this research wasconducted while the first author was a Research Fellow at Banco de España. The views expressedare those of the authors and should not be attributed to the Banco de España.†Department of Financial Economics, BI Norwegian Business School, Nydalsveien 37, 0484 Oslo,

paul.ehling@bi.no‡London Business School, Regent’s Park, London, NW1 4SA, cheyerdahllarsen@london.edu

Abstract

We study stock market correlations in an equilibrium model with heterogeneous risk aver-sion. Investor heterogeneity causes countercyclical variations in the level and the volatilityof aggregate risk aversion. At times of high volatility of aggregate risk aversion, which is acommon factor in stock returns, we see high stock return correlations. In equilibrium correla-tions comove with expected excess returns, standard deviations and volatility of trade. Thecalibrated model matches average industry correlations and changes in correlations from busi-ness cycle peaks to troughs and replicates the cyclical dynamics of expected excess returnsand standard deviations. Model implied aggregate risk aversion explains average industrycorrelations, expected excess returns, standard deviations and the volatility of turnover.

Keywords: Equity Return Correlations; Heterogeneous Risk Aversion; Volatility of Turnover;Cyclical Dynamics of Stock Price Moments

JEL Classification: G10; G11

1 Introduction

Equity return correlations increase during contractions and decrease in expansions. Further,correlations comove with average stock return volatility and expected excess returns as shownin Figure 1. The change in correlations, volatilities and expected returns during economywide contractions appear large and can, therefore, cause significant portfolio rebalancing.Indeed, the dynamics of turnover volatility shown in Figure 1 appear consistent with briskportfolio rebalancing during contractions. But what economic mechanism ties correlationsto stock return volatilities, expected returns and turnover volatility?

In this paper we consider a general equilibrium model that accounts for high stock re-turn correlations in recessions and that produces comovement of correlations with volatility,expected excess returns and volatility of trade just as in Figure 1. The main mechanismgenerating the stylized facts about stock return correlations is heterogeneity in risk aversion.When consumption falls, investors with low risk aversion sell stocks or the market portfolioto more risk averse investors. Over the course of trading, stock prices fall to stimulate de-mand by investors with high risk aversion as they require higher expected returns to holdstocks. Underlying the dynamics, we find that aggregate risk aversion, a “common factor” inasset prices, rises when prices fall or when investors with high risk aversion buy. Thus, wesee counter-cyclical asset-pricing dynamics: correlations but also expected excess returns,conditional volatilities, and conditional volatility of trade rise when prices fall. Importantly,we show that the model quantitatively matches unconditional moments plus the changes incorrelations, stock return volatilities and expected returns and their comovement over thebusiness cycle.

The cyclical properties of stock return correlations depend on both the level and thevolatility of aggregate risk aversion in the economy. We label these economic mechanismsthe risk aversion level effect and the risk aversion volatility effect. To understand the riskaversion level effect, consider a setting with uncorrelated dividends. A positive dividendshock to one stock, all else equal, will increase its price and thus its realized return. Therealized return has two components, a positive dividend effect and a negative discountingeffect, i.e., the discount rate increases. The discount rate increases since investors requirea higher expected return to compensate for the increase in exposure to that stock via itsincreased dividend. For other stocks, all else equal, the discounting effect is positive as theirmarket prices of risk decrease. Hence, all stocks experience positive returns after one positivedividend shock to one stock. Cochrane et al. (2008) study this market clearing mechanismwith a logarithmic representative investor. We show that the impact of the discounting effectincreases in risk aversion. From this it follows that the higher the risk aversion, the lower

the net return on the stock that experiences the positive dividend shock, and the higher thereturn on all other stocks. Therefore, the higher is the risk aversion the higher are stockreturn correlations.

To understand the risk aversion volatility effect, consider a setting with two types ofinvestors: high risk aversion and low risk aversion. When consumption declines, investorswith high risk aversion hold a larger fraction of the economy, i.e., investors with low riskaversion sell shares.1 Therefore, aggregate risk aversion increases, which then increasescorrelations due to the risk aversion level effect. Although aggregate risk aversion is boundedfrom above by the largest risk aversion in the economy, correlations in an economy withheterogeneous investors are significantly higher than correlations obtained in a homogeneousinvestor economy populated by the investor with the largest risk aversion. This effect oncorrelations relates to the slope of the consumption sharing rule (Dumas (1989)). A decreasein aggregate consumption causes a steepening of the sharing rule, which causes an increasein the volatility of aggregate risk aversion. An increase in the volatility of aggregate riskaversion leads to an increase in the volatility of all market prices of risk, which causes anincrease in the volatility of all discount rates. When the volatility of discount rates increases,then the volatility of stock returns increases too. Moreover, when the volatility of aggregaterisk aversion increases, which is a common factor in stock returns and volatilities, then stockreturn correlations also increase.

The sharing rule is implemented by trade in the market portfolio and in a riskless security.Trade constitutes a distinct observable that provides information about investor heterogene-ity. We solve for the volatility, or quadratic variation, of portfolio policies as a measure oftrading intensity, and put its implications to the test. Endogenous variation in risk aversionand the trading implications set our work apart from Chue (2005),2 who employs the Camp-bell and Cochrane (1999) model with a representative investor to study correlations.3 By

1Cochrane (2008) argues for a risk aversion explanation, as evidenced by the amount of deleveraging andforced selling taking place within capital markets, as the cause of the daily volatility of 75 percent of suchactivity occurring in October 2008. Cochrane (2011), in his presidential address to the American EconomicAssociation, says: “... all of these theories are really about discount rates, risk bearing, risk sharing, and riskpremiums” by which he essentially says that risk sharing must be a key factor in asset price moments.

2Other differences between Chue (2005) and our work include: His model is calibrated to a stylized two-country economy using data from emerging markets; Chue (2005) does not empirically study the model.Further, note that he assumes away market clearing, hence the two-trees effect of Cochrane et al. (2008) isturned off and there is no risk aversion level effect in the model.

3Aydemir (2008), Dumas et al. (2003), Pavlova and Rigobon (2008), and Ribeiro and Veronesi (2002)model co-movements in international stock market returns. Aydemir (2008) extends the model in Chue(2005) to an international setting and studies correlations with perfect and imperfect risk sharing. Dumaset al. (2003) find, in a representative investor framework, that the level of correlations can be matched,and so there is no international excess correlation puzzle. Pavlova and Rigobon (2008) study stock prices,exchange rates and the correlations of stock prices with multi-goods. In their model, spill-over effects arisebecause of binding portfolio constraints. Ribeiro and Veronesi (2002) analyze fundamental country processes

2

testing implications of trading,4 we set our work apart from other models that study het-erogeneity in risk aversion but do not empirically study the relation between aggregate riskaversion or model implied habit and the volatility of turnover in the data.5 Moreover, ourprediction that stock market conditional correlations, expected excess returns, conditionalstandard deviations and conditional volatility of portfolio policies contain a common factoror principal component —namely aggregate risk aversion— set our work apart from othermodels that study heterogeneity in risk aversion as well as from empirical work6 that studyrelations between correlations, returns, turnover and volatilities without a structural model.

In addition to preference heterogeneity, we also employ ratio habit preferences, see Abel(1990) and Chan and Kogan (2002). As with standard power utility, countercyclical varia-tions of endogenous quantities with the ratio habit preferences result from preference het-erogeneity. Importantly, ratio habits replicate the level of the risk-free rate, the aggregatestock market mean excess return, and the aggregate stock market return standard deviation.There are, however, two other equally important reasons why we use ratio habit preferences.First, ratio habit preferences allow us to utilize relative consumption, consumption in excessof habit, as a stationary state variable. Second, since relative consumption is procyclical, itdefines good and bad states of nature. Another convenient feature of ratio habit preferencesis that they lead to a stationary wealth distribution. Yet, the ratio habit preferences are notcrucial for our prediction that stock market expected excess returns, conditional correlations,conditional standard deviations and conditional volatility of portfolio policies contain a prin-cipal component that relates to time varying risk aversion. The same mechanism is present,but has not been studied before, in any consumption-based variant of the heterogeneous riskaversion model in Dumas (1989).

We solve the model with several dividend paying stocks or trees, Cochrane et al. (2008)and Martin (2011). Parametrized examples match conditional stock market return corre-lations. Correlations are asymmetric, high in bad states of the economy but low in good

that are jointly affected by an unobservable global business cycle factor. Time variation in correlationsof asset returns arises from the learning activity of the representative investor. Another related work isBuraschi et al. (2010), they study a Lucas orchard with heterogeneous beliefs about future earnings withmultiple consumption goods. Their counter-cyclical disagreement factor explains volatility and correlationrisk premium in the cross-section and time-series of individual stocks and portfolios of stocks.

4The model, or any continuous time model without frictions, has no implications for the “level” of trade.5Papers addressing asset pricing with preference heterogeneity include Bhamra and Uppal (2009), Bhamra

and Uppal (2011), Chan and Kogan (2002), Cvitanic et al. (2012), Dumas (1989), Garleanu and Panageas(2010), Longstaff and Wang (2008), Wang (1996), Weinbaum (2009), and Zapatero and Xiouros (2010). Wenote that all mentioned papers study an economy with only one risky security.

6Lamoureux and Lastrapes (1990) argue that volume explains standard deviations. Gallant et al. (1992)show that trading volume has a positive relation with absolute volatility changes. Goetzmann et al. (2005),Longin and Solnik (1995), and Moskowitz (2003) argue that correlations or covariances and standard devia-tions move together.

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states of nature. This feature of the model is consistent with Longin and Solnik (2001), whoargue that correlations increase in bear markets, but not in bull markets.7 Our calibrationsmatch the average change in industry and market correlations from booms to recessions. Thecalibrations also match the cyclical variation in the market return, in standard deviations ofthe market return, in average industry returns and in average industry standard deviations.

To test the model, we examine industry stock market correlations, expected excess re-turns, and standard deviations and quadratic variations of industry turnover and find positiverelations. Further, the four series show, as in the model, a negative relation with relativeconsumption. Our tests show that calibrated relative consumption explains the first princi-pal component of industry stock market correlations, expected excess returns, and standarddeviations and quadratic variations of industry turnover with an R-squared of up to fortypercent. To our knowledge, such a relation between these four sets of financial time serieshas not been described yet.

2 The Economy

This section introduces a continuous-time exchange economy defined on the time span [0, T ],in which N risky securities and one riskless security are traded.8

Risky securities pay dividends, δ, in the form of a single consumption good. The dividendsfollow geometric Brownian motions

dδi(t) = δi(t)(µδidt+ σ>δidW (t)

)(1)

with drift rates µδi and diffusion coefficients σδi ∈ RN , where δi(0) > 0, > denotes thetranspose, and i = 1, ..., N . Let the positive definite N ×N matrix σδ contain the diffusionterms of each dividend process. In similar manner, the N × 1 vector µδ contains the driftterms of each dividend process. The N -dimensional Brownian motion, W , defined on afiltered probability space (Ω,F , P, Ft) completes the description of dividend dynamics.9

7See Bollerslev et al. (1988), Erb et al. (1994), Ledoit et al. (2003), Moskowitz (2003) and Goetzmannet al. (2005) for evidence on time variation in return correlations and especially for correlations moving overthe business cycle. See also Ang and Chen (2001) for other related empirical work on correlations.

8Our setup allows for finite and infinite horizon economies (provided that additional conditions are placedon the primitives of the economy).

9The probability space is defined over the time horizon [0, T ], where Ω is the state space, F denotesthe σ-algebra, P represents the probability measure, and the information structure F(.) is generated by thenatural filtration with FT = F . In the remainder, all random variables appearing in equalities or inequalitiesare in the almost surely sense relative to P .

4

Aggregate consumption is the sum of the N dividend streams, i.e.,

C(t) =N∑i=1

δi(t). (2)

To make a distinction in our calibrations between aggregate consumption and dividends, weemploy the N -th dividend stream to help replicate the time-series properties of aggregateconsumption.

The dynamics of aggregate consumption are given by

dC(t) = C(t)(µC(t)dt+ σC(t)>dW (t)

)(3)

whereµC(t) = sδ(t)

>µδ and σC(t) = σ>δ sδ(t) (4)

and dividend shares, which are state variables, are given by

sδ(t) = (sδ1(t), sδ2(t), ..., sδN (t))> =

(δ1(t)

C(t),δ2(t)

C(t), ...,

δN(t)

C(t)

)>. (5)

By Ito’s lemma, dividend share processes follow

dsδi(t) = sδi(t)((µsi(t)− σsi(t)>σC(t)

)dt+ σsi(t)

>dW (t))

(6)

whereµsi(t) = µδi − sδ(t)>µδ and σsi(t) = σδi − σ>δ sδ(t). (7)

2.1 Financial Markets

Stock price processes follow

dSi(t) + δi(t)dt = Si(t)(µi(t)dt+ σi(t)

>dW (t))

(8)

with boundary conditions Si(T ) = 0. The locally risk-free asset, B(t), a zero net supplycontract, yields an instantaneous rate of return equal to r(t). The price of B(t) follows

dB(t) = r(t)B(t)dt (9)

with B(0) = 1.

5

2.2 Investors

Preferences are described by ratio habits, Abel (1990) and Chan and Kogan (2002),

Uj (C,X) = E0

[∫ T

0

e−ρtuj(Cj(t), X(t))dt

](10)

where ρ > 0, u represents the instantaneous utility function, C stands for the consumptionrate, and X denotes the external economy-wide living standard or habit.

The instantaneous utility function is given by

uj (Cj(t), X(t)) =1

1− γj

(Cj(t)

X(t)

)1−γj(11)

where γ measures the local curvature of the utility function, i.e., the relative risk aversionparameter. Investors have either low risk aversion, j = L, or high risk aversion, j = H. Theaverage historical standard of living evolves accordingly to

x(t) = x(0)e−λt + λ

∫ t

0

e−λ(t−u)c(u)du (12)

where x(t) = log(X(t)) and c(t) = log(C(t)), see Chan and Kogan (2002). The parameter λgoverns the dependency of the living standard, x(t), on past aggregate consumption.

In this setting, relative consumption, ω(t) = c(t) − x(t), instead of aggregate consump-tion, serves as another state variable besides the dividend shares. By Ito’s lemma, relativeconsumption follows a mean-reverting process

dω(t) = λ (ω(t)− ω(t)) dt+ σC(t)>dW (t) (13)

with long-run mean

ω(t) =µC(t)− 1

2σC(t)>σC(t)

λ. (14)

We emphasize that in states of nature with current consumption in excess (short) ofrecent past consumption, relative consumption is in excess (falls short) of its long-run mean.The pro-cyclical characteristics of ω(t) allow to define recessions (booms) in our calibrationsthat are analogous, in a probabilistic sense, to recessions (booms) in the data.

2.3 Equilibrium

Conditional on endowments and preferences, equilibrium is a collection of allocations andprices such that individuals’ consumption profiles are optimal and markets clear. We assume

6

complete markets and solve for the central planner problem or aggregate investor in stateby state and time by time form10

u(C(t), X(t), t) = maxCL(t),CH(t)

ae−ρt

1

1− γL

(CL(t)

X(t)

)1−γL+ (1− a) e−ρt

1

1− γH

(CH(t)

X(t)

)1−γH

s.t. CL(t) + CH(t) = C(t). (15)

Heterogeneous investors optimally share consumption risk. As shown in the followingproposition, the shape of the sharing rule depends on the degree of preference heterogeneityand the weight on investors in the objective function of the aggregate investor.

Proposition 1. Optimal consumption allocations are

CL(t) = f(t)C(t) and CH(t) = (1− f(t))C(t) (16)

where f is defined through

f(t) =

(a

1− a

) 1γL

e

(γHγL−1

)ω(t)

(1− f(t))γHγL (17)

and where a denotes the weight on investor type L in the objective of the aggregate investor.

It is apparent from Proposition 1 that the consumption share, f , is only a function ofthe state variable ω (or of aggregate consumption if habits are turned off). Moreover, theconsumption share converges to zero when ω approaches minus infinity and to one when itapproaches infinity. Therefore, the less risk averse investor, L, dominates in very good statesof nature, while the more risk averse investor, H, dominates in bad states of the economy.

The next proposition defines aggregate risk aversion as the consumption share weightedaverage of individual risk aversion.

Proposition 2. The coefficient of relative risk aversion for the aggregate investor is

R (t) =1

f(t) 1γL

+ (1− f(t)) 1γH

; (18)

the relative prudence of the aggregate investor is given by

P(t) = (1 + γL)

(R(t)

γL

)2

f(t) + (1 + γH)

(R(t)

γH

)2

(1− f(t)). (19)

10A sufficient condition for the market to be complete is that the stock price diffusion matrix is invertible foralmost all states and times. We verify numerically that the market is complete for our choice of parameters.For general results on completeness in continuous time economies see Anderson and Raimondo (2008).

7

The above proposition shows that aggregate risk aversion is bounded in between therisk aversion of the two types of investors. It also shows that the coefficient of relative riskaversion is countercyclical, it is high in bad states of nature and low in good states of nature.Aggregate relative prudence, however, is not bounded in between the prudence of the twotypes of investors inhabiting the economy.11 It can be shown that the diffusion coefficient ofaggregate risk aversion is R(t) (1 +R(t)− P(t))σC(t). We highlight that aggregate relativeprudence considerably drives the volatility of aggregate risk aversion. In particular, if theaggregate relative prudence equals 1+R(t), i.e., the relative prudence with standard CRRA,then risk aversion is constant.12

Proposition 3. Investors’ consumption dynamics evolve according to

dCj(t) = Cj(t)(µCj(t)dt+ σCj(t)

>dW (t))

(20)

where

µCj(t) =

(R(t)

γj

)µC(t) +

(1− R(t)

γj

)λω(t)

+1

2

[(1 + γj)

(R(t)

γj

)− P(t)

](R(t)

γj

)σC(t)>σC(t), (21)

σCj(t) =

(R(t)

γj

)σC(t). (22)

From Proposition 3, we see that diffusion coefficients of individual consumption growthdepend on the ratio of aggregate relative risk aversion to individual relative risk aversion.As aggregate risk aversion is bounded in between the risk aversion of the two investor types,investors with low risk aversion have higher consumption volatility than investors with highrisk aversion. Yet, investors optimally share risk such that marginal utilities equate stateby state and time by time. Because investors with high risk aversion have higher curvatureof the utility function than investors with low risk aversion, marginal utilities equate only ifvolatility of consumption is higher for investors with low risk aversion.

The fourth proposition shows that equilibrium quantities depend directly, or indirectlyvia aggregate risk aversion, on consumptions shares and relative consumption.

11See Wang (1996) for a discussion of the consequences of this result for the risk-free interest rate.12Aggregate relative prudence approaches relative prudence in a CRRA economy when the consumption

share approaches zero or one. For other values of the consumption share, the volatility of the aggregaterelative risk aversion is non-zero.

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Proposition 4. In equilibrium, the state price density is given by

ξ(t) = e−ρt(f(t)

f(0)

)−γL (C(t)

C(0)

)−1e(1−γL)(ω(t)−ω(0)) (23)

and the risk-free rate is

r(t) = ρ+R (t)µC(t) + (1−R(t))λω(t)− 1

2R (t)P(t)σC(t)>σC(t). (24)

Market prices of risk are given by

θ(t) = R (t)σC(t). (25)

Key quantities in the dynamics of stock return correlations, volatilities and expectedexcess returns are the market prices of risk. From the proposition, we see that marketprices of risk are determined by the product of aggregate risk aversion (R) and aggregateconsumption risk (σC). Aggregate risk aversion depends only on the state variable ω (oraggregate consumption) via optimal consumption allocation, which uniquely identifies goodversus bad times. Only variations in optimal consumption allocations generate excess orendogenous stock return correlations and volatilities. All endogenous variations in the marketprices of risk are captured by R. Finally, the amount of risk in the economy, as measuredby σC , depends only on dividend shares and not on the state of the economy, ω.

In similar manner, stock price diffusion coefficients depend on aggregate risk aversiontogether with innovations in aggregate consumption risk embedded in the habit process(first term), innovations between current and future market prices of risk (second term), andfundamentals (third term), i.e., dividend diffusion coefficients.

Proposition 5. In equilibrium, the stock price diffusion coefficients are

σi(t) =Et

[∫ Ttξ(u)δi(u)

([R(u)− 1]λ

∫ ute−λ(t−v)σC(v)dv

)du]

Et

[∫ Ttξ(u)δi(u)du

]+Et

[∫ Ttξ(u)δi(u) (θ(t)− θ(u)) du

]Et

[∫ Ttξ(u)δi(u)du

] + σδi . (26)

The first two terms of Proposition 5 can be interpreted as discount factor news, related tohabit and market prices of risk, respectively, while the last term represents cash flow news.

The next proposition presents optimal wealth and portfolio policies including hedging

9

terms13 that insure against shifts in the investment opportunity set.

Proposition 6. Equilibrium wealth allocations, Y , and portfolio policies, π, are

Yj(t) =1

ξ(t)Et

[∫ T

t

ξ(u)Cj(u)du

]and πj(t) =

(σ(t)>

)−1 [θ(t)Yj(t) +

ψj(t)

ξ(t)

](27)

where

ψj(t) =

(1

γj− 1

)Et

[∫ T

t

ξ(u)Cj(u)R(u)

[σC(u)− λ

∫ u

t

e−λ(t−v)σC(v)dv

]du

]. (28)

The above proposition illustrates that investors’ hedging terms depend on realizations ofaggregate risk aversion and aggregate consumption volatility in the future. Hedging termsalso relate to habit via its dependency on the path of aggregate consumption volatility (theintegral between t and u).14

2.4 Correlations, Returns, Volatility, and the Quadratic Variation

of Portfolio Policies

This subsection introduces the main endogenous economic variables that we study in themodel and the data: Instantaneous equity returns, dR(t), evolve according to

dRi(t) =dSi(t) + δi(t)dt

Si(t)= µi(t)dt+ σi(t)

>dW (t). (29)

The conditional variance, of security i’s instantaneous return, is defined as follows

V art(dRi(t)) = ‖σi(t)‖2 dt where ‖σi(t)‖ =√σi(t)>σi(t). (30)

The conditional covariance between security i and k, with k = 1, .., N , is given by

Covt (dRi(t), dRk(t)) = σi(t)>σk(t)dt. (31)

Then, we calculate the conditional correlation15 between security i and security k as

Corrt (dRi(t), dRk(t)) =σi(t)

>σk(t)

‖σi(t)‖ ‖σk(t)‖. (32)

13See Cox and Huang (1989) and Detemple et al. (2003) and the references therein.14In Detemple et al. (2003) the interest rate and market prices of risk drive the hedging terms. We can

re-write Proposition 6 in terms of these quantities. However, in equilibrium the interest rate and marketprices of risk depend on aggregate risk aversion, prudence and consumption volatility.

15In the remainder, we use the terms “conditional correlation” and “correlation” interchangeably.

10

The trading activity of investors separate this economy from other economies that modelstock market return correlations with a representative investor. Since trading volume orturnover in continuous-time economies is infinite, we employ the quadratic variation orvolatility of portfolio policies as a measure of trading intensity.16

In general form, the “dollar” denominated dynamics of portfolio policies follow

dπj(t) = [.] dt+ σπj(t)>dW (t) (33)

with drift and diffusion terms obtainable from Equations 27-28. The fraction of stock i heldby investor j evolves according to

d

(πi,j(t)

Si(t)

)=

(πi,j(t)

Si(t)

)([.] dt+ σπi,j

Si

(t)>dW (t)

)(34)

whereσπi,j

Si

(t) = σπi,j(t)− σSi(t). (35)

Market clearing implies thatπi,L(t)

Si(t)+πi,H(t)

Si(t)= 1. (36)

Applying Ito’s lemma to both sides of Equation 36 leads to

πi,L(t)

Si(t)σπi,L

Si

(t) +πi,H(t)

Si(t)σπi,H

Si

(t) = 0. (37)

From Equation 37, we see that in an economy with two types of investors πi,L(t)

Si(t)σπi,L

Si

(t) fullycaptures the degree of trading intensity in stock i. Hence, we use the relative quadraticvariation of portfolio policies

RQVi(t) =

√(πi,L(t)

Si(t)

)2

σπi,LSi

(t)>σπi,LSi

(t) (38)

to measure the second moment of trade in stock i or in any combination of stocks, i.e., aportfolio or an index. Appendix A contains additional details of Equation 38.

We note that changing the above asset structure will change the level of RQV . However,as long as assets or trees pay out dividends and the economy stays dynamically completethen the cyclical dynamics of RQV are not affected by the asset structure.

16Grossman and Zhou (1996) and Longstaff and Wang (2008), among others, also employ the quadraticvariation of portfolio policies to measure trading intensity.

11

3 Correlations with Two Trees

This section presents illustrative numerical examples of the effects of risk aversion and het-erogeneity in risk aversion on correlations in a series of economies with two trees.17

We solve the economies through Monte Carlo simulations. Details of the simulationsare found in Appendix B. Our plots show correlations as function of the share of the firstdividend process and the state of the economy. For the case without habit formation, weidentify the state of the economy with the consumption share of L investors, f . Note thata low value of f indicates low aggregate consumption (Dumas (1989)). With ratio habitpreferences, the state of the economy is identified through relative consumption, w.

Each economy contains two i.i.d. dividend processes or trees with µδ = 0.02 and σδ =

0.05.18 We set starting values for dividends to 1, the maturity of the economy to 50 years,the weight on L investors in the aggregate investor objective to 0.5,19 the rate for the timepreference of both investor types to 1% while the habit persistence, λ, is set at 0.1.

3.1 Homogeneous Preferences

Consider an economy populated by a representative investor with (i) logarithmic utility, (ii)power utility with a risk aversion coefficient of 3, or (iii) power utility with a risk aversioncoefficient of 10. For each economy, we study two cases: with and without habit formation.

The three top plots in Figure 4 show the correlation without habit formation. In the topleft plot, the aggregate investor is logarithmic. Although dividend processes are uncorrelated,stock returns show positive correlations with a value up to 0.029. Apparently, the correlationreaches its maximum when the dividend processes contribute 50% to aggregate consumption.

The top middle and top right plots show the effect of risk aversion on correlations. Holdingall other values constant, correlations are higher the higher is aggregate risk aversion. Forexample, correlations reach 0.05 in the economy with a risk aversion coefficient of 3 but reach0.086 in the economy with a risk aversion of 10.20

The top right plot in Figure 4 suggests that correlations are bimodal. We note that17The Internet-Appendix presents examples for the influence of other fundamentals on correlations.18Annualized consumption expenditure grew by an average rate of 2 percent (continuously compounded)

with a 3.5 percent standard deviation between 1890 and 2007 (Source: Robert Shiller’s web page). Two i.i.d.dividend streams with µδ = 0.02 and σδ = 0.05 roughly match these values.

19Heterogeneous investor models usually set weights on investors in the aggregate utility function. Chang-ing the risk aversion of one of the investors while keeping the weight on the investors constant implies thatendowments in the market portfolio change. Keeping endowments constant slightly alters the shape of cor-relations by moving the location of the peak. However, the maximum correlation in a particular economy isnot affected by changing endowments or weights.

20Cochrane et al. (2008) consider a two stock economy with a logarithmic investor. Their Figure 7 isanalogous to the three plots in the top row of Figure 4.

12

this particular feature of the correlation function is not due to the heightened risk aversion.Even the logarithmic economy exhibits this feature, although we cannot detect it in the plotsimply because the drop in correlations at the 50% dividend share is tiny.21

What is the intuition for positive correlations? Consider an economy with two uncor-related dividends. The net effect of a positive shock to dividend 1 on the price and returnof stock 1 is positive. The return can be decomposed into a positive dividend effect and asmaller but negative discounting effect. The discounting effect results from an increase inthe market price of risk. The market price or risk increases as a consequence of increasedexposure to shocks of dividend 1, i.e., aggregate consumption depends more on dividend 1risk than prior to the shock. For the second asset there is only a strictly positive discountingeffect as its associated market price of risk decreases. The same logic goes through withmany assets. Hence, all assets experience positive returns after a positive dividend shock todividend 1, which explains why stocks with uncorrelated dividends show correlated returns.

More formally, an application of Equation 25 to the economies in Figure 4 leads to thefollowing expression for the market price of risk22

θ(t) = γσC(t) (39)

whereθ1(t) = γsδ1(t)σδ1 and θ2(t) = γ (1− sδ1(t))σδ2 . (40)

A positive shock to dividend 1 increases its share, sδ1(t), and thus its own market price ofrisk but decreases the market price of risk of the second dividend or other dividends.

Figure 5 shows that the discounting effect for all returns, from one dividend shock,increases in risk aversion. What is the intuition for this —risk aversion level— effect? First,the larger the risk aversion, the larger is the negative discounting effect, and the smaller isthe net return of the asset which experiences a positive dividend shock, first part of Equation40. Second, the larger the risk aversion, the larger is the positive discounting effect, and thelarger is the return on the second asset (other assets), second part of Equation 40. Hence, thehigher the risk aversion, the higher is the correlation between assets returns. We conjecturethat correlations converge to one when risk aversion approaches infinity.23

The bottom plots of Figure 4 show correlations when the habit process is turned on. Forthe logarithmic case, however, equilibrium is independent of the external standard of living.24

21A brief explanation for the bimodal correlations is found in the Internet-Appendix.22In general, θi denotes the market price of risk for Brownian motion i, and θ = (θ1, ..., θN ) stands for the

N -dimensional Brownian market price of risk.23Numerical examples support this conjecture. However, proving it goes beyond the goals of this paper.24With log utility we have u(C,X) = log

(CX

), thus the marginal utility (which is proportional to the state

13

Therefore, the correlation is identical to the correlation with logarithmic investor withouthabit formation, bottom left plot versus top left plot. The bottom middle and bottom rightplots show the effect of the habit process. The procyclical variation in the correlations —low correlations when the habit level is low and vice versa— is inconsistent with the data.While ratio habit preferences with homogeneous investors cannot generate countercyclicalvariations in correlations and other asset pricing moments, nevertheless, correlations increasein the level of risk aversion plus habit significantly increases correlations. One last observationwe make is that, as in the case without habit formation, correlations reach a local minimumwhen the dividend share of dividend 1 equals 0.5.

3.2 Heterogeneous Preferences

We now consider the following risk aversion pairs: log−3 and log−10 but keep all otherparameters constant. The plots in Figure 6 show the correlation from these heterogeneouspreference economies. We see from the top plots, without habits, that stock market corre-lation increase in the bad state of nature, that is, correlations are the larger the lower therealization of aggregate dividends is. From the top left plot, we learn that the maximumcorrelation is higher in economies where the difference in risk aversion is high. This effect oncorrelations is noteworthy, as correlations in the log−3 economy range from 0.05 to slightlyabove 0.2 while in the log−10 economy correlations reach almost 0.8.

Investors with high risk aversion hold a larger fraction of the economy when aggregateconsumption declines. Since aggregate risk aversion is consumption share weighted, it in-creases after a decline in aggregate consumption, top middle-left plot in Figure 7. Heightenedrisk aversion increases correlations due to the risk aversion level effect. Aggregate risk aver-sion, however, cannot explain the entire increase in correlations. For instance, correlationsin the top middle plot of Figure 4 with aggregate risk aversion set at 3 reach 0.05, whilecorrelations in the top left plot of Figure 6 with risk aversion pair set at log−3 reach 0.2.

Why are correlations significantly larger in a log−3 economy than the correlations ob-tained in a homogeneous investor economy populated by an investor with risk aversion of3? With heterogeneous investors, the marginal utility of the aggregate investor varies due tochanges in aggregate consumption and due to changes in consumption shares of individualinvestors. When small changes in consumption imply large changes in consumption shares,then endogenous variations are high. This effect on correlations, and other endogenousquantities, is driven by the slope of the sharing rule. In economies with high difference inrisk aversion, the maximum slope of the sharing rule is steeper than in economies with low

price density) is u′(C,X) = 1C and does not depend on the habit level X.

14

difference in risk aversion. A steeper sharing rule implies higher volatility of aggregate riskaversion, which again implies more volatile market prices of risk. As the risk aversion of therepresentative investor is a common factor in all market prices of risk the correlation canreach higher values in economies with more heterogeneity in risk aversion. Hence, we callthis the risk aversion volatility effect.

The bottom left plot of Figure 7 shows correlations between stock 1 and stock 2 obtainedin an economy with risk aversion pairs log−3. Figure 7 also shows correlations between stock1 and stock 2 obtained in an economy with homogeneous risk aversion in which aggregate riskaversion corresponds to aggregate risk aversion in the heterogeneous investor economy. Thedifference in correlations between the homogeneous economy and the heterogeneous economycorresponds to endogenous variations. We see this from the top plots in Figure 7. The topleft plot shows the consumption sharing rule between investors, while the middle-left plotshows the slope of the consumption sharing rule. We note that the shapes of the slope ofthe sharing rule and the difference between homogeneous and heterogeneous correlations areidentical. Therefore, endogenous variations in equilibrium quantities, which we obtain byintroducing heterogeneous investors, lead to the increase in correlations above the correlationin a corresponding homogeneous investor economy.25

With ratio habit preferences, bottom plots of Figure 6, correlations range from below0.4 to approximately 0.6 in the log−3 economy and from 0.5 to almost 1.0 in the log−10

economy. Hence, habits significantly increase correlations.

3.3 On the Relation between Stock Return Correlations, Standard

Deviations, Expected Returns and Volatility of Turnover

Empirical literature on correlations, on the one hand, argues that correlations and standarddeviations move jointly over the business cycle.26 On the other hand, Longin and Solnik(2001) suggest that the results in the literature can be spurious. Therefore, we ask fortestable implications of the model for stock market return correlations and other quantities.

25For low levels of aggregate consumption, correlations drop in the bad state of nature, bottom left plotof Figure 7. Our calibrations, in the next section, indirectly address whether we should expect a fall incorrelations in extreme business cycle troughs.

26Erb et al. (1994) and Ledoit et al. (2003) find that correlations are higher during recessions. Longin andSolnik (1995) find that correlations rise in periods of high volatility. Moskowitz (2003) shows that covariancesand volatilities relate to NBER recessions. Goetzmann et al. (2005) find that correlations and volatilities peakduring market crashes. Fama and French (1989) and Ferson and Harvey (1991) show that expected excessreturns increase during economic contractions and peak near business cycle troughs. Harrison and Zhang(1999) and Campbell and Diebold (2009) also conclude that expected excess returns are countercyclical.Schwert (1989) argues that stock market volatility is higher in recessions than in booms. Hamilton and Lin(1996) argue that stock market volatility and the business cycle co-move.

15

The plots in Figure 7 show the fraction of aggregate consumption consumed by H in-vestors, the slope of the sharing rule, aggregate risk aversion, the derivative of risk aversionwith respect to consumption, the conditional correlation between stock 1 and stock 2, theexpected excess return on the market portfolio, the standard deviation of the market andrelative quadratic variation, RQV , of the most risk averse investors’ portfolio. We see fromthe top left plot that the consumption share of the investors with high risk aversion decreasesin aggregate consumption. The slope of the consumption sharing rule, middle-left top plot,reflects essentially the intensity of consumption shifts across investors per unit of aggregateconsumption. Note that the risk aversion of the representative investor (top middle-rightplot) and the sensitivity of the representative investor’s risk aversion to aggregate consump-tion shocks are similar to the sharing rule and the slope of the sharing rule. That is, when theslope of the sharing rule is high, then the volatility of aggregate risk aversion is high. Further,stock return correlations (bottom left plot), expected excess returns of the market portfolio(bottom middle-left plot), return volatilities of the market portfolio (bottom middle-rightplot), and relative quadratic variation of portfolio policies (bottom right plot) all peak whenthe volatility of risk aversion peaks. This can be understood in the following way; whenthe slope of the sharing rule is steep, then a small shock to aggregate consumption causesconsumption allocations to shift by a large amount. The shift in consumption allocationsleads to large changes in the risk aversion of the representative investor, and consequently tolarge changes in the market prices of risk. Importantly, the volatility of the market prices ofrisk is high in times when the slope of the sharing rule is steep. The high volatility of marketprices of risk leads to higher volatility of the equilibrium discount rates, and thus to morevolatile stock returns. Note that all market prices of risk, as given by Equation 25, depend onaggregate risk aversion, and thus all discount factors become more volatile (high stock returnvolatility) and more correlated as all move with aggregate risk aversion, which is a commonfactor. Expected excess returns increase due to higher volatility of stock returns (quantityof risk) and higher price of risk, which is driven by higher risk aversion of the representativeinvestor. The bottom left plot shows the relative quadratic variation. We see that relativequadratic variation peaks jointly with the slope of the sharing rule, volatility of aggregaterisk aversion, stock return volatilities and correlations. Thus, the trading intensity is high intimes when the slope of the sharing rule is steep. The plots in Figure 8 show these relationswith ratio habit preferences for the correlation between stock 1 and stock 2, the expectedexcess return on the market portfolio, the standard deviation of the market portfolio andthe relative quadratic variation of stock 1.

16

4 Calibrations

In this section we calibrate the model economy to address the following questions. First,can the model match the average level of correlations as well as other standard asset pricingmoments? Second, and more importantly, can the model match the change in correlationsas well as the change in other asset pricing moments over the business cycle? Further,our calibration addresses the question whether the intuition developed in Section 3 carriesthrough to settings with many risky securities. Finally, one might wonder whether the riskaversion volatility effect matters when shocks to dividends do not affect consumption much.

We construct our sample from the CRSP files for the period January 1927 to December2009. We employ all firms, surviving and non-surviving, that appear on CRSP. Industryportfolios are derived from Kenneth French’s industry classification. For each industry port-folio, we calculate aggregate dividends and returns. Dividends are adjusted for inflationand population growth. Stock returns are adjusted for realized inflation. For aggregate percapita real consumption, we employ data from Robert Shiller’s website. The risk-free rate isobtained from Kenneth French’s website.

As we are interested in the average return correlation between the ten industry portfolios,it is convenient to set homogeneous parameters for industry dividends.27 To accommodatethe empirical regularity that aggregate dividend growth is more volatile than, and imper-fectly correlated with, per capita real consumption growth, we employ another dividendstream in addition to the ten industries. We set the dividend share of the eleventh tree tomatch the average dividend/consumption ratio over the period January 1927 to December2009.28 Table 1 contains the parameters for consumption and dividend processes in the datatogether with the corresponding values of the calibrations. To illustrate the role played bypreference heterogeneity in explaining the business cycle properties of stock return correla-tions and other asset pricing moments, we consider two different calibrations of the model:One calibration with homogeneous risk aversion and one with heterogeneous risk aversion.Our choices for the preference parameters are reported in Table 1.

We choose the persistence of the habit level, λ, to match the persistence of the price-dividend ratio in the data. The risk aversion pair, 30 and log, together with the utility weight,a, are set to match unconditional asset pricing moments and changes of these moments over

27Formally, we assume Et[dδiδi

]= Et

[dδjδj

], V art

[dδiδi

]= V art

[dδjδj

],Covt

[dδiδi,dδjδj

]V art

[dδiδi

] = ρδ, where i 6= j in

Covt [.], for i, j = 1, .., 10 and δ1(0) = δ2(0) = ... = δ10(0). This simplification reduces the number of samplepaths required in the Monte Carlo simulations as we can exploit symmetry. A calibration based on industrydividend streams that match industry portfolio dividends from the CRSP/Compustat files are available inthe Internet-Appendix.

28Expected growth of the eleventh tree is set at 0.019970 while its standard deviation is set at 0.025846.

17

the business cycle. The homogeneous risk aversion economy cannot jointly match expectedexcess returns and average industry return correlations. We match correlations at the costof having too low returns. The subjective discount factor, ρ, is chosen to match the level ofthe risk-free rate.

Table 2 shows unconditional asset pricing moments of the calibrated models.29 We notethat the heterogeneous and the homogeneous investor economies reproduce the correlationand standard deviation of average industry returns. The level of the risk-free rate is slightlyabove that in the data for both economies, while the model volatility of the risk-free rate is toohigh, especially for the homogeneous investor economy. The heterogeneous investor economygenerates higher excess returns than the corresponding homogeneous investor economy, whileit still matches the other asset return moments. The autocorrelation of the price-dividendratio is matched in both economies. Overall, the performance of the homogeneous modelis comparable to that of the heterogeneous model in terms of replicating unconditionalmoments.

Next we study conditional asset pricing moments. We compute the average industryreturn correlation, standard deviation, risk free-rate and expected excess return conditionalon the NBER business cycle indicator.30 To compare the model to the data, we require acorresponding measure of business cycles in the model. We calculate the unconditional re-cession probability based on the NBER business cycle dates in the data, i.e., over the sampleperiod January 1927 to December 2009 with a total of 996 months. Based on 211 recessionmonths, the unconditional recession probability is 21.18%. To find a corresponding recessionprobability in the model, we simulate the distribution of ω from the calibrated model to backout the ω that correspond to the unconditional recession probability. In our calibration, thiscorresponds to a value of 0.1046 for ω. Hence, we compute conditional asset pricing momentsin recessions by integrating over all ω’s less than 0.1046. Conditional asset pricing momentsin booms make up the remaining distribution of ω’s. Table 3 shows the results from thisexercise. We see that the homogeneous investor economy fails to replicate the changes incorrelations, standard deviations and excess returns over the business cycle. This followsfrom the fact that in the homogeneous investor economy the market price of risk is constantand volatility is pro-cyclical. The heterogeneous investor economy is capable of simultane-ously reproducing changes in correlations, standard deviations and expected excess returns

29Strictly speaking, we compute moments conditional on dividend shares. However, dividend shares arenon-stationary with log-normal dividends. Yet, simulating the model over a 100 years period, allowingdividend shares to evolve over time, produces comparable results. For a multiple tree economy with stationarydividend shares see Menzly et al. (2004).

30Expected excess returns at time t are measured as realized average return from t to t + 3. Results arerobust with one or five years of data ahead, although the difference between booms and recession shrink atthe five year horizon.

18

over the business cycle. In particular, one lesson we learn from the calibration exercise isthat the model matches simultaneously unconditional and business cycle moments. We notethat the heterogeneous investor economy produces changes in the risk-free rate over businesscycles that appear large relative to the data. Yet, the homogeneous investor economy pro-duces an even larger countercyclical change in the risk-free rate. More importantly, in theheterogeneous investor economy the volatility of the risk-free rate evolves countercyclicallyand the data support this prediction.31

Table 3 shows that the heterogeneous risk aversion model produces counter-cyclical be-havior of correlations, standard deviations and expected returns. To further examine themodel’s ability to capture the dynamic properties of correlations, standard deviations andexpected returns, we simulate 100 paths of 1000 months of prices from the model. Next,using simulated prices, we estimate for each path a multivariate GARCH (DVEC(1,1)) andcompute 3-year ahead average returns. Similarly, we estimate a multivariate GARCH usingthe actual data. We run regressions of average correlations (Av. CORR), average 3-yearahead excess market returns (Av. EXR) and average standard deviations (Av. STDV) onrelative consumption, ω, for returns from the data and the simulation. To calculate relativeconsumption in the data, we employ consumption data from 1889 to 2009. We assume thatω is in its steady state in 1889. Then we back it out from the data using the Euler discretiza-tion of its dynamics. As the consumption data comes at annual frequency, we interpolate ωto get a monthly series, which is shown in Figure 2. Figure 2 also shows that aggregate riskaversion increases during NBER recessions. Further, model implied aggregate risk aversionstays below 10 over the period January 1927 to December 2009 except at the end of the GreatDepression. We note that changes in aggregate risk aversion, ∂R(t)/∂exp (ω(t)), mirror thedynamics of ω.

Table 4 shows the results from the data and model based regressions. All regressionsshow the expected negative sign with highly significant coefficient estimates. Next, adjustedR-squared are similar in the model and the data with the exception of average correlationsthat shows somewhat lower R-squared in the data. Slope coefficients for the data and themodel exhibit similar magnitudes for correlations and expected returns, but are lower forthe average standard deviation in the model data.32

In the model the price-dividend ratio increases in ω. This relation suggests that the31The GARCH(1,1) volatility of the risk-free rate shows the following correlations: with industry stock mar-

ket correlations (0.42), with expected excess returns (0.18), with standard deviations (0.41), with quadraticvariations of industry turnover (0.68), and relative consumption (-0.29).

32The corresponding results for quadratic variations of industry turnover, estimated from a GARCH(1,1)model based on log changes in turnover, are −0.6031 for the coefficient estimate with a Newey-West correctedt-statistics of −6.5560 and adjusted R-squared of 0.1946.

19

results in Table 4 can be replicated with the price-dividend ratio. These regressions areincluded in the Internet-Appendix.

To shed further light on the ability of the model to replicate changes in correlationsand standard deviations over the business cycle, we calculate model implied values usingactual consumption data. We feed the model with realized habit adjusted consumption overthe period 1927 to 2009. Again, it is convenient to keep dividend shares fixed over theentire period. The left plot of Figure 3 shows model implied correlation together with thedata.33 We learn that the model matches the level of the average industry return correlationover the period well. Importantly, the model replicates the high correlation levels during theearly thirties and the 2007-2009 recession. The correlation coefficient between model impliedaverage industry correlation and the data equals 0.4, which appears high given that we onlyemploy aggregate consumption data in the calibrated model. The right plot of Figure 3shows the model implied return standard deviation together with the standard deviation inthe data. Again, the model is successful in capturing the spike in volatility during the earlythirties and the increased volatility over 2007-2009. The correlation coefficient between themodel implied standard deviation and the data is 0.73.34

Finally, we compare the distribution of aggregate risk aversion and prudence from our twoinvestor setting to the case with a continuous distribution of investor types with a generalizedBeta distribution. The economy with a continuous distribution of risk aversion coefficientsyields aggregate risk aversion and prudence that are similar to the economy described inTables 1 - 3. Therefore, our simplified calibration with two investors can be generalized toa calibration with a continuous distribution of investor types. Results from this exercise arefound in the Internet-Appendix.

5 Empirics

In this section, we empirically assess the explanatory power of model implied relative con-sumption, ω, on the first principal component of industry correlations, excess returns, volatil-ity and the quadratic variation of turnover. Evaluating the performance of the model in this

33As relative consumption, ω, has annual frequency, we calculate the annual correlation as the within yearaverage correlation based on the monthly GARCH estimates. We follow the same procedure for the standarddeviation. Alternatively, we use monthly correlations together with interpolated relative consumption. Theresulting correlation between monthly model implied correlations and the data is 0.32. The estimationis robust to using exponentially weighted moving average (EWMA) to estimate correlations and standarddeviations, and the resulting correlation between the data and model using monthly frequency is 0.36.

34The correlation coefficient between model implied expected excess return of the market and the threeyear forward looking excess return of the market is 0.34. The corresponding correlation coefficients for therealized real risk-free rate and the price dividend ratio are 0.31 and 0.23, respectively.

20

way is important for several reasons. For example, above, we stress that the volatility oftrade constitutes an observable that provides information about investor heterogeneity. Inour calibrations, however, we do not study the quadratic variations of portfolio trade simplybecause it is difficult to compare the portfolio data from the model to real data.

Before we present the principal component analysis and the predictive regressions, webriefly summarize additional, untabulated, evidence in favor of the model. We calculate timevarying average pairwise industry correlations and average industry standard deviations byestimating a multivariate GARCH model, see Bollerslev et al. (1994). We use annualized3-year ahead continuously compounded average industry and market excess returns as proxyfor expected returns. 1-year ahead as well as 5-year ahead excess returns yield similar re-sults. As a proxy for the quadratic variations of portfolio policies in the model economy weestimate a GARCH(1,1) on the log-changes of industry turnover and the aggregate market.From Figure 1, which shows the average of the industry series, we see significant increases inthe time series in recessions or in proximity to recession periods. Averaged series show statis-tically significant positive relations, based on two-sided p-values at 1% level, with the NBERbusiness cycle indicator. Relative consumption from the heterogeneous investor economyshows a negative relation with the NBER business cycle indicator with significance at 1%level. Further, the series show positive relations with each other and negative relations withrelative consumption, consistent with a heterogeneous investor version of the model. Thesepositive (negative) relations are statistically significant at 1% level with two-sided p-values.

We calculate the first principal component of the forty-five correlation series (PCACORR), the ten series of 3-year ahead excess returns (PCA EXR), the ten series of standarddeviations (PCA STD) and the ten series of quadratic variations of turnover (PCA QV) sep-arately. To calculate the first principal component of all the series, we compute the averageof the four sets of series to reduce the impact of the forty-five correlation series and obtainfrom the averages the first principal component (PCA TOTAL). First, we regress the firstprincipal component of these four series onto relative consumption. Second, we regress thefirst principal component from all the series, PCA TOTAL, onto relative consumption.35

Table 5 shows the results from these principal component regressions. All regressioncoefficients show negative sign consistent with a heterogeneous investor version of the model.Further, all coefficient estimates for relative consumption shows highly significant Newey-West corrected t-statistics. The adjusted R-squared range from 14.06% to 40.95%. Ourresults regarding excess returns are essentially unchanged if we correct the nominal short

35The first principal component explains 54%, 86%, 87% and 71% of the variation in the 45 correlationcoefficients, 10 standard deviations, 10 quadratic variations of turnover and 10 three years ahead excessreturns. The first principal component explains 51% of the variation of the averages of the four series.

21

rate with expected inflation instead of realized inflation.36 Overall, we conclude that signs ofthe coefficients as well as the explanatory power of the regressions support our theory. Again,using the price-dividend ratio instead of ω leads to comparable results. These regressionsare shown in the Internet-Appendix.

Further empirical evidence that supports the model in this paper are found in theInternet-Appendix. In particular, we form portfolios sorted on size, book-to-market, andmomentum and show that ω explains Av. CORR, Av. EXR, and Av. STDV of these sorts.Finally, we perform standard in-sample and out-of-sample predictive regressions with rela-tive consumption that go beyond our Av. EXR regressions and find strong support for themodel.

6 Conclusions

We study stock market correlations in an equilibrium model. We identify two effects: First,correlations increase in (aggregate) risk aversion. Second, heterogeneity in risk aversionproduces excess correlation via endogenous variability in the state price density or aggregaterisk aversion. Endogenous variations in our examples and calibrations are at last one orderof magnitude larger than the stand alone effect of the level of risk aversion.

The model gives some advice for portfolio choice and risk management problems. In-vestors, portfolio managers and risk managers might ask, “Are stock return correlations sig-nificantly different over the business cycle?” The data and the model seem to suggest thatthe answer is “yes.” In fact, the model in this paper suggests something else, namely that theaffirmative answer to the above question is difficult to interpret in isolation. Yes, correla-tions and volatilities rise in recessions, however, recessions forecast high returns. Therefore,it is important to understand how correlations vary jointly with expected stock returns,volatilities and other potentially relevant time series.

In other words, the upshot of the model is that counter-cyclical variations in aggregate riskaversion or ratio habit, which are driven by variations in aggregate consumption, determinethe counter-cyclical variations of stock market correlations, expected excess returns, standarddeviations and of quadratic variations of turnover.

Our calibrations as well as our empirical investigations support the model predictions.This is despite the fact that measures of expected excess returns and turnover are noisy. Inaddition, there exist other trading motives than heterogeneity in risk aversion. Some of thesemotives, for example heterogeneity in beliefs or in time preferences, might at times movein opposite direction to the risk aversion trading motive. Other trading motives may even

36Relative consumption and the real short rate show statistically significant negative relation.

22

systematically move in opposite directions to the risk aversion trading motive. Even withthese obstacles, that bias against our model based regression specifications, this study showsthat heterogeneity in risk aversion is at least one of the important drivers of counter-cyclicaldynamics in financial time series.

23

A Quadratic Variation of Portfolio Policies

We compute the diffusion coefficients of portfolio policies from its dynamics given by

dπj(t) = µπj(t)dt+ σπj(t)dW (t). (A.1)

Below we ignore dt terms as we are only interested in the local volatility of portfolio policies:the N ×N matrix σπj . Recall optimal portfolio policies from Proposition 6:

πj(t) =(σ(t)>

)−1 [θ(t)Yj(t) +

ψj(t)

ξ(t)

]. (A.2)

From Equation A.2, the optimal portfolio policy is a function of stock price diffusion coef-ficients (σ), market prices of risk (θ), wealth (Yj), hedging terms (ψj) and the state pricedensity (ξ). Thus, if we know the dynamics of these quantities we can calculate σπj(t) bya standard application of Ito’s lemma to Equation A.1. The dynamics of ξ can be found inthe proof of Proposition 6 in the Internet-Appendix. Wealth dynamics are given by

dYj(t) =(r(t)Yj(t) + πj(t)

> (µ(t)− r(t)I)− Cj(t))dt+ πj(t)

>σ(t)dW (t). (A.3)

Hence, we have that σYj = πj(t)>σ(t). The dynamics of θ(t), σ(t) and ψj(t) are given by

dθ(t) = µθ(t)dt+ σθ(t)dW (t)

dσi(t) = µσi(t)dt+ σσi(t)dW (t)

dψj(t) = µψj(t)dt+ σψj(t)dW (t), (A.4)

where σθ, σσi and σψj areN×N matrices. Recall the market prices of risks: θ(t) = R(t)σC(t).Thus, we need to compute the dynamics of R(t) and σC(t). We obtain

dR(t) = d

(−uCC (C(t), X(t), t)

uC (C(t), X(t), t)C(t)

)= [.] dt+

((uCC (C(t), X(t), t)

uC (C(t), X(t), t)

)2

C(t)− uCC (C(t), X(t), t)

uC (C(t), X(t), t)− uCCC (C(t), X(t), t)

uC (C(t), X(t), t)C(t)

)×C(t)σ>CdW (t)

= [.] dt+R(t) (1 +R(t)− P(t))σ>CdW (t); (A.5)

24

dσC(t) = d(σ>δ sδ(t)

)= σ>δ dsδ(t)

= σ>δ Isδ(t) (νs(t)dt+ σs(t)dW (t))

= µσC (t)dt+ σσC (t)dW (t) (A.6)

whereµσC (t) = σ>δ Isδ(t)νs(t) and σσC (t) = σ>δ Isδ(t)σs(t) (A.7)

and where

dsδ(t) = Isδ(t) (νs(t)dt+ σs(t)dW (t)) with νs(t) =

µs1(t)− σs1(t)>σC(t)

.

.

µsN (t)− σsN (t)>σC(t)

, (A.8)

which is the compact form of Equations 6-7. Thus, by Ito’s lemma we have that

dθ(t) = d (R(t)σC(t))

= [.] dt+[R(t) (1 +R(t)− P(t))σC(t)σC(t)> +R(t)σσC (t)

]dW (t). (A.9)

From Proposition 5, the stock price diffusion coefficients are

σi(t) = θ(t) +Qi(t)

ξ(t)Si(t)+ σδi where Qi(t) = Et

[∫ T

t

ξ(u)δi(u)h(t, u)du

]. (A.10)

Note that we can write this as

Qi(t) = Et

[∫ T

0

ξ(u)δi(u)h(t, u)du

]−∫ t

0

ξ(u)δi(u)h(t, u)du

= QMi (t)−

∫ t

0

ξ(u)δi(u)h(t, u)du (A.11)

where QMi stands for the martingale part of Qi. Clark-Ocone’s theorem renders

dQMi (t) = σQi(t)dW (t) where σQi(t)

> = Et

[∫ T

0

Dt

(ξ(u)δi(u)h(t, u)>

)du

], (A.12)

25

in which σQi(t) is a N ×N matrix. Calculating the Malliavin derivative, leads to

Dt

(ξ(u)δi(u)h(t, u)>

)= δi(u)h(t, u)>Dtξ(u) + ξ(u)h(t, u)>Dtδi(u) + ξ(u)δi(u)Dth(t, u)>

= ξ(u)δi(u)h(t, u)h(t, u)> + ξ(u)δi(u)σδih(t, u)>

+ξ(u)δi(u)g(t, u). (A.13)

Using the above expressions and the following relation

d (ξ(t)Gi(t)) = ξ(t)Si(t) (σi(t)− θ(t))> dW (t) (A.14)

leads to

dσi(t) = [.] dt+(σθ(t) + σQi(t) + (θ(t)− σi(t)) (θ(t)− σi(t))>

)dW (t)

= [.] dt+ σσi(t)dW (t). (A.15)

Next we derive the diffusion coefficient of ψj. We have that

ψj(t) = Et

[∫ T

0

ξ(u)Cj(u) (h(t, u) +Hj(t, u)) du

]. (A.16)

Factoring out the martingale part of Equation A.16 and using the Clark-Ocone theoremyields

σψj(t)> = Et

[∫ T

0

Dt (ξ(u)Cj(u) (h(t, u) +Hj(t, u)))

]= Et

[∫ T

t

ξ(u)Cj(u)h(t, u)h(t, u)> + ξ(u)Cj(u)gj(t, u)

du

]+Et

[∫ T

t

ξ(u)Cj(u)h(t, u)Hj(t, u)> + ξ(u)Cj(u)Gj(t, u)

du

]. (A.17)

B Technical Details of Monte Carlo Simulations

Below, we describe the numerical procedures employed to solve for equilibrium quantities.The model is solved using Monte-Carlo simulations. State variables are simulated forward

using an Euler scheme with 10,000 sample paths and 1,000 time steps. We use antitheticsampling to reduce sampling variance. For each time step, we compute optimal allocationsof habit adjusted consumption between the two investors by solving the central plannerproblem. The sharing rule is solved by Newton’s method. Derivatives of optimal allocations

26

are computed using finite differences. Time integrals are calculated by using the trapezoidrule with 1,000 steps.

We benchmark our code in the following way: The two trees version of the model withlogarithmic utility function is available in closed form, Cochrane et al. (2008). Further, wesolve 2, 3 and 4 stock versions of our economy with quadrature methods. First, our threecodes (closed form, quadrature, Monte-Carlo) yield identical results in a two-stock economywhen preferences are logarithmic and investors do not exhibit habits. Second, our Monte-Carlo simulations coincide with the quadrature method for 2, 3 and 4 stock economies.Especially, Monte-Carlo simulations and the quadrature methods imply and yield the sameresults independently of the assumed pair of risk aversion and the choice of the utilityfunctions (standard power preferences versus power preferences with ratio habits).

Monte-Carlo errors are not crucial for our purposes. Nevertheless, we have checkedwhether model implied quantities such as the expected excess return, the risk-free rate orvolatilities vary substantially across runs. Overall, we find that model implied statistics areaccurately estimated by the Monte-Carlo simulations.

In a similar vein, all the plots reported in the paper are based on 1,000 sample paths.This is because one cannot make a distinction of our plots from plots based on a largernumber of runs.

27

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31

Table 1: Calibration — Parameters. This table summarizes moments of consumption,aggregate industry dividends, and average industry dividends as well as relations betweenconsumption, aggregate dividends, and average industry dividends for the period January1927 to December 2009 and corresponding moments from a heterogeneous and a homoge-neous investor model calibration. The table also shows the preference parameters employedin the heterogeneous and the homogeneous investor model calibration. A utility weight, a,of 0.02 is equivalent to a wealth share of 0.074 in the steady state.

Data ModelHeterogeneous Homogeneous

Consumption and Dividend MomentsMean consumption growth 0.020 0.020 0.020Standard deviation of consumption growth 0.030 0.030 0.030Mean aggregate dividend growth 0.024 0.024 0.024Standard deviation of aggregate dividend 0.082 0.082 0.082Consumption-aggregate dividend correlation 0.539 0.539 0.539Aggregate dividend-consumption ratio 0.030 0.030 0.030Average industry mean dividend growth 0.024 0.024 0.024Average standard deviation of industry dividends 0.133 0.148 0.148Average industry dividend correlation 0.229 0.229 0.229

Preference ParametersRisk aversion, high (γH) - 30 7Risk aversion, low (γL) - 1 7Utility weight (a) - 0.020 -Subjective discount factor (ρ) - 0.035 0.010Degree of habit history dependence (λ) - 0.130 0.130

32

Table 2: Calibration — Asset Pricing Moments. This table summarizes moments ofthe market portfolio, average industry portfolio, risk-free rate, and price-dividend ratio forthe period January 1927 to December 2009 and corresponding moments from a heteroge-neous and a homogeneous investor model calibration. Returns are annualized from monthlyfrequency. Model risk-free rate is r(t), see Equation 4, while the log-return of the marketportfolio is µM(t)− 1

2σ2M(t), see Equation 8, where M denotes the market portfolio. ACF(1)

denotes the first-order autocorrelation coefficient.Data Model

Heterogeneous HomogeneousExpected excess return of market 0.079 0.038 0.019Standard deviation of market 0.190 0.238 0.215Average industry correlation 0.719 0.744 0.730Risk-free rate 0.006 0.011 0.011Standard deviation of risk-free rate 0.019 0.038 0.045Average price-dividend ratio of market 30.0 39.1 69.4Standard deviation of log price-dividend ratio of market 0.412 0.364 0.315ACF(1) of log price-dividend ratio of market 0.874 0.871 0.877

33

Table 3: Calibration — Asset Pricing Moments over the Business Cycle. Thistable summarizes unconditional and conditional (booms and recessions) moments of threeyear ahead excess return and standard deviation of the market portfolio, average industryportfolio correlation, and risk-free rate for the period January 1927 to December 2009 andcorresponding moments from a heterogeneous and a homogeneous investor model calibration.Returns are annualized from monthly frequency. Model risk-free rate is r(t), see Equation4, while the log-return of the market portfolio is µM(t)− 1

2σ2M(t), see Equation 8, where M

denotes the market portfolio. In the data, recessions are defined by NBER recession dates.In the calibrated model, recessions have the same unconditional probability as in the data.

Data ModelHeterogeneous Homogeneous

Expected Excess Return of MarketAverage 0.079 0.038 0.019Boom 0.071 0.025 0.019Recession 0.110 0.081 0.019Recession minus boom 0.040 0.056 0.000

Standard Deviation of MarketAverage 0.190 0.238 0.215Boom 0.156 0.211 0.216Recession 0.280 0.335 0.211Recession minus boom 0.124 0.124 -0.004

Average Industry CorrelationAverage 0.719 0.744 0.730Boom 0.655 0.711 0.732Recession 0.812 0.864 0.723Recession minus boom 0.157 0.153 -0.009

Risk-Free RateAverage 0.006 0.011 0.011Boom -0.001 -0.005 -0.007Recession 0.031 0.065 0.071Recession minus boom 0.032 0.070 0.077

34

Table 4: Calibration — Regressions. This table summarizes OLS regression results (in-tercept, coefficient estimate and adjusted R-squared) of relative consumption as explanatoryvariable for average of industry market correlations (Av. CORR), 3-year ahead expectedexcess returns (Av. EXR), and standard deviations (Av. STDV). Newey-West correctedt-statistics are in parentheses. Correlations and standard deviations are estimated usinga DVEC(1,1) model. Relative consumption in the data is linearly interpolated from theheterogeneous investor model calibration employing annual consumption data from RobertShiller’s web page. The regressions in the data columns use 996 monthly observations withdata ranging from January 1927 to December 2009. The regressions in the model are basedon the parameters in Table 1. We simulate 100 paths of 1000 months each. For every pathwe calculate the average correlations, 3-year ahead expected excess returns and standarddeviations. The reported results are averages over the 100 sample paths.

Av. CORR Av. EXR Av. STDVData Model Data Model Data Model

Intercept 0.7649 0.7989 0.1866 0.1203 0.2939 0.2728(28.8627) (154.6527) (6.4946) (2.8024) (9.0453) (115.626)

Relative consumption, ω -0.4574 -0.3052 -0.6740 -0.5775 -0.6048 -0.1716(-2.8642) (-7.9585) (-4.0504) (-2.2547) (-3.2582) (-7.0651)

Adjusted R-squared 0.1317 0.3155 0.1254 0.0739 0.3954 0.3049

Table 5: Empirics — Principal Component Analysis. This table summarizes OLS re-gression results (intercept, coefficient estimate and adjusted R-squared) of relative consump-tion as explanatory variable for the first principal component of industry market correlations(PCA CORR), 3-year ahead expected excess returns (PCA EXR), standard deviations (PCASTDV), quadratic variations of industry turnover (PCA QV), and the first principal com-ponent of the four means of the respective series (PCA TOTAL). Newey-West correctedt-statistics are in parentheses. Industry market correlations and standard deviations are cal-culated using a DVEC(1,1) model. Quadratic variations of industry turnover are estimatedby a GARCH(1,1) model based on log changes in turnover. Relative consumption is linearlyinterpolated from the heterogeneous investor model calibration employing annual consump-tion data from Robert Shiller’s web page. The regressions use 996 monthly observations withdata ranging from January 1927 to December 2009.

PCA CORR PCA EXR PCA STDV PCA QV PCA TOTALIntercept 0.5387 0.3617 0.3232 0.4423 0.1687

(2.9547) (3.6869) (3.0444) (6.7586) (4.7019)Relative consumption, ω -3.3680 -2.2556 -2.0205 -2.7656 -1.0524

(-3.0384) (-4.0211) (-3.3327) (-7.9762) (-4.8871)Adjusted R-squared 0.1406 0.1501 0.4095 0.2392 0.2719

35

Figure1:

MeanIndustry

Correlation

s,Returns,

StandardDeviation

s,an

dQuad

raticVariation

ofTurnover.

Thisfig

ureshow

splotsof

averag

epa

irwiseindu

stry

correlations

from

amultivariateGARCH

mod

el,averag

ean

nualized

3-year

aheadcontinuo

usly

compo

undedmarketexcess

returns,

averag

eindu

stry

stan

dard

deviations,also

from

themultivariate

GARCH

mod

el,an

daverag

equ

adraticvariations

ofindu

stry

turnover

areestimated

byaGARCH(1,1)mod

elba

sedon

log

chan

gesin

turnover

forthesamplepe

riod

1927

to20

09usingKenneth

French’s

indu

stry

classification.

Grayshad

edareas

deno

teNBER

recessions.Sa

mpleba

sedon

theCRSP

/Com

pustat

files,u

sing

PERMNO.

Correlations

1930

1940

1950

1960

1970

1980

1990

2000

2010

0.4

0.5

0.6

0.7

0.8

0.9

Excess Returns

1930

1940

1950

1960

1970

1980

1990

2000

2010

−0.

4

−0.

20

0.2

0.4

Standard Deviation

1930

1940

1950

1960

1970

1980

1990

2000

2010

0.1

0.2

0.3

0.4

0.5

0.6

Quadratic Variation Turnover

1930

1940

1950

1960

1970

1980

1990

2000

2010

0.1

0.150.

2

0.250.

3

0.350.

4

36

Figure2:

Mod

elIm

plied

RelativeRatio

Hab

itan

dAgg

regate

RiskAversion.Thisfig

ureshow

splotsof

heterogeneou

sinvestor

mod

elim

pliedratioha

bit,ω

(t),ag

gregaterisk

aversion

,R(t

),an

dchan

gesin

aggregaterisk

aversion

,∂R

(t)/∂exp

(ω(t

)).

Grayshad

edareasdeno

teNBER

recessions.

1930

1940

1950

1960

1970

1980

1990

2000

2010

−0.

1

−0.

050

0.050.

1

0.150.

2

0.250.

3ω

(t)

1930

1940

1950

1960

1970

1980

1990

2000

2010

−12

0

−10

0

−80

−60

−40

−20020

R(t

)∂

R(t

) / ∂

exp

( ω(t

) )

37

Figure3:

Mod

elIm

plied

Correlation

san

dStandardDeviation

sversustheData.

Thisfig

ureshow

splotsof

heteroge-

neou

sinvestor

mod

elim

pliedaveragepa

irwiseindu

stry

correlations

andstan

dard

deviations

andthecorrespo

ndingda

taseries

over

thesamplepe

riod

1927

to20

09usingKenne

thFrench’sindu

stry

classification.

Grayshad

edareasdeno

teNBER

recessions.

Sampleba

sedon

theCRSP

/Com

pustat

files,u

sing

PERMNO.

Correlations

1930

1940

1950

1960

1970

1980

1990

2000

0.450.

5

0.550.

6

0.650.

7

0.750.

8

0.850.

9

0.95

Dat

aIm

plie

d by

Het

erog

eneo

us In

vest

or M

odel

Standard Deviations

1930

1940

1950

1960

1970

1980

1990

2000

0.1

0.150.

2

0.250.

3

0.350.

4

0.450.

5

38

Figure4:

Correlation

swithHom

ogen

eousPreferences.

Thisfig

ureshow

sthecond

itiona

lreturn

correlationbe

tween

stock1an

dstock2as

afunction

ofdividend

share,s δ

1(0)

=δ 1

(0)/

(δ1(0

)+δ 2

(0)),s

eeEqu

ation6,

oras

afunction

ofdividend

sharean

dmod

elim

pliedha

bit,ω,s

eeEqu

ation12

.The

figurecontains

sixplotswiththefollo

wingrisk

aversion

,γ,forL

and

Hinvestors,respectively:log−log,3−

3,10−

10(top

row

plots)

withstan

dard

power

preferencesan

dlog−log,3−

3,10−

10withratioha

bitpreferences(bottom

row

plots).Investorstimepreference,ρ

,issetat

0.01

.Hab

itpe

rsistence,λ,issetat

0.1.

Bothstocks

have

identicald

ividenddrift

anddiffu

sion

coeffi

cients.The

drift

issetat

0.02

while

thediffu

sion

coeffi

cientis

0.05

.Dividends

areun

correlated.The

horizonof

theecon

omy,T,issetat

50years.

39

Figure5:

Correlation

san

dRiskAversion.Thisfig

ureshow

sequity

returnsdu

eto

apo

sitive

dividend

shockto

dividend

onein

astylized

econ

omywithtw

ostocks.The

return

ofstock1—gray

shad

edarea—

increasesin

theow

npo

sitive

dividend

shockbu

tdecreasesin

thediscou

ntingeff

ect—white

area—

dueto

atw

otreeseff

ect,

seeCochran

eet

al.(

2008

).R

1deno

tes

theneteff

ect.

The

discou

ntingeff

ect—R

2—

onthesecond

stockis

positive.The

dividend

effectan

dthediscou

ntingeff

ect

causepo

sitive

correlation.

The

correlation,

viathediscou

ntingeff

ects,increases

inrisk

aversion

.

R1

R1

R2

0 Ret

urns

Ret

urn

due

to o

wn

(pos

itive

) div

iden

d sh

ock,

div

iden

d ef

fect

Ret

urn

due

to o

wn

(incr

ease

) pric

e of

risk

shoc

k, d

isco

untin

g ef

fect

Net

eff

ects

from

an

incr

ease

in δ

1

γ=1

R1

R2

γ=10

γ≈∞

R2

Ret

urn

due

to o

wn

(dec

reas

e) p

rice

of ri

sk sh

ock,

dis

coun

ting

effe

ct

40

Figure6:

Correlation

swithHeterog

eneousPreferences.

Thisfig

ureshow

sthecond

itiona

lreturn

correlationbe

tween

stock1an

dstock2as

afunction

ofdividend

sharean

dconsum

ptionshareofL

investors,

Propo

sition

1,or

asafunction

ofdividend

sharean

dmod

elim

pliedha

bit.

The

figurecontains

four

plotswiththefollo

wingrisk

aversion

,γ,forLan

dH

investors,

respectively:log−

3an

dlog−

10(top

row

plots)

withstan

dard

power

preferencesan

dlog−

3an

dlog−

10withratioha

bit

preferences(bottom

row

plots).Allotherfund

amentals

areas

inFigure4.

00.

20.

40.

60.

81

0

0.5

1

0

0.1

0.2

0.3

0.4

s δ 1(0)

Log

−−

− 3

f(0)

Correlation Corr0(dR1(0),dR2(0))

00.

20.

40.

60.

81

0

0.5

1

0

0.2

0.4

0.6

0.8

s δ 1(0)

Log

−−

− 1

0

f(0)

Correlation Corr0(dR1(0),dR2(0))

00.

20.

40.

60.

81

−0.

5

0

0.5

1

0.4

0.5

0.6

0.7

s δ 1(0)

Log

−−

− 3

ω(0

)

Correlation Corr0(dR1(0),dR2(0))

00.

20.

40.

60.

81

−0.

50

0.5

1

0.4

0.6

0.81

s δ 1(0)

Log

−−

− 1

0

ω(0

)

Correlation Corr0(dR1(0),dR2(0))

41

Figure7:

RiskAversionVolatilityEffect.

Thisfig

ureshow

splotsof

theconsum

ptionshareof

investorswithhigh

risk

aversion

,H,slop

esof

thesharingrules,

therisk

aversion

oftherepresentative

agent(R

),thenega

tive

ofthederivative

oftherisk

aversion

oftherepresentative

agent(−

∂R∂C),

thecond

itiona

lreturn

correlations

betw

een

stock

1an

dstock

2in

aheterogeneou

sinvestor

econ

omyan

dan

equivalent

homog

eneous

econ

omy(γL

=γH

=R

(t)),expe

cted

excess

return

onthe

marketpo

rtfolio

,con

dition

alstan

dard

deviationof

themarketpo

rtfolio

,the

relative

quad

raticvariationof

thepo

rtfolio

ofH

investorsas

function

sof

aggregatedividend

s,C

(0).

Riskaversion

coeffi

cients

areγL

=logan

dγH

=3.

Allotherfund

amentals

areas

inFigure4. 0

510

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

C(0

)

CH(0) / C(0)

05

100

0.050.1

0.150.2

0.250.3

0.350.4

C(0

)

-∂ (CH(0) / C(0)) / ∂ C(0)

05

101

1.52

2.53

C(0

)

Risk Aversion of Representative Investor, R(0)

05

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

C(0

)

-∂ R(0) / ∂ C(0)0

510

0

0.050.1

0.150.2

0.250.3

0.35

C(0

)

Correlation Corr0(dR1(0),dR2(0))

H

eter

ogen

eous

Hom

ogen

eous

05

101.

52

2.53

3.54

4.55

x 10

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C(0

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05

100.

035

0.03

6

0.03

7

0.03

8

0.03

9

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0.04

1

0.04

2

0.04

3

0.04

4

C(0

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Stdev0(dRM(0))

05

100

0.00

2

0.00

4

0.00

6

0.00

8

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C(0

)

RQVH(0)

42

Figure8:

Empirical

Prediction

swithRatio

Hab

itPreferences.

The

topleft

figureshow

sthecond

itiona

lreturncorrela-

tion

sbe

tweenstock1an

dstock2,

thetoprigh

tfig

ureshow

stheexpe

cted

excess

return

onthemarketpo

rtfolio

,the

bottom

left

figureshow

sthecond

itiona

lvarianceof

themarketpo

rtfolio

,and

thebo

ttom

righ

tfig

ureshow

stherelative

quad

raticvariation

ofthepo

rtfolio

ofH

investors.

Riskaversion

coeffi

cients

areγL

=logan

dγH

=3.

Allothe

rfund

amentals

areas

inFigure4.

43