University of Southern California Dissertations and Theses

Asset prices and trading in complete market economies with heterogeneous agents

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Asset prices and trading in complete market economies with heterogeneous agents

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Content ASSET PRICES AND TRADING IN COMPLETE MARKET ECONOMIES WITH
HETEROGENEOUS AGENTS
by
Costas Xiouros
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BUSINESS ADMINISTRATION)
May 2009
Copyright 2009 Costas Xiouros
Dedication
To my family
ii
Acknowledgments
I would like to foremost express my gratitude to my supervisor and chair of my thesis
committee professor Fernando Zapatero for his support and guidance over the years of my
doctoral work. I would also like to express my appreciation to the rest of my committee
professors Christopher Jones, Antonios Sangvinatsos and Michael Magill for their sugges-
tions and constructive criticism.
iii
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables vi
List of Figures vii
Abstract ix
Chapter 1: Introduction 1
Chapter 2: A Heterogeneous Risk-Aversion Economy with External Habit For-
mation 9
2.1 Introduction 9
2.2 The Model 11
2.2.1 Aggregate Uncertainty 12
2.2.2 Financial Markets 13
2.2.3 Heterogeneous Preferences with External Habit 14
2.2.4 Financial Equilibrium 16
2.3 The Representative Agent Equivalent Economy 21
2.3.1 Preferences and Equilibrium 21
2.4 Agent Distribution and the Variation ofγ pq 23
2.5 Habit Process and Asset Prices 27
2.5.1 The Stochastic Discount Factor 29
2.5.2 Asset Prices 32
2.6 Quantifying The Effect of Agent-Heterogeneity 35
2.6.1 Calibration Procedure 35
2.6.2 Analysis 39
2.7 Conclusion 52
Chapter 3: A Heterogeneous Beliefs Economy and an Explanation of Market
Prices 54
iv
3.1 Introduction 54
3.2 A General Economy with Heterogeneous Beliefs 59
3.2.1 Uncertainty, Agents and Beliefs 60
3.2.2 Complete Financial Markets 63
3.2.3 Preferences and Endowments 66
3.3 Complete Market Equilibrium with Heterogeneous Beliefs 69
3.4 Equilibrium Asset Prices 76
3.4.1 Discounted Prices 77
3.4.2 The Equity Premium 79
3.5 The Economy 80
3.6 Calibration 87
3.6.1 Homogeneous Economy with Uncertainty 88
3.6.2 Heterogeneous Beliefs Economy 95
3.6.3 Evidence of Convergence in Beliefs 99
3.7 Conclusion 101
Chapter 4: Prices and Trading in an Economy with Differences of Opinions 104
4.1 Introduction 104
4.2 The Model 112
4.2.1 Uncertainty, Information and Beliefs 113
4.2.2 Financial Market 121
4.2.3 Preferences and Endowments 122
4.3 Equilibrium 124
4.3.1 Financial Market Equilibrium 124
4.3.2 Recursive Characterization 128
4.3.3 Intermediate Period 130
4.3.4 Asset Prices and Portfolio Holdings 133
4.4 Asset Price Variation and Trading 137
4.4.1 The Stock Price 138
4.4.2 Price, Return V olatility and V olume 141
4.4.3 Price Changes and V olume 146
4.5 Empirical Evidence 151
4.6 Conclusion 157
References 160
Appendices 168
A Appendix to Chapter 2 168
B Appendix to Chapter 3 173
C Appendix to Chapter 4 178
v
List of Tables
2.1 Model parameterizations 38
2.2 Price and return statistics 41
2.3 Autocorrelations of price-dividend ratio 42
2.4 Autocorrelations of excess returns 42
2.5 Long-run predictability regressions 46
3.1 Aggregate consumption growth parameters 89
3.2 Model parameters 91
3.3 Quarterly Statistics (1947Q1-2008Q1) 93
4.1 Model configuration 137
4.2 Contemporaneous correlations 154
vi
List of Figures
2.1 Fitted cross-sectional distribution of agents at ¯ ω 25
2.2 Variation in the representative agent risk aversion 27
2.3 Variation in the conditional volatility of ˜ m 32
2.4 Data and model 1 implied time series of price-dividend ratio and risk-free
rate 43
2.5 Model parametrization 1 45
2.6 Data and model 2 implied time series of price-dividend ratio and risk-free
rate 48
2.7 Model parametrization 2 50
2.8 Data and model 3 implied time series of price-dividend ratio and risk-free
rate 51
2.9 Model parametrization 3 53
3.1 Scatter plot of the price-dividend ratio with the fitted conditional volatility
of consumption growth. The correlation is 0.78. 57
3.2 Time series of the state vector. 90
3.3 Homogeneous economy log price dividend ratio and risk free rate. 92
3.4 Homogeneous economy log price dividend ratio 93
3.5 Homogeneous economy risk-free rate 94
3.6 Homogeneous economy expected excess return 95
3.7 Homogeneous economy conditional mean and volatility of expected excess
return 96
3.8 Heterogeneous economy log price dividend ratio and risk free rate. 97
vii
3.9 Heterogeneous economy conditional mean and volatility of expected excess
return. 98
3.10 Heterogeneous economy log price dividend ratio. 99
3.11 Heterogeneous economy risk-free rate. 100
3.12 Heterogeneous economy expected excess return. 101
3.13 Model and data heterogeneity of beliefs 102
4.1 Price changes and volume changes (annual data) 105
4.2 Price cycles and volume cycles (annual data) 106
4.3 Price changes vs. volume changes (monthly data) 107
4.4 Price cycles vs. volume cycles (monthly data) 108
4.5 Log-price divided by the aggregate endowment (η 0.5). 139
4.6 Expected stock return (η 0.5). 141
4.7 Conditional return volatility (η 0.5). 143
4.8 Expected turnover (η 0.5). 144
4.9 Model correlation between price changes and volume changes 148
4.10 Model correlation between price changes and volume 149
4.11 Model correlation between consumption growth and market returns 150
4.12 Price-earnings ratio and trading 152
4.13 Price-earnings ratio and trading 153
4.14 Price-earnings ratio and trading 156
4.15 Return volatility and trading 157
4.16 Dispersion in beliefs and price-earnings ratio 158
viii
Abstract
This thesis examines how and to what extend certain types of heterogeneity of agents in an
economy with complete financial markets can explain the variation in aggregate prices and
the volume of trade that we observe. There are a number of characteristics of the financial
markets that have been particularly puzzling researchers; most important of which are the
fact that the level of the market is on average low in comparison to interest rates, it varies a
lot over time while the fundamentals of the economy do not and the fact that agents trade
too much to be justified by the standard view that we hold for the economy. One clue that
is found in the data is that the amount of trading is related to prices in a couple of ways
like volatility or whether the market is moving upwards or downwards. This implies that
the market is moved partly by the same forces that make agents to trade and at the basis of
it is that agents are heterogeneous. I consider two types of heterogeneities in three separate
studies. I first analyze an economy where agents differ in their risk preferences and I find
with a realistic set of parameters that it cannot be a noticeable driver of the phenomena that
I examine. In the other two studies I look at economies where agents differ in their beliefs
about the future. These two studies yield a number of positive results: (i) high variation in
the level of the prices can be explained by variations in the level of heterogeneity of beliefs
and in particular the phenomenal increase in prices over the 1990s, (ii) price characteris-
tics can be connected with trading volume through sentiment risk which determines how
volatile individual beliefs are and (iii) both heterogeneity of beliefs as well as sentiment
ix
risk could possibly explain the low level of prices. The studies are mostly theoretical but
some empirical support is provided.
x
Chapter 1
Introduction
The theory of asset pricing strives to understand and identify the main factors that deter-
mine prices in financial markets both in the cross-section as well as over time. This thesis
concentrates on the aggregate stock market and looks at prices in the time dimension. As
the title reveals, this thesis tries to see whether the fact that investors are heterogeneous
in certain aspects plays a significant role in determining asset prices. In reality prices are
determined like in any other market which is by matching the supply and demand of finan-
cial assets between different economic agents. It is obvious that agent heterogeneity and
variations in demand and supply produces trading. However, it is still unknown whether
these variations in demand and supply are in the center of the price determination process
or whether prices are mainly driven by the current economic conditions that are common
to everyone.
The standard dynamic asset pricing theory based on Lucas (1978) has neglected agent
heterogeneity and has concentrated on drawing a direct link between macroeconomic fac-
tors and asset prices. However, in this attempt it has been largely unsuccessful since in the
process of shedding light on the matter it has generated a number of asset pricing puzzles.
This failure could be coming either from the representative agent assumption, the simplistic
preferences considered or even the assumption of rationality. All three channels are being
investigated and all three lines of research have interesting results to show. In this thesis I
investigate two types of heterogeneity namely of risk preferences in chapter 2 and of beliefs
in chapters 3 and 4. Even though the heterogeneity of beliefs obviously touches upon the
1
rationality assumption I put constraints on the settings so that on average the economy is
correct.
Before I outline the three studies I need to further explain the open questions that these
studies deal with. The stock market can be simply viewed as a claim to the future aggregate
dividends. Therefore, its level should be determined by the expectations of discounted
future dividends. Variations in the level of prices should therefore be due to variations either
in expectations about future dividend growth or variations in discount rates. However,
the volatility puzzle raised by Shiller (1981) and Leroy and Porter (1981) states that the
variability of prices cannot be explained in an economy where discount rates are close to
constant and the dividend growth is largely unforcastable. In a rational setting where agents
anticipate correctly the possible future movements of the aggregate dividend it would mean
that the level of the market today should be a predictor of what the future dividend growth
is on average. Therefore, the only rational explanation is that prices move due to changes
in discount rates even though they do not appear to vary much over time. The theoretical
justification came from Campbell and Shiller (1988) that showed that even small changes
in discount rates can explain the variation of prices, if discount rates are persistent enough.
A discount rate for a future risky cash flow is roughly the risk free rate plus the compen-
sation for risk (excess return) for holding a claim to this uncertain cash flow. Therefore, if
prices move due to changes in the discount rate then it should be either due to changes in the
risk-free rates or changes in the excess return or a combination of both. Evidence however
provided by Famma and French (1988), Campbell and Shiller (1988) or Campbell (1991)
among others, suggest that the variations must be driven by the excess return because the
price-dividend ratio, which is the measure of the level of the market, appears to forecast
excess returns and not interest rates. Consequently, the quest has transformed into finding
the persistent process that is responsible for the variation in expected excess returns which
is the compensation required by agents to hold the aggregate risk carried by the market.
2
The expected excess return of the market is determined by the amount of risk of the
market times the price of risk or the attitude towards risk. Since what agents care about is
their consumption the amount of risk is the covariance of the market returns with consump-
tion growth while the price of risk is the risk aversion with respect to consumption growth.
One could possibly say that the volatility of the market which is known to vary as shown
for example by Schwert (1989) and Bollerslev, Chou, and Kroner (1992) could be driving
the level of the market. However, the problem is that we do not know what drives the
changes in the volatility and second these variations are not enough to explain the variation
of the price-dividend ratio that I observe. The same can be said about the consumption risk.
All these findings have lead Campbell and Cochrane (1999) to explain the variation in the
level of prices by changes in the risk-aversion of the economy. In their model like in many
other popular models of habit formation agents instead of actual consumption care about
consumption over a common external habit that moves slowly. In turn the variations in the
risk-aversion are assumed to be counter-cyclical with respect to their consumption surplus.
The model of Campbell and Cochrane (1999) is able to match a number of market
statistics like the variation of prices with constant interest rates and hence the predictability
of excess returns but it does not enlighten us as to what the actual driving force behind the
variation in the price of risk is. It only assumes that when consumption is low with respect
to habit the risk aversion of the economy increases drastically and falls when consumption
becomes high in relation to habit. The counter-cyclical risk aversion causes the risk premia
to be counter-cyclical and hence the prices to be pro-cyclical. One possible explanation
is that a counter-cyclical risk-aversion in the market is produced because the economy is
populated with agents that are heterogeneous in their risk aversion. Then because of the
more aggressive investment behavior on the part of the more risk tolerant agents, there
is wealth transfer between more risk tolerant agents to more risk averse agents when the
economy moves from a peak of a business cycle to a trough. This way the effective risk
3
aversion of the economy moves counter-cyclically. This is shown by Chan and Kogan
(2002) and the authors claim that a significant variation in the risk premia can be generated
in an economy in this way.
In chapter 2 of this thesis I look at a similar economy to that of Chan and Kogan (2002)
but I arrive at the opposite conclusion which is that the variation in risk-premia in such
an economy is so small that it would not be noticeable in the variation of the level of
prices. The main differences between my economy and the economy of Chan and Kogan
(2002) is first that I match the model distribution of agents that are heterogeneous in their
risk-aversion to a distribution estimated in Kimball, Sham, and Shapiro (2008) through
survey responses and hence I only allow a realistic level of heterogeneity. The second main
difference is in the assumed persistence in the external habit which is needed to produce
persistence in the variation of the level of prices. While I assume the same persistence as in
Campbell and Cochrane (1999) to match the persistence in the price-dividend ratio, Chan
and Kogan (2002) assume more persistence which also results into much more variation in
the level of prices. The variation in the level of prices hence is mostly due to variations in
the risk-free rate which is counterfactual.
There are several obstacles that do not allow us to accept that the reallocation of wealth
among agents with different risk-aversion could be an important driving force behind
prices. A significant variation in the risk aversion of the economy requires an unrealisti-
cally high level of heterogeneity across agents. Otherwise we cannot expect differences in
the sharing of the aggregate risk to be substantial that would lead to significant reallocation
of wealth over time. Secondly, even with an unrealistically high level of heterogeneity we
still need a significant level of aggregate risk to share, a level substantially higher than what
we observe in the data. The final obstacle in such a story is related to the most well known
puzzle of asset pricing namely the high equity premium. As noted by Mehra and Prescott
(1985), Kocherlakota (1996), Hansen and Jagannathan (1991) and others, the aggregate
4
risk in the economy which is described by the volatility of aggregate consumption growth
is very low. Consequently, the standard homogeneous agent economy predicts a very low
equity premium which is about the same as the average equity premium of the heteroge-
neous economy. With such a low average equity premium even a high level of variation in
the risk-aversion will not produce noticeable effects.
The relative success of the Campbell and Cochrane (1999) specification in addressing
the volatility and the equity premium puzzles depends on the assumption that the risk-
aversion is high on average and it additionally has particularly high variation. In terms of
economic theory however we still walk in the dark as to what causes the risk-premia to be
so high, persistent and volatile. In chapter 3 I look at a second kind of heterogeneity which
is in terms of beliefs about future consumption growth. The qualitative and quantitative
results that come from such an economy are much more promising in understanding the
puzzling characteristics of the financial markets. The main reason for this is that in an
economy with even a moderate level of heterogeneity of beliefs agents’ investments differ
substantially. Therefore, even though the aggregate risk is small the individual risks are
much higher and this adds additional volatility into the markets. This additional riskiness
causes discount rates to rise and all prices to lower. If further the level of heterogeneity is
persistent then the volatility puzzle can be explained since the market will be much more
volatile than the risk-free rate.
Heterogeneity of beliefs has also been found as for example in David (2008) to increase
risk-premia because of the additional volatility induced. In chapter 3 we identify one more
reason why such a setting can increase the compensation required by agents to hold the
market instead of the risk-free asset. On average the risk-averse agents of the economy
are afraid of states in the future for which there is high disagreement because this means
that in those states wealth will be very concentrated. If those states can be regarded as
non-ordinary in which case the disagreement among agents would possibly increase then
5
the market would be low and hence a bad hedge against the risk that heterogeneity will be
high in the future.
The model presented is able to generate a significant equity premium but still low when
compared to the data. In matching the first two moments of the price-dividend ratio as
well as its persistence the model predicts a higher average risk-free than what is found in
the data. However, the time variation in the level of the market due to variations in the
level of heterogeneity produces one more appealing prediction. One puzzling period for
the markets was the phenomenal increase in prices over the 1990s. In fact these 10 years
of data have been enough to raise doubts of whether excess returns are actually predictable
as pointed by Lettau and Ludvigson (2001) and Goyal and Welch (2003). Further doubts
have been cast over predictability by Ang and Bekaert (2007) or Boudoukh, Richardson,
and Whitelaw (2008). One reason that has been proposed for this increase by Lettau, Lud-
vigson, and Wachter (2008) was that macroeconomic risk fell. However, we know already
that it is difficult to produce significant variation in the level of prices by variations in the
amount of macroeconomic risk let alone to explain such a phenomenal increase. What
I propose in chapter 3 is that it is quantitatively possible that such an increase in prices
was caused due to a decrease in the level of heterogeneity over that period possibly due
to a decrease in macroeconomic risk. Evidence from differences in professional macroe-
conomic forecasts indicate a decrease of dispersion in forecasts over this period. Hence,
it is possible that a decrease in the level of heterogeneity of beliefs was one of the factors
responsible for the phenomenal increase in prices during the 1990s.
One other possible factor has been proposed by Geanakoplos, Magill, and Quinzii
(2004) where the authors look at the issue of variation in prices and predictability under
a different kind of heterogeneity. They show that changes in the demographics of the U.S.
economy could have been a main determinant of the level of prices in the post war period
due to the fact that agents at different ages have different investment needs. For example
6
they propose that the increase in prices during the 1990s was because the baby-boomers
reached middle age and started investing for their retirement and in this way pushed up
prices. Similarly, cycles in baby births that cause cycles in the demographics produce also
cycles in prices and make them appear predictable.
Both stories offer a possible explanation as to why the predictability sought for example
by Cochrane (2008) cannot be strongly detected in the data. The argument behind the
predictability and the variation in prices due to changes in expected returns derives from
the view that the price-dividend ratio is a stationary variable moving around some long
term mean. However, if among other things a variable that is not expected to behave in
a particularly stationary fashion, such as the level of heterogeneity, is a factor of the level
of the price-dividend ratio then we it may have made the series look non-stationary in the
short history that we observe.
The story of belief heterogeneity that I pursue might be able to become a good candidate
for solving the main asset pricing puzzles if in the same setting the high equity premium is
explained as well. The last study of this thesis which is presented in chapter 4 deals with
another dimension of belief heterogeneity which is what we call as sentiment risk. This risk
refers to how much the level of heterogeneity will change in the future. When sentiment
risk is high then beliefs are volatile and this causes the stock price to be even lower. If now
sentiment risk is counter-cyclical then the stock is a bad hedge against this risk and agents
would require an additional compensation for holding it.
Even though sentiment risk is promising in producing a sizeable equity premium, in
chapter 4 I deal with another aspect of such an economy where beliefs are heterogeneous
and sentiment risk is time-varying. One attractive feature of belief heterogeneity when
compared with other types of heterogeneities is that it predicts substantial amount of trad-
ing. The lack of trading in the standard asset pricing models even when preferences are
heterogeneous, as for example in Judd, Kubler, and Schmedders (2003), or information
7
is disperse, as in Milgrom and Stokey (1982), produces another puzzle of the financial
markets. However, understanding trading activity can also help us understand asset prices
because the data show a connection between the two. In particular high volatility in the
market is associated with high volume and also volume is higher when the market goes up.
The question that was posed at the beginning of the introduction as to whether heterogene-
ity affects the determination of prices finds an indirect answer in these observations. The
scope of the last chapter is to find a connection between prices and volume and through this
connection further our understanding about both.
The connecting link in the model presented in chapter 4 is the time-varying sentiment
risk. When sentiment risk is high beliefs are volatile and this produces a lot of trading as
well as high volatility in the market. At the same time high sentiment risk causes prices to
be low first because it predicts high level of disagreement in the future as well as possibly
because of an increased risk premium. The advantage of this model is that if this is a
significant channel through which the level of prices vary over time it produces a robust
prediction namely the amount of trading by which we are able to quantify it.
Unlike heterogeneity in preferences, an economy with heterogeneous beliefs produces
qualitatively interesting and quantitatively important effects by which we might be able to
expand our understanding of the financial markets. Of course when we deal with differ-
ences in beliefs we need to be careful as to what exactly we allow because such theoretical
arguments can easily become untenable once the assumptions become unrealistic. This
obviously requires for us to measure the level and understand the origin of this type of
heterogeneity. Despite these pitfalls, I believe that this thesis through its positive results
warrants significant optimism for understanding better prices and trading in the financial
markets and that further theoretical as well as empirical work should be done in this line of
research.
8
Chapter 2
A Heterogeneous Risk-Aversion
Economy with External Habit
Formation
2.1 Introduction
I study a model similar to that of Chan and Kogan (2002), although in discrete time. With a
detailed study of the mechanics of the stationary equilibrium of this heterogeneous agents
economy, I derive explicitly the time-varying risk-aversion of the representative agent and
I analyze its properties. I find that, although the counter-cyclical pattern of Campbell and
Cochrane (1999) is accurate, many more assumptions are needed in order to have more
than just a marginal effect on asset price dynamics. In particular, I show that enough time-
varying risk aversion fails to obtain as the result of aggregation in an economy of rational
agents with standard preferences and different risk-attitudes. Even when I inflate the level
of heterogeneity and increase the risk in the economy to levels that gives us the ability to
predict an average equity premium close to the average excess return of the past 75 years,
I am not able to produce enough predictability. For the baseline model for which the level
of heterogeneity is calibrated to fit the estimated distribution of Kimball et al. (2008) the
underlying consumption risk is not enough to even predict an asset pricing behavior that is
clearly different, either in patterns or in levels, from a homogeneous agents economy.
9
Chan and Kogan (2002) assume a highly persistent and slow moving external habit, as
well as a particularly high level of heterogeneity in risk-aversion. The representative agent
formulation reveals that if either of these assumptions is missing, a substantially varying
Sharpe ratio does not obtain. There are certain problems with such a slow moving state; (i)
its effects will be seen over much longer periods than a business cycle and, (ii) it predicts a
high variability in the risk-free rate which gives rise to a high-term premium as opposed to a
high risk premium and (iii) it predicts higher persistence in price-dividend ratios than what I
find in the data. Campbell and Cochrane (1999) also assume a slowly varying state variable,
namely the surplus consumption (current aggregate consumption over the external habit),
but with the additional feature of a highly time-varying conditional variance, that as I show
is related to the risk-aversion of the economy. In this study I let the persistence in the price-
dividend ratio to guide us in the selection of the persistence of the habit process. The study
of Campbell and Cochrane (1999) is very useful because it identifies the main features that
a successful asset pricing model needs to have in order to explain all the aforementioned
empirical facts. For risk-aversion heterogeneity to have an impact on prices I would need
a different source of variability in the wealth distribution across agents. For example, in
the overlapping generations model of Gˆ arleanu and Panageas (2008), heterogeneity of risk
aversion does have an impact because the re-allocation of wealth is considerably more
drastic across time.
The literature on heterogeneity of risk aversion has a long tradition in finance. Dumas
(1989) solves numerically a model with two agents, one of them with logarithmic utility.
Wang (1996) considers also a two agent economy and concentrates on the dynamics of bond
prices. Pirani (2004) focuses on the dynamics of wealth distribution among two agents with
Epstein and Zin preferences. Bhamra and Uppal (2007) show that completing the market in
an economy populated with heterogeneous agents might increase the stock price volatility
substantially. Kogan, Makarov, and Uppal (2007) show that in a two agent economy with
10
borrowing constraints the Sharpe ratio can be high while at the same time having a low
risk-free rate. In this study I consider an arbitrary number of agents with “catching up with
the Joneses” preferences where markets are dynamically complete. Using both analytical
results as well as computing the exact equilibrium of several economies I find that in the
absence of any frictions or incompleteness in the market the effect of heterogeneity is
potentially minimal. In effect, representative agent economies can approximate well a
certain family of heterogeneous agent economies.
The rest of the study is structured as follows. In section 2.2 I describe the heteroge-
neous agents economy and solve for the competitive equilibrium. In section 2.3 I consider
a representative agent economy that is homeomorphic in its pricing implications with the
heterogeneous agents economy of section 2.2. I derive an expression for the stochastic risk
aversion of the representative agent and analyze its properties. In section 2.4 I parameterize
the distribution of agents and fit it to the distribution estimated by Kimball et al. (2008).
Using the results of sections 2.2 and 2.3, I derive analytically the stochastic risk-aversion
of the economy. In section 2.5 I assume a particular process for habit and examine the-
oretically the asset pricing behavior. An extensive quantitative analysis of the effects of
heterogeneity is carried out in section 2.6. I conclude in section 2.7.
2.2 The Model
I consider a version in discrete time, but more general, of the infinite horizon endowment
economy of Chan and Kogan (2002). I chose discrete time instead of continuous time in
order to allow for more general specifications of the uncertainty of the economy that are yet
numerically tractable. In the model, uncertainty is driven by an exogenous state that follows
a time homogeneous Markov process. The exogenous state is perfectly observable to all
agents in the economy. Financial markets are dynamically complete, in the sense that the
equilibrium asset structure spans the one period ahead uncertainty at every possible state
11
of nature. There is a single perishable good, and agents exhibit power utility preferences
with external habit formation, in the style of the “catching up with the Joneses” preferences
of Abel (1990a). I present two versions of the model: In the first version the economy is
populated by a number of different types of agents with possibly different coefficients of
relative risk aversion; in the second version, I replace the heterogeneous agents with a rep-
resentative agent with a stochastic coefficient of relative risk aversion (as in Campbell and
Cochrane (1999)); I then introduce an expression for the stochastic risk aversion coefficient
of the representative agent parameterized by the primitives of the multiple agent economy
and derive the rule that makes the two economies equivalent.
As in Chan and Kogan (2002), catching up with the Joneses preferences are not only
attractive from an economics point of view (there are some influential studies that assume
this type of preferences, especially Campbell and Cochrane (1999)), but they also yield
a stationary equilibrium such that the wealth distribution follows a time-homogeneous
Markov process. The problem of standard power utility preferences is that, in the limit,
wealth is accumulated by the least risk-averse agent.
2.2.1 Aggregate Uncertainty
The single source of uncertainty in the economy is growth in the aggregate endowment.
I denote aggregate endowment by Y , with y
t
log pY
t
q. As it is customary, I model the
dynamics of the logarithm of the growth process, which for now I assume to be normal iid,
y
t
y
t 1
μ σǫ
t
, t ¥ 0. (2.1)
where ǫ
t
i.i.d. N p0,1 q and y
0
is given. This simple structure will allow us to compare
the results to those of Campbell and Cochrane (1999) and Chan and Kogan (2002).
12
2.2.2 Financial Markets
I assume that financial markets are dynamically complete, that is, at any point in time the
equilibrium asset structure locally spans the one period ahead uncertainty. To keep the
model as simple as possible, I consider only one dividend-paying asset, the market security
(or simply market), and the risk-free asset. I denote the price of the market byP
m
, and the
dividend it pays by D
t
(d log pD q). I assume that dividend growth is the result of the
shock that drives aggregate endowment growth, and an independent normal shock,
d
t
d
t 1
μ
d
σ
d
̺ǫ
t
a
1 ̺
2
ǫ
d
t
, (2.2)
whereǫ
d
i.i.d.N p0,1 q is independent ofǫ, and̺ represents the correlation between div-
idend growth with consumption growth. This specification of the dividend growth nests the
simple case in which the market security pays the totality of the aggregate endowment. For
simplicity and in order not to introduce an additional risk factor I assume that the dividend
paid by the market is part of the aggregate endowment. The price of the market asset is its
fundamental complete markets value. The return on the market is denoted withR
m
. There
is also a risk-free security, with a returnR
f
, the risk-free rate. R
e
is the excess return of the
market over the risk-free rate. In addition, I implicitly assume (I don’t need a formal char-
acterization for the results) that there is a continuum of zero net supply contingent claims
written on the aggregate endowment, so that markets are dynamically complete.
I also assume a process p
t
to represent the relative “price” of the consumption good
(or pricing kernel), so that the price of the market is the present value of future dividends
priced at p. The one period risk free rate is derived from the price of a claim to a unit of
consumption next period.
13
2.2.3 Heterogeneous Preferences with External Habit
There is a set of infinitely lived agents, Γ. For now I assume that Γ is a compact set of
positive values. All agents have the same type of time and state separable preferences,
U pc,X |γ q E
0
‚
t ¥0
δ
t
u pc
t
,X
t
|γ q, (2.3)
where δ P p0,1 q is the common subjective discount factor and c
t
is consumption at time
t. X
t
, with x
t
log pX
t
q, is the external habit, common to all agents. The external habit
is an indicator of contemporaneous and/or past aggregate consumption. I will discuss its
specification later.
The running utility is drawn from the “catching up with the Joneses” literature and is
given by,
u pc,X |γ q c
1 γ
X
γ ρ
1
1 γ
, γ P Γ.
γ is the coefficient of relative risk aversion of the agent, possibly different across agents, so
that different types are characterized by theirγ. Letτ be the inverse ofγ, i.e., the coefficient
of relative risk tolerance. Due to the homotheticity and time-separability properties of
preferences, aggregation results hold for agents with the same type γ. Therefore, for the
purpose of the paper I only need to specify the initial wealth distribution across the different
types, which I denote byθ
0
pγ q. More precisely,θ
t
pγ q denotes the proportion of wealth held
by agents with typeγ at timet, and therefore,
»
Γ
θ
t
pγ qdγ 1, t ¥ 0.
Throughout the study I will be usingγ andτ interchangeably, so that when I writeθ pγ q or
θ pτ q I imply the same distribution.
14
The parameter ρ is common to all agents and determines the relative effect that the
external habit has on the marginal utility of each agent. The derivative of the marginal
utility of consumption with respect to the external habit is given by,
Bu
c
pc,X |γ q
BX
pγ ρ qc
γ
X
γ ρ 1
.
Since I would like to have a negative externality for all agents, I impose the restriction
that ρ ⁄ min
γ PΓ
γ. With a negative externality, an increase in the level of habit increases
the value that each agent places on consumption. I also note that the smaller the habit
parameter is, the bigger is the effect of the habit on the marginal utility.
As noted by Chan and Kogan (2002), these preferences ensure that the curvature of the
value function with respect to wealth is the same as that of the utility function with respect
to consumption and the relative risk aversion w.r.t. wealth is still given by the parameter
γ; this is the case because the multiplicative external habit does not affect the curvature
of the value function. However, Campbell and Cochrane (1999) assume a different utility
specification whereby the external habit affects the risk aversion of the agent. Campbell
and Cochrane (1999) specify the process for the so called surplus consumption ratio (
c X
X
)
instead of that of the habit. Hence, I can consider the Campbell and Cochrane (1999) model
as a special case of ours in which there is only one type of agent with risk aversion coeffi-
cientγ,ρ is equal to zero andx logX follows the process specified for the consumption
surplus ratio of that study. The risk-aversion of the representative agent in that particular
case is always equal toγ unlike what is implied by the preference assumptions of Campbell
and Cochrane (1999).
15
2.2.4 Financial Equilibrium
As it will become clear later in this section, it is convenient to introduce the following
variable
ω
t
y
t
x
t
, (2.4)
which I will call endowment/habit ratio, for obvious reasons. The dynamics of ω depend
on the dynamics of y, given by (2.1), and the specification -not yet provided- of x. The
results hold for a large class of specifications of x. I just need that x grows on average at
the same rate as y and that x is a functional of current and/or past aggregate endowment
y
s
, s ⁄ t,. Thereforeω is a stationary Markov process, and I treat ω as the state variable.
For example, in the continuous time model of Chan and Kogan (2002), x is a weighted
average of past aggregate endowment, and the resultingω is a stationary Markov process.
Finally, I denote by ¯ ω the unconditional average of the state which, in order to simplify
notation, I assume is also the initial state of the economy.
Some of the following derivations are a discrete-time version of the results in Chan and
Kogan (2002). I include them for completeness. The main difference with Chan and Kogan
(2002) is that they use as initial condition the weights the social planner gives to the utility
of each type (γ). Ideally, I would want to use as initial condition the wealth distribution
across types, which has a clear economic interpretation. Furthermore, at the average state,
characterized by ¯ ω, the equilibrium distributions of wealth and consumption are almost
identical.
1
I then choose to use as initial condition the distribution of consumption in the
average state, as a proxy for the distribution of wealth. As I show later, this allows us to
derive the representative agent risk aversion function in closed form.
1
As I show later, I compute numerically and very efficiently the equilibrium, including the resulting
distribution of wealth across types.
16
At any timet, an agent typeγ holds a positive proportion of the aggregate wealthθ
t
pγ q.
Since I have complete markets, the budget constraint of each agent can be expressed as
a single intertemporal budget constraint. At the initial period the intertemporal budget
constraint of an agent of typeγ is,
E
0
‚
t ¥0
δ
t
p
t
c
t
pγ q ⁄ θ
0
pγ qp
0
pP
m
0
Y
0
q,
where pp
t
,t ¥ 0 q is the equilibrium consumption price process.E
t
is the expectation opera-
tor conditional on the consumption growth process up to timet or, alternatively, conditional
on the history of the endowment habit ratio since, for a given specification of the external
habit processx, I can infer the endowment process from the endowment habit ratio process.
Therefore, it suffices to say that the information set is the endowment process and the initial
value of the endowment habit ratio. Define also
z
t
logp
t
ρx
t
, (2.5)
which can be interpreted as a normalized and stationary pricing kernel. From now on I
will refer to it simply as “the pricing kernel.” Given the pricing kernel process, the external
habit process, and the initial price of the dividend-paying asset, agents optimally choose
their consumption plan in order to maximize their utility. The following proposition char-
acterizes the optimal consumption allocation as a proportion of the aggregate endowment,
α
t
pγ q c
t
pγ q{Y
t
.
Proposition 2.1. The optimal consumption allocation of an agent type γ is characterized
by,
α
t
pγ q λ pγ qexp
z
t
γ
ω
t
, (2.6)
17
where λ pγ q
1 {γ
is the Lagrange multiplier of the intertemporal budget constraint, and is
given by,
λ pγ q θ
0
pγ q
E
0
‚
t ¥0
δ
t
p
t
Y
t
E
0
‚
t ¥0
δ
t
p
t
Y
t
exp
z
t
γ
ω
t
. (2.7)
The optimality condition (2.6) is the same as equation (8) in Chan and Kogan (2002).
However, in my case, the Lagrange multiplier λ pγ q is endogenously determined, given
the initial wealth distribution. It is well known that there is one-to-one mapping between
the equilibria resulting from each set of conditions, but this subtle point has important
quantitative implications, as I will see later on, due to the fact that the distribution of wealth
across types is a key factor to determine the equilibrium. For example if I increase the
wealth held by the most extreme types, then the stochastic risk aversion of the economy is
more volatile.
A financial equilibrium is a normalized pricing kernel process, tz
t
,t ¥ 0;z
0
0 u
and a set of consumption allocation processes of ratios of the aggregate endowment,
tα
t
pγ q,t ¥ 0;γ P Γu, such that consumption allocations satisfy the optimality conditions
of the agents, and the consumption good market clears at all times. I have the following
corollary.
Corollary 2.1. In equilibrium, the pricing kernel is a function of the endowment/habit ratio
and the initial wealth distribution, and is characterized by the following equation,
1 »
Γ
λ pγ qexp
z
γ
ω
dγ, (2.8)
whereλ pγ q is given by (2.7).
From equation (2.8) it is not feasible to derive z in closed form. However, it can be
computed numerically with very high accuracy after I discretize the distribution of types
18
and the state variable, and solve a large system of equations (a similar method is provided
in Judd et al. (2003), while my method is outlined in Appendix A).
One alternative approach that allows us to derive an expression forz and use it solve for
the risk aversion of the representative agent in closed form, is as follows. Instead of assum-
ing the initial wealth distribution of types I assume that I know the initial (at the average
state) consumption distribution. As I have argued before, at the average state is almost
identical to the distribution of wealth (as I can verify using the numerical method sketched
before). In addition, from an empirical point of view, the distribution of consumption across
types is as easily observable as the distribution of wealth.
I introduce an auxiliary concept. For a given endowment/habit ratio ω, I define the
probability measureP
ω
pτ q that assigns to agents of type γ 1 {τ probabilities equal to
their equilibrium consumption share. Furthermore, letE
ω
denote the expectation operator
under this probability measure. Then, from Proposition 1, this corollary follows,
Corollary 2.2. The probability measure at the average state is given by,
P
¯ ω
pτ q θ
0
pτ q
E
0
‚
t ¥0
δ
t
p
t
Y
t
E
0
‚
t ¥0
δ
t
p
t
Y
t
e
τzt p ωt ¯ ω q
, (2.9)
and, therefore, the following relation holds,
exp pω
t
¯ ω q E
¯ ω
rexp p τz
t
qs, t ¥ 0. (2.10)
I first note thatP
¯ ω
is very close to θ
0
pτ q, since the fraction on the right-hand side of
(2.9) is close to one. This is due to the fact that, for anyτ,e
τzt p ωt ¯ ω q
does not vary much
in equilibrium and it is “centered” around one, its value at the average state.
In addition, the right-hand side of (2.10) is the moment generating function of τ and
hence, for certain distributions it is known, and it is straightforward to derivez. However,
19
in general, it is straightforward to approximate with high precision the pricing kernel by
discretizing the distribution and using numerical integration methods. When agents are
identical, the pricing kernel is linear in the state,z pω q γ pω ¯ ω q.
In the following lemma, I can analyze the properties of the pricing kernel with the help
of the probability measure I just introduced.
Lemma 2.1. The pricing kernel is a continuous function of the state with a negative first
derivative and positive second derivative,
z
1
pω q 1
E
ω
pτ q
, (z2)
z
2
pω q E
ω
pτ
2
q E
ω
pτ q
2
E
ω
pτ q
3
. (z3)
Furthermore,
lim
ω 8
z pω q 8 , lim
ω 8
z
2
pω q 0,
lim
ω 8
z
1
pω q min
γ PΓ
γ, lim
ω 8
z
1
pω q max
γ PΓ
γ.
As a natural extension of what happens in an economy populated by a single agent,
equation (z2) shows that the slope of the pricing kernel is negative and equal to the inverse
of the weighted average risk tolerance in the economy. If time was continuous |σz
1
pq|
would correspond to the price of risk where σ is the volatility of consumption growth.
Bhamra and Uppal (2007) as well as Gˆ arleanu and Panageas (2008) also show that the price
of risk is determined by the weighted harmonic average of the risk-aversion of the agents.
Furthermore, from (z3), the curvature of the pricing kernel depends on the dispersion of the
risk tolerance types, which from (z2) implies that pricing kernel and average risk tolerance
in the economy are more volatile when the dispersion of types is higher. This is due to the
20
fact that the more variability there is in types, the more extreme the investment positions of
the agents are, and this leads to bigger changes in the cross-sectional wealth distribution.
2.3 The Representative Agent Equivalent Economy
In this section I construct a representative agent economy with state-dependent risk-
aversion parameter that is equivalent to the heterogeneous agent economy. I say that
two economies are equivalent when they have the same aggregate endowment process,
financial market structure and pricing kernel process. I will derive an expression for the
stochastic risk-aversion parameter of the representative agent and study its properties. The
risk-aversion of the representative agent provides a natural measure of risk-aversion in the
heterogeneous agent economy.
2.3.1 Preferences and Equilibrium
In this economy, there is a representative agent with stochastic risk-aversion coefficient that
depends on the stateω. The representative agent has the following utility,
U
r
pc,x q E
0
‚
t ¥0
δ
t
c
1 γ pωt q
t
pX
r
t
q
γ pωt q ρ
1
1 γ pω
t
q
,
where the external habit is X
r
t
X
t
e
¯ ω
, for all t ¥ 0. With this change in the definition
of the habit process, the stochastic risk-tolerance of the representative agent is equal to the
average (using the probability measureP
ω
) risk-tolerance of the economy at the average
state. With the standard definition, this would be true at the state ω 0. Since I would
like the risk-aversion of the representative agent to be a good representation of the risk-
aversion in the economy, this choice seems more appropriate. As in the heterogeneous
agents economy, I require a negative externality from the habit, and for that I need the habit
parameter to be always less than the risk-aversion parameter, i.e. ρ ⁄ min
ω
γ pω q.
21
For a consumption price process pp
t
,t ¥ 0 q, the representative agent maximizes the
previous utility subject to the intertemporal budget constraint. Financial equilibrium exists
when there is a pricing kernel process pz
r
t
,t ¥ 0;z
r
0
0 q such that the consumption
process pY
t
,t ¥ 0 q is optimal for the representative agent. The following proposition states
the result.
Proposition 2.2. The equilibrium pricing kernel of the representative agent economy is a
function of the state, and is characterized by the following equation,
1 exp rz
r
t
γ pω
t
qpω
t
¯ ω qs, t ¥ 0. (2.11)
For the representative agent economy to be equivalent to the heterogeneous agent econ-
omy, it suffices to have the same equilibrium pricing kernel processes in the two economies.
Hence, I require thatz pω q z
r
pω q for allω. This gives rise to the following corollary.
Corollary 2.3. Letz pω q be the equilibrium pricing kernel of the heterogeneous agent econ-
omy. Then the representative agent economy is an equivalent economy if the stochastic risk
aversion of the representative agent is the following continuous function of the state,
γ pω q $
’
&
’
%
z pω q
ω ¯ ω
, ω ¯ ω
z
1
p¯ ω q 1
E
¯ ω
pτ q
, ω ¯ ω
(2.12)
I point out that γ p¯ ω q can be set to any value.
2
, but the choice in (2.12) makes the
stochastic risk aversion function continuous at ¯ ω. From the properties of the pricing kernel
function I can derive certain properties of the risk-aversion function. I summarize them in
the following corollary.
2
In equation (2.11) I have the term γ pω
t
qpω
t
¯ ω q, therefore it does not matter what γ p¯ ω q is, since it is
always multiplied by zero.
22
Corollary 2.4. The stochastic risk aversion functionγ pω q, characterized in (2.12), has the
following first and second derivatives,
γ
1
pω q 1
ω ¯ ω
γ pω q 1
E
ω
pτ q
0, (γ1)
γ
2
pω q 1
ω ¯ ω
r 2γ
1
pω q z
2
pω qs. (γ2)
Furthermore,
lim
g 8
γ
1
pω q 0, lim
g 8
γ
2
pω q 0
lim
g 8
γ pω q min
γ PΓ
γ, lim
g 8
γ pω q max
γ PΓ
γ.
The negative first derivative of the risk-aversion function establishes the counter-
cyclicality of risk aversion in the economy. The risk aversion of the economy moves from
the highest level to the lowest as the endowment habit ratio goes from minus infinity to
plus infinity. Intuitively, more risk-tolerant agents invest more in the stock market and,
therefore, end-up wealthier in good states (high endowment/habit ratio) and poorer in bad
states.
2.4 Agent Distribution and the Variation ofγ pq
As I have explained before, it is straightforward to find the equilibrium pricing kernel
numerically. However, for my analysis, it is convenient to make the following parametric
assumption aboutP
¯ ω
pτ q: I assume that at the average state ¯ ω the risk-tolerance is gamma
distributed, with parametersκ andθ,
τ p¯ ω q Gamma pκ,ϑ q,
23
so thatE
¯ ω
pτ q κϑ andV
¯ ω
pτ q κϑ
2
, whereV denotes the variance. With this assump-
tion, I can derive the pricing kernel, and hence the risk-aversion function, in closed form.
Kimball et al. (2008) through survey responses, construct an empirical distribution of risk-
tolerance, and then fit it to a log-normal distribution. The log-normal distribution does not
have a moment-generating function, thus I choose the closely related gamma distribution.
3
I fit the gamma distribution by minimizing the overall distance between the distribution
of risk-aversion implied by the log-normal distribution and that implied by the gamma
distribution. In this exercise I am making the implicit assumption that the wealth or the
consumption share of an agent is independent of his/her relative risk aversion coefficient
at the average or initial state. Figure 2.1 plots the cumulative and density functions of the
log-normal distribution of Kimball et al. (2008), and the fitted gamma distribution. I note
that they are very close to each other but the gamma distribution exhibits a fatter tail. This
implies slightly higher mean and standard deviation of risk aversion. Let ¯ γ and ¯ ν denote,
respectively, the inverse of the average risk-tolerance and the standard deviation of risk-
tolerance, both at the average state. The estimated parameters imply that ¯ γ 5.17 and
¯ ν 0.13.
Given the parametric assumption about the cross-sectional consumption distribution of
types I have the following corollary.
Corollary 2.5. Let us defineη p¯ γ¯ ν q
2
. When the cross-sectional dispersion of types (risk-
tolerance) at the average state is gamma distributed with mean1 {¯ γ and standard deviation
¯ ν, and weight for each type given by the consumption share, the equilibrium pricing kernel
is
z pω q ¯ γ
exp r p ω ¯ ω qη s 1
η
.
3
It is straightforward to verify numerically that the results are insignificantly different when I assume the
log-normal distribution instead.
24
Figure 2.1: Fitted cross-sectional distribution of agents at ¯ ω
10
−1
10
0
10
1
10
2
0
0.2
0.4
0.6
0.8
1
Relative risk−aversion (log scale)
Cumulative distribution functions

Gamma
Log−normal
10
−2
10
−1
10
0
10
1
10
2
10
3
0
1
2
3
4
5
6
x 10
−3
Relative risk−aversion (log scale)
Density functions

Gamma
Log−normal
Given the estimated risk-tolerance distribution log pτ q N p 1.84,0.73 q, by Kimball
et al. (2008) I fit the gamma distribution by minimizing the distance between the two
distribution functions,
min
κ,θ
»
8
0
rF
1
pτ q F
2
pτ qs
2
dτ
where F
1
and F
2
are the log-normal and gamma distribution functions. The integral
was approximated numerically. The estimated parameters where κ 2.2474 and
θ 0.0860.
I point out that as the level of heterogeneity in the economy ¯ ν tends to zero, the pricing
kernel tends to ¯ γ pω ¯ ω q, as was noted in the discussion of Corollary 2.2. I now can
study the variability of the risk-aversion of the representative agent. First, I recall from
25
(2.12) that the risk-aversion of the representative agent is equal to ¯ γ. In addition, I define
the following function,
h pω q γ pω q
¯ γ
,
that is, the ratio of the risk-aversion coefficient of the representative agent in stateω, given
in (2.12), to the risk-aversion coefficient in the average state ¯ ω. It represents the coefficient
of variation (I will call it “multiplier”) of risk-aversion in a given state, with respect to the
average state. Using the result of corollary 2.5, I can express it in closed form. Figure 2.2
plotsh for three different values of ¯ ν, the value empirically estimated using the distribution
of Kimball et al. (2008), twice this value, and one half this value; the value of ¯ γ is the one
estimated given the distribution in the same study. The function is plotted for deviations
of the average state ranging between 1 to 1. As I will argue later, this kind of range
is unrealistically large. It is more natural to expect that the deviations will be less than
0.5 in absolute value: a deviation of 0.5 implies that the aggregate consumption surplus
ratio is higher than the average by around65%. From figure 2.2 I observe that the possible
variation in the coefficient of risk-aversion for the representative agent is very small. Unless
I assume a level of heterogeneity twice as much as the estimated level, the risk-aversion
of the representative agent is not expected to deviate from the average risk aversion by
more than 20% at any time. Even for the most extreme case I consider, the risk-aversion
of the economy doubles only when the economy is deep in recession. This plot is a first
indication that the effect of risk-aversion heterogeneity on asset prices, and particularly on
risk-premia, is potentially small.
In an economy with rational investors the risk-attitude of the representative agent can be
time-varying due to two reasons. The first one, which drives the risk-aversion coefficient in
this model, is the evolution of the cross-sectional wealth distribution. Unless the variation
in the state vector that determines the cross-sectional wealth distribution is very high, the
wealth reallocation across time cannot have a substantial effect on asset prices, as I have
26
Figure 2.2: Variation in the representative agent risk aversion
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
Function h
Deviations from the average state

0.5
1
2
Function h is the multiplier of the risk-aversion of the representative agent as the
economy deviates from the average state. For the three plots I multiply the estimated
standard deviation of risk-tolerance with the corresponding number while the average
risk-tolerance is the one estimated.
already seen. A second possible reason for time-variation in the risk-aversion coefficient
of the representative agent is time-variation in the individual risk-aversion coefficients. In
order to have a high variation in the risk-preferences of the representative agent I would
also need the individual preferences to be moving together, and in the same direction. In
particular the entire distribution needs to be moving up and down with the state variable.
2.5 Habit Process and Asset Prices
In order to solve for asset prices, I need to specify a process for the external habit. Follow-
ing Chan and Kogan (2002) and a good part of the literature that uses “catching up with the
27
Joneses” preferences, I assume that the habit is a weighted average of the previous habit
and the previous aggregate endowment level,
x
t 1
λy
t
p1 λ qx
t
. (2.13)
From (2.13) the endowment/habit ratio, the state variable, follows a mean-reverting pro-
cess,
ω
t 1
ω
t
λ pω
t
¯ ω q σǫ
t 1
. (2.14)
The unconditional mean of the state variable is ¯ ω μ {λ, and the unconditional volatility is
σ
ω
σ {
a
λ p2 λ q. The reversion rate parameterλ is particularly important in this model
because it determines the likely “range” in which the state variable moves through the
dependence ofσ
ω
onλ. The smallerλ is, the bigger is the unconditional variance. Figure
2.2 shows that risk-aversion of the representative agent varies more the larger the “range”
of the state variable is. Therefore, asλ decreases, the potential effect of agent heterogeneity
on asset prices becomes larger. When the state of the economy moves further away from its
average state the wealth allocation tends to be concentrated on the more or less risk-averse
agents, depending on whether the deviation is negative or positive, respectively. Hence,
with a smaller rate of reversion (equivalent to higher persistence in the state variable) larger
reallocations of wealth and, therefore, variations in the risk-aversion of the economy, are
possible. However, these large swings in risk-aversion require a long time. For example,
Campbell and Cochrane (1999) use a reversion rate value of 0.13 which implies a half-life
of around 5 years,
4
while Chan and Kogan (2002) use values that imply half-lives of 12
years for their heterogeneous agents economy, and around 17 years for their single agent
4
The half-life is the time required for the deterministic version of the process to cover half of the distance
to the unconditional mean. It is given by log p2 q{log p1 λ q
28
economy. Persistence in the state variable translates into persistence in the price-dividend
ratio. It is natural therefore to select this parameter in order to match the price-dividend
ratio persistence implied by the model to the persistence observed in the data.
2.5.1 The Stochastic Discount Factor
The fundamental price of an asset is the expected value of the discounted future dividends.
LetM
t 1
(m log pM q) denote the one-period stochastic discount factor between periods
t andt 1. Sincep
t
denotes the price of a unit of consumption in periodt, the stochastic
discount factor is,
M
t 1
δ
p
t 1
p
t
, t ¡ 0.
Using equation (2.5) and the other assumptions, I have the following corollary:
Corollary 2.6. Assume that the habit process is as in (2.13) and the cross-sectional dis-
tribution of types (risk-tolerance) with respect to their consumption share at the average
state is gamma distributed, with mean 1 {¯ γ and standard deviation ¯ ν. The equilibrium one
period log stochastic discount factor is conditionally log-normally distributed,
m
t 1
log pδ q ρλω
t
¯ γ
η
e
η pωt ¯ ω q
1 e
λη pωt ¯ ω q σηǫ
t 1
.
When agents are homogeneous, i.e. ¯ ν 0,
m
t 1
log pδ q ρλω
t
¯ γ pω
t 1
ω
t
q.
I observe that the stochastic discount factor of both the standard Lucas tree and the
model of Campbell and Cochrane (1999) are particular cases of the stochastic discount fac-
tor given in corollary 2.6. Campbell and Cochrane (1999) assume homogenous agents, but
29
their state variable, the consumption surplus (x using my notation) satisfies some specific
dynamics that, Chan and Kogan (2002) argue, might be obtained as the result of simpler
dynamics and agents with heterogeneous risk-aversion.
5
One of my objectives is to study
this point further. For example, Campbell and Cochrane (1999) assume that their state vari-
able is counter-cyclical, and that would explain the counter-cyclicality of the risk premium.
In our model (as in Chan and Kogan (2002)), the equilibrium risk-aversion of the repre-
sentative agent turns out to be negatively related to the state of the economy (γ
1
pω q 0).
However, I have also showed that the possible variation of the risk-aversion of the repre-
sentative agent (figure 2.2) is relatively modest for realistic parameter values, and it seems
difficult to argue that it can explain the variation in the risk-premium observed in the data.
I next elaborate further on this point.
The main driving forces of Campbell and Cochrane (1999) are: (i) the persistence in
the consumption surplus ratio, which produces the persistence in price dividend ratios and
the variability of stock expected returns; (ii) the counter-cyclical conditional volatility of
the state variable (their consumption surplus ratio or in my case of x). In Campbell and
Cochrane (1999), the varying conditional volatility is chosen so that it fixes the risk-free
rate at a certain level, and the entire variation in expected returns translates into variation
in risk premia. In my model the persistence of the state variable is the result of assuming
persistence in habit, but the varying conditional volatility of the stochastic discount factor
is endogenous and related to the variation in risk-aversion. For simplification purposes, I
5
In the special case of Campbell and Cochrane (1999)x is the state-variable and notω, since it is Markov
stationary, following the process,
x
t 1
x
t
λ px
t
¯ x q ˜
φ px
t
qǫ
t 1
,
whereǫ is the aggregate endowment growth innovation and
˜
φ px q is some given function. Hence I have,
ω
t 1
ω
t
μ λ px
t
¯ x q ¯ γ
1 ˜
φ px
t
q
σǫ
t 1
. Sinceρ 0 in this special case I obtain the same stochastic discount factor.
30
assumee
σηǫ
1 σηǫ and introduce ˜ m, an approximation to the true stochastic discount
factor,
˜ m
t 1
log pδ q μγ p¯ ω
t 1
q rγ p¯ ω
t 1
q ρ sλω
t
rγ pω
t
q γ p¯ ω
t 1
qs pω
t
¯ ω q
¯ γσ r1 φ pω
t
qsǫ
t 1
(2.15)
where
φ pω
t
q η p1 λ qp¯ ω ω
t
qh p¯ ω
t 1
q, (2.16)
and ¯ ω
t 1
E
t
rω
t 1
s. Since ση is a very small number, the approximation is indeed very
good. ˜ m is conditionally normally distributed with an endogenously varying conditional
volatility equal to ¯ γ r1 φ pω
t
qsσ. The conditional volatility of ˜ m has the same form as the
conditional volatility of the pricing kernel of Campbell and Cochrane (1999). The only dif-
ference is that the function φ pω q in Campbell and Cochrane (1999) is exogenously given,
and it varies considerably more than ours, as I show next. In figure 2.3 I plot the function
φ for three different levels of agent heterogeneity. The label of each line is the number that
multiplies the value of the risk-tolerance standard deviation ¯ ν estimated by Kimball et al.
(2008). To get the same persistence in the state variable as Campbell and Cochrane (1999)
I setλ 0.13. The sensitivity functionφ in Campbell and Cochrane (1999) ranges in value
from around 50 to 0. Clearly, the level of variation that can be generated endogenously in
my economy is substantially smaller, implying that my economy will not be able to predict
substantial variation in risk-premia. In addition, the level of the conditional volatility is
quite small, and therefore this economy cannot predict the high equity premium observed
in the data. Unless the consumption risk were substantially higher, and the level of hetero-
geneity in the economy significantly bigger than the estimate of Kimball et al. (2008), it
31
is unlikely that a substantial part of the observed variation in risk-premia can be explained
with risk-preference heterogeneity.
Figure 2.3: Variation in the conditional volatility of ˜ m
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Function φ
Deviations from the average state

0.5
1
2
The conditional volatility of the approximate pricing kernel is,
¯ γ r1 φ pω qsσ.
For the three plots I multiply the estimated standard deviation of risk-tolerance with the
corresponding number while the average risk-tolerance is the one estimated. Parame-
ter λ was set to 0.13 for these plots.
2.5.2 Asset Prices
The assets of interest are the risk-free bond, that pays a unit of consumption next period,
and the infinitely lived market security, that pays the dividend process (2.2). The price of
the risk-free bond is
P
f
pω
t
q E
t
e
m pωt,ω
t 1
q
.
32
The price of the market security is increasing in the dividend, but the price-dividend ratio,
that I denotePD is stationary,
PD pω
t
q E
t
e
m pωt,ω
t 1
q d
t 1
dt
pPD pω
t 1
q 1 q
.
In the appendix I explain how to compute the prices numerically.
The continuously compounded risk-free rate is the negative log of the bond price. Using
(2.15) I can derive a good approximation,
˜ r
f
t
log pδ q μγ p¯ ω
t 1
q rγ p¯ ω
t 1
q ρ sλω
t
rγ p¯ ω
t 1
q γ pω
t
qs pω
t
¯ ω q ¯ γ
2
σ
2
r1 φ pω
t
qs
2
. (2.17)
The model of Campbell and Cochrane (1999) explains the observed low volatility of the
interest rate by assuming that the precautionary savings term is inversely proportional to the
habit term. In fact the conditional volatility of the pricing kernel of their model is derived
by making the risk-free rate constant. The habit term refers to the incentive to postpone
consumption when consumption is high today with respect to habit. The precautionary
savings term refers to the incentive to save less when the real risk in the economy tomorrow
is low. In my model these two terms are the most significant and also inversely proportional
to each other. However, the habit term dominates the precautionary savings term unless the
level of heterogeneity in the economy is very high, and the fundamental risk of the economy
is significantly higher than in reality. I show this next.
In (2.17), both rγ p¯ ω
t 1
q ρ s and rγ p¯ ω
t 1
q γ pω
t
qs pω
t
¯ ω q are always posi-
tive. The variability of the interest rate in (2.17) comes mostly from the habit term,
rγ p¯ ω
t 1
q ρ sλω
t
, increasing in ω
t
and, possibly, from the precautionary savings term
¯ γ
2
σ
2
r1 φ pω
t
qs
2
, decreasing in ω
t
. The intertemporal substitution term μγ p¯ ω
t 1
q does
not vary much compared to the other terms, even when the level of heterogeneity is high.
33
The term rγ p¯ ω
t 1
q γ pω
t
qs pω
t
¯ ω q varies even less since the change in γ pω
t
q from t to
t 1 is very small.
The habit term says that when consumption increases with respect to habit, agents want
to save more in order to increase their future consumption, and this puts downward pressure
on the interest rate. The precautionary savings term is quadratic in the “expected” risk
aversion in the economy next period since φ pω
t
q is linear in γ p¯ ω
t 1
q. The precautionary
savings term’s variation depends on the risk of the economy as given byσ and the variability
of φ pω
t
q. I recall that φ pω q depends on h pω q, whose variability depends on both the level
of heterogeneity in the economy and the risk of the economy that drives the variation in
ω. When consumption increases with respect to the habit, the “expected” risk-aversion in
the economy decreases, and this has a positive impact on the interest rate. Since these two
terms work in opposite directions, the only way in which a decrease of the variability of
the risk-free rate happens is if the precautionary savings part varies enough to offset part of
the variation coming from the habit incentive. This is possible only when the risk is high
and the level of heterogeneity in the economy is also high.
Using the expression for ˜ m I can also approximate well the maximum possible Sharpe
ratio in the economy. The maximum Sharpe ratio, that I denote SR, is obtained when an
asset (or portfolio of assets) is conditionally perfectly correlated with consumption growth.
Since ˜ m is conditionally normally distributed, the maximum Sharpe ratio, which is derived
from the usual Euler equation on excess returns, takes the following simple form,
SR pω
t
q a
e
γ
2
σ
2
r1 φ pωt qs
2
1.
From this expression I see how the function φ pω
t
q relates directly to the variation in the
price of risk. I also observe that the unconditional average of the Sharpe ratio is not affected
substantially by the variation ofφ, and hence by the level of heterogeneity in the economy.
34
2.6 Quantifying The Effect of Agent-Heterogeneity
In this part of the study I try to assess quantitatively up to what extent risk-aversion hetero-
geneity can explain the value of financial variables observed in the economy. In particular,
I study the price of risk, the equity premium, the risk-free rate, the price dividend ratio and
the conditional volatility of stock returns. The main economic implications of the model
come from two key elements. First, the persistent habit, which is able to produce persis-
tence in the price dividend ratio. Second, the endogenously generated varying conditional
volatility of the stochastic discount factor due to the heterogeneity in risk-preferences. The
main question I try to address with this model is whether risk-preference heterogeneity can
produce enough variation in the price of risk so as to be able to explain the low variability
in the risk-free rate, the variation in equity premia and the long-run predictability of excess
returns. After I calibrate the heterogeneous-agent economy, I compare its predictions to a
homogeneous-agent -but otherwise identical- economy. This exercise allows us to quantify
the marginal effect resulting from the heterogeneity of risk-aversion. I next explain the
methodology I follow to calibrate my model to real data.
2.6.1 Calibration Procedure
As I have discussed, the price impact of risk-aversion heterogeneity depends on several
parameters. First and foremost, it depends on the level of heterogeneity, measured by
the standard deviation of risk-tolerance at the average state. More heterogeneity induces
agents to take more extreme positions, and this leads to higher variability in the cross-
sectional wealth and consumption distributions, and hence higher variability in the risk-
aversion of the representative agent of the economy. I parameterize these distributions
using the empirical findings of Kimball et al. (2008). The second channel through which
agent heterogeneity affects prices is the unconditional volatility of the state. If the state
of the economy is very volatile, then there is more wealth re-distribution over time and,
35
therefore, more volatility in the risk-aversion of the economy. The unconditional volatility
of the state variable depends on the persistence parameterλ and the volatility of consump-
tion growth. To estimate the mean and standard deviation of aggregate consumption I use
NIPA data on real consumption growth between1930 and2005. The persistence parameter
is not directly observable, but can be selected so as to match the persistence it induces in
price-dividend ratios: I estimate λ by fitting the model implied price-dividend ratio auto-
correlation function (up to lag 7) to the autocorrelation parameter found in the data. The
price-dividend ratio data I use is the annual series of Boudoukh, Michaely, Richardson, and
Michael (2007), which includes common share repurchases from cash flow statements.
Both the persistence parameterλ and the habit parameterρ have a strong effect on the
unconditional volatilities of the price-dividend ratio and the risk-free rate. In the calibration
exercise I include both volatilities. I have already listed the source for the price-dividend
ratio. For the (real) risk-free rate I take the yield of the 3-month Treasury bill after subtract-
ing the realized inflation provided in NIPA. The risk-free rate volatility for the full sample
is slightly over3%. The post-war period, during which inflation was more predictable, the
risk-free rate volatility drops to a bit less than 2%. For this reason, I put a smaller weight
on the risk-free rate volatility in the calibration exercise.
The last parameter I need to calibrate is the subjective discount factorδ. This parameter
affects the average level of the price-dividend ratio, as well as the average risk-free rate.
Since I am unable to fit both of them at the same time, following the emphasis of the
literature, I exclude the average risk-free rate and focus on the average price-dividend ratio.
This underscores the inability of the model to explain the average excess return found in
the data. As it will be shown, in order to produce an excess return high enough, I need
to assume a volatility of consumption growth significantly larger than the one estimated
36
from the data. As I have explained, such an additional real risk amplifies the effect of risk-
aversion heterogeneity. For that reason, in the calibration exercise I include the average
excess return, but with a small weight.
The model solves for the price-dividend ratio,pd, and risk-free rate,r
f
, endogenously.
Then, for a given set of parameters, I take the observed consumption growth series and the
estimated value for λ and generate a time series for ω. I use this time-series to compute
the model-implied time-series for pd and r
f
. From the price-dividend time-series and the
dividend growth time-series I derive the market returns.
I calibrate the model for three different sets of parameters that I henceforth call models
(or parameterizations) 1, 2 and 3. These models differ in the parameter values that explain
consumption growth and dividend growth, as well as the parameters that characterize the
distribution of agents. Models 1 and 2 assume the distribution of agents estimated using the
distribution of Kimball et al. (2008), with ¯ γ 5.17 and ¯ ν 0.13. Models 1 and 3 assumes
the estimated consumption growth parameters, and that dividend and consumption are the
same. Model 2 uses instead the estimated statistics of the dividend growth process, which
displays a smaller average than that of consumption growth (2.40% to 3.12%) and sub-
stantially higher volatility (13.73% to2.18%); its correlation with the consumption growth
time-series is 0.09. The objective of the third model is to produce a sizable equity pre-
mium, similar to the one observed in the economy; I choose the parameters of this model
with that objective in mind: I assume that volatility of consumption growth is 2.5 times
the estimated level, and that the standard deviation of risk-tolerance is twice as high as that
estimated using Kimball et al. (2008).
37
The parameters values ofλ,ρ andδ minimize the following objective,
J
j
pλ,ρ,δ q rμ
j
pPD q μ pPD qs
2
rσ
j
pPD q σ pPD qs
2
7
‚
i 1
acf
i
j
pPD q acf
i
pPD q
2
w
e
rμ
j
pR
e
q μ pR
e
qs
2
w
f
σ
j
pR
f
q σ pR
f
q
2
, (2.18)
for each modelj. μ
j
px q andσ
j
px q denote the model implied (for the observed consumption
growth), average and standard deviation, respectively, of the time series of a variable x.
acf
i
denotes the autocorrelation for lag i. μ, σ and acf denote the data estimated values.
In table 2.1 I collect the values of the parameters for each model both those assumed and
Table 2.1: Model parameterizations
Data 1 2 3
μ 3.1197 3.1197 3.1197 3.1197
σ 2.1834 2.1834 2.1834 5.4584
μ
d
2.3937 3.1197 2.3937 3.1197
σ
d
13.7272 2.1834 13.7272 5.4584
̺ 0.0898 1.0000 0.0898 1.0000
λ
0.1203 0.1203 0.0960
¯ γ 5.1739 5.1739 5.1739 5.1739
¯ ν 0.1289 0.1289 0.1289 0.2579
ρ
-3.1654 -3.1899 1.4488
δ
0.8404 0.8383 0.9642
¯ ω 0.2593 0.2593 0.3250
σ
ω
0.0459 0.0459 0.1277
half life 5.4077 5.4073 6.8683
The parameters with
were chosen by minimizing an objective function that measures
the squared errors in certain quantities between the data and those of the model implied
time-series. The quantities were the mean, standard deviation and autocorrelation
function of the price-dividend ratio, the standard deviation of the risk-free rate and the
mean equity premium. The model implied time-series of the price-dividend ratio and
the risk-free rate were generated by using the true consumption and dividend growth
series.
38
those computed according to (2.18). The estimates for models 1 and 2 are almost identical.
I estimate the persistence parameterλ as explained before, and I find it to be 0.12, similar
to the value in Campbell and Cochrane (1999). The persistence of the price-dividend ratio
in the model implied time-series (not shown) is in fact slightly higher than the persistence
in the data. Model 2 requires a slightly higher (in absolute value) habit parameter value
in order to capture the unconditional volatility of the price-dividend ratio. Due to the low
correlation of the dividend growth with consumption growth, the systematic risk of the
market security is effectively lower in model 2 than in 1. In model 3 the resulting value
forλ is smaller (more persistence), whileδ is higher than for models 1 and 2. This is due
to the fact that higher risk and higher heterogeneity both induce higher variability in the
price-dividend ratio. For the same reason the value of ρ is much higher in model 3 since
less variability in the price-dividend ratio need to be produced by the habit effect.
2.6.2 Analysis
The main purpose of this section is to study whether time variation in the risk-aversion
of the economy coming from agent heterogeneity can produce the asset pricing facts that
Campbell and Cochrane (1999) focus on. These facts are the high and counter-cyclical
equity premium, the high variability and persistence of the price-dividend ratio, the low
volatility of the risk-free rate and the price-dividend ratio’s ability to predict long-run excess
returns. I show first that under the baseline calibration of the model the impact of risk-
aversion heterogeneity is negligible. The reason is twofold: First the consumption risk is
small, and even with a persistent state variable, the wealth reallocation in the economy is
minimal. Second, the level of risk-aversion heterogeneity as estimated by Kimball et al.
(2008) is again quite small and the wealth reallocation does not lead to significant changes
in the risk-aversion of the economy.
39
One might argue that the fundamental risk is higher than consumption risk and that
maybe the level of heterogeneity is not well captured by the empirical study of Kimball
et al. (2008). I increase both considerably and look at the new predictions. The model
performs better when compared to the data, and heterogeneity has a noticeable effect on
prices. However, the risk-free rate is still high on average and quite volatile, the greater
part of the equity premium is still term-premium and the predictability of long-run excess
returns does not come close to that found in the data.
In the analysis, I first study price and return unconditional statistics (table 2.2). For
each model I generate two sets of results: (i) model statistics, that correspond to the long-
run unconditional values as time tends to infinity; (ii) historical simulation statistics that
correspond to the sample values estimated from the model implied time-series. I emphasize
that the model implied time-series is generated using the model solutions of the pd-ratio
andr
f
-rate and the historical data on consumption and dividend growth. These time-series
are shown for each model in figures 2.4, 2.6 and 2.8 respectively.
I then consider autocorrelations of the pd-ratio and the excess return, as well as the
pd-ratio predictability regressions (tables 2.3, 2.4 and 2.5). These statistics are the result
of simulating the economy 1000 times. For each simulation I generate 75 years of annual
artificial data which corresponds to the frequency and length of the real data. The reported
statistics are the averages of the estimated values of each simulation. Finally I analyze
the model implied conditional price and return statistics in figures 2.5, 2.7 and 2.9, each
corresponding to one of the three parameterizations.
Pricing effect of risk-aversion heterogeneity
The top panel of table 2.2 compares unconditional price and return statistics of the data
and model 1. In the heterogeneous economy, the variability, as well as the level of the
price-dividend ratio, is fitted relatively well. The average market return is predicted at a
40
Table 2.2: Price and return statistics
Parametrization 1
Heterogeneous ec. Homogeneous ec.
Data Model Hist.sim. Model Hist.sim.
E pR
f
q 0.5667 7.2238 6.0343 7.2261 6.0234
E pR
e
q 7.6821 1.9119 2.5181 1.9116 2.4963
σ pR
e
q 20.0214 17.2009 20.3598 17.2248 20.2110
σ pR
f
q 3.8288 4.9264 5.1531 4.9418 5.0990
E pSR q 0.3849 0.1112 0.1154 0.1110 0.1154
log pE pPD qq 3.1928 3.2118 3.2765 3.2120 3.2763
log pσ pPD qq 1.9715 2.0222 1.9062 2.0299 1.9090
̺ py
t 1
y
t
,r
m
t 1
q 0.0839 0.9548 0.5903 0.9552 0.5889
Parametrization 2
E pR
f
q 7.4124 6.2175 7.4146 6.2066
E pR
e
q 0.1017 2.3392 0.1014 2.3175
σ pR
e
q 22.1841 20.3683 22.2019 20.2204
σ pR
f
q 4.9490 5.1771 4.9644 5.1228
E pSR q 0.0064 0.1071 0.0064 0.1070
log pE pPD qq 3.2114 3.2762 3.2116 3.2760
log pσ pPD qq 2.0234 1.9071 2.0311 1.9100
̺ py
t 1
y
t
,r
m
t 1
q 0.7076 0.5906 0.7081 0.5892
Parametrization 3
E pR
f
q 3.7433 3.0993 4.3756 3.0271
E pR
e
q 6.1415 6.3984 5.7333 5.6413
σ pR
e
q 20.9119 24.7746 21.1446 20.0496
σ pR
f
q 3.4194 4.5910 4.7691 4.7444
E pSR q 0.2937 0.2395 0.2711 0.2642
log pE pPD qq 3.1970 3.2818 3.1606 3.2417
log pσ pPD qq 1.9573 1.8995 2.1081 1.9781
̺ py
t 1
y
t
,r
m
t 1
q 0.9293 0.5901 0.9695 0.5949
The historical simulations were generated by using the true consumption and dividend
growth series. The homogeneous economy of each parametrization sets the level of
heterogeneity to zero by setting ¯ ν 0.
bit more than 1% higher than what is found in the data. The model completely misses
the average risk free rate, predicted at around 7.22%; the historical simulation average
(explained before) is around 6% while the data average is less than 1%. This is the result
41
Table 2.3: Autocorrelations of price-dividend ratio
Lag
1 2 3 5 7
Data 0.88 0.75 0.65 0.46 0.28
Heter.ec.1 0.83 0.68 0.56 0.36 0.22
Homog.ec.1 0.83 0.68 0.56 0.36 0.22
Heter.ec.2 0.83 0.68 0.56 0.37 0.23
Homog.ec.2 0.83 0.68 0.56 0.37 0.23
Heter.ec.3 0.86 0.74 0.64 0.46 0.32
Homog.ec.3 0.85 0.72 0.61 0.43 0.30
The autocorrelation function for each model was estimated by averaging the sample
estimates of 1000 simulations of the same length as the data.
Table 2.4: Autocorrelations of excess returns
Lag
1 2 3 5 7
Data -0.06 -0.20 0.03 -0.05 0.02
Heter.ec.1 -0.01 -0.02 -0.01 -0.02 -0.01
Homog.ec.1 -0.01 -0.02 -0.01 -0.02 -0.01
Heter.ec.2 -0.01 -0.01 -0.02 -0.01 -0.01
Homog.ec.2 -0.01 -0.01 -0.01 -0.01 -0.01
Heter.ec.3 -0.04 -0.03 -0.03 -0.02 -0.01
Homog.ec.3 -0.01 -0.01 -0.02 -0.01 -0.01
The autocorrelation function for each model was estimated by averaging the sample
estimates of 1000 simulations of the same length as the data.
of the inability of the model to generate enough conditional volatility for the stochastic
discount factor that would explain both the high equity premium along with the low risk-
free rate. The historical simulation captures the return variability but the model predicts a
number around 3% lower than the data return variability. The risk-free rate, on the other
hand, is more volatile in the model by a bit more than 1%. From the previous results, it
is not surprising that the model predicts a very small average Sharpe ratio of around 0.11
while the data average is more than three times higher. The results from the homogeneous
42
economy of the first parametrization are almost identical to these. The differences are
negligible and, therefore, the two economies cannot be separated based on such data. The
endogenous variation in the volatility of the stochastic discount factor (functionφ and figure
2.3) generated by the variation in the risk-aversion of the economy is so small that it cannot
be detected either in the risk-free rates, the risk premia, or the return volatilities.
Figure 2.4: Data and model 1 implied time series of price-dividend ratio and risk-free rate
1930 1940 1950 1960 1970 1980 1990 2000
2
2.5
3
3.5
4

recessions
data
het
hom
1930 1940 1950 1960 1970 1980 1990 2000
−15
−10
−5
0
5
10
15
20
25

recessions
data
het
hom
The model implied time-series of the price-dividend ratio and the risk-free rate were
generated by using the true consumption and dividend growth series.
Figure 2.4 shows the model 1 implied time-series of the pd-ratio and r
f
-rate for both
the heterogeneous agents and the homogeneous agents economies. From the theoretical
43
and empirical results I have obtained so far it is no surprise that the two time-series are
indistinguishable. The two economies are also indistinguishable when it comes to the auto-
correlation coefficients shown in the top panels of tables 2.3 and 2.4. There is a slight dif-
ference in the predictability regressions in the top panel of table 2.5. The data predictability
starts with a modest value of around7% increases monotonically with horizon and reaches
a level of 37%. For the homogeneous economy the predictability starts at 0.13% and goes
up to 5.81%, while for the heterogeneous economy these numbers range from 0.15% to
5.87%.
In figure 2.5 I plot some conditional statistics. The state variable is shown within four
standard deviations from the unconditional mean. The risk-aversion of the representative
agent varies little, as it ranges from slightly less than5 to only5.4. The risk-aversion coef-
ficient for the homogeneous economy is 5.17. For this reason, the curvature in the pricing
kernel is almost unnoticeable. From lemma 2.1 I know that the slope of the pricing ker-
nel is equal to the consumption-weighted harmonic average risk-aversion in the economy.
Similarly, the price-dividend ratio, as well as the risk-free rate of the homogeneous and
heterogeneous economies of model 1, are once more almost indistinguishable.
The curvature of the price-dividend ratio function is closely related to the conditional
return volatility. When the price-dividend ratio is flatter around the expected future state,
then the return volatility is smaller. Since heterogeneity in the economy makes thepd-ratio
slightly flatter during expansions and slightly steeper during recessions, the conditional
volatility of returns becomes slightly more counter-cyclical with heterogeneity. The reason
why thepd-ratio behaves in this way is the following. Whenω increases, the price of con-
sumption today drops in relation to the future, and agents want to postpone consumption.
Hence all asset prices increase. However, the discount rates decrease at a decreasing rate,
since the risk-aversion of the economy decreases as the state of the economy improves, and
this makes thepd-ratio flatter when the price of consumption is low.
44
Figure 2.5: Model parametrization 1
0 0.1 0.2 0.3 0.4 0.5
−1
−0.5
0
0.5
1
Pricing Kernel (z)
ω
0 0.1 0.2 0.3 0.4 0.5
4.9
5
5.1
5.2
5.3
5.4
Representative Agent Risk−Aversion (γ)
ω
0 0.1 0.2 0.3 0.4 0.5
2
2.5
3
3.5
4
4.5
Log of Price/Dividend Ratio
ω
0 0.1 0.2 0.3 0.4 0.5
0.1
0.105
0.11
0.115
0.12
0.125
Conditional Sharpe Ratio
ω
0 0.1 0.2 0.3 0.4 0.5
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
Conditional Excess Return
ω
%
0 0.1 0.2 0.3 0.4 0.5
−20
−10
0
10
20
30
Risk−free Rate (γ)
ω
%
0 0.1 0.2 0.3 0.4 0.5
15
15.5
16
16.5
17
17.5
18
18.5
Conditional Return Volatility
ω
%
0 0.1 0.2 0.3 0.4 0.5
−10
0
10
20
30
40
Conditional Market Return
ω
%
The range of the state variable is four standard deviations around its mean.
45
Table 2.5: Long-run predictability regressions
Parametrization 1
Data Heterogeneous ec. Homogeneous ec.
Horizon coef. std.er. R
2
coef. std.er. R
2
coef. std.er. R
2
1 -0.18 (0.05) 6.86 -0.03 (0.05) 0.15 -0.03 (0.05) 0.13
2 -0.30 (0.10) 11.29 -0.02 (0.10) 1.34 -0.02 (0.10) 1.32
3 -0.41 (0.12) 18.66 -0.01 (0.13) 2.48 -0.01 (0.13) 2.44
5 -0.61 (0.16) 29.07 -0.01 (0.20) 4.35 -0.01 (0.20) 4.30
7 -0.82 (0.21) 36.82 -0.03 (0.24) 5.87 -0.03 (0.24) 5.81
Parametrization 2
1 -0.02 (0.07) 0.01 -0.02 (0.07) 0.00
2 -0.00 (0.12) 1.22 -0.00 (0.12) 1.21
3 0.01 (0.17) 2.38 0.01 (0.17) 2.37
5 0.01 (0.25) 4.24 0.01 (0.25) 4.21
7 0.00 (0.30) 5.68 0.00 (0.31) 5.65
Parametrization 3
1 -0.08 (0.05) 2.23 -0.03 (0.06) 0.12
2 -0.10 (0.09) 3.50 -0.02 (0.10) 1.42
3 -0.12 (0.13) 4.98 -0.01 (0.15) 2.73
5 -0.16 (0.18) 7.53 -0.01 (0.22) 4.79
7 -0.20 (0.22) 9.67 -0.01 (0.27) 6.40
Regressions were run with log price-dividend ratio as the predictive variable and j-
period realized excess log returns on the left hand side,
r
e
t j
β
0
β
1
pd
t
ε
t j
.
The regression estimates for each model are the averages of the sample estimates of
1000 simulations of the same length as the data. The standard errors were corrected
using a GMM procedure and the Newey-West weighting scheme.
Finally I look at the variation in the Sharpe ratio. For the homogeneous agent economy
the variation is only due to the conditional correlation between the market returns and con-
sumption growth. The maximum Sharpe ratio in an economy where consumption growth
is iid normal is constant. For model 1, the Sharpe ratio of the heterogeneous economy
becomes more counter-cyclical, but only marginally, since it decreases from around0.12 to
0.105, reflecting again the small variability in the risk-aversion of the economy.
46
Abnormally high risk and level of heterogeneity
I next show that even when I multiply the consumption risk estimated from the data by two
and a half times, and I double the level of heterogeneity found by Kimball et al. (2008)
(parametrization 3), the heterogeneous agents economy is not able to offer significantly
better predictions than the homogeneous agents economy. First, I consider the statistics
(bottom panel of table 2.2), and I observe that heterogeneity increases the equity premium
by about 0.5%. This parametrization manages to produce a sizeable equity premium of
more than 6%, not very far away from the data, which is 7.7%. It also predicts a sizeable
average Sharpe ratio of almost 0.3, while the average and volatility of the price-dividend
ratio, as well as the volatility of the equity premium, are close to the true values. However,
the homogeneous agents economy predicts an equity premium almost as high as the het-
erogeneous agents economy owing to the fact that the biggest part of the equity premium
in this model is in fact term-premium rather than risk-premium. As Abel (1999) explains,
the term- premium arises when the future discount rates are volatile, which is the case in
my model. Furthermore, from the historical simulation statistics it might be argued that the
homogeneous agents economy is as close, if not closer, to the data as the heterogeneous
agents economy. With the exception of the average excess return and the volatility of the
risk-free rate the rest of the statistics look marginally better for the homogeneous agents
economy.
With respect to the model unconditional statistics, the average risk-free drops from
4.37% to3.74% when I introduce heterogeneity. Even though this is a noticeable difference
it is still clearly different from the data. I emphasize again that the model is unable to fit
the average price-dividend ratio and the average risk-free rate simultaneously, even for this
parametrization. With this parametrization, heterogeneity also has a noticeable impact on
the volatility of the risk-free rate. The estimated data variability of the risk-free rate is
3.8%, given that this sample includes periods of unusual volatility. For example the post
47
Figure 2.6: Data and model 2 implied time series of price-dividend ratio and risk-free rate
1930 1940 1950 1960 1970 1980 1990 2000
2
2.5
3
3.5
4

recessions
data
het
hom
1930 1940 1950 1960 1970 1980 1990 2000
−15
−10
−5
0
5
10
15
20
25

recessions
data
het
hom
The model implied time-series of the price-dividend ratio and the risk-free rate were
generated by using the true consumption and dividend growth series.
war sample volatility of the risk-free rate is only around2%. Hence, the3.4% predicted by
the heterogeneous agents economy of model 3 still seems high.
The main difference between the heterogeneous agents and the homogeneous agents
economies relates to the variation in the price of risk, as shown in figure 2.9. While it is
almost flat for the homogeneous agents economy, when I introduce heterogeneity it goes
from around 0.6 down to almost 0.1. However, this quantity is not observable. What
48
is observable, and is relevant to the variation of the equity premium is the long-run pre-
dictability of thepd-ratio. The lower panel of table 2.5 shows that the heterogeneous econ-
omy does not produce a significantly greater level of predictability. For example the R
2
for the 7-year horizon is just less than 10% for the heterogeneous agents economy, while
it is almost 6.5% for the homogeneous agents economy. In addition, in the lower panel
of table 2.4 I observe that the heterogeneity does indeed have a noticeable impact on the
counter-cyclicality of the excess returns, but still does not yield the level of mean-reversion
shown in the data.
Additional findings
Figures 2.4, 2.6 and 2.8 show in the top panel the actual price-dividend ratio along with
the model implied time series for both the heterogeneous and homogeneous economies.
The first thing I observe is that the model implied time series of price-dividend ratio fol-
lows closely the true data. This stems from the high correlation between the time-series
of ω generated from the true consumption growth process and the price-dividend ratio.
The real evidence here is that positive consumption growth shocks are related to positive
shocks in the price-dividend ratio, that persist for many years. The habit process assumed
in this economy “captures” this behavior because of its persistence and its effect on the dis-
count rates. When consumption is high in relation to habit the demand for financial assets
increases and the subsequent increase in prices is expected to persist because it will take
time for the habit to adjust to the increased consumption. The model, however, is unable
to follow the true series during the period between 1930 and the first recession shown, as
well as during the stock market boom of the 1990’s. The negative consumption growth of
the1930’s is not associated with an analogous decline in prices, while the prices during the
1990’s go up very fast, while consumption growth was around its average level. Despite
49
Figure 2.7: Model parametrization 2
0 0.1 0.2 0.3 0.4 0.5
−1
−0.5
0
0.5
1
Pricing Kernel (z)
ω
0 0.1 0.2 0.3 0.4 0.5
4.9
5
5.1
5.2
5.3
5.4
Representative Agent Risk−Aversion (γ)
ω
0 0.1 0.2 0.3 0.4 0.5
2
2.5
3
3.5
4
4.5
Log of Price/Dividend Ratio
ω
0 0.1 0.2 0.3 0.4 0.5
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Conditional Sharpe Ratio
ω
0 0.1 0.2 0.3 0.4 0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Conditional Excess Return
ω
%
0 0.1 0.2 0.3 0.4 0.5
−20
−10
0
10
20
30
40
Risk−free Rate (γ)
ω
%
0 0.1 0.2 0.3 0.4 0.5
19
20
21
22
23
24
25
Conditional Return Volatility
ω
%
0 0.1 0.2 0.3 0.4 0.5
−10
0
10
20
30
40
Conditional Market Return
ω
%
The range of the state variable is four standard deviations around its mean.
50
this, it seems that a slowly moving habit process, along with a free utility parameter on
habit, is able to fit the time variability in aggregate prices to some extent.
Figure 2.8: Data and model 3 implied time series of price-dividend ratio and risk-free rate
1930 1940 1950 1960 1970 1980 1990 2000
1
1.5
2
2.5
3
3.5
4

recessions
data
het
hom
1930 1940 1950 1960 1970 1980 1990 2000
−10
0
10
20
30

recessions
data
het
hom
The model implied time-series of the price-dividend ratio and the risk-free rate were
generated by using the true consumption and dividend growth series.
In practice, the stock market pays a dividend that has a low correlation with the aggre-
gate endowment. I take this into consideration in parametrization 2. I first observe from
the unconditional statistics of table 2.2 and the model implied time-series in figures 2.4 and
2.6 that with a small change in the calibrated parameters this model is able to fit the data
51
as well as model 1. This model falls apart in its long-run behavior relating to the equity
premium.
2.7 Conclusion
Campbell and Cochrane (1999) consider a stochastic discount factor that can explain a
number of well documented properties of asset prices. It is central to their model the
assumption that risk-aversion is stochastic and counter-cyclical, however, for their model
to be successful, they need risk-aversion to have a very high variation in a very broad
range. Chan and Kogan (2002) show that a model with multiple agents with heterogeneous
risk-aversion can produce the counter-cyclical pattern of the stochastic discount factor of
Campbell and Cochrane (1999), as well as a varying Sharpe ratio.
In this study I consider a model similar to that in Chan and Kogan (2002), I derive
explicitly the risk-aversion coefficient of the representative agent and find that the variation
required by the stochastic discount factor of Campbell and Cochrane (1999) is unlikely
to be produced by such a model with reasonable (as observed in the economy) parameter
values.
From the analysis, it appears that the sensitivity function of Campbell and Cochrane
(1999) has to proxy for something other than the risk-aversion of the economy. A promising
alternative would be to examine asset prices as they are formed on expectations or beliefs
of investors about the underlying risks in the economy. The way they update these beliefs
or the uncertainty they have about the conditional distribution might be able to explain why
investors require such high compensation for every unit of ex-post risk they take and why
the price of risk appears to vary so radically across time.
52
Figure 2.9: Model parametrization 3
−0.5 0 0.5 1
−3
−2
−1
0
1
2
3
4
5
Pricing Kernel (z)
ω
−0.5 0 0.5 1
3
4
5
6
7
8
9
Representative Agent Risk−Aversion (γ)
ω
−0.5 0 0.5 1
1
1.5
2
2.5
3
3.5
4
4.5
5
Log of Price/Dividend Ratio
ω
−0.5 0 0.5 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Conditional Sharpe Ratio
ω
−0.5 0 0.5 1
0
5
10
15
20
25
30
Conditional Excess Return
ω
%
−0.5 0 0.5 1
−20
−10
0
10
20
30
40
50
Risk−free Rate (γ)
ω
%
−0.5 0 0.5 1
0
10
20
30
40
50
Conditional Return Volatility
ω
%
−0.5 0 0.5 1
−20
0
20
40
60
80
Conditional Market Return
ω
%
The range of the state variable is four standard deviations around its mean.
53
Chapter 3
A Heterogeneous Beliefs Economy and
an Explanation of Market Prices
3.1 Introduction
In a recent study Lettau et al. (2008) propose that the unprecedented increase in asset prices
during the last decade of the twentieth century has been due to a structural shift in macroe-
conomic risk. A decrease in macroeconomic risk causes a decrease in equity premium and
if the risk-free rate stays unaltered then the prices might increase substantially. Even though
there is evidence
1
that the equity premium has declined over the past years, the argument
proposed presupposes that the level of the conditional volatility of consumption growth
directly determines the level of the equity premium. We know however that in a standard
representative agent model consumption risk is very low to be able to explain either the
level or the variations of the equity premium.
In this study I provide an alternative explanation, based on heterogeneity of beliefs,
about not only the run up in prices but also to a significant extent the joint evolution of
the risk-free rate and the market dividend yield without the need to systematically explain
the high equity premium in the sense of a high expected excess return. Famma and French
(2002) in fact reach the conclusion “... that the average stock return of the last half-century
is a lot higher than expected”. Similar evidence has been found by Soderlind (2008). There-
fore, I pursue the explanation that the US market was a lucky market and that the factor
1
Jagannathan, McGrattan, and Scherbina (2000), Famma and French (2002) and references therein.
54
responsible for this realization was belief heterogeneity. In particular, I argue that the stock
market returns were high and the phenomenal increase in the price dividend ratio especially
during the 1990’s was due to a significant convergence in beliefs.
I first show theoretically that heterogeneity in beliefs causes asset prices to be dis-
counted. Similarly Jouini and Napp (2006b), Jouini and Napp (2006a) and Jouini and
Napp (2007) show in a more general setting that when heterogeneous beliefs are aggre-
gated an additional time-varying discount factor arises. In order to understand the pricing
implication that prices are depressed with heterogeneity, consider a particular asset that has
a positive payoff only in one future state. Let us further suppose that the opinions of two
agents about the probability of this state start diverging. The one agent becomes more opti-
mistic and the other agent more pessimistic by the same amount. Then the optimistic agent
will want to buy more of the asset while the pessimistic agent to sell some of his current
holdings. As they start exchanging the asset their positions become more risky. For each
agent there are two competing pricing effects. The optimistic agent values the asset more
now but the additional risk in his portfolio makes him value the asset less. If he is suf-
ficiently risk-averse
2
the additional riskiness effect is greater and causes a price decrease.
The pessimist agent on the other hand values the asset less with his new opinion. As he
starts giving up some of the asset his valuation increases but the additional riskiness causes
a decrease in price. Eventually, the new equilibrium price will be lower if the agents are
sufficiently risk-averse. These two effects cancel each other out for agents with logarithmic
utility preferences and hence the price is unaffected by dispersion in beliefs.
The main contribution of this study is to quantify this discounting effect and show that
it could have possibly been the driver behind the history of the price-dividend ratio. The
additional assumption required is that agents believe that this heterogeneity is persistent
which means that the effect on long-lived securities is much greater than the risk-free rate.
2
In this study I consider power utility and sufficiently risk-averse in this case means more risk-averse than
logarithmic utility as also shown by Varian (1985).
55
But despite this persistence, it is possible that during the 1990’s the expectations about
the economy converged significantly. As people were gaining confidence that their beliefs
were becoming more aligned with time they started bidding up the prices up to the point
where this convergence of beliefs stopped. This convergence in beliefs might have been
due to a downward shift in macroeconomic risk which then explains the high correlation
between consumption risk and the price-dividend ratio as shown by Lettau et al. (2008).
I provide similar evidence that the price-earnings ratio is strongly negatively correlation
with the conditional volatility of aggregate consumption growth. Figure 3.1 shows the
scatter plot of the price-dividend ratio imputed from the CRSP index returns against the
fitted values of the conditional volatility of real aggregate consumption growth obtained
from NIPA tables. I used quarterly date from 1947 to 2007. The negative correlation is
impressive and statistically is almost -0.8. In order to explain this observation I assume that
the level of heterogeneity is an increasing function of this conditional volatility. The line
of thought is that the level of macroeconomic risk affects the level of uncertainty about the
mean consumption growth and in turn the uncertainty produces the level of heterogeneity.
This assumption, however, is not needed to explain the increase in prices during the 1990’s.
The idea that asset prices decrease with the level of belief heterogeneity in the econ-
omy is opposite to the prevailing opinion in the asset pricing literature. Miller (1977) with
a static setting in mind argues that when there is divergence of opinions and short selling
is not allowed, the price of an asset is likely to reflect the most optimistic agents. Harrison
and Kreps (1978) go a step further and show that in a dynamic setting with risk-neutral
agents the price in equilibrium will include a speculative component. The rationale behind
it is that agents may be willing to pay more than their shadow valuation
3
because there is
a positive probability that in the future they will be able to sell the security to agents with
higher shadow valuation than their own. Scheinkman and Xiong (2003) and Scheinkman
3
I use the term shadow value for an agent to mean the equilibrium price in an economy where the agent
was the only agent in the economy.
56
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.5
4
4.5
5
5.5
6
fitted σ (%)
pd
Figure 3.1: Scatter plot of the price-dividend ratio with the fitted conditional volatility of
consumption growth. The correlation is 0.78.
and Xiong (2006) build on the same idea that with short-selling constraints and limited
supply of shares the assets have an embedded option of reselling it at a higher future price.
The value of this option then increases with belief heterogeneity. This result however is
due to the risk-neutrality assumption and it holds as long as agents are less risk-averse than
a myopic agent. However, it is believed that economic agents are more risk-aversion than
a logarithmic utility agent and further there is evidence by Doukas, Kim, and Pantzalis
(2006) that stocks with higher dispersion in beliefs are priced at a discount.
4
I also offer
evidence here that the heterogeneity in professional macroeconomic forecasters predictions
has decreased over time which is consistent with the predictions of our model. Gallmeyer
and Hollifield (2008) find that even with short-selling constraints the effect of belief dis-
persion on asset prices depends on the intertemporal elasticity of substitution.
4
Diether, Malloy, and Scherbina (2002) provide the opposite evidence that the expected stock returns are
higher for stocks with high dispersion of analysts beliefs.
57
The asset pricing literature on belief heterogeneity is quite rich. This is simply because
it is quite evident either directly or indirectly through differences in investment strategies
5
and trading
6
that with respect to the conditional probability distribution of prices different
investors hold different beliefs and this heterogeneity does not vanish. However it is still
unclear as to the extend in which heterogeneity of beliefs matter for prices and in what
way. In complete market models Detemple and Murthy (1994), Zapatero (1998), Basak
(2005) and Basak (2000) show that divergence of opinions puts downward pressure on the
risk-free rate which in turn implies a higher risk premium. The possible effect identified
is the higher volatility in individual consumption growth processes. In this study I derive
the interest rate in closed form and I examine the relative importance between the effect of
increased volatility and the discount effect due to the aggregation of beliefs. I show that
possibly the effect of increased volatility through higher precautionary savings might not
be enough to cause an overall decrease in the interest rates since the the discount effect is
bigger. The increased volatility in individual consumption also affects the equity premium.
The main finding however is that the effect on equity premium even though significant it is
not enough to predict an excess return of more than 2%. The way the equity premium is
affected in this economy is different than in the model of David (2008) where agents are
assumed to be less-risk averse than a myopic agent.
Jouini and Napp (2006a) resort to pessimism and doubt in order to have an impact
on the equity premium.
7
In my analysis I find that the additional volatility does not have a
significant impact on the equity premium. David (2008) in a similar frictionless economy is
5
One way to approach the heterogeneity of beliefs is to take the different demand functions of different
groups of investors as given or alternatively consider myopic agents as in Ciarella, Dieci, and Gardini (2006)
or Brock and Hommes (1998)
6
Models that relate trading to heterogeneous beliefs include Harrison and Kreps (1978), Harris and Raviv
(1993), Wang (1994), Kandel and Pearson (1995), Hong and Stein (2003), Scheinkman and Xiong (2003)
and Cao and Yang (2008) among others
7
Pessimism refers to the underestimation of the good states of nature and doubt in overestimating the
volatility.
58
able to generate a significant effect on the equity premium by assuming a low risk-aversion
which results to aggressive investment and significant volatility increase.
8
Other studies
that look at asset pricing implications of heterogeneous beliefs include Abel (1990b), Wang
(1993), Basak (2005) and Buraschi and Jiltsov (2006).
The study is organized as follows. Section 3.2 presents a general structure of a com-
plete market economy with heterogeneous beliefs. In section 3.3 I derive the equilibrium
conditions and in section 3.4 I look at the pricing implications at a theoretical level. I close
the economy with specific assumptions about beliefs in section 3.5. In section 3.6 I cali-
brate the economy and look at the quantitative impact of belief heterogeneity. Section 3.7
concludes.
3.2 A General Economy with Heterogeneous Beliefs
Modeling belief heterogeneity has always been hard. First we do not really understand
the origin of this kind of heterogeneity and second we have not found a way to model it
without incorporating some form of irrational behavior so that belief heterogeneity does not
vanish through learning. If agents share the same information about the economy and its
fundamental structure that drives the asset prices, then rational agents will obviously have
their opinions converge since they will soon learn whatever is possible to learn about the
underlying structure and state of the economy and their priors will vanish.
9
One possible
explanation of the fact that beliefs do not seem to converge is that there is a significant
mass of irrational, or boundedly rational agents that manage to survive in the economy
8
Kurz and Motolese (2001) show that endogenous uncertainty arising from belief heterogeneity in con-
junction with correlated beliefs between optimists and pessimists is able to reconcile a high equity premium
with a low risk-free rate while explaining also the high stock price volatility.
9
In private information economies prices are fully revealing as shown by Milgrom and Stokey (1982)
unless I introduce some form of incompleteness as in Shalen (1993) and Wang (1993).
59
and hence do not have the incentive to learn.
10
A second explanation has to do with lack
of information. If the structure of the underlying risk in the economy evolves faster than
the release of relevant information then it is possible that the beliefs of agents can be both
rational and still divergent as shown by Kurz (1994a) and Kurz (1994a)). However, this
only gives a mathematical justification to belief heterogeneity.
In my model the heterogeneity of beliefs is assumed to be a function of the uncertainty
in the economy and in turn the uncertainty is assumed to be a function of the level of
macroeconomic risk. Therefore, instead of modeling directly the beliefs of the agents, I try
to look at the pricing implications given that there is divergence of opinions in the market.
However, this setup proxies for heterogeneity in the interpretation of new information in the
economy as for example in Harris and Raviv (1993) and Kandel and Pearson (1995). Every
period there are changes to the fundamentals of the economy but available information
are not fully revealing and therefore agents may interpret them differently. Therefore the
assumption that I will make later is that the more uncertainty there is the more disagreement
there is among agents about the interpretation of new information. For now however, I
start with a general economy where beliefs are heterogeneous. I first conduct a qualitative
analysis of the pricing effects of belief heterogeneity and then I derive the equilibrium
conditions that will be used to close the economy with further assumptions.
3.2.1 Uncertainty, Agents and Beliefs
I consider a probability space pΩ,F,P q in which the uncertainty of the economy lies. Any
given element of the set Ω describes an entire history for the economy. Consider also the
set of natural numbersN t0,1,... u.
10
Kogan, Ross, Wang, and Isterfield (2007) provide a model with only final period consumption where
irrational agents have a probability of survival.
60
Assumption 3.1. Time is discrete and infinite and is denoted by t P N. The aggregate
uncertainty is driven by the vector s pg,ǫ q P S of independent random processes. g
denotes the natural logarithm of the growth process of the aggregate endowment Y (y log pY q) of a single perishable good. ǫ is a random variable that affects the beliefs of the
agents in the economy. The history ofs which is denoted bys
t
ts
1
,...,s
t
u is observable
and the associated information is denoted by the filtration tF
t
,t PN u.
All agents in the economy have the public information and none of them have any private
information. The true probability distributionP is not known and therefore agents need
to form their own beliefs about the generating process of the economic variables. The
additional uncertaintyǫ is only used for completeness to justify the divergence of opinions
and does not affect the results. I think of this innovation as information other than the
history of aggregate endowments that potentially affects the future distribution of aggre-
gate endowment growth. For example it could contain information about macroeconomic
indices, technological shocks, changes in monetary and fiscal policies, international mon-
etary and political crises, oil shocks etc. Hence, as I stated already I assume that the belief
heterogeneity in the economy is sustained due to the difference in interpretation of publicly
available information.
I model beliefs in a quite general way. For this I need to define the setM
1
of strictly
positive martingales defined on the probability space, adapted to the filtration tF
t
,t PN u
and with initial value one. The following corollary allows us to define beliefs.
Corollary 3.1. Given a martingaleξ PM
1
there exist a probability measureQ equivalent
toP such that for all0 ⁄s t 8,
ξ
t
ξ
s
dQ p|F
s
q
dP p|F
s
q
Ft
61
It is clear from the above corollary that beliefs can be represented by a martingale
process of the set M
1
. The ratio ξ
t k
{ξ
t
would therefore represent certain conditional
beliefs held at some timet about the path of the economy over the nextk periods. Learning
would then be represented by a process ξ of which the variation of this ratio decreases
with timet irrespective of the horizonk and vanishes eventually. Convergence on the other
hand between two different beliefs would mean that these ratios become with time more
“correlated” with each other, until they become identical. Hence,ǫ is a process that will be
used to prevent this convergence.
This is a non-standard way to define beliefs since in order to reconstruct a set of beliefs I
need to know the processξ under the objective probability measureP which I have assumed
that I do not know. The first reason is technical and it has to do with ensuring that the
probability measures considered are equivalent to the true probability measure as well as
equivalent to each other. In equilibrium this guarantees that for all finite times there is
zero probability of an agent loosing all his or her wealth. The second reason is that such
a definition renders the analysis much easier since I isolate the effect of heterogeneity.
Two different martingale processes indicate the relation between two sets of beliefs which
is what I need to analyze the effect of heterogeneity without worrying about the actual
beliefs. Finally, the whole analysis is correct under any equivalent probability measure that
I decide to use.
A belief process ξ being adapted to the filtrationF
t
implies that these beliefs are held
both with respect to the aggregate endowment growth in the economy as well as the process
ǫ. This not only allows for heterogeneity in beliefs about the growth of the economy but
also in beliefs about the level of belief heterogeneity in the future. Even though this is a
realistic assumption later on I restrict to cases where agents agree onǫ.
62
Assumption 3.2. There is a continuum of agents of mass one that is represented by the set
I r0,1 s. Belief heterogeneity is described by a function,
ξ :I M
1
from the set of agents to the set of beliefs.
In the case where all agents have homogeneous beliefs thenξ pi q is the same process for all
i PI and in the case where everyone has the correct beliefs thenξ pi q is the constant process
for all agents. Given therefore a process ξ
i
the beliefs of agent i are given by dP
i
ξ
i
dP
for all possible future periods.
11
3.2.2 Complete Financial Markets
The financial structure of the economy is not explicitly modeled because first I assume that
markets are dynamically complete and second I am only interested to look at the prices of
certain financial securities while I am not interested in portfolio holdings. In particular I
am interested in the joint evolution of the price of the market security and the risk-free rate.
Assumption 3.3. There exists a continuum of both short-lived and infinitely lived assets
that span the local uncertainty of the economy. The aggregate endowment is paid as divi-
dends by the market security.
The above assumption about the aggregate dividend growth process of the market was made
for simplicity and in order to be able to compare my results with other studies that make the
same assumption. Any other assumption can be made as long as no additional aggregate
uncertainty is introduced. For example it is straight-forward to modify the model when I
assume that the aggregate dividend growth is linear in the aggregate endowment growth
11
From the rest of the paper I will use the notationξ
i
to denoteξ pi q.
63
plus an additional independent shock. Such an assumption would potentially decrease the
correlation of the market returns with consumption growth. This is however not needed in
this model as the variation in the heterogeneity of beliefs that is independent of consump-
tion growth will decrease this correlation.
Market completeness and equilibrium no arbitrage prices imply that given the objective
probability measureP there exists a uniqueF
t
-adapted pricing kernel process pp
t
,t ¥ 0 q,
given ap
0
, that prices all assets. p
t
is collinear with the marginal utility of consumption at
timet of a hypothetical agent with the correct beliefs. Hence, for the market portfolio that
pays the aggregate dividend, the ex-dividend equilibrium price is given by,
P
t
E
t
‚
k PN
δ
k
p
t k
p
t
Y
t k
(3.1)
whereE
t
rs denotes the expectation operator under the objective probability measure, con-
ditional on the information setF
t
. δ is a preference parameter and it denotes the common
subjective discount factor.
12
Market completeness and equilibrium also imply that for any equivalent probability
measureP
i
, believed by an agent i, there exists a unique (up to a factor) pricing kernel
process pp
i
t
,t ¥ 0 q by which the agent prices the assets. This is the same as saying that
in equilibrium all agents agree on the prices of the Arrow-Debreu securities at time 0. An
12
In order for prices to have their fundamental values I assume the transversality condition
lim
t 8
E
δ
t
p
t
P
t
0
holds.
64
agent’s price of an Arrow-Debreu security that pays one unit of consumption at some future
dates
t
is given byp ps
t
qdP ps
t
q.
13
. Therefore in equilibrium it has to be that,
p
i
t
p
t
ξ
i
t
1
, pi,t q PI N. (3.2)
Since p
i
t
is (up to a factor) the marginal utility of consumption at time t of agent i in
equilibrium, relation (3.2) implies that the marginal rates of substitution between any two
states are different for two agents that have different beliefs. One might think that this is
a result of some kind of incompleteness in the market due to the heterogeneity in beliefs.
However, in an incomplete market setting agents do not agree on all Arrow-Debreu security
prices while here agents do agree on all prices as it is assumed from (3.2).
Even though the marginal rates of substitution vary cross-sectionally, the allocation is
Pareto optimal when considering the subjective probability measures of the agents. This
implies that the social planner considers that fact the agents have heterogeneous beliefs.
However, the allocation is not Pareto optimal under the objective probability measure but
this is only due to the assumption that there is lack of information in the market and the
available information is not used similarly by all agents. Lack of information refers to the
assumption that agents do not know the exact process that drives the aggregate risk. Hence,
the allocation would cease to be Pareto optimal in case complete information was revealed
to the economy but only due to the fact that the initial incompleteness of information was
assumed to give rise to differences in beliefs. Otherwise, if agents held the same beliefs,
even in the case of incomplete information, the allocation would still be Pareto optimal
under complete information and given some wealth reallocation. The wealth reallocation
would be needed because under different probability measures asset prices are different. Of
course if agents were allocated a fixed proportion of the aggregate endowment then there
would not be any need for wealth reallocation.
13
Note that any process that is adapted to the filtrationF
t
can be written as a function of the historys
t
.
65
3.2.3 Preferences and Endowments
The preferences assumed are standard time and state separable preferences with external
habit similar to that of Abel (1990a).
14
Assumption 3.4. Agents are utility maximizers with time and state separable preferences
over consumption processes as follows:
U
i
pc,X q E
i
0
‚
t PN
δ
t
u pc
t
,X
t
q
, (3.3)
where
u pc,X q c
1 γ
1 γ
X
γ η
(3.4)
andη ⁄ γ andγ ¡ 0. X denotes the common external habit. The log of the external habit
is formed according to
x
t 1
λ
x
x
t
p1 λ
x
qy
t
, t ¥ 0,
whereλ
x
P p0,1 q.
The habit process imposes an externality on all agents of the economy because it affects
their marginal utilities. The marginal utility of consumption is given by,
u
c
pc,X q c
γ
X
γ η
.
14
Habit formation preferences have been extensively explored in the literature in various forms. Signifi-
cant contributions include Ryder and Heal (1973), Sundaresan (1989), Constantinides (1990), Detemple and
Zapatero (1991), (Gali, 1994) and Hindy, Huang, and Zhu (1997).
66
Essentially agent preferences are over their consumption surpluses over the common habit.
Simply for interpretation purposes I assumed thatη γ and therefore the derivative of the
marginal utility with respect to the level of the habit,
Bu
c
pc,X q
BX
pγ η qc
γ
X
γ η 1
,
is positive. Therefore a higher level of habit for a given consumption level induces agents
to consume more and invest less.
The preference assumption of external habit was made for several reasons. First exter-
nal habit produces potentially enough time-variability in discount rates and hence poten-
tially enough variability in price-dividend ratios in the absence of belief heterogeneity.
When heterogeneous beliefs are introduced the variability of the price-dividend ratio can
match the variability found in the data. In a homogeneous beliefs setting the observed per-
sistence in the price-dividend ratio can easily be generated in a model with external habit
as shown for example in Campbell and Cochrane (1999).
Secondly, in an economy where the expectations about the future vary, CRRA pref-
erences without habit produce a counter intuitive prediction, namely that more optimistic
expectations are associated with lower prices. This is an obvious outcome of the fact that
in power utility preferences the intertemporal elasticity of substitution is determined by the
risk aversion parameter which is normally higher than one. The higher is the risk-aversion
the lower is the elasticity of intertemporal substitution since agents have a higher propensity
to smooth consumption inter-temporally. So when agents are optimistic about the future,
in an attempt to smooth consumption inter-temporally they want to borrow more and this
increases interest rates and decreases prices. However, with the inclusion of a persistent
habit this relation can reverse if the habit parameter pγ η q is sufficiently high and the
habit does not follow a close to unit root process.
67
In order to see the mechanism behind such an effect I need to examine what happens
to the current interest rate when habit is introduced. A high consumption growth today
increases consumption over habit today as well as tomorrow because habit is not fast in
adjusting to increased consumption. In this argument we can think of tomorrow’s con-
sumption growth as fixed. An increase in consumption over habit decreases the price of
consumption today. However, the decrease in today’s price of consumption is bigger than
tomorrow’s decrease because habit does increase by p1 λ
x
q. Therefore p1 λ
x
q gives the
wedge by which today’s price of consumption increases in relation to tomorrow’s increase.
This effect decreases interest rates and the decrease is higher the smallerλ
x
is. Since, now
an increase in consumption growth today decreases the interest rate through habit then a
higher expected consumption growth will decrease the expectation of tomorrow’s interest
rate. Now, if habit is sufficiently persistent then optimism about consumption growth will
decrease the expectation of not only tomorrow’s interest rate but also the expectation of
many interest rates in the future. This causes current prices of long-lived securities like
stock prices to decrease.
Assumption 3.5. Each agent is endowed with initial holdings of the dividend generating
financial assets. This translates to proportion of the aggregate endowment initially allo-
cated to each agent that is represented by
θ
0
:I r0,1 s,
such that,
»
I
θ
0
pdi q 1.
68
For a given pricing kernel process p
i
believed by an agent i and given market complete-
ness, an agent’s sequence of budget constraints collapses to a single inter-temporal budget
constraint,
E
i
‚
t ¥0
δ
t
c
i
t
p
i
t
⁄E
i
‚
t ¥0
δ
t
θ
0
pdi qY
t
p
i
t
.
Using (3.2) I can rewrite the budget constraint as,
E
‚
t ¥0
δ
t
c
i
t
θ
0
pdi qY
t
p
t
ξ
i
t
⁄ 0 (3.5)
after switching to the objective probability measure.
3.3 Complete Market Equilibrium with Heterogeneous
Beliefs
Given the assumptions I can derive the optimal consumption process for each agent and
once I define the equilibrium concept I can derive the equilibrium conditions. By equilib-
rium conditions I mean the endogenous aggregate state vector of the economy and its law
of motion that are needed in order to determine prices of financial securities.
Since the economy grows, the analysis is simplified if I express the relevant variables
in terms of the aggregate endowment. I define therefore α
t
pdi q c
i
t
{Y
t
to denote the
consumption proportion and ω
t
log pY
t
{X
t
q to be the log of the endowment habit ratio,
henceforth referred to as endowment surplus. ω is a state variable and it follows the process,
ω
t 1
λ
x
ω
t
g
t 1
. (3.6)
69
Below I derive the optimal consumption proportion process for each agent. For ease
of notation and since the pricing kernel process is unique up to a scaling factor, I fix p
0
exp p ηx
0
γω
0
q.
Lemma 3.1. Given the price processp, an agenti that maximizes (3.3) subject to the budget
constraint (3.5) has an optimal consumption process that is adapted toF
t
and satisfies the
following first order condition,
α
t
pdi q α
0
pdi q
p
t
ξ
i
t
1 {γ
exp
ηx
t
γ
ω
t
. (3.7)
α
0
pdi q is the initial consumption proportion and is optimally chosen according to
α
0
pdi q E
0
‚
t PN
δ
t
p
t
Y
t
pξ
i
t
q
1
θ
0
pdi q
E
0
‚
t PN
δ
t
p
t
Y
t
pξ
i
t
q
1
pp
t
{ξ
i
t
q
1 {γ
exp
ηx
t
γ
ω
t
(3.8)
Let us now define the concept of equilibrium which is standard.
Definition 3.1. An equilibrium is a set ofF
t
-adapted processes for the pricing kernelp pp
t
,t PN q and the consumption allocationsα tpα
t
pdi q,t PN qu
i PI
such that:
(i) given the pricing kernel processp, for each agenti PI,α pdi q maximizes his utility (3.3)
given his budget set (3.5), and
(ii) in every period the consumption good market clears,
»
i PI
α
t
pdi q 1, t PN (3.9)
With this concept of equilibrium I am assuming that all agents have structural knowledge in
the sense that they can associate observed prices with the exogenous state of the economy
and agents only differ in their beliefs about the probabilities of the draws of nature. This
70
concept is different than the concept of Rational Belief Equilibria of Kurz (1994a) and Kurz
(1994b) by which agents do not have structural knowledge and their beliefs are different
with respect to the conditional distribution of prices.
Note from the market clearing condition of equilibrium that in any period t the con-
sumption proportion function α
t
is a probability distribution over the set of agentsI. For
the rest of the study I will denote any integral with respect to the consumption distribution
across agents in some period t withE
t
.
15
Since the beliefs ξ is a function over the set of
agents thenα
t
is a distribution over the beliefs in some periodt. Further, since bothξ
t
and
α
t
areF
t
-adapted it means that,
(i) α
t
and the primitive functionξ, that describe the distribution across beliefs in period
t and how these believes will evolve, along with
(ii) ω
t
the surplus ratio and its law of motion,
carry all the relevant information about the economy. In fact the surplus ratio and the
consumption distribution will be the economy’s state vector as will be made clear once I
characterize the equilibrium.
Before I characterize the equilibrium I will introduce some variables that relate to
the aggregation of beliefs. Aggregation results regarding beliefs date back to Rubinstein
(1975), Verrecchia (1979) and Varian (1985). More recent results are those of Calvet,
Grandmont, and Lemaire (2004), Jouini and Napp (2006a) and Jouini and Napp (2007).
This aggregation will provide us with further insight about the heterogeneity of beliefs and
will help us analyze the price effects. I will also show that the economy is equivalent to a
representative agent economy with a certain beliefs process and a modified habit process.
15
For example the market clearing condition in some periodt can be trivially written asE
t
1 1.
71
Definition 3.2. Letα
0
be the equilibrium consumption allocation in the initial period. The
aggregated beliefs process is defined as
˜
ξ
t
E
0
ξ
i
t
1 {γ
γ
. (3.10)
The belief compensator is defined as
b
t
logE
t
˜
ξ
t 1
˜
ξ
t
, (3.11)
whereb
0
0 and the consensus beliefs process is defined sequentially by
ξ
r
t 1
e
bt
˜
ξ
t 1
˜
ξ
t
ξ
r
t
, (3.12)
whereξ
r
0
1.
Let us now clarify what these processes are and in order to do this I use the next corollary.
For convenience I introduce the notation ξ
t,t 1
ξ
t 1
{ξ
t
, to represent the one period
conditional beliefs in periodt. I remind that for an agenti and a states,ξ
i
t,t 1
ps q is the ratio
of his conditional probability about the states over the true probability.
Corollary 3.2. In equilibrium the aggregated beliefs process satisfies the following rela-
tion,
˜
ξ
t,t 1
E
t
ξ
i
t,t 1
1 {γ
γ
. (3.13)
This corollary says that the conditional aggregated beliefs is equal to the ratio of the aggre-
gated beliefs for a path up to the period t 1 over the aggregated beliefs for the same
path up the period t. This is a property of every belief process. However, when there
72
is divergence of beliefs the aggregated beliefs is not a martingale process because due to
Lyapunov’s inequality, whenγ ¡ 1
˜
ξ
t,t 1
⁄E
t
pξ
i
t,t 1
q
and therefore sinceE
t
pξ
i
t,t 1
q 1 for all agents,E
t
p
˜
ξ
t,t 1
q ⁄ 1. Using the belief compen-
sator processe
bt
, I re-scale the aggregated beliefs to construct the consensus beliefs. As I
will see later on, the belief compensator will have the greatest impact on asset prices and
will be my major additional pricing element. I postpone the discussion about its economic
meaning for later on until I look at the equilibrium prices. For now I only note that in
the absence of belief heterogeneity or in the case where γ 1, b
t
is identically zero and
therefore the aggregated beliefs are also the consensus beliefs. Further I note thatb
t
varies
across time when the level of heterogeneity varies. So far I have been using the term “level
of heterogeneity” loosely. For any given future state s the variability of ξ
i
t,t 1
ps q across
agents (using the consumption distributionα
t
for weights) is the level of heterogeneity for
that state. The higher the variability the lower is
˜
ξ
t,t 1
ps q whenγ is greater than one. There-
fore for a givenγ ¡ 1, b
t
is a transformation of the average of
˜
ξ over all future states and
hence itself can be used as a measure of the level of heterogeneity. Finally, the consensus
beliefs ξ
r
is obtained once I divide the aggregated beliefs by their mean e
b
. Since e
b
is
related to the average heterogeneity across states thenξ
r
t,t 1
ps q is the relation of the level of
heterogeneity in state s to the average heterogeneity. A value higher than one means that
for the given states agents agree more than they agree on average. ξ
r
t,t 1
ps q also increases
when agents believe that the probability of the particular state is higher than the true.
Now I can characterize the equilibrium. For existence of time-homogeneous Markov
equilibria in settings of heterogeneous agents one can refer to Duffie, Geanakoplos, Mas-
Colell, and McLennan (1994).
73
Corollary 3.3. Let α
0
be the equilibrium consumption allocation in the initial period and
let p
0
exp p ηx
0
γω
0
q. Then in equilibrium the pricing kernel is sequentially deter-
mined by
p
t 1
p
t
˜
ξ
t,t 1
exp r η p1 λ
x
qω
t
γ pω
t 1
ω
t
qs, (3.14)
and the equilibrium consumption allocations are described by,
α
t 1
pdi q α
t
pdi q
ξ
i
t,t 1
L
˜
ξ
t,t 1
1
γ
, i PI. (3.15)
The equilibrium consumption process (3.15) is quite intuitive. An agent increases her
consumption in states that she believes to be more probable when compared to the aggre-
gated beliefs. The optimal consumption relation also indicates that the individuals’ con-
sumption processes are more volatile when the beliefs are more disperse, while they are
less volatile when the risk-aversion is higher. When agents are more risk-averse they are
more cautious to invest according to their beliefs because this implies higher variability
across their consumption plan. David (2008) exploits the impact of the risk-aversion on the
volatility of consumption and hence on the marginal utilities, by choosing a risk-aversion
between 0 and 1. Such a low risk-aversion parameter generates substantial variability in
the pricing kernel but not necessarily a higher equity premium as I will show later.
The law of motion of consumption is independent of the consumption surplus and this is
simply due to the fact that the habit is common to everyone and the preference parameters
are the same. The consumption surplus however, does affect prices as I observe from
(3.14) and it depends clearly on pγ η q.
16
The rate of change in the pricing kernel (which
is the price of the Arrow-Debreu security that pays a unit of consumption next period at
the given state divided by the probability of the state) depends on the consumption surplus,
16
The term η p1 λ
x
qω
t
γ pω
t 1
ω
t
q can be re-written as γg
t 1
p1 λ
x
qpγ η qω
t
.
74
the consumption distribution of beliefs and the exogenous state. Therefore for a given
economy the equilibrium state vector is z pω,α q. Further note that the equilibrium is
path dependent because beliefs are path dependent. The law of motion for the economy’s
state vector is denoted by,
z
t 1
L pz
t
,s
t 1
q. (3.16)
I remind that the conditional beliefs across all agentsξ
i
t,t 1
are a function of the exogenous
state s
t
by assumption. This means that if I know the path of the economy up to and
including time t I know the conditional beliefs about next period of every agent. The
equilibrium is then described by the initial consumption allocation and the law of motion
L. The law of motion is given by the primitive belief function ξ, the exogenous process
(3.6) and the endogenous process (3.15). The initial consumption allocation pins down the
equilibrium.
Proposition 3.1. Let D pI q be the set of distribution functions over the set I. Let also
p
t
p pa,s
t
q, Y
t
Y pY
0
,s
t
q, x
t
x px
0
,s
t
q and ξ
i
t
ξ pi,s
t
q. Define the functional
F :D pI q R
|I |
, where | | denotes the cardinality of a set, by
F pa,j q »
j
0
E
0
‚
t PN
δ
t
p
t
Y
t
pξ
i
t
q
1
E
0
‚
t PN
δ
t
p
t
Y
t
pξ
i
t
q
1
pp
t
{ξ
i
t
q
1 {γ
exp
ηx
t
γ
ω
t
θ
0
pdi q, i PI.
Then the equilibrium is determined by the solution to the following functional equation,
F pa q a,
in which caseα
0
a.
75
3.4 Equilibrium Asset Prices
The heart of every asset pricing model is its stochastic discount factor (SDF) since with it
we can price all assets in the economy. The price of an assetj that pays a (risky) stream of
future cash flowsD
j
is given by,
P
j,t
E rM
t,t 1
pP
j,t 1
D
j,t
q|F
t
s (3.17)
where the equilibrium SDF is given by,
M
t,t 1
δ
p
t 1
p
t
. (3.18)
The price of a risk-free bond that pays a unit of consumption next period is denoted byQ
t
and P
t
gives the price of the market security. Once I have derived the equilibrium pricing
kernel it is straightforward to express the SDF under the objective probability measure.
I have already noted that in equilibrium it is given by some function M pz
t
,s
t 1
q of the
endogenous state vector and the exogenous shocks. From equation (3.14) and the definition
of the consensus beliefs the equilibrium SDF is given by
M
t,t 1
δξ
r
t,t 1
exp r γg
t 1
p1 λ
x
qpγ η qω
t
b
t
s. (3.19)
where I have used thatx
t 1
x
t
p1 λ
x
qω
t
. The expression of the stochastic discount
factor simplifies to its usual form when all agents in the economy have the correct beliefs
since both the consensus belief process as well as the belief compensator vanish from the
expression. The expression also accommodates an economy with homogeneous but possi-
bly erroneous beliefs in which case only the belief compensatorb vanishes. If I would like
to associate such a homogeneous beliefs economy with a heterogeneous beliefs economy I
76
would have to derive the additional variation coming fromb from a modified habit process.
The following corollary is straight forward:
Corollary 3.4. The heterogeneous agent economy can be represented as a homogeneous
agent economy with a modified habit processx
r
such that,
x
r
t 1
x
r
t
p1 λ
x
qω
t
b
t
pγ η q
,
and beliefs given byξ
r
.
The importance of corollary 3.4 is not in deriving an equivalent representative agent econ-
omy but in noting the additional elements that appear when we introduce heterogeneous
beliefs. We can see that an economy with heterogeneous beliefs introduces two new ele-
ments. The first one is a time varying discount factor b which reflects the level of belief
heterogeneity in the economy over the next period at a given point in time. This element
results in discounted prices when compared to a homogeneous beliefs economy. The sec-
ond element is the additional risk factor as given by ξ
r
and in principle can affect the
risk-premium, but let us examine these two new elements in turn and their possible con-
nection. The only thing I need to mention in order to be able to connect the two effects is
that a higher value forγ increases the discount factorb and the variability ofξ
r
.
3.4.1 Discounted Prices
The following corollary formalizes the concept of discounted prices.
Corollary 3.5. Consider an asset j that pays a future stream of cash flows. The shadow
price for an agenti, denoted byP
i
j
, is the equilibrium price of the asset in a homogeneous
beliefs economy with beliefsξ
i
, in which case the SDF is given by,
M
i
t,t 1
δξ
i
t,t 1
exp r γg
t 1
p1 λ
x
qpγ η qω
t
s.
77
Then, when γ ¡ 1 and when beliefs are heterogeneous, the equilibrium asset prices are
strictly less than the consumption weighted average of individual shadow prices,
P
j,t
E
t
P
i
j,t
.
The main result of this study is to show that quantitatively this discounting in prices due to
heterogeneity is very significant and therefore variations in the level of heterogeneity causes
prices to vary across time. In fact when heterogeneity is sufficiently high the equilibrium
price can be less than the shadow price of the most pessimistic agent. Prices become dis-
counted because investors are afraid that their different investment behaviors might lead
them to states with very low levels of wealth. These are under their assessment low proba-
bility events but they are still afraid of them because they would be hit really hard by those
states. Agents are afraid of their individual bad states more than they value their good states
due to being more risk averse than a myopic agent (γ ¡ 1).
This result has a clear implication about the economy. Note first that the belief com-
pensator affects all prices at the same time regardless of their riskiness because it reflects
the reluctance of the investors to transfer wealth to the next period. During periods of great
divergence of beliefs we should observe low prices while prices should increase when
beliefs are aligned. As I will show with the calibration of the economy that this can explain
the behavior of the aggregate price-dividend ratio that seems to be non-stationary. It appears
that since the second world war the price-dividend ratio has a slight upward trend and this
could be a result of a slow convergence of beliefs. With the same token I can explain the
dramatic increase in the price-dividend ratio during the 1990’s if we accept that the beliefs
of the investors for some reason were aligned. The reason seems to be that the level of
the macroeconomic risk decreased significantly and this probably lead to a decrease in the
level of heterogeneity as I will show later on. This explanation does not require that beliefs
converged to the true probability distribution but that the majority of the investors were
78
having the same expectations about the prospects of and riskiness in the economy and they
were aware of this. Consequently, they were not afraid that they would end up in a state
with a very low level of wealth and they were bidding up the prices as they were gaining the
confidence that such an event was not possible. The hit of the recession lead to a significant
decrease in prices probably because the recession caused the beliefs to diverge.
I have so far talked about the effect of heterogeneity on asset prices as a whole but
not about their relative valuation. In particular I am interested in the relative valuation of
the market security and the price of the risk-free bond. This depends on the divergence of
beliefs not only with respect to the next period but also with respect to all future periods,
because the market security is infinitely lived and pays dividends every period. When
changes in belief heterogeneity are persistent then long-lived securities like the market
security will be more sensitive than short-lived securities and therefore variations in the
price-dividend ratio can be explained without excessive variation in the risk-free rate.
3.4.2 The Equity Premium
The expected excess return on the market over the risk-free rate, which I denote with R
e
,
can be affected by the heterogeneity of beliefs in a number of ways. From (3.17) I derive
the standard expression of equilibrium excess returns, E rMR
e
s 0, and expanding it
using the equilibrium expression of the SDF I have the following:
E pR
e
q C pe
γg
,R
e
q
E pe
γg
q
C pξ
r
,e
γg
R
e
q
E pe
γg
q
whereC denotes the covariance under the objective probability measure. The first part is
the standard expression which apart from the possible additional variability in returns it is
not affected. I therefore examine the second part. I remind that ξ
r
is a random variable
with average value of one. It has higher values for states that on average agents believe that
their probability is higher than the true. It also has higher values for states that agents agree
79
more than average. The equity premium can increase with belief heterogeneity through the
second part due to the following reasons:
(i) ξ
r
can be positively correlated with consumption growthg. This could happen either
because agents are optimistic or because agents agree more about the probabilities
of the good states and disagree more about the bad states. None of these reasons
can be realistically assumed to cause equity premia to be consistently higher than a
homogeneous economy.
(ii) ξ
r
can be negatively correlated withR
e
. The optimism argument applies equally here.
I do expect however forξ
r
to be negatively correlated with future prices because it is
probable that states about which there is high disagreement (ξ
r
is low) are also states
in which the belief heterogeneity about the future is high (and therefore P low). In
the reduced form model presented in the next section I do have this effect.
The risk aversion parameterγ plays a double role with respect to the equity premium.
When agents are less risk averse their investment behavior is more radical and the variabil-
ity of the additional risk-factor ξ
r
increases. Therefore, if everything else stays the same
the equity premium should increase. The second effect is related to the second point above.
With higher risk-aversion prices become more discounted with heterogeneity and therefore
the effect of the correlation betweenξ
r
andR
e
increases.
3.5 The Economy
In this study I depart from the usual approach of modeling belief heterogeneity by which
the beliefs of the different agents are assumed explicitly. If I assumed a particular gener-
ating process of which its structure is known by the agents and that they are rational and
update their beliefs in a Bayesian fashion then their beliefs will very soon converge. This is
however not what we observe in reality. There is an inherent difficulty in sustaining belief
80
heterogeneity in a model while not departing from rationality. The issue is not that agents
are not rational, but in reality the generating process is much more complex than can be
assumed in a model and hence the available information is not able to exclude many “sets”
of beliefs to be held by investors. Additionally, investors’ information set is not only the
history of consumption growth but their beliefs are also influenced by other types of infor-
mation like technological shocks, wars, the price of oil, anything that can affect aggregate
production in an economy and in a possibly non-stationary way.
Therefore, instead I assume that agents in equilibrium hold different beliefs and this
divergence does not vanish in time. Now that I have derived the general equilibrium condi-
tions I can model directly the equilibrium law of motion for the divergence of beliefs. The
following assumptions are about the functionξ but do not explicitly characterize it. These
assumptions will refer to the conditional distribution of g held by agents at any period t.
Letf
i
t
pg q denote the density function of one period consumption growth believed by agent
i at timet and letf
t
pg q denote the true and unknown density function.
Assumption 3.6. Agents disagree only about the probability distribution of consumption
growth and know the true dynamics ofǫ,
ξ
i
ps
t 1
q
ξ
i
ps
t
q
ξ pi,g
t 1
,s
t
q
wheres
t 1
ps
t 1
,s
t
q.
Then I have that,
ξ pi,g
t 1
,s
t
q f
i
t
pg
t 1
q
f
t
pg
t 1
q
.
Note that the time subscript denotes the dependence of the beliefs on the observed history
s
t
but the opinions differ only with respect to the consumption growth. This assumption
is made for simplicity since if I assumed otherwise the equilibrium state vector would
become too large too allow us to approximate the price functions accurately. It would
81
have been realistic to assume otherwise since if beliefs change over time then beliefs about
the evolution of beliefs might differ as well. After all this is what Keynes implied when
he talked about beauty contests. Such an additional assumption would introduce further
riskiness in the economy.
Let us now continue with the assumptions aboutf
i
t
and its evolution.
Assumption 3.7. In any periodt PN each agent believes that the aggregate consumption
growth for the next period is conditionally normally distributed
g
t 1
|F
t
N pμ
i
t
,σ
2
t
q
where pμ
i
t
,σ
t
q areF
t
-measurable. Agents are certain aboutσ
t
but are uncertain about the
conditional mean,
P
i
: μ
t
|F
t
N pμ
i
t
,u
2
t
q
Note that I do not assume any particular process for consumption growth and therefore do
not care about f
t
. For pricing it is irrelevant anyway since only beliefs matter. Later on
when I need to express unobserved quantities like the equity premium I will use the prob-
ability measure of the consensus beliefs. Further note that agents are assumed to disagree
only about the mean of aggregate consumption growth. It would be interesting to see what
is the effect of heterogeneity of beliefs about the second moment but I leave this for future
research.
The variance term u
2
denotes the uncertainty in the economy about the conditional
mean and it is common to all agents. This uncertainty adds further risk to the economy
since agents discount the future according to the following probability distribution,
P
i
: g
t 1
|F
t
N pμ
i
t
,σ
2
t
u
2
t
q.
82
With assumption 3.7 I have reduced the belief heterogeneity to just differences in opin-
ions about the conditional mean of macroeconomic risk. Hence, I can aggregate agents
according to their types and define a new probability measure,
˜ α
t
pμ q E
t
1 tμ
i
t
μ u. (3.20)
1 tA u denotes the indicator function that takes the value of 1 in the subset A of the set of
agents and0 otherwise. With this notation I can now define the cross-sectional distribution
of beliefs.
Assumption 3.8. The initial consumption distribution ˜ α
0
of beliefs about means across the
set of agents is given by
˜ α
0
: μ N pμ
0
,ν
0
q.
With the above assumption I parameterize belief heterogeneity in some period t with ν
t
the consumption weighted volatility of the beliefs about the mean. Homogeneity of beliefs
implies thatν
0
0. With this assumption I can characterize the consensus beliefs for time
0 or any other period where the distribution of beliefs is normally distributed.
Lemma 3.2. The consensus beliefs about consumption growth is normally distributed,
N
μ
t
,σ
2
t
u
2
t
ν
2
t
γ
.
The belief compensator is given by,
b
t
pγ 1 qlog
d
1 ν
2
t
γ pσ
2
t
u
2
t
q
.
From this lemma we see very clearly the two pricing effects of belief heterogeneity and
what they depend on. The belief compensator depends first linearly on the cautiousness
of the agents as parameterized by pγ 1 q. There is also a counter effect that appears in
83
the termν
2
t
{γ pσ
2
t
u
2
t
q which represents the fact that when agents are more cautious they
speculate less and they have less to fear about their beliefs being wrong. This explains
further why the additional risk in the economy is inversely proportional to γ. Finally the
divergence of beliefs naturally increases both the discounting term as well as the risk of the
economy.
In such a setting where the consensus beliefs about aggregate consumption growth is
normally distributed I can express the interest rater
f
log pQ q in closed form. This will
be the equilibrium interest rate since I will assume that in equilibrium the beliefs continue
to have this form in all periods.
Corollary 3.6. The equilibrium interest rate is given by,
r
f
t
log pδ q γμ
t
p1 λ
x
qpγ η qω
t
pγ 1 qlog
d
1 ν
2
t
γ pσ
2
t
u
2
t
q
1
2
γ
2
σ
2
t
u
2
t
ν
2
t
γ
.
First I note that the standard expression about the interest rate is obtained in the case of
homogeneity and no uncertainty,ν u 0. Then I note that the effect of the heterogeneity
variable ν depends on the beliefs about the volatility σ. This is natural since in the case
where ν is very low in comparison to σ then agents have a considerable disagreement
only in states that are far away from the mean. Essentially, what describes the level of
heterogeneity in the economy is the ratioν {σ.
Before I complete the set of assumptions about the beliefs let us look at a specific simple
economy where the law of motion is derived explicitly in equilibrium. In this economy
there is no extra shockǫ to the beliefs but are in fact dogmatic.
84
Example 3.1. Agents have dogmatic beliefs, i.e. μ
i
t
μ
i
t P N. Agents have common
beliefs about σ which can be time varying. Then in equilibrium the distribution of beliefs
about the mean continues to be normal,
α
t
: μ
i
N pμ
t
,ν
t
q
with time varying moments,
μ
t 1
ν
2
t 1
ν
2
t
μ
t
ν
2
t 1
γσ
2
t
g
t 1
,
1
ν
2
t 1
1
ν
2
t
1
γσ
2
t
,
t PN.
It is quite interesting to first note that the dogmatic agents are represented in equilibrium
by a consensus consumer that behaves as a bayesian updating agent. This represents the
fact that in equilibrium agents that have beliefs that are far away from the average long-term
consumption growth increasingly loose their wealth. Eventually the agents that happen to
have the correct beliefs about the long-term mean are the only ones who survive. The rate
of convergence depends on the risk-aversion parameter and it is faster the lower isγ. This
is because then agents are less cautious and therefore those agents with the wrong beliefs
loose their wealth faster.
Further if all agents are perfectly rational in the sense that they update their beliefs in the
same way then the convergence will be even faster. However, the reality is that economic
agents do hold different beliefs and these differences do not vanish. For this reason I assume
that the additional informationǫ is interpreted differently across agents and does not allow
beliefs to converge. However, I depart from the usual approach whereby we would need to
model the beliefs of every agent and I assume instead the law of motion of the distribution
of beliefs in equilibrium which I do next.
85
Assumption 3.9. The average beliefs about the mean and volatility of consumption growth
evolve according to,
μ
t 1
λ
μ
μ
t
p1 λ
μ
qg
t 1
,
σ
2
t 1
λ
σ
σ
2
t
p1 λ
σ
qpμ
t
g
t 1
q
2
.
whereλ
j
P p0,1 q,j P tμ,σ u. Uncertainty and belief heterogeneity in equilibrium are given
by,
ν
t
σ
1 κ
t
,
u
t
φ
u
σ
t
,
whereκ P p0,1 q.
The autoregressive processes ofμ
t
and σ
t
reflect the adaptation of beliefs in the economy
as a whole. The economy recognizes that the conditional moments of consumption growth
change and the average beliefs follow these simple processes and only depend on g and
the previous average beliefs. Even though individual beliefs are not modeled implicitly it
is assumed that each agent i updates his beliefs about the current mean of consumption
growth according to,
E
i
pμ
t 1
|F
t 1
q λ
μ
E
i
pμ
t
|F
t
q p1 λ
μ
qg
t 1
ε pǫ
t 1
,i q
The constant updating weight λ
μ
is consistent with Bayesian updating and the assump-
tion that the uncertainty is a constant fraction of macroeconomic risk. In fact, this puts a
restriction on the parameterφ
u
since,
λ
μ
σ
2
t
σ
2
t
u
2
t
,
86
and thereforeφ
u
a
1 {λ
μ
1.
The individual error terms ε pǫ
t 1
,i q is what keeps the heterogeneity variable ν
t
from
going to zero. Every period the level of heterogeneity changes three times. First it decreases
because wealth is transferred to agents whose predictions were more aligned with the real-
ization of g. It decreases even more because they observe g and this makes them update
their beliefs about the previous conditional mean. Finally, ǫ
t 1
carries information about
the change in the conditional mean and it is implicitly assumed that is interpreted differ-
ently and therefore causesν to increase on average.
The assumption thatν
t
is an increasing function ofσ
t
is a natural one since it is realistic
to assume that the uncertainty is increasing in the macroeconomic risk and in turn the level
of heterogeneity is increasing in the uncertainty. The important assumption however for
the results is that effective heterogeneity as it is expressed by the ratioν
t
{σ
t
is an increas-
ing function of σ
t
. Through this assumption I will be able to generate a high correlation
between the price level and macroeconomic risk.
With the last assumption the model is complete. The economy and its equilibrium are
parameterized by the preference parameters pδ,γ,η q the autoregressive parameters pλ
j
,j P
tx,μ,σ uq and the parameters pφ
ν
,κ q. The endogenous state vector is,
z pω,μ,σ q
and the law of motion is described by that of ω as given by (3.6) and by assumption 3.9.
The equilibrium interest rate is then given by corollary 3.6.
3.6 Calibration
Theoretically, the introduction of heterogeneous beliefs affects both the level of prices as
well as the equity premium. While the overall effect on the equity premium is unclear,
87
the effect on the level of prices is straightforward, namely that when heterogeneity rises
prices decrease. I have further argued that when changes in the level of heterogeneity
are persistent, then the level of the stock market is more sensitive to those changes than the
risk-free rate is. The calibration exercise that will be presented will show that quantitatively
the important asset pricing element is the variations in the discounting factor and that the
increase in prices during the last decade of the 20th century can be explained by a decrease
in the level of belief heterogeneity.
For the calibration exercise I use quarterly data of consumption growth, the market
index return and the real risk-free rate from the first quarter of 1947 until the last quarter
of 2007. t 0 corresponds to the start of the data period. The data that I use are first the
real aggregate quarterly consumption growth of non-durables and services as obtained from
NIPA tables. I then obtain the nominal 3-month T-bill yield from the Fama risk-free rates
file of the CRSP monthly treasuries database. The real rate is obtained after deflating the
nominal with the realized inflation, measured by the CPI index obtained from the Bureau
of Labor Statistics. The market price-dividend ratio is imputed from the quarterly CRSP
market index returns including and excluding dividends.
3.6.1 Homogeneous Economy with Uncertainty
The first parameters that need to be chosen are the updating weightsλ
μ
andλ
σ
which reflect
how the average beliefs about the conditional mean and conditional volatility of aggregate
consumption growth are adapted. Since beliefs have not been modeled I have chosen to use
the maximum likelihood estimates assuming the particular time-series model. In addition
to these two parameters I also estimate the initial values forμ andσ for the time period. The
estimates are shown in table 3.1. The conditional volatility appears to be quite persistent
with a value of0.94 whereas the conditional mean shows significantly less persistence with
a value of 0.76. The time series of these two conditional moments during the time period
88
Table 3.1: Aggregate consumption growth parameters
μ
0
(%) σ
0
(%) λ
μ
λ
σ
log likelihood
g 0.3316 0.7895 0.7551 0.9403 -970.1422
(0.5027) (0.1825) (0.0706) (0.0166) .
Maximum likelihood estimates of the process
g
t 1
|F
t
∼ N pμ
t
,σ
t
q
where,
μ
t 1
λ
μ
μ
t
p1 λqg
t 1
,
σ
2
t 1
λ
σ
σ
2
t
p1 λ
σ
qpg
t 1
μ
t
q
2
.
The parameters were estimated using quarterly aggregate consumption data of non-
durables and services from the first quarter of 1947 to the last quarter of 2007.
are shown in the top two panels of figure 3.2. The bottom panel of the same figure shows the
consumption surplus variableω y x, given the selected parameter value forλ
x
0.95.
The parameterλ
x
was chosen in order to match the persistence in the price-dividend ratio
in the date, which is0.93. It is very interesting to note that the estimated persistence of the
price-dividend ratio is very close to the estimated persistence of the conditional volatility
of consumption growth.
The fitted time-series of the conditional volatility of consumption growth was plotted
against the contemporaneous price-dividend ratio. The correlation between the two series
is impressive and the sample linear correlation is 0.78. This correlation however can-
not be explained in a homogeneous economy with varying beliefs and uncertainty. I first
calibrate such an economy by trying to match the evolution of the price-dividend ratio.
Table 3.2 shows the parameter values for both the homogeneous and heterogeneous econ-
omy. The risk-aversion parameterγ was chosen to be 4, a realistic value for asset pricing.
The subjective discount factorδ and the habit parameterη were chosen to match the level
89
1950 1960 1970 1980 1990 2000
5
10
15
x 10
−3 μ
1950 1960 1970 1980 1990 2000
2
4
6
8
10
x 10
−3 σ
1950 1960 1970 1980 1990 2000
0.14
0.16
0.18
0.2
0.22
ω
Figure 3.2: Time series of the state vector.
and variability of the price-dividend ratio until the end of 1980’s. The initial consumption
surplusω
0
was set at its long-term average value.
Figure 3.3 shows the data and the model implied time-series for the price-dividend ratio
and the risk-free rate. I observe that the model is able to match the price-dividend ratio quite
well until the end of the1980’s. Then, despite the decrease of both the macroeconomic risk
as well as the uncertainty the price-dividend ratio stays around the same levels. The risk-
free rate implied by the model is significantly higher than the data since the model is unable
to generate enough equity premium.
Table 3.3 compares the sample period price statistics of the data with the model implied
statistics. First I note that the model is able to match the first lag autocorrelation of the
price-dividend ratio at 0.93 but it is not able to match the fourth lag autocorrelation. The
90
Table 3.2: Model parameters
Homogeneous Heterogeneous
δ 0.9475 1.087
γ 4.0 4.0
η -4.0 -4.0
κ 0.07
λ
x
0.95 0.95
λ
μ
0.7551 0.7551
λ
σ
0.9403 0.9403
μ
0
(%) 0.3316 0.3316
σ
0
(%) 0.7895 0.7895
ω
0
0.4119 0.4119
The parameters pλ
μ
,λ
σ
,μ
0
,σ
0
q where estimated using maximum likelihood as indi-
cated in table 3.1. λ
x
was chosen to match the persistence in the price-dividend ratio.
δ andη where chosen to match the level and variability of the price-dividend ratio until
the end of the 1980’s.
average risk-free rate is much higher for the model while the variability is a little smaller.
The volatility of the price-dividend ratio implied by the homogeneous economy is about
half of the data sample volatility and this is attributed to the fact that the model is not able
to explain the significant increase in prices after the end of 1980’s. Figure 3.4 shows two
plots of the price-dividend ratio over different combinations of the state vector. In the left
plot I fix the conditional mean at the data sample average and vary the other two parameters.
I see clearly that the effect of the conditional volatility on the price-dividend ratio is very
small contrary to what I see in the data. Neither the level of macroeconomic risk nor the
level of uncertainty cause any significant variation in the stock price.
On the right plot of figure 3.4 I fix the conditional volatility to the sample average
and vary the conditional mean and the consumption surplus. These two state variables
almost equally affect the price-dividend ratio in a natural way. When the economy is more
optimistic, prices increase as well as when the current consumption is high in relation to
habit. To understand this behavior I need to see the effect of these two state variables on
91
Figure 3.3: Homogeneous economy log price dividend ratio and risk free rate.
1950 1960 1970 1980 1990 2000
4
4.5
5
5.5
6
Log price−dividend ratio

recessions
data
model
1950 1960 1970 1980 1990 2000
−3
−2
−1
0
1
2
3
Risk−free rate

recessions
data
model
The model time series were generated using the time series of the state vector as shown
in figure 3.2.
the risk-free rate as shown in the right plot of figure 3.5. When the consumption surplus
is high agents want to transfer wealth to the next period which causes the risk-free rate to
fall. The opposite happens when the conditional mean increases in which case agents want
to borrow. However, when the conditional mean is high it means that the expected future
consumption surplus is high and therefore the future interest rates are low. This causes the
current stock price to increase since habit is persistent. The left plot of the same figure
shows that the macroeconomic risk does not affect the interest rate much and for the same
reason neither the price-dividend ratio. In fact I see that such a model of homogeneous
beliefs even with the inclusion of uncertainty that is proportional to the level of the risk
cannot explain the variation of the price-dividend ratio.
92
Table 3.3: Quarterly Statistics (1947Q1-2008Q1)
Data Homogeneous Heterogeneous
μ ppd q 4.8163 4.5617 4.7908
σ ppd q 0.4365 0.2296 0.4237
μ pr
f
q (%) 0.2450 1.9684 1.0633
σ pr
f
q (%) 0.8651 0.6864 0.8043
ρ ppd,r
f
q 0.1317 0.0661 -0.3828
ACF
pd
p1 q 0.9305 0.9348 0.9641
ACF
pd
p4 q 0.9136 0.6806 0.8120
The model implied statistics are generated using the observed history of consumption
growth. μ pxq and σ pxq denote the sample mean and standard deviation of variable x.
ρpx,y q denotes the sample correlation and ACF
x
pn q is the autocorrelation of x for
lagn.
Figure 3.4: Homogeneous economy log price dividend ratio
0.05
0.1
0.15
0.2
0.25
0.2
0.4
0.6
0.8
1
4
4.2
4.4
4.6
4.8
5
5.2
5.4
ω
pd (μ=0.82373%)
σ (%) 0.05
0.1
0.15
0.2
0.25
−0.5
0
0.5
1
1.5
2
3.8
4
4.2
4.4
4.6
4.8
5
5.2
ω
pd (σ=0.4885%)
μ (%)
The conditional volatility does affect on the other hand the equity premium, as shown in
the left plot of figure 3.6. Figure 3.6 shows the homogeneous economy’s expected excess
93
Figure 3.5: Homogeneous economy risk-free rate
0.05
0.1
0.15
0.2
0.25 0.2
0.4
0.6
0.8
1
−2
−1
0
1
2
3
4
5
6
σ (%)
r
f
(%) (μ=0.82373%)
ω
0.05
0.1
0.15
0.2
0.25 −0.5
0
0.5
1
1.5
2
−5
0
5
10
μ (%)
r
f
(%) (σ=0.4885%)
ω
return for different combinations of the state vector. As I see the expected excess return
is not so much affected by the other state variables. In fact the endowment habit ratio
has almost no effect whereas the conditional mean of consumption growth determines the
level of the term premium through variations in the risk-free rate. The model implied time
series of the conditional equity premium as shown in figure 3.7 starts from a value around
1.6% annually and steadily declines down to around 0.5% annualized and is driven by the
decrease of the conditional volatility of consumption growth. The conditional volatility of
returns on the other hand, which is shown in the lower panel of 3.7, is driven by all three
variables.
94
Figure 3.6: Homogeneous economy expected excess return
0.05
0.1
0.15
0.2
0.25
0.2
0.4
0.6
0.8
1
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
ω
E(R
e
) (%) (μ=0.82373%)
σ (%) 0.05
0.1
0.15
0.2
0.25
−0.5
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
0.25
ω
E(R
e
) (%) (σ=0.4885%)
μ (%)
3.6.2 Heterogeneous Beliefs Economy
In order to see the impact of belief heterogeneity I keep the same parameters as for the
homogeneous economy except from the subjective discount factor. I chose the subjective
discount factor and the parameter κ, that determines the sensitivity of the level of hetero-
geneity in the economy to the macroeconomic risk, in order to match the time series mean
and volatility of the price-dividend ratio. The subjective discount factor was calibrated at
a value greater that one. This was necessary due to the fact that the rest of the parame-
ters were taken from the homogeneous economy. Otherwise a stronger habit (by makingη
more negative) would require a smallerδ.
Table 3.3 shows the model implied time series statistics and it is noted that the mean and
volatility of the price-dividend ratio were matched quite well. The first lag autocorrelation
is slightly higher than in the data while the fourth lag autocorrelation is at 0.81 instead of
95
Figure 3.7: Homogeneous economy conditional mean and volatility of expected excess
return
1950 1960 1970 1980 1990 2000
0
0.1
0.2
0.3
0.4
0.5
Conditional expected excess return (%)
1950 1960 1970 1980 1990 2000
6
7
8
9
10
11
Conditional return volatility (%)
The model time series were generated using the time series of the state vector as shown
in figure 3.2.
0.91. However, the autocorrelation shows an improvement compared to the homogeneous
economy because now the autocorrelation is also affected by the autocorrelation of the
macroeconomic risk. The model average of the risk-free rate is significantly higher than in
the data, 1.06% as opposed to 0.25% but at the same time significantly lower than the one
generated by the homogeneous economy which is1.97%. This is a result of the impact on
the equity premium as well as the increase in the price-dividend from 1990 and onwards.
The main result of the study is shown in the top panel of figure 3.8 and the left panel of
figure 3.10. The model implied time series of the price-dividend ratio captures quite well
the data since the sensitivity of the price-dividend ratio to a decrease in the consumption
risk is sufficient to explain the phenomenal increase in the prices during the last decade
96
Figure 3.8: Heterogeneous economy log price dividend ratio and risk free rate.
1950 1960 1970 1980 1990 2000
4
4.5
5
5.5
6
Log price−dividend ratio

recessions
data
model
1950 1960 1970 1980 1990 2000
−3
−2
−1
0
1
2
3
Risk−free rate

recessions
data
model
The model time series were generated using the time series of the state vector as shown
in figure 3.2.
of the 20th century. The left panel of figure 3.10 shows the model implied variation of
the price-dividend ratio for a fixed value of μ and the complete range of values of the
other two state variables. Comparing the variation induced by the conditional volatility
of consumption growth as predicted by the model with the data as shown in figure 3.1 I
observe that the model does a very good job replicating this behavior.
Finally, looking at the predicted equity premium as shown in figure 3.12, the hetero-
geneity of beliefs as modeled in this study does have a significant impact on the expected
excess return with values that reach above 2% annually. The main avenue through which
the equity premium is affected is again related to the discounting effect. Due to the assump-
tion of normality and in addition the assumption that agents disagree about the conditional
97
Figure 3.9: Heterogeneous economy conditional mean and volatility of expected excess
return.
1950 1960 1970 1980 1990 2000
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Conditional expected excess return (%)
1950 1960 1970 1980 1990 2000
7
8
9
10
11
12
13
14
Conditional return volatility (%)
The model time series were generated using the time series of the state vector as shown
in figure 3.2.
mean of consumption growth, the largest disagreement among agents is on the tails of the
distribution, which means concerning large absolute values of consumption growth. Now
large future values of consumption growth either positive or negative will increase the level
of heterogeneity in the next period and therefore decrease the future prices. Hence, there
is a high negative correlation between the endogenous risk-factor ξ
r
and the future stock
returns. I remind here thatξ
r
is low in states where there is a lot of disagreement. There-
fore, the market is a bad hedge for the states were agents’ consumption is low. Note that
even though different agents have different states of low consumption, the stock has a low
return in all of those states.
98
Figure 3.10: Heterogeneous economy log price dividend ratio.
0.05
0.1
0.15
0.2
0.25
0.2
0.4
0.6
0.8
1
3.5
4
4.5
5
5.5
6
σ (%)
pd (μ=0.82373%)
ω
0.05
0.1
0.15
0.2
0.25
−0.5
0
0.5
1
1.5
2
4
4.5
5
5.5
ω
pd (σ=0.4885%)
μ (%)
The only drawback concerning the equity premium of this model is that it is increasing
in the price-dividend ratio. This is mostly because of the particular function chosen that
relates the level of heterogeneity of beliefs with the level of consumption risk. In particular
it becomes more and more sensitive to changes in consumption risk as consumption risk
decreases. The higher sensitivity increases in absolute value the negative correlation of
the endogenous risk-factor with the market returns and hence the equity premium. The
particular function was only chosen due to its simplicity and it is not crucial to the results
of this study.
3.6.3 Evidence of Convergence in Beliefs
In order to see whether there is any evidence concerning the evolution of belief hetero-
geneity in the economy concerning future consumption growth I looked at the Survey of
Professional Forecasters provided by the Federal Reserve Bank of Philadelphia. In one of
99
Figure 3.11: Heterogeneous economy risk-free rate.
0.05
0.1
0.15
0.2
0.25 0.2
0.4
0.6
0.8
1
−4
−2
0
2
4
6
σ (%)
r
f
(%) (μ=0.82373%)
ω
0.05
0.1
0.15
0.2
0.25
−0.5
0
0.5
1
1.5
2
−6
−4
−2
0
2
4
6
8
μ (%)
r
f
(%) (σ=0.4885%)
ω
the surveys the professional forecasters are asked to predict the next quarter real consump-
tion growth. From this survey I compute the cross-sectional standard deviation of forecasts
for every period provided. Data are available from the third quarter of 1981 until today.
This time series turned out to be quite volatile and for this reason I extracted the trend
from this series using the Hodrick-Prescott filter. The trend was then plotted along with
the time-series of ν of the calibrated heterogeneous agent economy. The plot is shown in
figure 3.13. Even though the heterogeneity of forecasts by the professional forecasters is
not considered to be representative of the heterogeneity of believes in the economy due
to strategic behavior,
17
the data does provide significant evidence that during this period
beliefs about consumption growth converged significantly.
17
See for example Ottaviani and Sørensen (2006) and references therein.
100
Figure 3.12: Heterogeneous economy expected excess return.
0.05
0.1
0.15
0.2
0.25
0.2
0.4
0.6
0.8
1
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
σ (%)
E(R
e
) (%) (μ=0.82373%)
ω
0.05
0.1
0.15
0.2
0.25
−0.5
0
0.5
1
1.5
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ω
E(R
e
) (%) (σ=0.4885%)
μ (%)
3.7 Conclusion
In this study I show that the driving force of heterogeneous beliefs is very important in
shaping asset prices. For example I successfully show through my model economy that
the phenomenal increase in the aggregate price-dividend ratio during the 1990’s can be
explained by an alignment in the beliefs of the various investors. During this time the
macroeconomic risk as given by the conditional volatility of consumption growth declined
significantly causing in this way a decrease in the heterogeneity of beliefs in the economy.
Hence, agents became confident that all investors had the same prospects in mind about
the economy which implies that they did not fear individual bad states and this led to a
significant increase in the level of prices.
101
Figure 3.13: Model and data heterogeneity of beliefs
1940 1950 1960 1970 1980 1990 2000 2010
0
10
20
30
40
50
60
70
Cross−sectional volatility of beliefs (ν)
%

model
professional forecasters
Model time-series of cross-sectional volatility of beliefs, ν and the Hodrick-Prescott
filter trend of the time-series of the cross sectional volatility of one quarter predictions
on real consumption growth from the Survey of Professional Forecasters as provided
by the Federal Reserve Bank of Philadelphia.
The risk premium in my economy is also significantly increased due to an endogenous
risk factor that arises due to the different investment behavior. This new risk factor is nega-
tively correlated with stock returns because when the heterogeneity of beliefs is positively
autocorrelated then the stock pays poorly at states where the heterogeneity is high. In those
states consumption is low for many agents and hence a bad hedge for their risk.
The link that connects consumption risk with asset price behavior has been particularly
elusive, so much that significant doubts have been cast over the canonical asset pricing
model. The results of this study however show that the quantitative behavior of asset prices
predicted by the model is significantly improved once I introduce differences in beliefs.
102
If we are therefore to understand asset markets we must understand how and why beliefs
differ and how they evolve over time.
103
Chapter 4
Prices and Trading in an Economy with
Differences of Opinions
4.1 Introduction
The amount of financial trading in any given day is extraordinary. In 2007 the monthly
share turnover in the New York Stock Exchange (NYSE) ranged from 11% to 20% with
an average of 14%.
1
Yet the asset pricing literature is unable to shed light on this fact.
Understanding the amount of trading is quite important by itself. However, even from a
purely asset pricing perspective understanding the determinants of financial trading is very
important since prices and volume have been found to be amply related with each other.
This study focuses on explaining the highly significant positive correlation between
changes in the average level of prices and changes in the amount of trading. Figure 4.1 is
similar to figure 3 of Hong and Stein (2007) and displays the real percentage change in the
Standard and Poor’s (S&P) composite index along with the percentage change in the NYSE
share turnover from 1901 to 2003. The remarkable correlation between the two series is
0.52. The percentage changes where considered in order to remove time-trends from the
data. Another way to remove the trend is by using the Hodrick-Prescott (HP) filter and
the resulting plot is shown in figure 4.2. The correlation is virtually unaltered. About the
same highly significant correlation is also exhibited by monthly data from January 1964
to August 2008. Figure 4.3 shows a scatter plot between real percentage changes in the
1
New York Stock Exchange Factbook, (http://www.nyxdata.com).
104
Figure 4.1: Price changes and volume changes (annual data)
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
−60
−40
−20
0
20
40
60
%
Annual Price and Trade Relation

Change in NYSE Turnover
Change in Real S&P Comp. Index Price
The plot shows the annual percentage changes in the S&P Composite index level
adjusted for inflation along with the annual percentage changes in the NYSE turnover.
The correlation between the series is 0.5257. The data cover the period from 1900 to
2003. Price date where obtained from Rober Shiller’s website and turnover data from
the NYSE factbook.
monthly dollar volume of NYSE against real percentage changes on the S&P composite
index. The statistical correlation is close to0.3. When trend is removed using the HP filter
the correlation between the two series increases to about0.56 and the corresponding scatter
plot is shown by figure 4.4.
The empirical regularity already mentioned along with the positive correlation between
volume and volatility were the first and most notable regularities uncovered by the empir-
ical literature. Karpoff (1987) offers an early literature overview that looks at these two
effects.
2
Both of these regularities as well as other empirical facts like excess volatility
2
Further studies about the positive relation between volume and volatility are those of Schwert (1989)
and Gallant, Rossi, and Tauchen (1992). Other empirical studies that analyze the dynamic relation between
trading and returns for both the cross-section and the aggregate stock market include Campbell, Grossman,
and Wang (1993), Llorente, Michaely, Saar, and Wang (2002), Chordia, Huh, and Subrahmanyam (2007) and
Griffin, Nardari, and Stulz (2007). Lo and Wang (2000) offers some further list of references.
105
Figure 4.2: Price cycles and volume cycles (annual data)
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
−60
−40
−20
0
20
40
%
Annual Price and Trade Relation

HP filtered NYSE Turnover
HP filtered Real S&P Comp. Index Price
The plot shows the standardized deviations from a trend of the S&P Composite index
level adjusted for inflation along with standardized deviations from a trend of the
NYSE turnover. The trend was estimated using the Hodrick-Prescott filter and the
deviations were standardized with the trend. The correlation between the two series is
0.5269. The data is annual cover the period from 1900 to 2003.
and volatility persistence, are explained in a general equilibrium model of differences of
opinions and risk averse agents. The main elements of my theoretical study is the risk-
preference assumption that agents are more risk-averse than a logarithmic agent and a time-
varying sentiment risk, which are new in this literature. In this study I make a distinction
between the belief dispersion and sentiment risk in the market. While the first refers to the
current beliefs about the future macroeconomic risk, sentiment risk determines how differ-
ently agents may interpret new information. The connection between the two is that a high
sentiment risk is expected to lead to high dispersion in beliefs in the future.
The study makes a number of contributions in the literature of dynamic models of
trading activity as well as the pure asset pricing literature. First, I provide a natural general
equilibrium framework that creates a continuous need for agents to trade. Secondly, with
106
Figure 4.3: Price changes vs. volume changes (monthly data)
−60 −50 −40 −30 −20 −10 0 10
−10
−5
0
5
10
Change in $ Volume of Trade (%)
Change in S&P Comp. Index Price (%)
Monthly Price and Trade Relation
This is a scatter plot of percentage changes in the level of the S&P Composite index
adjusted for inflation against real percentage changes in the dollar volume of trade in
NYSE. The correlation is 0.2858 and the data is monthly and cover the period from
January 1965 to May 2008.
the introduction of what I call sentiment risk and with my realistic preference assumption I
create a direct link between price characteristics and volume, that are found to be strongly
related by empirical studies, in a way consistent with many seemingly unrelated empirical
regularities. Further, within this framework I give an argument that can explain the positive
correlation between price changes and volume. Finally, even though it is not the focus of
this study, the model predicts that the sentiment risk is a significant risk asset pricing factor.
The seminal study by Milgrom and Stokey (1982) shows that in a complete market
economy with asymmetric information (and even different utility functions) agents do not
107
Figure 4.4: Price cycles vs. volume cycles (monthly data)
−35 −30 −25 −20 −15 −10 −5 0 5 10
−10
−5
0
5
10
HP filtered Volume of Trade
HP filtered Real S&P Comp. Index Price
Monthly Price and Trade Relation
The scatter plot shows the standardized deviations from an estimated trend of the real
level of the S&P Composite index against the standardized deviations from an esti-
mated trend of the NYSE dollar volume. The trends were estimated using the Hodrick-
Prescott filter. The correlation is 0.5625. Data is monthly and cover the period from
January 1965 to May 2008.
trade beyond the first period.
3 ,4
The no-trade result has two main components. First, in a
market with no frictions disparate information is aggregated and revealed by equilibrium
prices. Secondly, the aggregated information is interpreted in the same way by agents.
3
Another important no-trade theorem is that of Judd et al. (2003). They show that in an economy with
preference heterogeneity and dynamic completeness agents do not trade beyond the first period. The required
assumption is that the transition matrix of the exogenous state is of full rank. One could relax this assumption
and predict trading in case for example when labor income risk is not perfectly correlated with aggregate
endowment risk. In this case agents would need to adjust their portfolio because their income risk demands
different hedging at different periods. However, the predicted trading is minimal and cannot be expected to
be related to prices. A similar no-trade theorem in continuous time and the conditions for non-trivial trading
volume are shown by Berrada, Hugonnier, and Rindisbacher (2007)
4
Blume, Coury, and Easley (2006) show that the no-trade theorem fails to hold when markets are incom-
plete since new arrival of information offers new opportunities for risk-sharing.
108
These two components have given birth to much of the literature that tries to explain trad-
ing. The first stream of research introduces frictions in the market through noise or liquid-
ity traders whose action first impedes the aggregation of information and further introduces
volatility and trading through their own non-informational motives.
5
Wang (1994) explains
the positive contemporaneous covariance between volume and absolute price changes in
a model with asymmetric information where the informed investors enjoy private invest-
ment opportunities. For this reason the uninformed investors are unable to extract all the
information from prices and require a significant drop in price for them to buy the stock.
Therefore trading is accompanied by absolute price changes but the correlation per se is
not examined. The predictions in the noise trading literature hold so much as there is
non-informational trading. Additionally, Wang (1994) points out that it is unlikely that
non-informational trading is the main driver of the trading volume I observe.
The second line of research in which this study belongs, assumes difference in opinions
that comes either explicitly or implicitly through relaxing the assumption of uniform inter-
pretation of available information.
6
Harris and Raviv (1993) and Scheinkman and Xiong
(2003) offer a similar explanation to trading and volatility in models with risk-neutral
agents and short selling constraints. Essentially, always the most optimistic agent holds
the stock and the stock changes hands when the relative valuations reverse. Under their
specific settings price changes are likely to be associated with trading since more valuation
reversals will indicate volatility which will also be associated with trading. This expla-
nation however seems unsatisfactory for the volume volatility relation firstly because the
5
See for example Grossman and Stiglitz (1980), Hellwig (1980), Diamond and Verrecchia (1981), Kyle
(1985), Admati and Pfleiderer (1988), Grrundy and McNichols (1989), Foster and Viswanathan (1990), Foster
and Viswanathan (1993), Long, Shleifer, Summers, and Waldmann (1990), Kim and Verrecchia (1991a), Kim
and Verrecchia (1991b), Shalen (1993) and Wang (1994).
6
Examples of this literature are Harrison and Kreps (1978), Varian (1985), Varian (1989), Harris and
Raviv (1993), Kandel and Pearson (1995), Hong and Stein (2003), Scheinkman and Xiong (2003) and Cao
and Yang (2008).
109
preference assumption is not realistic and the predicted trading behavior is too extreme.
7
Further they produce a counterfactual prediction that stock ownership is concentrated. The
remark of Gallant et al. (1992) that “(t)here seems to be no model with dynamically opti-
mizing heterogeneous agents that can jointly account for major stylized facts - serially
correlated volatility, contemporaneous volume-volatility correlation, and excess kurtosis
of price changes,” appears to be a still fair description of the current state of research on
trading.
The failure of this literature to uncover the connection between prices and volume
seems to be firstly due to the simplistic preference assumptions. Despite the conventional
wisdom, trading and especially pricing predictions are sensitive to the preference assump-
tion when beliefs are heterogeneous. Whereas risk-neutrality predicts a bubble component
when opinions differ, the more realistic assumption of a constant relative risk aversion
higher than one predicts the exact opposite, i.e. that prices decrease when belief hetero-
geneity increases, especially when heterogeneity is persistent as I have seen in chapter 3.
The other convenient preference assumption of constant absolute risk aversion (CARA)
also exhibits major shortcomings in this context both in terms of prices as well as trading.
The CARA utility predicts that the price impact only depends on the beliefs and not on the
relative wealth of every agent because their positions are independent of their wealth. The
reason I am able to use CRRA preferences is because the equilibrium is accurately approx-
imated using a computational method developed to address complete market equilibria of
heterogeneous agent economies as it is outlined in appendix B.
The economic story that emerges from the model and is able to explain the positive
correlation between price changes and volume relies on the mentioned pricing effect and on
the assumption that sentiment risk in the market is either negatively or positively correlated
with macroeconomic risk. Agents trade mainly for two reasons: (i) when their opinions
7
In the continuous time setting of Scheinkman and Xiong (2003) without transaction costs trading volume
is infinite.
110
change and (ii) when their relative wealth changes. When opinions diverge agents trade
and prices decrease, while when opinions converge agents trade again but prices increase
this time. These two effects on average cancel each other out and therefore the first reason
does not predict a particular positive or negative correlation between price changes and
volume. The positive or negative correlation comes from the second source of trading
depending on how sentiment risk is related with macroeconomic risk. Without getting
into the details of the argument I only mention that trade in this case happens when the
realization of the uncertainty does not favor the agent who happens to load more heavily on
the market portfolio due to the disagreement in the market and needs to unload his position.
Whether prices will increase or decrease in this case depends on the specific assumption
about the correlation between sentiment risk and macroeconomic risk. Two distinct cases
are considered that both predict a positive correlation between price changes and volume
and I choose between them using other pricing implications.
One significant contribution of the study is that, irrespective of the correlation of senti-
ment risk with macroeconomic risk, it establishes a clear link between volume and prices
through sentiment risk. In this way I am able to explain the connection between volatility
and volume, the excess volatility, volatility persistence and the “leverage effect”, that is
the negative correlation between price levels and conditional volatility of returns. The eco-
nomic story behind the volatility volume relation is simple. When sentiment risk is high the
expected reallocation in the economy as well as the expected change in disagreement is sig-
nificant and this predicts a high trading activity. Further, this reallocation as well as changes
to disagreement creates significant volatility in the market due to the discounting effect that
I already mentioned. Direct evidences for the connection between disagreement and vol-
ume is offered by Ziebart (1990) that documents a positive relation between the volume
and the absolute change in the mean forecast of analysts. Further, Bessembinder, Chan,
and Seguin (1995) find that increases in the S&P500 Index futures’ open interest which is
111
considered to be caused by an increase in dispersion in beliefs is associated with higher
trading volumes. In this study I also offer empirical evidences that confirm all significant
predictions of the model in relation to known empirical price and volume regularities. The
model also generates some new predictions in relation to the connection between beliefs
and prices for which I provide some supporting evidence.
The rest of the study is structured as follows: Section 4.2 outlines the model econ-
omy and section 4.3 derives the equilibrium conditions and proves its Markovian structure.
Section 4.4 looks at the impact of opinion differences on trading. With the inclusion of a
varying sentiment risk I am able to explain the volatility volume relation, excessive volatil-
ity and volatility persistence. With a further assumption on the correlation of sentiment
risk with macroeconomic risk I am able to offer two distinct cases where price changes are
positively correlated with volume. In section 4.5 I offer empirical evidence that support all
significant predictions of the model. Section 4.6 concludes.
4.2 The Model
I consider a model of an endowment economy with a single consumption good where the
aggregate consumption growth represents the only source of fundamental risk. Agents in
this economy are risk-averse and differ in their opinions about the distribution of macroeco-
nomic risk. The belief heterogeneity is supported by the assumption that agents differ in the
way they interpret publicly available information. Kandel and Pearson (1995) offers such
evidence. Classical economic thought through perfect rationality assumes that economic
agents know the true data generating process and therefore common information does not
allow them to differ in the way they assess future uncertainty. No matter how convenient as
a modeling assumption it may be, it is hard to imagine anyone being able to establish this
empirically. The fact that I am unable to pin down the heterogeneity in beliefs should not
prevent us from trying to see its economic implications.
112
One way to rationally explain belief heterogeneity is by considering that agents in real-
ity do not know the true data generating process. Then it would be natural to assume that
agents can have a range of models of different types and complexities that are statistically
indistinguishable. Then any agent in forming his beliefs about the future, needs to rely on
one or maybe several of those models with certain weights. If I add to this decision process
costs of collecting and processing all available information for picking the best possible
model every period, I arrive at a point where the economic agent might rely randomly on
one of competing models. Therefore, with this simple chain of thought I have arrived at the
assumption concerning the differential interpretation of public information without even
resorting to behavioral factors like varying confidence or sentiment that would affect how
agents might transform information to beliefs.
8
4.2.1 Uncertainty, Information and Beliefs
Time is discrete and infinite, t 0,1,2,.... During every time step I introduce an inter-
mediate point at which agents acquire relevant information and trade. The intermediate
periods are denoted witht
. It is assumed that agents do not consume during the interme-
diate periods. The reasons for the introduction of the intermediate period is for the agents to
receive relevant information for the economy. At every intermediate period the underlying
structure of the economy changes and beliefs may diverge due to different interpretations
about available information concerning this change. I denote withF
t
all information about
observable and unobservable quantities in the economy up and including periodt. WithP
I denote the true probability measure which will be assumed to be unknown.
There is one consumption good in the economy and y
t
denotes the aggregate endow-
ment of the consumption good in period t. Let y
0
be given. The aggregate endowment
8
For models of overconfidence and market sentiment one can look at Barberis, Shleifer, and Vishny
(1998), Daniel, Hirshleifer, and Subrahmanyam (1998) and Gervais and Odean (2001).
113
is paid by the only productive stock in the economy with price P
t
and uncertain divi-
dend growth g
t 1
y
t 1
{y
t
. The dividend growth which represents the macroeconomic
uncertainty takes two possible values tg
h
,g
l
u where g
h
¡ g
l
. I therefore consider no
labor income in the economy. The probability of the high growth state, which is con-
sidered to represent the fundamentals of the economy, is assumed to be time varying. Let
π
t
P pg
t 1
g
h
|F
t
q denote the true probability of the high growth state in periodt 1
given the path of the economy until and including the intermediate period t
. I assume
that the underlying probability changes in time according to the following autoregressive
process,
π
t
φ
1
π
t
p1 φ
1
qǫ
π
t
, (4.1)
where φ
1
P p0,1 q and ǫ
π
is an iid process that takes values from t1,0 u with equal proba-
bilities. Note that the assumed process ensures that π will always be in the set p0,1 q and
has an unconditional average of0.5. In fact the unconditional moments of dividend growth
are independent of the process of the underlying probability due to the independence of the
shocks ǫ
π
and g. LetE andV denote the expectation and variance operators respectively
under the probability measureP.
There are two agents in the economy indexed by i 1,2. Agents do not observe
the underlying probability π or the shocks ǫ
π
. However, it is assumed that they know the
form of the process (4.1) along with the parameter φ
1
. Their information setF
a
includes
the past shocks g
t
and information signals tǫ
i
u
i 1,2
that are believed to carry information
about the true shock ǫ
π
. I denote withP
i
the probability measure of each agent i. Note
that a probability measure different than the objective measure needs to be assumed in
order to have heterogeneity of beliefs in the economy. This is because in the model the
differences of opinions at any point in time is not assumed to be due to differences in
information sets but due to differences in the interpretation of the same informationF
a
.
114
The signalsǫ
i
are meant to represent different information about the economy, for example
two macroeconomic indicators.
9
Every agent then once he observes the new information
will have to assess them in order to form his new opinion about the current state of the
economy. The two agents differ in the weights they give to these two macroeconomic
signals. The indexingi indicates that agenti believes that the macroeconomic signalǫ
i
on
average carries more information than ǫ
j
about the true change in the economy, which is
represented in this model by the shockǫ
π
.
In order to formulate the way that agents form their opinions about macroeconomic
uncertainty I need to specify their beliefs both about π and about the informativeness
of macroeconomic signals tǫ
1
,ǫ
2
u. In any whole period t I assume that agents hold
beliefs about the current true probability of the high state, according to a beta distribution
Beta pπ
i
t
τ
i
t
, p1 π
i
t
qτ
i
t
q,
dP
i
pπ
t
x |F
a
t
q x
π
i
t
τ
i
t
1
p1 x q
p1 π
i
t
qτ
i
t
1
»
1
0
u
π
i
t
τ
i
t
1
p1 u q
p1 π
i
t
qτ
i
t
1
du
dx. (4.2)
Therefore, the prediction of agent i is E
i
pπ
t
|F
a
t
q π
i
t
and his uncertainty about this
estimate is given by the variance V
i
pπ
t
|F
a
t
q π
i
t
p1 π
i
t
q{pτ
i
t
1 q. The variable τ
i
therefore is related to the precision of the beliefπ
i
and indicates their confidence. Infinite
confidence would imply that an agent is certain about the current true probability. This
study is not about confidence but about disagreement. The effect of confidence determines
only the weight he puts on new information relative to his current beliefs. For simplicity and
without loss of generality I assume that the initial beliefs for the two agents are identical.
I have already said that in every intermediate period when the true probability changes,
the two agents observe new information in the form of two signals or macroeconomic
indicators tǫ
1
,ǫ
2
u. Eachǫ
i
takes values from the set t0,1 u just likeǫ
π
. Each agent has his
9
To be more precise the innovationsǫ
i
should be regarded as changes to macroeconomic indicators.
115
own beliefs as to the informativeness of these two pieces of information. In order to form
their beliefs about the shock to π agents need to have certain beliefs about how these two
pieces of information are generated given the true state of nature. In particular I assume
that given someρ P p0,0.5 q each agenti believes that,
P
i
pǫ
i
,ǫ
j
|ǫ
π
q 1 ρ
2
1 tǫ
i
ǫ
π
u ρ
2
1 tǫ
j
ǫ
π
u, (4.3)
where 1 tA u is the indicator function that takes the value of one if statement A is true
and zero otherwise. ρ is related to the level of disagreement in the interpretation of new
information or what I will call later sentiment risk and for now I assume it to be constant.
Scheinkman and Xiong (2003) consider something similar but they interpret the differential
interpretation of signals as overconfidence. My approach and interpretation is different and
this is justified by the specifics of assumption (4.3). For example in their model agents
only consider one of the signals as they know that the other signal is uninformative. Here
I assume that both pieces of information are relevant but since agents do not know their
true informativeness they need to subjectively put relative weights. It is in this subjective
approach that they disagree.
Let us examine closely assumption (4.3). First thing I need to note is that when the
signals agree agents have no basis for disagreement no matter how they weight the signals.
Further, they believe that there is zero probability for both signals to be wrong. When
signals disagree ρ determines the disagreement between the agents when interpreting the
new information. There is no disagreement only ifρ is equal to0.5.
Lemma 4.1. Each agenti P t1,2 u believes that signali is more informative than the other
signal by p1 {2 ρ q in the sense that,
P
i
pǫ
i
ǫ
π
|ǫ
π
q 1 ρ
2
, and P
i
pǫ
j
ǫ
π
|ǫ
π
q 1 ρ
2
.
116
Further, the two signals are believed to be unconditionally independent,
P
i
pǫ
j
|ǫ
k
q P
i
pǫ
j
q 1
2
,
wherei,j P t1,2 u andk j.
Quantity p1 {2 ρ q indicates the level of disagreement about new information between the
agents and I call it sentiment risk. From lemma 4.1 I see than whenρ is (close to) zero, i.e.
when sentiment risk is high, then each agent thinks that one of the pieces of information is
(almost) fully informative while the other signal is (almost) not informative at all. If ρ is
1 {2 and therefore there is no sentiment risk then both agents consider both signals equally
informative. The following lemma makes this point even clearer.
Lemma 4.2. For each agent i P t1,2 u the expectation about the true shock ǫ
π
given the
signals tǫ
1
,ǫ
2
u and givenρ P p0,0.5 q is given by,
E
i
pǫ
π
|ǫ
i
,ǫ
j
q p1 ρ qǫ
i
ρǫ
j
.
The conditional variance is,
V
i
pǫ
π
|ǫ
i
,ǫ
j
q ρ p1 ρ q1 tǫ
i
ǫ
j
u.
Now I see from lemma 4.2 how ρ plays the role of weights on the different pieces of
information and through its deviation from 1 {2 how it generates sentiment risk. I only
assume that the two agents put the opposite weights on the two pieces of macroeconomic
information and further that ρ is time varying. The specific assumption about the time
variation ofρ will be stated next but I would like first to clarify how the variableρ affects
individual beliefs through new information. I note that when the signals agree agents are
certain about the true shock to the probability of the good state regardless of ρ and the
117
uncertainty as given by the conditional variance is zero. When the signals disagree then the
perceived informativeness depends on the value of ρ. When ρ has its maximum possible
value 1 {2 and sentiment risk is at its lowest then the signals are believed to be equally
informative and hence jointly non-informative since the variance attains its maximum value
and the conditional expectation aboutǫ
π
is equal to its unconditional average.
I assume that the disagreement about new information as determined byρ is time vary-
ing. In particular I assume thatρ follows a simple autoregressive process,
ρ
t 1
φ
ρ
ρ
t
p1 φ
ρ
qǫ
ρ
t 1
, (4.4)
whereφ
ρ
P p0,1 q andǫ
ρ
is an identically distributed process that takes values from the set
t0,0.5 u. Further the shock ǫ
ρ
is contemporaneous to and is allowed to be correlated with
consumption growthg,
P pǫ
ρ
0.5 |g g
h
q P pǫ
ρ
0 |g g
l
q η. (4.5)
This time variability in sentiment risk can be thought of as capturing several things; (i)
time variability in the uncertainty in the economy, (ii) variations in the amount of infor-
mation available, (iii) sentimental factors and (iv) variations in the level of dispersion of
information in the economy, that is variations in the degree to which the macroeconomic
indicators disagree. The two signals with a constantρ cannot capture these issues and hence
with the last assumption I am able to add another realistic and as it turns out an important
dimension to the heterogeneity of beliefs. What this assumption does is to allow the sen-
sitivity of beliefs to change through time and therefore control the speed by which beliefs
either diverge or converge. With this assumption the quantity p1 {2 ρ q becomes what I
call the sentiment risk which connects asset pricing characteristics and trading volume. It
will be the main factor driving the level of prices and the return volatility and the expected
118
volume of trade. Finally, the assumption concerning the correlation of the innovations of
ρ with consumption growth will be the one of two determinants for predicting a positive
correlation between price changes and volume.
I have already assumed that the posterior distribution of each agent at the end of every
whole periodt isBeta pπ
i
t
τ
i
t
, p1 π
i
t
qτ
i
t
q. However, if I add to this distribution a discrete ran-
dom variable as it is the case with the shockǫ
π
the resulting distribution does not belong to
the family of beta distributions. For this reason I make the following simplifying assump-
tion. After the signals are observed in period t
the posterior distribution of each agent
concerningπ
t
isBeta pπ
i
t
τ
i
t
, p1 π
i
t
qτ
i
t
q where,
π
i
t
φ
π
π
i
t
p1 φ
π
qE
i
pǫ
π
t
|F
a
t
q, (4.6)
andτ
i
t
is given by,
π
i
t
p1 π
i
t
q
τ
i
t
1
φ
2
π
π
i
t
p1 π
i
t
q
τ
i
t
1
p1 φ
π
q
2
V
i
pǫ
t
|F
a
t
q. (4.7)
Essentially, what is assumed is that once the new information is available the posterior
distribution remains to be beta and the uncertainty of the beliefs is determined by adding
the uncertainty from the previous beliefs and the uncertainty of the new information.
The two variablesπ
i
andτ
i
that determine the belief of agenti and his confidence serve
two different purposes. The probability π
i
is used by the agent to form his expectations
about the future. The precision variableτ
i
determines how the new information will alter
the mean estimate of the probability. In intermediate periodsτ
i
t
does not affect howπ
i
t
is
formed because the new posterior is aboutπ
t
while the previous posterior was aboutπ
i
t
and it only affects the new precision. The new precisionτ
i
t
is only used during the whole
periods where the realization of aggregate consumption growthg
t 1
is observed and agents
119
update their beliefs aboutπ
i
t
in Bayesian fashion. Therefore, since the beta distribution is
a conjugate prior to a bernoulli random variable the posterior distribution is also beta with,
π
i
t 1
p1 κ
i
t 1
qπ
i
t
κ
i
t 1
1 tg
t 1
g u, (4.8)
The precision variable is given by,
τ
i
t 1
τ
i
t
1, (4.9)
and the updating weight is hence given byκ
i
t 1
1 {τ
i
t 1
. Equation (4.7) can be modified
and written in terms ofκ,
κ
i
t 1
κ
i
t
1 κ
i
t
φ
2
π
π
i
t
p1 π
i
t
q
π
i
t
p1 π
i
t
q
p1 φ
π
q
2
V
i
pǫ
t
|F
a
t
q
π
i
t
p1 π
i
t
q
. (4.10)
In summary the beliefs are updated in the intermediate periods once the signals are
observed as given by (4.6) and during the whole periods according to (4.8) whereκ is given
by (4.10). The shocks that drive the economy are denoted by z
t 1
pǫ
1
t
,ǫ
2
t
,g
t 1
,ǫ
ρ
t 1
q
and letZ t1,0 u
2
tg
h
,g
l
u t0.5,0 u. The information at any whole period t is the
history of shocksz,F
a
t
ptz
k
,k 1,...,t uq whereσ denotes the sigma-algebra. In the
intermediate periods I haveF
a
t
tǫ
1
t
,ǫ
2
t
u,F
a
t
. I close this subsection by clarifying
how an agent forms his expectation in timet about the next period.
Lemma 4.3. Letf pz q be a function of the vector of shocksz. Then given the beliefs of an
agenti in some periodt,
E
i
rf pz q|F
a
t
s ‚
z PZ
P
i
pz |F
a
t
qf pz q
where,
P
i
pz |F
a
t
q P pǫ
ρ
|g qP
i
pg |F
a
t
qP pǫ
1
,ǫ
2
q.
120
Note that the conditional probability forg only depends on the prediction about the under-
lying probability; for exampleP
i
pg
h
|F
a
t
q π
i
t
and similarly forg
l
and not on the confi-
dence τ
i
. Further note from lemma 4.3 that agents only disagree on macroeconomic risk
and not on the non-fundamental uncertainty. Despite this it turns out that agents need to
share the non-fundamental risks as well.
At this point I make a certain simplification in the way agents form their opinions in
order to decrease the number of state variables and constitute the equilibrium computable.
The beliefs of an agent can be summarized by π
i
and κ
i
where κ
i
in any period depends
on its previous value. Therefore, I need to include both in the state vector. Alterna-
tively I remove the dependence on the previous value and substitute κ
i
with a function
κ pπ
i
,ρ,ǫ
i
,ǫ
j
q. For computations this functions is estimated by regressing κ
i
linearly on
the specified variables using simulated data with anR
2
higher than 90%. The quantitative
results are insignificantly affected by this adjustment.
4.2.2 Financial Market
During every sub-period there are two shock realizations and each shock takes two possible
values. In the second sub-period I haveg andǫ
ρ
and in the first I have the signals tǫ
1
,ǫ
2
u.
Therefore, in order to have complete financial markets the economy needs to offer at least
4 independent financial assets during every sub-period. Out of these four shocks agents
disagree only in terms of the probabilities of the fundamental shock g. One might think
that since agents do not disagree about the non-fundamental (or extraneous) risks in the
economy they have no need to share those risks and hence there is no need for more than
two assets. It turns out that they do need to trade on these risks because they affect their
wealth through their beliefs. These risks affect their wealth differently because their beliefs
are different and hence it is optimal to share those risks differently every period. The
reasons for this result will be discussed later once I derive the equilibrium variables.
121
The first and the most important asset in this study is the dividend paying stock whose
price divided by the aggregate endowment is denoted by P and is in unit net supply.
The other three assets are zero net supply contingent claims. In particular they are one
sub-period Arrow-Debreu securities that pay the aggregate endowment only in one of
immediate future states. For the first sub-period the contingent claims pay the previous
aggregate endowment. For example the claim for state p1,0 q in period t pays y
t
only if
pǫ
1
t
,ǫ
2
t
q p1,0 q and zero otherwise. For the first sub-period it is of no significance for
which three states there are contingent claims since the two shocks are independent of each
other and independent of anything else. However, the relation between trading and prices
as I will see does depend on which states contingent claims exist for the second sub-period.
I consider two cases: (i) case h where there is no contingent claim for state pg
h
,0.5 q
and (ii) case l where there is no contingent claim for state pg
l
,0 q. The state of shock ǫ
ρ
is immaterial. What matters is only the macroeconomic risk state. I call them case h or l
because in each corresponding case the stock becomes a vehicle to invest in either the high
growth state or the low growth state depending on the case. For example in case h when an
agent is more optimistic he invests more in the stock whereas in case l the more optimistic
agent invests less in the stock. The optimistic agent in a given state is the agenti such that
π
i
¡ π
j
, i.e. the one who is more optimistic about high consumption growth. The trading
behavior of these two cases is fundamentally different and I examine them both.
4.2.3 Preferences and Endowments
Agents have standard time and state separable power utility preferences over consumption
streamsc
i
pc
i
t
,t P t0,1,... uq with external habit similar to Gali (1994),
U pc
i
,y q E
i
#
8
‚
t 0
δ
t
u pc
i
t
,y
t
q
F
a
0
+
, (4.11)
122
whereδ P p0,1 q is the subjective discount factor and
u pc,y q y
γ
1
γ
2
$
’
&
’
%
ln pc q, ifγ
1
1,
c
1 γ
1
1 γ
1
, o/w.
, (4.12)
whereγ
1
¡ 0 is the coefficient of relative risk aversion andγ
2
is the habit parameter. When
γ
1
γ
2
then the preferences reduce to the usual power utility preferences without habit.
The external habit is the contemporaneous aggregate consumption and it is introduced only
for the following reason: Whenγ
2
is equal to one and beliefs are homogeneous then equi-
librium prices are constant regardless of the value ofγ
1
. Therefore, by forcing this param-
eter to be equal to one the resulting price volatility comes only from belief heterogeneity
and in this way I can quantify its effect.
As it is made clear from the preference assumption, agents only consume during whole
periods. In the intermediate periods preferences do not change apart from the exclusion
of the utility of consumption in the previous whole period. Therefore in an intermediate
periodt
preferences are given by,
E
i
#
8
‚
k 1
δ
k
u pc
i
t k
,y
t k
q
F
a
t
+
. (4.13)
Essentially, the intermediate period is considered to be a decision made by the agents once
they have consumed which is when they receive the new information. This way I can
examine the trading and asset pricing behaviors in the absence of changes in real allocation.
For simplicity and without loss of generality I assume that initially agents are endowed
with equal proportion of the dividend paying stock. I denote portfolio holdings at the end of
a periodt withθ
i
t
. Therefore at the beginning of time agents start with half of the dividend
paying stock and no holdings of the contingent claims.
123
4.3 Equilibrium
This section is notational intensive and I hope the reader can bear with us. I need to first
derive the conditions of a financial market sequential equilibrium and it is convenient at
first to not consider the intermediate periods but instead a different asset structure that
dynamically completes the market every whole period. I can do this since any sequen-
tial equilibrium with complete markets is also an Arrow-Debreu equilibrium. From the
sequential equilibrium conditions I need to show that the equilibrium is time-homogeneous
Markovian in terms of prices, consumption as well as holdings and derive the pricing con-
ditions for the whole periods. Once the recursive equilibrium is in hand I derive the price
conditions for the intermediate periods. I finally return to the assumed asset structure and
derive the equilibrium portfolio holdings.
There are several important points to look for in this section. With respect to prices I
show qualitatively why prices are discounted when opinions differ. In terms of trading I
derive in closed form the portfolio allocations and trading for the special case of logarithmic
utility. Further, I need to point out that both asset prices as well as portfolio holdings derive
from one equilibrium function which is the individual wealth as a function of the state of
the economy.
4.3.1 Financial Market Equilibrium
For this section it is easier to substitute the asset structure with only one period z pǫ
1
,ǫ
2
,g,ǫ
ρ
q contingent claims with prices p
t
pz q p pz |F
a
t
q, t 0,1,.... An assetp
t
pz q
is the price in terms of consumption in periodt of the contingent claim that pays the entire
wealth in period t 1 if the shock z realizes. This is equivalent to saying that a fraction
of the contingent claim p
t
pz q pays only in the state z the same fraction of the aggregate
endowment as well as the same fraction of all the contingent claims in that state. The
contingent claims are in unit net supply.
124
Instead of4 assets available every half period agents trade only when they consume on
16 contingent claims that are potentially required to complete the markets and allow agents
to share the risks efficiently given their beliefs. The fact that I introduce all possible contin-
gent claims does not mean that they are always necessary. In certain cases the equilibrium
positions in some assets are always zero or some assets are redundant.
Further, letp
t
pp
pz |F
a
t
q,z PZ q denote the vector of prices of the contingent claims
in periodt. The budget constraint of agenti in periodt is given by,
c
i
t
θ
i
t
p
t
⁄θ
i
t 1
pz
t
qW
t
(4.14)
where θ
i
t
pθ
i
pz |F
a
t
q,z P Z q is the portfolio holdings of contingent claims of agent i in
the end of periodt andW
denotes the entire wealth in the economy. The entire wealth in
any periodt is given by the prices of the contingent claims and the aggregate endowment,
W
t
y
t
z PZ
p
t
pz q. With the introduced notation I can define the equilibrium.
Definition 4.1. A financial market equilibrium is a process of portfolio holdings
tpθ
1
t
,θ
2
t
q,t 0,1,... u and a process of contingent claim prices tp
t
,t 0,1,... u such
that:
(i) Financial markets clear every periodt 0,1,...,
θ
1
t
pz q θ
2
t
pz q 1, z PZ.
(ii) For each agenti,
θ
i
P argmax
θ
U pc
i
,y q
s.t. c
i
t
θ
i
t 1
pz
t
qW
t
θ
i
t
p
t
, t 0,1,...
125
Speculative bubbles and Ponzi schemes are excluded from this equilibrium even though
I have not assumed explicitly the required conditions. A financial equilibrium with com-
plete markets is well known to generically deliver equilibria with efficient allocations. It is
equivalent to a social planner equilibrium with stochastic weights as for example in Basak
(2005).
Before I characterize the equilibrium allocations and equilibrium prices I introduce two
new quantities. I define α c
1
{y to be the consumption proportion of the first agent
which means that in equilibrium the consumption proportion of the second agent is given
by p1 α q. Further I define the following key variable,
q pz |F
a
t
q α
t
P
1
pz |F
a
t
q
1 {γ
1
p1 α
t
qP
2
pz |F
a
t
q
1 {γ
1
γ
1
. (4.15)
which is the generalized weighted mean of the individual beliefs about next period’s shock
realization. Note first that when agents have the same beliefs then q pz |F
a
t
q P
i
pz |F
a
t
q
for i 1,2 and therefore it is independent of the risk-aversion parameter γ
1
. Otherwise,
q pz |F
a
t
q ¿ α
t
P
1
pz |F
a
t
q p1 α
t
qP
2
pz |F
a
t
q when the risk aversion parameterγ
1
1. The
quantityq pz q depends both on the level of heterogeneity of beliefs about the statez as well
as the parameterγ
1
and I look at them in turn.
A known property (that can be shown using Jensen’s inequality) is that givenF
a
t
(i.e.
fixing the set of beliefs and consumption allocations) in some periodt, Bd pz |F
a
t
q{Bγ
1
⁄ 0
with the equality holding only when agents agree on their beliefs. As a result the quantity
q decreases as agents become more risk-averse given that they disagree. In particular, it
is also known that the mentioned properties imply that lim
γ
1
0
q pz q max
i
tP
i
pz qu and
similarly lim
γ
1
8
q pz q min
i
tP
i
pz qu. Therefore, when γ
1
1, q represents the average
beliefs in the economy but asγ
1
decreases towards0,q represents more and more the views
of the most optimistic agents. The opposite happens when γ
1
increases from 1 to infinity.
Note also that this happens state by state since the quantity q pz q is state specific. This
126
means for example, that asγ
1
increases, the economy becomes more and more pessimistic
overall since for every stateq leans towards the more pessimistic views for the given state.
The effect of the differences of opinions on the variableq pz q is made through the fol-
lowing remark.
Remark 4.1. Let α P p0,1 q, π P p0,1 q and let the beliefs of the two agents be π
1
π Δ {α 1 andπ
2
π Δ {p1 α q ¡ 0 for someΔ ¡ 0, so that the weighted average
of beliefs is constant and equal toπ. Let thenq be given by,
q απ
1 {γ
1
p1 α qπ
1 {γ
2
γ
I then have that
Bq
BΔ
» 0 whenγ ” 1.
For the typical case where agents are more risk-averse than a myopic logarithmic agent the
level of heterogeneity of beliefs for a given state decreases the value ofq and this decrease
is greater the bigger is the value of γ
1
. The quantity q is a key variable because it affects
directly the prices as I see from the following lemma.
Lemma 4.4. In equilibrium the consumption allocation of the first agent follows the pro-
cess,
α
t 1
α
t
P
i
pz
t 1
|F
a
t
q
q pz
t 1
|F
a
t
q
1 {γ
1
andα
0
1 {2. The equilibrium prices are given by,
p
t
pz
t 1
q δW
t 1
g
γ
2
t 1
q pz
t 1
|F
a
t
q.
The pricesp
t
pz q are linear in the quantityq. Hence, whatever applies toq in terms of how
it is affected by the level of heterogeneity of beliefs or the risk-aversion parameterγ
1
also
applies to the prices. This is the first element of the main argument of this study and states
127
that ceteris paribus at times of high dispersion in beliefs prices are depressed and therefore
prices increase as the opinions converge.
4.3.2 Recursive Characterization
In order to characterize the equilibrium portfolio holdings I need to first formulate the equi-
librium in recursive form. This means that the equilibrium must be Markovian in terms of a
finite dimensional state vector with known law of motion. Looking at the equilibrium con-
ditions in lemma 4.4 I see thatα
t 1
as well as the ratiop
t
pz
t 1
q{W
t 1
are functions of the
previousα
t
and the beliefs of the agents,P
i
pz
t 1
|F
a
t
q. Note that the ratio p
t
pz
t 1
q{W
t 1
,
which I denote with p
t
pz
t 1
q, is the price of an asset in period t that pays a unit of con-
sumption in state z
t 1
. From lemma 4.3 I see that these beliefs are a function of π
i
t
and
the external shock z
t 1
. Therefore, α
t 1
is a function of α
t
the probabilities π
i
t
, i 1,2,
and the shock z
t 1
. The state vector needs also to have the variableρ
t
since it is required
for the law of motion of the beliefs. The state vector of the economy is hence given by
s pα,π
1
,π
2
,ρ q and letS denote the set of all possible states. Therefore I can denote
q pz
t 1
|F
a
t
q withq pz
t 1
|s
t
q andπ
i
t
π
i
ps,ǫ
1
,ǫ
2
q. The next lemma looks at the recursivity
of the equilibrium consumption allocations.
Lemma 4.5. In equilibrium the consumption allocation of the first agent is independent of
the shockǫ
ρ
since
α ps,z q α
1
α p1 α qξ ps,ǫ
1
,ǫ
2
,g q
1 {γ
1
,
whereξ ps,ǫ
1
,ǫ
2
,g
h
q π
2
ps,ǫ
1
,ǫ
2
q{π
1
ps,ǫ
1
,ǫ
2
q and similarly forξ ps,ǫ
1
,ǫ
2
,g
l
q.
Lemma 4.5 states that agents change their consumption allocation in time only in states
that they disagree on their probability. The shock ǫ
ρ
changes only how the future beliefs
will be affected. The variableξ is the ratio of beliefs of the two agents and it depends on
the states and the signals pǫ
1
,ǫ
2
q since the signals affect the beliefs.
128
Let further the functionL denote the law of motion of the state vector, i.e. s
1
L ps,z
1
q.
The law of motion of the state is given by the equilibrium relation forα in lemma 4.5 and
the equations that describe how the individual beliefs change in time. Now let w
i
t
be the
wealth of agenti in some periodt normalized by the aggregate endowmenty
t
. Then from
the budget constraint (4.14) that is satisfied with equality in equilibrium and the equilibrium
prices as given in lemma 4.4 I get the following equilibrium relation:
α
t
δ
‚
z
t 1
PZ
g
1 γ
2
t 1
q pz
t 1
|F
a
t
qw
1
t
pz
t 1
q w
1
t
.
The notationw
1
t
pz
t 1
q makes explicit the state I refer to in period t 1. Hence, the equi-
librium is Markovian if the wealth can be written as a function of the state vector which is
the result of the next lemma.
Lemma 4.6. If δg
1 γ
2
⁄ 1 g P tg
h
,g
l
u, then there exist a unique function w ps q that
satisfies
w ps q α δ
‚
z PZ
g
1 γ
2
q pz |s qw ps
1
q,
for alls PS, wheres
1
L ps,z q. It takes the formw ps q αA ps q{p1 δ q and in the special
case whereγ
1
γ
2
1 I have thatA ps q 1 s PS. The price of a unit of consumption
for each state next period is given byp ps,z q δg
γ
2
q pz |s q.
Note thatw ps q, where once mores pα,π
1
,π
2
,ρ q, is the equilibrium wealth (normalized
by the aggregate endowment) of an agent with optimal consumption proportion α, and
beliefs pπ
1
,κ
1
q while the other agent’s beliefs are given by pπ
2
,κ
2
q. So in order to get the
wealth of the other agent I only need to re-order the state variables. I denote the equilibrium
aggregate wealth normalized by the aggregate endowment withW , which is given by,
W ps q w ps q w p˜ s q, (4.16)
129
where ˜ s p1 α,π
2
,π
1
,ρ q. Therefore W is derived by the fundamental equilibrium
functionw.
Since the individual wealth and the aggregate wealth are recursive in the state s then
the portfolio holdings of the contingent claims are also recursive. For a given state s the
portfolio holdings for the first agent are given byθ ps q tθ ps,z q,z PZ u, where,
θ ps,z q w pL ps,z qq
W pL ps,z qq
.
From the equilibrium functionw ps q I can derive all equilibrium prices and the portfolio
holdings that depend on the particular asset structure that I assume. But first I look at the
agent decisions in the intermediate period and how they price assets.
4.3.3 Intermediate Period
I have proven that the financial equilibrium can also be represented as a recursive equilib-
rium. Thus, I can write the optimization problem of the individual as a stationary dynamic
programming problem. I retain the assumption of 16 contingent claims which renders the
intermediate trading period unnecessary. However, I can still derive the asset prices in the
intermediate period and how they are related to the prices in the whole periods which is the
goal of this sub-section.
LetV
0
pθ |y,s q be the value function in the whole period given the states and aggregate
endowment y and given portfolio of contingent claims θ. I remind that agents trade on
contingent claims that each pays the aggregate wealth in one specific state the next period.
130
Let also V
1
pθ |y,s,ǫ
1
,ǫ
2
q be the value function in the intermediate periods. The dynamic
problem in whole periods is then given by,
V
0
pθ
i
0
|y,s q max
θ
i
1
ps q
u pc
i
,y q δE
i
V
1
θ
i
1
ps q
y,s,ǫ
1
,ǫ
2
s.t. c
i
θ
i
0
W ps qy ‚
z PZ
θ
i
1
ps,z qp ps,z qW pL ps,z qqyg.
(P
0
)
Let us explain the optimization problem (P
0
) and its notation. First θ
i
0
denotes the
holdings of agent i of the contingent claim that pays the aggregate wealth in the cur-
rent state s. Therefore, his current wealth is given by θ
i
0
W ps qy. Remember that W
is the aggregate wealth normalized by the aggregate endowment. The control vector
θ
i
1
ps q tθ
i
1
ps,z q,z P Z u is the new portfolio to be acquired. The budget constraint
(4.14) has the term
z PZ
θ
i
t
pz qp
t
pz q. But each termp
t
pz q is equal to the price of the asset
that pays a unit of consumption for statez,p ps
t
,z q, times the aggregate wealth in that state,
W pL ps
t
,z qqyg, and note that the aggregate endowment next period isyg.
The dynamic optimization problem in the intermediate periods is similarly given by:
V
1
pθ
i
1
ps q|y,s,ǫ
1
,ǫ
2
q max
θ
i
2
ps q
E
i

V
0
θ
i
2
ps,z q
yg,L ps,z q
ǫ
1
,ǫ
2
(
s.t. 0 ⁄
‚
z PZ
θ
i
1
ps,z q θ
i
2
ps,z q
p ps,ǫ
1
,ǫ
2
,z qW pL ps,z qqyg.
(P
1
)
The new portfolio to be composed is denoted by the vector θ
i
2
ps q while p ps,ǫ
1
,ǫ
2
q is the
vector of contingent claim prices in the intermediate period.
The budget constraints of both optimization problems are satisfied with equality at the
optimal solution. However, for problem (P
0
) consumptionc
i
is used as a substitution vari-
able while for problem (P
1
) I use the Lagrange multiplier approach. The next lemma gives
the equilibrium prices.
131
Lemma 4.7. Given states, the equilibrium Lagrange multiplier of agent i’s intermediate
period problem is given byu
1
pα
i
y,y q{λ ps,ǫ
1
,ǫ
2
q where λ ps,ǫ
1
,ǫ
3
q ¡ 0. The equilibrium
price of a unit of consumption for each state next period is given by:
p ps,ǫ
1
,ǫ
2
,z q δλ ps,ǫ
1
,ǫ
2
qg
γ
2
q pz |s,ǫ
1
,ǫ
2
q.
In lemma 4.7 the variableq pz |s
t
,ǫ
1
t
,ǫ
2
t
q is the generalized weighted average of the indi-
vidual beliefs as defined in (4.15) but in the intermediate state, i.e. q pz |F
a
t
q. The equi-
librium variableλ ps,ǫ
1
,ǫ
2
q is a “free” variable that only determines the unit of prices and
it does not affect either real allocations or equilibrium portfolios. In the whole periods
the numeraire good is consumption but since in the intermediate periods there is no con-
sumption the level of prices is undetermined. However, for the analysis it is natural to set
λ ps,ǫ
1
,ǫ
2
q such that the prices change only if the beliefs change and hence the variableq
changes. Therefore I setλ ps,ǫ
1
,ǫ
2
q 1 ps,z q PS Z.
Even though there are16 assets in the intermediate period only4 of them have positive
value since q pz |s,ǫ
1
,ǫ
2
q is equal to zero for all z that the shocks pǫ
1
,ǫ
2
q are different.
With the equilibrium prices and using the budget constraint in the intermediate period I
can express the wealth of the first agent standardized by the aggregate endowmenty of the
previous whole period:
w ps,ǫ
1
,ǫ
2
q δ
‚
z PZ
g
1 γ
2
q pz |s,ǫ
1
,ǫ
2
qw ps
1
q, (4.17)
wheres
1
L ps,z q. In the special case whenγ
1
γ
2
1 and heterogeneity of beliefs does
not matter forq, the wealth of an agent in the whole periods is simply given byα {p1 δ q
as derived in lemma 4.4. As a result in the intermediate periods the wealth is given by
w ps,ǫ
1
,ǫ
2
q δα {p1 δ q. This decrease in wealth is only to account for the consumed
good, otherwise it is unaffected by any change in beliefs in the economy.
132
4.3.4 Asset Prices and Portfolio Holdings
I return to the originally assumed asset structure to look at the equilibrium portfolio hold-
ings. I remind that every period or sub-period agents are able to invest in the dividend
paying stock which is in unit net supply and in three zero net supply contingent claims, that
pay the aggregate endowment in the next period or intermediate period, in only one state
and in different states each. The contingent claims available in the whole periods that expire
in the sub-periods pay the aggregate endowment of the previous period. The ordering of
the shocks as well as the ordering of the holdings is pǫ
1
,ǫ
2
q P tp1,1 q, p1,0 q, p0,1 q, p0,0 qu
for the intermediate periods and pg,ǫ
ρ
q P tpg
h
,0.5 q, pg
h
,0 q, pg
l
,0.5 q, pg
l
,0 qu for the whole
periods. I also remind that I consider two different asset structures in the second sub-period
that I call case h and case l. In case h the three contingent assets are for the last three states,
i.e. pg
h
,0 q, pg
l
,0.5 q and pg
l
,0 q and in case l the three contingent assets are for the first three
states.
The price of the stock in states normalized by the aggregate endowment (i.e. the price-
dividend ratio) is denoted byP ps q and in sub-periods byP ps,ǫ
1
,ǫ
2
q. Henceforth I refer to
the price dividend ratio simply as stock price. I start with an important result that shows
the significance of preference assumptions in explaining the relation between price changes
and the volume of trade.
Proposition 4.1. When agents have logarithmic preferences with no external habit, i.e.
γ
1
γ
2
1, then the stock price is constant,
P ps q P ps,ǫ
1
,ǫ
2
q δ
1 δ
, ps,z q PS Z.
In whole periods only the stock is required for trading since the equilibrium holdings for
the first agent are,
θ
1
ps q pα,0,0,0 q, s PS.
133
In the intermediate periods only one other asset is required that pays in stateg
l
since the
equilibrium holdings of the first agent in case h are,
θ
1
ps,ǫ
1
,ǫ
2
q α ps,ǫ
1
,ǫ
2
,g
h
q,0,ϑ,ϑ
, ps,z q PS Z,
where ϑ rα ps,ǫ
1
,ǫ
2
,g
l
q α ps,ǫ
1
,ǫ
2
,g
h
qs {p1 δ q. In case l the holdings are
p ϑ, ϑ,0,α ps,ǫ
1
,ǫ
2
,g
l
qq.
To understand why prices are constant even with heterogeneous beliefs in the special
case of proposition 4.1 I need to remember the behavior of the variable q. When γ
1
is
equal to one the heterogeneity of beliefs does not affect the variableq that determines the
state prices. What matters is only the weighted average of beliefs. But when I additionally
assume that γ
2
is also one then average beliefs do not matter either because the intertem-
poral marginal utility termg
γ
2
is offset by the increase in aggregate endowmentg which
is paid by the stock.
In proposition 4.1 consumption proportion α ps,ǫ
1
,ǫ
2
,g q refers to the next state of s
when the shocks realized are pǫ
1
,ǫ
2
,g q. The result that only the stock is necessary for trade
in the whole periods is not at all surprising after seeing that the stock price is constant
and therefore unaffected by beliefs. In the whole periods agents would need to trade if the
endogenous risk arising from the release of new information in the market pǫ
1
,ǫ
2
q needed
to be shared. This risk refers to the variation in the agents’ portfolio wealth from the release
of new information, but since such variation does not occur in this special case then there is
no risk to be shared. The trading that occurs during the sub-periods is to finance their new
investment plans once the new beliefs are formed. These new investment plans in turn since
the beliefs do not affect their wealth apart from through their consumption ratio, involve
only their consumption plan.
Let us now examine for each of the two cases h and l the trading that takes place
throughout a single period, from a states to the next.
134
h: When the new information arrives in the intermediate period agents update their
beliefs. If opinions differ the more optimistic agent will increase his holdings of
the stock by as much as he wants to increase his consumption in the high state from
the previous period. The next period there is only trade if the low state arrives in
which case the optimistic agents looses and has to give up part of his holdings in
order to eat. Being risk-averse the agent does not want to decrease his consumption
as much as the wealth in the economy decreased.
l: When the new information arrives and agent opinions differ the more pessimistic
agent will increase his stock holdings as much as he wants to increase his consump-
tion in the low state. When growth realizes trade occurs only in the high state since
then the previously pessimistic agent needs to give up some of his financial wealth to
eat.
The significant difference between the two cases is in the second round of trading. In
the first case trade occurs in the low state whereas in the other case trade occurs in the
high state. The specific trading behavior of the two cases will be the basis for predicting
a positive correlation between trading and price changes once I allow agents to be more
risk-averse than a logarithmic utility agent.
Once I relax the assumptions of proposition 4.1 and allow the preference parametersγ
1
and γ
2
to be different than one, then all assets become necessary for trade in all periods.
Both the individual beliefs as well as the heterogeneity of beliefs affect the wealth of agents
and hence agents need to hedge against the wealth risk they face. With the term wealth risk
I mean the variation in wealth that is uncorrelated with consumption. It is driven by the
term A ps q. This is an important implication of a model with both belief heterogeneity
and preferences other than logarithmic and is summarized in the following proposition.
For the next proposition I define the payoff matrices R ps q for the first sub-period and
R
k
ps,ǫ
1
,ǫ
2
q for the second sub-period where k th,l u refers to the two different cases
135
of asset structures considered. These payoff matrices are normalized with the aggregate
endowment. In the column of the stock price it has the price-endowment ratio P for the
different states pǫ
1
,ǫ
2
q for the first sub-period andP 1 inR
k
ps,ǫ
1
,ǫ
2
q again for the four
different states of pg,ǫ
ρ
q. For the contingent claims I only have1 in the corresponding state
and zero otherwise. I denote also withw ps q andw ps,ǫ
1
,ǫ
2
q the vector of wealth of the first
agent after the corresponding state.
Proposition 4.2. Whenγ
1
1 then agents need to trade in all securities to share efficiently
the non-fundamental wealth risk that arises. The equilibrium portfolio holdings of the first
agent are given by,
θ
1
px q R px q
1
w px q
forx P tps q, ps,ǫ
1
,ǫ
2
qu and ps,z q PS Z.
There are two distinct effects, one due to belief heterogeneity whenγ
1
1 and one due
to variations in the average beliefs when γ
2
1. I will restrict to cases where γ
2
1 in
order to see the effect of belief heterogeneity in the absence of any additional effect due to
variations in the average beliefs. Another reason why I restrict toγ
2
being one is to isolate
the effect of belief heterogeneity on stock return volatility.
The trading behavior on the stock does not change much when I move from the special
case of proposition 4.1 to the general case of proposition 4.2. The most notable difference
is that when agents are more risk-averse they invest less aggressively and therefore the
overall trading decreases. What changes significantly is the behavior of the stock price
since whenγ
1
increases the stock becomes more volatile with the variation in the dispersion
in beliefs. I remind that when opinions differ the state prices are depressed and hence the
same happens with the stock price. What remains to be seen is whether quantitatively this
136
variation is important and in what cases I can predict a positive correlation between trading
and price changes.
4.4 Asset Price Variation and Trading
I have so far analyzed the equilibrium theoretically and derived the trading dynamics when
agents have logarithmic preferences. In this section I will analyze quantitatively the asset
pricing behavior of this model and its relation with trading volume. I will also look at the
return volatility and its relation with trading volume as well. For this analysis I have chosen
a certain parameter configuration as shown in table 4.1.
Table 4.1: Model configuration
Preferences γ
1
γ
2
δ
5.00 1.00 0.99
g
t
log pg
h
q log pg
l
q φ
π
1.31 0.34 0.50
ρ
t
φ
ρ
η
0.95 p0,1 q
The preference parameterγ
1
was chosen to take the generally acceptable value of5. A
lower value would produce more trading due to more aggressive behavior and less volatility
for stock returns since prices become less sensitive to belief heterogeneity. The subjective
discount factorδ was set to0.99 only to control the level of prices. The second preference
parameter γ
2
was set to one in order to remove any price volatility due to other factors
apart from heterogeneity. The autocorrelation of the true probabilityφ
π
was chosen to be
1 {2. This value does not affect the two first moments of consumption growth. Its pricing
effects will be discussed later. The two consumption growth states were chosen to match
137
the mean and volatility of real quarterly consumption growth as obtained from the Bureau
of Economic Analysis NIPA tables for the period between 1947 to the end of 2007. The
autocorrelation of the process that determines sentiment risk φ
ρ
was set to 0.95 in order
to make the prices highly autocorrelated as observed in the data. The quarterly price-
dividend ratio obtained from the CRSP time-series of value weighted returns including and
excluding dividends or the quarterly price earnings ratio as obtained from Shiller’s data
exhibit an autocorrelation slightly higher than 0.95. About the same autocorrelation is
obtained in this model. Finally the correlation between the shocks to ρ and consumption
growth will be let to vary across the entire region of p0,1 q. The quantities that will be
shown are quarterly.
4.4.1 The Stock Price
In this model the stock price is affected by the belief heterogeneity through three distinct
avenues. First it is affected by the contemporaneous disagreement, i.e. the difference
betweenπ
1
andπ
2
, that directly affects current state prices. Secondly, it is affected by the
wealth dispersion across different agents as given by α. Even if agents do not disagree in
a given state, prices are still affected because the wealth dispersion allows for future dis-
agreement. Consequently the wealth dispersion depresses current prices through decreased
future prices. Finally, the stock price is affected by the variable ρ. When ρ is high it
implies that agents will interpret the new information similarly and as a consequence either
will make beliefs converge or will not allow them to diverge. The opposite happens when
ρ is low which predicts a high future dispersion in beliefs. Effectively the stock price is
decreasing in the variableρ. Further, the persistence ofρ will determine how sensitive the
stock price is toρ.
Figure 4.5 shows the logarithm of the price of the stock after being divided by the
aggregate endowment for various combinations of states of the economy. The price shown
138
Figure 4.5: Log-price divided by the aggregate endowment (η 0.5).
0
0.5
1
0
0.5
1
3.4
3.5
3.6
3.7
π
1
α=0.5, ρ=0.25
π
2
0
0.5
1
0
0.2
0.4
0.6
0.8
3
3.5
4
4.5
5
α
π
1
=π
2
=0.5
ρ
0
0.2
0.4
0.6
0.8
0
0.5
1
3
3.5
4
ρ
α=0.5
π
1
=π
2
0
0.5
1
0
0.5
1
3.5
4
4.5
5
α
ρ=0.25
π
1
=π
2
corresponds to either the log price-dividend ratio or more likely to the log price-earnings
ratio of an economy. I remind that the equilibrium price is a function of four state variables
the consumption proportion of the first agentα which is close to his wealth proportion. π
1
and π
2
are the current beliefs of the two agents about the probability of the high growth
state g
h
. Finally, ρ indicates the current level of the factor that determines disagreement
in the interpretation of public information. In the left top panel I keep α 0.5, ρ 0.5
and I vary π
1
and π
2
. I see that when these beliefs diverge the equilibrium price becomes
depressed but the effect is not particularly strong. In the left bottom panel I see the effect
ofρ and the beliefs of the agents when they agree while I keepα 0.5. I see thatρ affects
the price greatly as the price ranges form 3 to about4 when sentiment risk in the economy
decreases (i.e. the factor ρ increases). The effect of the beliefs, i.e. how optimistic the
economy is, is very small firstly due to the fact that the correlation η is set to 0.5. This
means that how optimistic the economy is does not affect how they form their views about
139
futureρ. If η is close to 0 which means that an optimism is accompanied by expectations
for low ρ then this would increase the prices. Even though this effect is not shown here,
in reality it is very small compared to the other effects. The other reason why optimism
does not affect much prices is because the beliefs are mostly affected by the information
tǫ
1
,ǫ
2
u and very little by the realizationg. If it was the opposite then the optimism would
be accumulative and would have a more significant impact on prices.
The two right panels show the effect of dispersion in wealth in the economy even when
agents agree. For example in the lower right panel I see that even if π
1
is equal to π
2
and
regardless of their common belief, higher dispersion in wealth causes prices to be depressed
simply because the wealth dispersion allows for future disagreement and therefore future
depressed prices. This effect becomes stronger when the sentiment factor increases, i.e. ρ
decreases. I need to note here even if I do not provide the visual confirmation the effect of
a different φ
π
. If it was higher than 0.5 then the effect of the differences in beliefsπ
1
and
π
2
would be greater and the effect of α and ρ smaller. By smaller or greater effect I only
mean the sensitivity or variability of the price due to variations in these factors. Now, these
effects are naturally due to the fact that ifφ
π
were chosen to be higher then the individual
beliefs would be more persistent and the effect of the public information tǫ
1
,ǫ
2
u on shaping
beliefs would be smaller.
Figure 4.6 shows the one period expected return of the stock for the same combinations
of states. Looking at this figure along with figure 4.5 I see that naturally the expected return
on the stock is decreasing in the level of the price. In particular lower prices and high
expected returns are associated with states where beliefs are disperse, wealth is disperse
and sentiment risk through the factor ρ is high. I only feel the need to note the reason
behind the not so obvious relation between high sentiment risk and high expected return.
This result is simply a result of the mean reversion in the variable ρ. So when sentiment
risk is high agents believe that future sentiment risk will be lower and therefore returns will
140
Figure 4.6: Expected stock return (η 0.5).
0
0.5
1
0
0.5
1
−5
0
5
10
15
π
1
α=0.5, ρ=0.25
π
2
0
0.5
1
0 0.1 0.2 0.3 0.4 0.5
−1
0
1
2
3
α
π
1
=π
2
=0.5
ρ
0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
−2
0
2
4
6
8
ρ
α=0.5
π
1
=π
2
0
0.5
1
0
0.5
1
−0.3
−0.2
−0.1
0
0.1
α
ρ=0.25
π
1
=π
2
be high. Similarly, the dispersion in beliefs affects expected returns. When there is high
dispersion in beliefs, beliefs tend to converge and therefore increase prices. I next look at
trading and its connection with the level of prices and the conditional return volatility.
4.4.2 Price, Return Volatility and Volume
Let us first look at the behavior of return volatility. I have said before that the stock price is
affected particularly by sentiment risk p1 {2 ρ q and the dispersion in wealth as indicated
byα. It is therefore natural to expect that the return volatility will be greatly determined by
these two factors. Looking at the right top panel of figure 4.7 I see that when sentiment risk
increases through the decrease of ρ from 1 {2 to zero, the conditional volatility of returns
moves from less than1% which is the fundamental volatility to a value almost fifteen-fold
when sentiment risk is at its highest. This result addresses the volatility puzzle which states
141
that the aggregate stock price and market returns should not be more volatile than the fun-
damental risk that they carry. However, the endogenous risk of belief heterogeneity which
would be natural to expect to be time-varying along with a realistic preference assumption
leads to such an increase in price and return volatilities. Further, the assumption that the
factorρ that determines the sentiment risk is persistent, increases its impact and therefore
the volatility of prices and additionally lends its persistence both to prices as well as volatil-
ities. Summing up, sentiment risk in this model is able to produce time variability in prices,
high volatility and volatility variation, associate high volatility with low prices and finally
lend its persistence to both of these quantities. All these predictions are well known to hold
empirically.
The dispersion in wealth α has also a significant impact on volatility and in particular
when sentiment risk is high. Again the reason is that even if agents agree currently when
in general disagreement is potentially high in the future, prices are sensitive and therefore
returns volatile when dispersion in wealth is high. The non-monotonic behavior observed
in the left two panels of figure 4.7 is due to the two agent assumption. When α is equal
to 0.5 then the dispersion in wealth can only decrease and the prices become more sensi-
tive around 0.25 when the wealth dispersion can both increase and decrease substantially.
Finally, the disagreement in current beliefs does not affect volatility that much unless it is
extreme in which case it is expected with great certainty that this disagreement can only
decrease and therefore prices increase substantially.
The basic story behind the volume of trade is quite simple. Agents are expected to trade
more the higher the probability that the dispersion in beliefs will change in the future. I
have to note that trade does not happen because beliefs are heterogeneous but because this
heterogeneity changes, either converges, diverges or beliefs reverse. The current positions
held by agents reflect their current beliefs and these will not change and will not produce
142
Figure 4.7: Conditional return volatility (η 0.5).
0
0.5
1
0
0.5
1
2
3
4
5
π
1
α=0.5, ρ=0.25
π
2
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0
2
4
6
8
10
α
π
1
=π
2
=0.5
ρ
0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
0
2
4
6
8
ρ
α=0.5
π
1
=π
2
0
0.5
1
0
0.5
1
0
1
2
3
4
α
ρ=0.25
π
1
=π
2
any trading unless the beliefs change and effectively their heterogeneity as well. There-
fore, I have to look for variables or states of the economy where the expected change in
belief heterogeneity is high and these will be the states of high expected volume of trade.
Figure 4.8 shows the expected turnover on the stock within the next period for the same
combinations of states that I have considered before for the price and return volatility. The
plots shown are for the case l where the stock becomes a vehicle for investing in the low
growth state. By this I mean that if one is pessimistic in relation to the other and wants to
increase his wealth in the low state and decrease in the high state then he will invest more
in the stock. I do not provide the figures for the other case but I explain the effect which
is minimal and does not alter the conclusions outlined here. The different cases will be
needed for the two different stories that can explain the positive correlation between price
changes and volume.
143
Figure 4.8: Expected turnover (η 0.5).
0
0.5
1
0
0.5
1
0
10
20
30
π
1
α=0.5, ρ=0.25
π
2
0
0.5
1
0 0.1 0.2 0.3 0.4 0.5
0
5
10
15
α
π
1
=π
2
=0.5
ρ
0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
0
10
20
30
ρ
α=0.5
π
1
=π
2
0
0.5
1
0
0.5
1
0
2
4
6
8
α
ρ=0.25
π
1
=π
2
The most important plot is the right top panel where I see the effect of the sentiment
risk. When ρ is low and sentiment risk high then I expect the new information in the
economy to cause agents’ beliefs to change significantly and in particular diverge when
the signals in the economy do not agree and converge if they agree and in both cases it
will cause trade. What is important is to note that the higher the sentiment risk the higher
the expected trade. The expected turnover is substantial and it goes up to 10% even in
the case when currently agents agree. If I see this plot along with the corresponding plot
of the conditional volatility of returns I observe a high correlation between the two. This
is because high potential disagreement implies that it is expected that belief heterogeneity
will change and through it prices as well. A clear and strong link is therefore established
between these two quantities using an element that was missing from the asset pricing
literature. Namely that sentiment risk, which is the rate by which beliefs change in the
economy, is time-varying and this leads to periods of high and low volume along with
144
high and low volatility. This effect now becomes quantitatively important once I take into
account something that has also been missing from the asset pricing literature on trading
namely the risk aversion.
The connection between volatility and volume can also be seen through the rest of the
factors. When dispersion in wealth is high I also observe high expected turnover for the
simple reason that holdings are disperse and all agents have the ability to change substan-
tially their position. Finally, when beliefs are disperse I also expect high turnover because
they are expected to converge. I see that the connection is very clear and strong, in that
both volume and volatility are always driven by the same factor whatever this is that can
cause the heterogeneity of beliefs to change. I also see that the periods of high volatility
and high expected turnover are also periods of low prices and high expected return.
As I have said, figure 4.8 shows the expected turnover for case l. In the other case where
the stock is a vehicle for investing in the low growth state then I have the following changes.
I have analyzed before after solving for the equilibrium holdings for the logarithmic utility
case, that there is more trading in the second sub-period in the low growth state for case
h and in the high growth state in case l. This is because in each case the agent who loads
on the stock market needs to unload his position when the opposite happens of what he
expects. So in the results of figure 4.8 I see higher expected turnover when the beliefs are
higher for stateg
h
. Therefore the same figures appear for case h when I reverse the axis of
π
1
andπ
2
and hence there is higher expected turnover when the beliefs about the high state
are lower.
For these results of turnover I have also set parameterη which is the correlation between
the shocks to sentiment risk and consumption growth to 0.5 which means no correlation.
The results presented however are negligibly affected by this parameter. This correlation
parameter only helps to explain the time-series joint behavior between price and volume
which I do next.
145
4.4.3 Price Changes and Volume
The main focus of this study is to explain the pervasive time-series positive correlation
between price changes in the aggregate and overall turnover. As I saw from the empiri-
cal evidence shown in figures 4.1 through 4.4 higher volume or an increase in volume is
associated with increases in prices. The model is flexible enough to provide two different
possible stories. I choose between the two on the basis of an additional pricing implication
namely the correlation between stock returns and consumption growth.
The model provides two rounds of trade in order to be able to separate the trading
due to merely changes in beliefs and due to wealth allocation changes. This distinction
was not made when I looked at the relation of volume with volatility because these two
affects are highly correlated in magnitude and magnitude was the scope of that analysis.
Here however I need to look at the correlation of these effects with price changes. I start
with the first sub-period where I only have changes in beliefs due to the new arrival of
information in the form of the signals tǫ
1
,ǫ
2
u. I have noted after lemma 4.7 that I assume
prices to change in the intermediate period only when beliefs change. This and due to the
fact that q pz |s,ǫ
1
,ǫ
2
q P pǫ
1
,ǫ
2
qq pz |s q I have that the expected change of prices for the
first sub-period is zero:
P
t
E pP
t
q.
I note once more that the above expectation is the same for all agents because they agree
on the joint unconditional distribution of the two signals. They only disagree on how to
interpret these signals when they differ. In the two states that the two signals agree and
hence their interpretation is the same their beliefs change towards the same direction and
the price more typically increases. The trading that results in these two states is minimal
because essentially their heterogeneity does not change. They add or subtract the same
quantity from their previous beliefs. In the other two states where they interpret the signals
146
differently their belief heterogeneity changes and typically increases and this causes trad-
ing. The increase in belief heterogeneity results into a decrease in price which is associated
therefore with price decreases. The first round of trade exhibits a high negative correlation
between price changes and volume. In order now the effect to be overall positive I need the
second round of trading to entail a positive correlation with greater magnitude.
For the second sub-period I need to distinguish between the two different cases of asset
structure that I consider. I remind however that regardless of the structure, significant
trading happens only in one of the growth states and in particular in the opposite state that
is expected by the agent who loads on the stock relative to his wealth. Let us first take case
h where the optimistic agent will always hold more stock than what his wealth commands.
If now the high growth state realizes then the agent will arrive at a state where he will have
increased his wealth. This will result in holding more of the stock but he has already bought
that part of the stock in the previous intermediate period. Therefore, the trading that occurs
is minimal as compared to the other state and is only due to the fact that his holdings on
the stock are not perfectly correlated with his wealth. Consumption and wealth are only
perfectly correlated with logarithmic utility. In the other state now, i.e. of low consumption
growth, the optimistic agent looses part of his wealth and he needs to unload his holdings
on the stock in order to eat. This results into trading. Therefore, higher turnover happens
in this case in low growth states. In order finally to have a positive correlation between
turnover and price changes I need to assume that the factorρ is negatively correlated with
consumption growth. A decrease in aggregate consumption growth would be associated
with a decrease in sentiment risk and therefore an increase in prices. Such an assumption
however results into a counterfactual prediction namely the negative correlation between
consumption growth and stock returns.
The second story assumes the asset structure of case l. Under this case I assume that
if an agent is optimistic about the economy relative to other agents will choose to invest
147
Figure 4.9: Model correlation between price changes and volume changes
0 0.2 0.4 0.6 0.8 1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
η
Corr(ΔP, ΔT), case high
0 0.2 0.4 0.6 0.8 1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
η
Corr(ΔP, ΔT), case low
Model correlation between price changes and turnover changes. Data were generated
using 100 simulations of 200 periods each. The dotted line shows the one-standard
deviation bounds.
in individual stocks or derivatives rather than the stock market. Therefore an optimistic
agent will shift his holdings from the market towards individual components that are more
correlated with consumption growth or for example call options on the market. If now the
good consumption state arrives the optimistic agent will have increased his wealth and this
will lead him to go back to holding the market and in higher proportions than before. This
leads to significant turnover on the stock. If further the high growth state is correlated with
an increase inρ, i.e. the parameterη being higher than1 {2, then this turnover is associated
with price increases. This is a more realistic assumption first because it predicts a positive
correlation between consumption growth and returns. Further it is natural to expect that
sentiment risk falls when the market goes up and likewise when the economy is doing well.
From the data I have obtained a correlation between price changes and volume that
ranges from around 0.3 to about 0.5. In order to see what the model predicts about this
148
quantity I run simulations of the model for each case and for different values forη. For each
parametrization I run 100 simulations of 200 whole periods each. I always start the econ-
omy from the state p0.5,0.5,0.5,0.25 q. Figure 4.9 shows the correlation between whole
period price changes and volume changes. By whole period I mean the price change fromt
tot 1 and for each whole period the volume is given by adding the turnover on the stock
in both sub-periods. Consistent with my explanation I see that for case h the correlation
between price changes and volume changes is decreasing inη and increasing inη for case
l.
Figure 4.10: Model correlation between price changes and volume
0 0.2 0.4 0.6 0.8 1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
η
Corr(ΔP,T), case high
0 0.2 0.4 0.6 0.8 1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
η
Corr(ΔP,T), case low
Model correlation between price changes and turnover. Data were generated using
100 simulations of 200 periods each. The dotted line shows the one-standard deviation
bounds.
The model correlation however does not become very positive but only up to slightly
greater than 0.1 in either case. If I look at the correlation between price changes and
volume per se which I do in figure 4.10 then these correlations increase up to 0.3 which
falls within the region observed in the data. I believe that this measure corresponds better
149
to the empirical evidence because the only reason that I used differences in the volume data
was in order to remove low frequency trend changes that are driven by other factors such
as trading technology, liquidity or other exogenous factors like wars. Similar correlations
were obtained also when I removed the trend using the Hodrick-Prescott filter. Neither of
these procedures is needed for the simulated data since the model equilibrium is stationary.
Figure 4.11: Model correlation between consumption growth and market returns
0 0.2 0.4 0.6 0.8 1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
η
Corr(G,R)
Model correlation between stock return and consumption growth. Data were generated
using 100 simulations of 200 periods each. The dotted line shows the one-standard
deviation bounds.
In order to choose one case as more plausible for the explanation behind this evidence I
use the pricing implication about the correlation of consumption growth with stock returns.
In figure 4.11 I plot the model correlation as I varyη. As I see this correlation is increasing
inη since an increase in sentiment risk increases prices. The post-war correlation between
the quarterly return on the value weighted Center of Research in Security Prices (CRSP)
index and the quarterly real consumption growth as obtained from NIPA tables is positive
and close to 0.25. It is interesting to note that the model does not predict a high correla-
tion between consumption growth and asset returns which is typically the case with asset
150
pricing models. This is obviously due to the fact that the model has other factors unrelated
to consumption growth like sentiment risk, dispersion in beliefs and dispersion in wealth
that affect the stock price. In order to predict the observed pro-cyclicality of equity prices
I need the assumption that sentiment risk is counter-cyclical. This could be due to behav-
ioral/sentimental factors as the model requires agents to become more critical and more
sensitive to new information during bad times. This could be in turn due to issues related
to nature that beings become more alert during bad times or under danger.
4.5 Empirical Evidence
The positive correlation between price changes and volume generated by the model
depends crucially on the assumption of the asset structure. It is not however restrictive
to the specific asset structure of one stock and three Arrow-Debreu stile securities. In its
essence the assumption is much more general but it requires one condition. When agents
differ in their beliefs they need to deviate from holding the market portfolio in proportion
to their wealth. The assumption is that the relatively optimistic agents deviate downwards
in the sense that they end up holding less proportion of the market portfolio than their
wealth and choose to speculate by investing more in specific stocks or going long certain
derivatives. Accordingly, on the net the relatively pessimistic agents end up increasing their
holding of the market portfolio for example by taking the short positions on the derivatives.
When now the market goes up the previously relatively optimistic agents increase their
wealth and buy back the market portfolio. Essentially therefore the argument is that the
positive correlation might be due to a difference in how easy it is to speculate for or against
the market. If this is true then one would expect that this correlation should vanish with the
development of the financial markets. The financial industry over the past 20 to 30 years
has revolutionized through the development of simple and complex derivative securities.
Hence, due to these recent developments investors can speculate for or against the market
151
with the same ease and their investment behavior may have caused the positive correla-
tion to go away. Such an evidence would not prove the argument put forward but it would
only provide some support I do however plot the time-series of the correlation between
price-changes and volume.
Figure 4.12: Price-earnings ratio and trading
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Running window Correl(Δ P,V)

NBER recessions
Yearly (20)
Monthly (60)
The plot shows the running window correlations of the percentage changes in the S&P
Composite index level adjusted for inflation along percentage changes in the NYSE
trading activity. For the yearly data the running window is20 years and for the monthly
data the running window is 5 years.
Figure 4.12 shows the20 year running window correlation of the annual price changes
with volume along with the5 year running window correlation of the monthly price changes
with volume. The data are the same used for the figures 4.1 and 4.3 respectively. As I see
the correlation does indeed start a steady decline around the beginning of the 1980’s and
reaches a value of zero soon after the year 2000. One other way to find support of my
argument is to look at the trading activity on call and put options. I should first observe
overall more trading volume on call options than put options and the difference should be
correlated with the running window correlations. Unfortunately, I only have a short history
152
of trading activity on call and put options. Figure 4.13 plots the monthly call/put ratio
as obtained from the Chicago Board Options Exchange from September 1995 to the end
of 2008. I also plot along the monthly running window correlation of price-changes with
volume over the same period. The two series are highly correlated especially after 1998
while the call/put ratio declines from a value above one and reaches one around the same
period that the correlation becomes zero.
Figure 4.13: Price-earnings ratio and trading
1996 1998 2000 2002 2004 2006 2008
0
0.2
0.4
0.6
Correlation = 0.43689

1996 1998 2000 2002 2004 2006 2008
1
1.5
2
2.5
Call/Put Ratio
NBER recessions
Running window 5 year monthly Correl(Δ P,T)
I plot the monthly call/put ratio which is the trading volume on call options over
the trading volume of put options over one month. The date was obtained from the
Chicago Board Options Exchange and cover the period from September 1995 to the
end of 2008. I also plot the 5 year running window correlation between monthly price
changes and trading volume.
Besides the positive correlation between price changes and volume, the model devel-
oped in this study has certain strong asset pricing predictions. These predictions are inde-
pendent of the correlation parameter η and are driven by the two main elements of this
model, namely risk-aversion of the investors and the time-varying and relatively persistent
sentiment risk. The predictions are:
153
i. V olume and volatility are positively correlated.
ii. The level of prices is negatively correlated with volume.
iii. V olatility and prices are inversely related.
In this section I examine these predictions empirically.
The main difficulty of this empirical analysis is to obtain a stationary measure of trad-
ing activity. I use from CRSP monthly data of turnover on all common stock traded at
the NYSE, NASDAQ and ASE for the period 1926 to 2008. The raw data, which is not
shown here, exhibits high variability in its low frequency trend. The overall trading activity
decreases substantially from around 20% per month to around 5% during and after World
War II. It starts a steady increase around the beginning of the1980’s and reaches again lev-
els above20% by2008. I then first remove the trend with the Hodrick-Prescott filter, using
a typical monthly smoothing parameter of 1440. However, the resulting time-series is still
highly non-stationary due to its changing variance. This variance is naturally connected to
the average level and, therefore, I divide the resulting series by the extracted trend. The
final series looks like white noise centered around zero. The measure for trading activity
is then taken to be the volatility of the final series at the given point. Higher variability
indicates higher trading activity during that period. The trading activity is estimated by
fitting a GARCH(1,1) process to the final series.
Table 4.2: Contemporaneous correlations
PE ratio Return vol. Turnover
PE ratio 1.0000
Return vol. -0.2457 1.0000
Turnover -0.4296 0.4605 1.0000
154
For the same period I also obtain the market S&P composite index log price-earnings
ratio from Robert Shiller’s web-page.
10
A time-series of return volatility is obtained using
the return on the value-weighted CRSP index and fitting an ARMA(1,0) model for the
mean and a GARCH(1,1) model for the volatility. I first compute the correlations between
the price-earnings ratio, the conditional return volatility and the turnover measure I just
discussed. This is shown in table 4.2. The results confirm all the predictions of the model
listed i. to iii. at the beginning of this section. It is interesting to note that not only
the predictions are confirmed with respect to the sign of the correlations but they are also
significant with respect to their magnitude. Figures 4.14 and 4.15 plot my turnover measure
compared to the price-earnings ratio and the return volatility, respectively. Looking at
figure 4.14 I note the particularly strong negative correlation between trading and the level
of prices from the beginning of the1960’s until today. The correlation during this period is
close to 0.7. Furthermore, the positive correlation between trading and volume is evident
from figure 4.15, in particular during periods of high volatility.
The evidence offered so far confirms the empirical predictions of the model about the
relation between prices and volume. In addition, I would like to study empirically the
connection the model establishes between some fundamentals and equilibrium prices. In
particular I would like to see if in fact belief dispersion and sentiment risk, as determined
by the variable ρ, affects prices in the way predicted by the model. For a proxy of the
dispersion in beliefs about macroeconomic fundamentals, I chose the cross-sectional stan-
dard deviation of individual mean forecasts of quarterly Survey of Professional Forecasters
provided by the Philadelphia Fed. The longer continuous series available is forecasts about
the nominal Gross Domestic Product -Gross National Product, prior to1992-.
I plot the raw series of the measure of belief dispersion along with the S&P compos-
ite index log price-earnings ratio. The plotted series offers strong support for the model
10
http://www.econ.yale.edu/ shiller/
155
Figure 4.14: Price-earnings ratio and trading
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
0
0.2
0.4
0.6
0.8
1
Correlation = −0.46639

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
1.5
2
2.5
3
3.5
4
S&P log price earnings
NBER recessions
Turnover measure
The plot shows the log price-earnings ratio of the S&P composite index along with the
turnover measure constructed from the monthly turnover series obtained from CRSP.
The data is monthly and cover the period from 1926 to 2008. The turnover measure
is the conditional volatility of the standardized deviations from the estimated trend of
the raw data. The trend was estimated using the Hodrick-Prescott filter.
predictions. From the figure I see clearly that the level of belief dispersion is negatively
correlated with the level of prices with a correlation of 0.60. The data also offers evi-
dence that sentiment risk affects prices in the way predicted by the model: In particular, a
high level of sentiment risk should be detected in the data by radical changes in the level of
heterogeneity. Therefore the volatility of the belief dispersion series should be an indication
for what I call sentiment risk in the model and is controlled by the factorρ. High sentiment
risk should be associated with low level of prices, and this is exactly what I observe in
the data. Periods where the belief heterogeneity is volatile, are also periods of low prices.
For example this relation is evident in the period from 1975 to 1990. After 1990 I see low
volatility in belief dispersion and high prices.
156
Figure 4.15: Return volatility and trading
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
0
10
20

1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
0
0.5
1
CRSP turnover measure
CRSP vw return volatility
The plot shows the fitted time series of the conditional volatility of returns on the value-
weighted CRSP index along with the turnover measure constructed from the monthly
turnover series obtained from CRSP. The data is monthly and cover the period from
1926 to 2008. The turnover measure is the conditional volatility of the standardized
deviations from the estimated trend of the raw data. The trend was estimated using
the Hodrick-Prescott filter. The fitted conditional volatility series was obtained after
estimated an ARMA(1,0) for the means and GARCH(1,1) for the volatilities model.
4.6 Conclusion
The classical asset pricing paradigm even though particulary successful in offering a
workhorse with which we progressively understand the interrelation between prices and
their evolution in time, this same framework has shed no light on trading volume. Conse-
quently, the theoretical asset pricing literature lacks a general framework within which we
can study and understand the crucial element of trading which appears to be so intimately
related with asset prices. Understanding the determinants of volume which is so pertinent
to asset prices will not only be important in understanding in general the workings of the
financial markets but it will also offer a new aspect of how equilibrium prices are shaped.
157
Figure 4.16: Dispersion in beliefs and price-earnings ratio
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
1
2
3
4

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
0
0.5
1
1.5
Belief dispersion
S&P log price earnings ratio
The plot shows the log price-earnings ratio of the S&P composite index along with
the dispersion in professional macroeconomic forecasts of one quarter ahead nominal
GDP. The data was obtained from the Survey of Professional Forecasters provided by
the Philadelphia Fed. The correlation between the two series is 0.60. The data cover
the period from the third quarter of 1968 until the last quarter of 2007.
This chapter is an attempt in this direction. Past literature has indicated that the most
promising modeling element towards explaining volume is belief heterogeneity which can
be generated due to opinion differences about common information as initially assumed by
Harris and Raviv (1993) and further developed by Scheinkman and Xiong (2003). With this
study I show that not only substantial volume can be generated in an economy where agents
hold different beliefs about the relevant macroeconomic risk but also that volume and its
time-series properties are directly connected to price variations. The economic reasoning
put forward by this chapter relies first on the asset pricing prediction that variations in
belief dispersion causes variations in the level of prices due to differences in individual
consumption processes. This prediction is a result of a realistic preference assumption of
power utility which also affects how agents trade. This result goes against the general
158
practice of using simplistic preference assumptions and points to the importance of the
preference assumption in understanding both prices and volume.
Further, the success of this model in explaining a range of asset pricing and volume
regularities depends on the assumption of a time-varying sentiment risk that determines
how different agents interpret the same public information. This factor is behind the time-
variation in prices because it predicts future belief dispersion as well as trading because it
predicts changes in the level of dispersion. It is made clear in this study that it is not the
belief heterogeneity that causes trade but changes to it and for this reason sentiment risk is
so important.
Due to the positive results of this study, further empirical and theoretical examination
of the constituents of this model needs to be conducted. For example, even though I offer
some preliminary evidence that possible time-variation in sentiment risk is associated with
variations in the level of prices a much deeper and more sophisticated study is required.
On the theoretical front an attempt should be made to endogenize this sentiment risk and
of-course to try to understand its origin. Finally, an interesting question that arises is how
the cross-section of prices as well as trading behaves in this context once I assume multiple
dividend paying assets.
159
References
Abel, A. B. (1990a). Asset prices under habit formation and catching up with the joneses.
American Economic Review Papers and Proceedings, 80, 38-42.
Abel, A. B. (1990b). Asset prices under heterogeneous beliefs: Implications for the equity
premium. (working study, Wharton School)
Abel, A. B. (1999). Risk premia and term premia in general equilibrium. Journal of
Monetary Economics, 43, 3-33.
Admati, A. R., & Pfleiderer, P. (1988). A theory of intraday patterns: volume and price
variability. Review of Financial Studies, 1(1), 3-40.
Ang, A., & Bekaert, G. (2007). Stock return predictability: Is it there? Review of Financial
Studies, 20(3), 651-707.
Barberis, N., Shleifer, A., & Vishny, R. (1998). A model of investor sentimen. Journal of
Financial Economics, 49, 307-343.
Basak, S. (2000). A model of dynamic equilibrium asset pricing with heterogeneous beliefs
and extraneous risk. Journal of Economic Dynamics and Control, 24, 63-95.
Basak, S. (2005, November). Asset pricing with heterogeneous beliefs. Journal of Banking
& Finance, 29(11), 2849-2881.
Berrada, T., Hugonnier, J., & Rindisbacher, M. (2007). Heterogeneous preferences and
equilibrium trading volume. Journal of Financial Economics, 83, 719-750.
Bessembinder, H., Chan, K., & Seguin, P. J. (1995). An empirical examination of infor-
mation, differences of opinion and trading activity. Journal of Financial Economics,
40(19), 105-134.
Bhamra, H. S., & Uppal, R. (2007). The effect of introducing a non-redundant derivative
on the volatility of stock-market returns. Review of Financial Studies. (forthcoming)
160
Blume, L., Coury, T., & Easley, D. (2006). Information, trade and incomplete markets.
Economic Theory, 29, 379-394.
Bollerslev, T., Chou, R. Y ., & Kroner, K. F. (1992). Arch modeling in finance: A review of
the theory and empirical evidence. Journal of Econometrics, 52(1-2), 5-59.
Boudoukh, J., Michaely, R., Richardson, M., & Michael, R. R. (2007). On the impor-
tance of measuring payout yield: Implications for empirical asset pricing. Journal of
Finance, 62, 877-915.
Boudoukh, J., Richardson, M., & Whitelaw, R. F. (2008). The myth of long-horizon
predictability. Review of Financial Studies, 21(4), 1577-1605.
Brock, W. A., & Hommes, C. H. (1998). Heterogeneous beliefs and routes to chaos in a
simple asset pricing model. Journal of Economic Dynamics and Control, 22, 1235-
1274.
Buraschi, A., & Jiltsov, A. (2006). Model uncertainty and option markets with heteroge-
neous beliefs. Journal of Finance, 61, 2841-2897.
Calvet, L., Grandmont, J. M., & Lemaire, I. (2004). Aggregation of heterogeneous beliefas
and asset pricing in complete financial markets (Tech. Rep.). CREST. (Working
Paper)
Campbell, J. Y . (1991). A variance decomposition of stock returns. Economic Journal,
101(157-179).
Campbell, J. Y ., & Cochrane, J. H. (1999). By force of habit: A consumption-based
explanation of aggregate stock market behavior. Journal of Political Economy, 107,
205-251.
Campbell, J. Y ., Grossman, S., & Wang, J. (1993). Trading volume and serial correlation
in stock returns. Quarterly Journal of Economics, 108, 905-939.
Campbell, J. Y ., & Shiller, R. J. (1988). The dividend-price ratio and expectations of future
dividends and discount factors. Review of Financial Studies, 1(3), 195-228.
Cao, H. H., & Yang, H. O. (2008, March). Differences of opinion of public information and
speculative trading in stocks and options. Review of Financial Studies. (forthcoming)
Chan, Y. L., & Kogan, L. (2002). Catching up with the joneses: Heterogeneous preferences
and the dynamics of asset prices. Journal of Political Economy, 110, 1255-1285.
Chordia, T., Huh, S. W., & Subrahmanyam, A. (2007). The cross-section of expected
trading activity. Review of Financial Studies, 20(3), 709-740.
161
Ciarella, C., Dieci, R., & Gardini, L. (2006). Asset price and wealth dynamics in a financial
market with heterogeneous agents. Journal of Economic Dynamics and Control, 30,
1755-1786.
Cochrane, J. H. (2008). The dog that did not bark: A defense of return predictability.
Review of Financial Studies, 21(4), 1533-1575.
Constantinides, G. (1990). Habit formation: A resolution of the equity premium puzzle.
Journal of Political Economy, 98(3), 519-543.
Daniel, K., Hirshleifer, D., & Subrahmanyam, A. (1998). Investor psychology and security
market under- and overreaction. Journal of Finance, 53, 1839-1885.
David, A. (2008, February). Heterogeneous beliefs, speculation, and the equity premium.
Journal of Finance, 63(1), 41-83.
Detemple, J. B., & Murthy, S. (1994). Intertemporal asset pricing with heterogeneous
beliefs. Journal of Economic Theory, 62, 294-320.
Detemple, J. B., & Zapatero, F. (1991, November). Asset prices in an exchange economy
with habit formation. Econometrica, 59(6), 1633-57.
Diamond, D. W., & Verrecchia, R. E. (1981). Rational aggregation in a noisy rational
expectations economy. Journal of Financial Economics, 9(3), 221-235.
Diether, K., Malloy, C., & Scherbina, A. (2002). Differences of opinion and the cross
section of stock returns. Journal of Finance, 57, 21132141.
Doukas, J. A., Kim, C. F., & Pantzalis, C. (2006). Divergence of opinion and equity returns.
Journal of Financial And Quantitative Analysis, 41(3), 573-606.
Duffie, D. (1996). Dynamic asset pricing theory. Princeton, New Jersey: Princeton Uni-
versity Press.
Duffie, D., Geanakoplos, J., Mas-Colell, A., & McLennan, A. (1994). Stationary markov
equilibria. Econometrica, 62, 745-781.
Dumas, B. (1989). Two person dynamic equilibrium in the capital market. Review of
Financial Studies, 2(2), 157-188.
Famma, E. F., & French, K. R. (1988). Dividend yields and expected stock returns. Journal
of Financial Economics, 22, 3-27.
Famma, E. F., & French, K. R. (2002). The equity premium. Journal of Finance, 57(2),
637-659.
162
Foster, F. D., & Viswanathan, S. (1990). A theory of the interday variations in volume,
variance, and trading costs in securities markets. Review of Financial Studies, 3(4),
593-624.
Foster, F. D., & Viswanathan, S. (1993). The effect of public information and competition
on trading volume and price volatility. Review of Financial Studies, 6(1), 23-56.
Gali, J. (1994). Keeping up with joneses: Consumption externalities, portfolio choice, and
asset prices. Journal of Money, Credit and Banking, 26(1), 1-8.
Gallant, R., Rossi, P., & Tauchen, G. (1992). Stock prices and volume. Review of Financial
Studies, 5(2), 199-242.
Gallmeyer, M., & Hollifield, B. (2008). An examination of heterogeneous beliefs with a
short-sale constraint in a dynamic economy. Review of Financial Studies. (forthcom-
ing)
Gˆ arleanu, N., & Panageas, S. (2008, April). Young, old, conservative and bold: The
implications of heterogeneity and finite lives for asset pricing. (Working Paper)
Geanakoplos, J., Magill, M., & Quinzii, M. (2004). Demography and the long-run pre-
dictability of the stock market. Brookings Papers on Economic Activity, 2004(1),
241-307.
Gervais, S., & Odean, T. (2001). Learning to be overconfident. Review of Financial Studies,
14, 1-27.
Goyal, A., & Welch, I. (2003). The myth of predictability: Does the dividend yield forecast
the equity premium? Management Science, 49(5), 639-654.
Griffin, J. M., Nardari, F., & Stulz, R. M. (2007). Do investors trade more when stocks
have performed well? evidence from 46 countries. Review of Financial Studies,
20(3), 905-951.
Grossman, S., & Stiglitz, J. (1980). On the impossibility of informationally efficient
markets. American Economic Review, 70, 393408.
Grrundy, B. D., & McNichols, M. (1989). Trade and the revelation of information through
prices and direct disclosure. Review of Financial Studies, 2, 495526.
Hansen, L. P., & Jagannathan, R. (1991). Restrictions on intertemporal marginal rates of
substitution implied by asset returns. Journal of Political Economy, 99, 225-262.
Harris, M., & Raviv, A. (1993). Differences in opinion make a horse race. Review of
Financial Studies, 6, 473-506.
163
Harrison, J. M., & Kreps, D. M. (1978, May). Speculative investor behavior in a stock
market with heterogeneous expectations. Quarterly Journal of Economics, 92, 323-
336.
Hellwig, M. (1980). On the aggregation of information in competitive markets. Journal of
Economic Theory, 22, 477-498.
Hindy, A., Huang, C., & Zhu, S. H. (1997). Optimal consumption and portfolio rules with
durability and habit formation. Journal of Economic Dynamics and Control, 21(2-3),
525-550.
Hong, H., & Stein, J. (2003). Differences of opinion, short-sales constraints, and market
crashes. Review of Financial Studies, 16(2), 487-525.
Hong, H., & Stein, J. (2007). Disagreement and the stock market. Journal of Economic
Perspectives, 21(2), 109-128.
Jagannathan, R., McGrattan, E. R., & Scherbina, A. (2000, Fall). The declining u.s. equity
premium. Federal Reserve Bank of Minneapolis Quarterly Review, 24(4), 3-19.
Jouini, E., & Napp, C. (2006a, September). Aggregation of heterogeneous beliefs. Journal
of Mathematical Economics, 42(6), 752-770.
Jouini, E., & Napp, C. (2006b, July). Heterogeneous beliefs and asset pricing in discrete
time: An analysis of pessimism and doubt. Journal of Economic Dynamics and
Control, 30(7), 1233-1260.
Jouini, E., & Napp, C. (2007). Consensus consumer and intertemporal asset pricing with
heterogeneous beliefs. Review of Economic Studies, 74, 1149-1174.
Judd, K. L., Kubler, F., & Schmedders, K. (2003, October). Asset trading volume with
dynamically complete markets and heterogeneous agents. Journal of Finance, 58(5),
2203-2217.
Kandel, E., & Pearson, N. D. (1995). Differential interpretation of public signals and trade
in speculative markets. Journal of Political Economy, 103(4), 831-872.
Karpoff, J. M. (1987, March). The relation between price changes and trading volume: A
survey. Journal of Financial and Quantitative Analysis, 22(1), 109-126.
Kim, O., & Verrecchia, R. E. (1991a, Autumn). Trading volume and price reactions to
public announcements. Journal of Accounting Research, 29(2), 302-321.
Kim, O., & Verrecchia, R. E. (1991b, Decemmber). Market reaction to anticipated
announcements. Journal of Financial Economics, 30(1), 273-309.
164
Kimball, M. S., Sham, C. R., & Shapiro, M. D. (2008, September). Imputing risk tolerance
from survey responses. Journal of the American Statistical Association, 103(483),
1028-1038.
Kocherlakota, N. (1996). The equity premium: it’s still a puzzle. Journal of Economic
Literature, 34, 42-71.
Kogan, L., Makarov, I., & Uppal, R. (2007, April). The equity risk premium and the
riskfree rate in an economy with borrowing constraints. Mathematics and Financial
Economics, 1(1), 1-19.
Kogan, L., Ross, S. A., Wang, J., & Isterfield, M. M. (2007, February). The price impact
and survival of irrational traders. Journal of Finance, 61(1), 195-229.
Kurz, M. (1994a). On rational belief equilibria. Economic Theory, 4, 859-876.
Kurz, M. (1994b). On the structure and diversity of rational beliefs. Economic Theory, 4,
877-900.
Kurz, M., & Motolese, M. (2001). Endogenous uncertainty and market volatility. Economic
Theory, 17, 497-544.
Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 53(6), 1315-
1335.
Leroy, S., & Porter, R. (1981). The present-value relation: Tests based on implied volatility
bounds. Econometrica, 49, 97-113.
Lettau, M., & Ludvigson, S. (2001). Consumption, aggregate wealth, and expected stock
returns. Journal of Finance, 56(3), 815-849.
Lettau, M., Ludvigson, S., & Wachter, J. (2008). The declining equity premium: What role
does macroeconomic risk play? Review of Financial Studies, 21(4), 1653-1687.
Llorente, G., Michaely, R., Saar, G., & Wang, J. (2002). Dynamic volume-return relation
of individual stocks. Review of Financial Studies, 15, 1005-1047.
Lo, A. W., & Wang, J. (2000). Trading volume: definitions, data analysis, and implications
of portfolio theory. Review of Financial Studies, 13(2), 257-300.
Long, J. B. D., Shleifer, A., Summers, L. H., & Waldmann, R. J. (1990). Noise trader risk
in financial markets. Journal of Political Economy, 98(4), 703-738.
Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica, 46, 1429-1445.
165
Magill, M., & Quinzii, M. (1996). Theory of incomplete markets (V ol. I). Cambridge,
Massachusetts: The MIT Press.
Mehra, R., & Prescott, E. (1985). The equity premium. Journal of Monetary Economics,
15, 145-161.
Milgrom, P., & Stokey, N. (1982). Information, trade and common knowledge. Journal of
Economic Theory, 12, 112-128.
Miller, E. M. (1977, September). Risk, uncertainty, and divergence of opinion. Journal of
Finance, 32, 1151-1168.
Ottaviani, M., & Sørensen, P. N. (2006). The strategy of professional forecasting. Journal
of Financial Economics, 81, 441-466.
Pirani, D. C. (2004, November). Effects of differences in risk aversion on the distribution
of wealth. Macroeconomic Dynamics, 8, 617-632.
Rubinstein, M. (1975, December). Securities market efficiency in an arrow-debreu econ-
omy. The American Economic Review, 65(5), 812-824.
Ryder, H. E., & Heal, G. M. (1973). Optimal growth and intertemporally dependent
preferences. The Review of Economic Studies, 40, 1-31.
Scheinkman, J., & Xiong, W. (2003). Overconfidence and speculative bubbles. Journal of
Political Economy, 111, 11831219.
Scheinkman, J., & Xiong, W. (2006). Asset float and speculative bubbles. Journal of
Finance, 61(3), 1073-1117.
Schwert, W. (1989). Why does stock market volatility change over time? Journal of
Finance, 44(5), 1115-1153.
Shalen, C. T. (1993). V olume, volatility, and the dispersion of beliefs. Review of Financial
Studies, 6(2), 405-434.
Shiller, R. (1981). Do stock prices move too much to be justified by subsequent changes
in dividends. American Economic Review, 71, 421-436.
Soderlind, P. (2008, March). C-capm without ex post data. Working Paper.
Sundaresan, S. M. (1989). Intertemporally dependent preferences and the volatility of
consumption and wealth. Review of Financial Studies, 2, 73-88.
166
Varian, H. R. (1985). Divergence of opinion in complete markets: A note. Journal of
Finance, 40(1), 309-317.
Varian, H. R. (1989). Divergence of opinion in financial markets. Financial Risk: Theory,
Evidence and Implications, Proceedings of the Eleventh Annual Economic Policy
Conference of the Federal Reserve Bank of St. Louis, 3-37. (KluIr, Boston)
Verrecchia, R. E. (1979, Septemmber). A proof of the existence of “consensus beliefs”.
Journal of Finance, 34(4), 957-963.
Wang, J. (1993). A model of intertemporal asset prices under asymmetric information.
Review of Economic Studies, 60, 249-282.
Wang, J. (1994). A model of competitive stock trading volume. Journal of Political
Economy, 102(1), 127-168.
Wang, J. (1996). The term structure of interest rates in a pure exchange economy with
heterogeneous investors. Journal of Financial Economics, 41, 75-110.
Zapatero, F. (1998). Effects of financial innovation on market volatility when beliefs are
heterogeneous. Journal of Economic Dynamics and Control, 22, 597-626.
Ziebart, D. (1990). The association between consensus of beliefs and trading activity
surrounding earnings announcements. Accounting Review, 65, 477-488.
167
Appendices
A Appendix to Chapter 2
Proofs
Below I provide the proofs of the propositions, lemmas, and corollaries found in the text.
The preference assumption is that agents have time and state separable power preferences
with external habit, and the running utility is given by
u pc,X |γ q c
1 γ
X
γ ρ
1
1 γ
.
where c denotes consumption. I denote by α the consumption proportion c {Y . Agents’
types are characterized by the initial distribution of wealthθ pγ q, which is a density function
over the set of typesΓ. The state variable of the economy is the aggregate endowment habit
ratioω y x, wherey andx are the natural logs ofY andX respectively. I also define
the pricing kernelz
t
log pp
t
q ρx
t
.
Proof of Proposition 2.1. Given the habit process pX
t
,t ¥ 0 q, the consumption price pro-
cess pp
t
,t ¥ 0 q, and the initial total wealth W
0
, the optimization problem of an agent of
typeγ with initial wealth allocationθ pγ q is given by,
max
pct pγ q,t ¥0 q
L pγ q E
0
‚
t ¥0
δ
t
u pc
t
,X
t
|γ q λ pγ q
γ
E
0
‚
t ¥0
δ
t
p
t
c
t
pγ q θ
0
pγ qp
0
W
0
Due to the preference assumptions, the optimal solution is interior, and is characterized by
the first order condition,
c
t
pγ q
X
t
γ
λ pγ q
γ
p
t
X
ρ
t
, t ¥ 0.
and the equality of the inter-temporal budget constraint. Using the definition ofz
t
and the
consumption proportion α, I arrive at the optimality condition (2.6). I then substitute in
the budget constraint, satisfied with equality, the optimal consumption share from (2.6), to
arrive at the equilibrium value forλ pγ q.
168
Proof of Corollary 2.1. Market equilibrium is obtained when the optimal consumption
share in each period as function of the type γ integrates to one. Given the optimality
condition (2.6) it is evident that the pricing kernel is a function of the endowment habit
ratio,z
t
z pω
t
q.
Proof of Corollary 2.2. The pricing kernel is normalized to have a value of zero at the
average state, z p¯ ω q 0. The initial state is assumed to be the average state and hence
expression (2.9) is easily derived from (2.6) at the initial state and (2.7). Since the equi-
librium pricing kernel is a function ofω, so is the optimal consumption share as given by
(2.6). At the average state it is given by
α p¯ ω,γ q λ pγ qe
¯ ω
.
Using the previous result, I expressλ pγ q in terms ofα p¯ ω,γ q P
¯ ω
p1 {γ q and substitute it in
(2.8) to arrive at (2.10). I have replaced the coefficient of risk-aversion with the coefficient
of risk-tolerance,τ 1 {γ.
Proof of Lemma 2.1. Differentiate expression (2.8) with respect toω to get
0 »
Γ
z
1
pω q
γ
1
λ pγ qexp
z pω q
γ
ω
dγ
and note thatP
ω
pτ q λ p1 {τ qexp p τz pω q ω q to derive (z2). Differentiate again with
respect toω to get,
0 »
Γ
z
2
pω q
γ
z
1
pω q
2
γ
2
2
z
1
pω q
γ
1
P
ω
p1 {γ qdγ
z
2
pω qE
ω
pτ q z
1
pω qE
ω
pτ
2
q 2z
1
pω qE
ω
pτ q 1.
Using the result forz
1
pω q and rearranging I get the second derivative ofz. Since the numer-
ator is a variance term then the second derivative is positive.
For the rest of the results I assume explicitly that the setΓ has only positive values and
is bounded above and below. Now, since the termE
ω
rexp p τz pω qqs is strictly positive and
finite for all finite and positive values ofz pω q, from (2.10) it has to be thatz tends to minus
and plus infinity, as ω tends to plus and minus infinity, respectively. Let γ
max
denote the
maximumγ of the setΓ, and consider the following ratio of optimal consumption shares,
α pω,γ
max
q
α pω,γ q
λ pγ
max
q
λ pγ q
exp
z pω q
1
γ
max
1
γ
Now sincelim
ω 8
z pω q 8 then from the previous ratio, I have to conclude that
lim
ω 8
α pω,γ q 0, γ γ
max
.
169
Hence, the consumption distribution collapses to a spike at the maximum γ. Similarly,
when the state tends to plus infinity, the distribution collapses to the minimumγ. Finally,
since in either case the variance tends to zero, then the second derivative tends to zero as
well.
Proof of Proposition 2.2. Given the habit process pX
r
t
X
t
e
¯ ω
,t ¥ 0 q, the consumption
price process pp
r
t
,t ¥ 0 q and the initial total wealth W
r
0
, the optimization problem of the
representative agent is given by,
max
pct,t ¥0 q
L
r
E
0
‚
t ¥0
δ
t
u
r
pc
t
,X
r
t
q λ
r
E
0
‚
t ¥0
δ
t
p
r
t
c
t
p
r
0
W
r
0
where
u
r
pc
t
,X
r
t
q c
1 γ pωt q
t
pX
r
t
q
γ pωt q ρ
1 γ pω
t
q
,
γ pω q is the stochastic risk-aversion of the representative agent, and ω is as before. Then,
the representative agent optimally consumes the aggregate endowment,, and the first order
condition is given by,
Y
t
X
t
γ pωt q
e
¯ ω rγ pωt q ρ s
λ
r
e
z
r
t
,
where z
r
t
log pp
r
t
q ρx
t
. To normalize z
r
to be zero at the average state the Lagrange
multiplier has to equale
ρ¯ ω
.
Proof of Corollary 2.3. From the definitions ofz
r
andz, the two pricing kernels are equal
in all states iff the consumption processes p
r
and p are equal. Therefore, the asset price
processes of the two economies are equal if z
r
z. Hence, from Proposition 2.2 I must
have that,
γ pω q z pω q
ω ¯ ω
, ω ¯ ω.
For continuity, I use l’Hˆ opital’s rule to defineγ pω q at the average state.
Proof of Corollary 2.4. Since
γ pω q z pω q
ω ¯ ω
,
the first and second derivatives are obtained using the first two derivatives ofz from Lemma
2.1. In order to prove thatγ
1
pω q is negative, I need to show that the expression in the square
bracketsγ pω q z
1
pω q » 0 whenω » ¯ ω. From the convexity of functionz and the fact that
z p¯ ω q 0, I have that
z pω q z
1
pω qpω ¯ ω q ¡ 0
The result then obtains after dividing by pω ¯ ω q, and noting that the inequality switches
when ω ¯ ω. The limits of γ pω q, γ
1
pω q and γ
2
pω q are straightforward applications of the
limit properties ofz.
170
Proof of Corollary 2.5. If the risk-tolerance at the average state is gamma distributed with
parameters pκ,ϑ q, from the moment generating function I know that,
E
¯ ω
rexp p τz pω qqs r1 ϑz pω qs
κ
.
Using the equilibrium condition (2.10) and rearranging, I get,
z pω q exp
ω ¯ ω
κ
1
ϑ
¯ γ denotes the inverse of the average risk-tolerance, and therefore is equal to pκϑ q
1
,and ¯ ν
2
denotes the risk-tolerance variance, given byκϑ
2
. Furthermore, I useη p¯ γ¯ ν q
2
to obtain
the result.
Proof of Corollary 2.6. From the definition ofz I have that,
m pω,ω
1
q log pδ q z pω
1
q z pω q ρ px
1
x q.
The habit process is given by,
x
1
λy p1 λ qx
and thereforex
1
x λω. The result is then obtained once I substitute in the expression
for z derived in Corollary 2.5. When ¯ ν 0 then it is obvious from (2.10) that z pω q ¯ γ pω ¯ ω q.
Computational Approach
I outline below the numerical method used to compute the price-dividend ratio and the
risk-free rate as functions of the state. The price-dividend ratio of the market security that
pays a dividend stream that grows according to (2.2) is given by
PD pω q δE
ω
e
z pω
1
q z pω q ρλω μ
d
σ
d
̺ǫ σ
d
?
1 ̺
2
ǫ
d
pPD pω
1
q 1 q
where ǫ and ǫ
d
are independent and normally distributed variables. The distributional
assumptions made here are not required, in general. The method can be adapted to any
arbitrary distribution. Let the law of motion forω be some known function,
ω
1
L pω,ǫ q.
Let
˜
δ δexp
μ
d
1
2
σ
2
d
p1 ̺
2
q
, and also let,
N pω,ǫ q exp rz pL pω,ǫ qq z pω q ρλω σ
d
̺ǫ s.
171
Then the price-dividend equation can be rewritten as follows:
PD pω q ˜
δ
»
8
8
ϕ pǫ qN pω,ǫ q rPD pL pω,ǫ qq 1 sdǫ,
whereϕ pǫ q is the standard normal density. I then discretize the state variable in a set ofn
values Ω and compute the n n transition matrix Π rπ pω,ω
1
qs based on L and ϕ. Let
PD be the vector of price-dividend ratios and letN be the n n matrix computed from
N pω,ǫ q. The price-dividend ratio equation can now be written simultaneously for all states
inΩ in the following matrix form:
PD ˜
δ rΠ N s pPD 1
n
q,
where denotes the element-wise matrix multiplication and1
n
is then dimentional unit
vector. After a matrix inversion the price-dividend ratio vector is computed according to,
PD ˜
δ rI
n
pΠ N qs
1
1
n
,
whereI
n
is then dimentional identity matrix.
The price of the risk-free bond that pays a unit of consumption the next period is com-
puted in a similar fashion. Let
M pω,ǫ q δexp pz pL pω,ǫ qq z pω q ρλω q
and letM denote the square matrix for all the combination of values in the setΩ. Then the
vector of bond price values is computed by
P
f
pΠ M q1
n
.
The risk-free rate is then computed asR
f
pω q 1 {PD pω q 1.
For the simulation part of the results I also approximate the price-dividend ratio func-
tion, as well as the risk-free rate function, with cubic-splines. The data used to estimate the
piece-wise polynomial coefficients were the computed vectorsPD andP
f
and the vector
of arguments Ω. For the computational results I use a specific assumption for the state
variable that gives the following law of motion,
ω
1
ω λ pω ¯ ω q σǫ,
whereσ is the volatility of consumption growth. Hence the state variable is unconditionally
normally distributed. The set Ω used was an equidistant grid of 251 values where the
minimum and the maximum values are8 standard deviations, left and right from the mean,
respectively.
172
B Appendix to Chapter 3
Proofs
Below I provide the proofs of the propositions, lemmas, and corollaries found in chapter
3. The preference assumption is that agents have time and state separable power utility
preferences with external habit, and the running utility is given by
u pc,X q c
1 γ
X
γ η
1
1 γ
.
where c denotes consumption. I let α pdi q c
i
{Y be the consumption proportion of the
marginal agent di. Agents differ in their beliefs denoted with pξ
i
t
,ξ
i
0
1 q which is the
Radon-Nikodym process with respect to the objective measure. I also use the notation
ω y x, wherey andx are the natural logs ofY andX respectively. I denote withp
t
the
pricing kernel process under the objective probability measure and I setp
0
exp p γω
0
ηx
0
q. For an agent with beliefsξ
i
the pricing kernel is given byp
i
t
p
t
{ξ
i
t
.
Proof of Lemma 3.1. Given the habit process pX
t
,t ¥ 0 q, the pricing kernel process
pp
t
,t ¥ 0 q, and the initial total wealth W
0
, the optimization problem of an agent i with
initial wealth allocationθ pdi q is given by,
max
pc
i
t
,t ¥0 q
L
i
E
i
0
‚
t ¥0
δ
t
u pc
i
t
,X
t
q λ
i
E
i
0
‚
t ¥0
δ
t
p
i
t
c
i
t
pγ q θ pdi qY
t
Due to the preference assumptions, the optimal solution is interior, and is characterized by
the first order condition,
c
i
t
X
t
γ
λ
i
p
t
ξ
i
t
X
η
t
, t ¥ 0.
and the equality of the inter-temporal budget constraint (3.5). Given the initial value ofp
0
I have that α
0
pdi q λ
i
and hence I get the rearranged first order condition (3.7). I then
substitute in the budget constraint, satisfied with equality, the optimal consumption share
from (3.7), to arrive at the equilibrium value forα
0
pdi q.
Proof of Corollary 3.2. From the first order condition (3.7) at two periodst ¥ 0 andt k,
k ¥ 0, I have that
α
t k
pdi q α
t
pdi q
ξ
i
t k
ξ
i
t
1 {γ
p
t k
p
t
1 {γ
exp
η
γ
px
t k
x
t
q pω
t k
ω
t
q
. (B.1)
173
Integrating both sides with respect to the set of agents I get
E
t
pξ
i
t,t k
q
1 {γ
» ξ
i
t k
ξ
i
t
1 {γ
α
t
pdi q z
t k
z
t
, (B.2)
wherez
t
p
1 {γ
t
exp
η
γ
x
t
ω
t
. The result is directly obtained from the derived gen-
eral relation (B.2) sincez
t
˜
ξ
1 {γ
t
t ¥ 0.
Proof of Corollary 3.3. Note that x
t 1
x
t
p1 λ
x
qω
t
and substitute in (B.2) for the
special case k 1 to obtain (3.14). Equation (3.15) is obtained after I substitute
˜
ξ
1 {γ
t
p
1 {γ
t
exp
η
γ
x
t
ω
t
in (B.1).
Proof of Proposition 3.1. I only need to recognize from the result,
p
t
˜
ξ
t
exp p ηx
t
γω
t
q, t ¥ 0
which is given by (3.14), that p
t
is a functional of the initial consumption distribution α
and the exogenous beliefs processesξ
i
. ThenF pα,i q is a function of the initial distribution
of consumption and exogenous processesy
t
,x
t
and pξ
i
t
,i PI q that are given. Hence using
lemma 3.1 I getF pα,j q ‡
j
0
α pdi q α pj q.
Proof of Corollary 3.5. Consider period t ¥ 0 and let pD
j
t,t k
,k ¥ 1 q be the stream of
future cash-flows for which asset j is a claim. Then the shadow valuation of agent i is
given by,
P
i
j,t
E
t
‚
k ¥1
M
i
t,t k
G
j,t k
(B.3)
whereM
i
t,t k
–
k ¥1
M
i
t k 1,t k
and
M
i
t k 1,t k
δξ
i
t k 1,t k
exp r γg
t k
p1 λ
x
qpγ η qω
t k 1
s.
that is given by (3.19) in the special case of a homogeneous beliefs economy with beliefs
ξ
i
. G
j
t,t k
D
j
t k
{D
j
t
is the cash-flow grow from periodt tot k. Integrate equation (B.3)
over the set of agent consumption distribution at timet and substitute in M the heteroge-
neous economy SDF:
E
t
P
i
j,t
E
t
‚
k ¥1
M
t,t k
G
j,t k
E
t
ξ
i
t,t k
˜
ξ
t,t k
But I already know thatE
t
ξ
i
t,t k
¥
˜
ξ
t,t k
with strict inequality when there exist heterogene-
ity of beliefs at least for one future period.
174
Proof of Lemma 3.2. Let the beliefs of agenti about one period consumption growth to be,
P
i
: g
t 1
|F
t
N pμ
i
t
,σ
2
t
u
2
t
q
and the consumption distribution across beliefs to be,
˜ α
t
: μ N pμ
t
,ν
2
t
q
Now let us aggregate beliefs and let f denote the probability density while ϕ denotes the
normal probability density function:
˜
f
t
pg
t 1
q »
f
t
pg
t 1
|μ q
1 {γ
˜ α pdμ q
γ
»
ϕ pg
t 1
|μ,σ
2
t
u
2
t
q
1 {γ
ϕ pμ |μ
t
,ν
2
t
qdμ
γ
a
2πγ pσ
2
t
u
2
t
q
γ
a
2π pσ
2
t
u
2
t
q
»
ϕ
g
t 1
|μ,γ pσ
2
t
u
2
t
q
ϕ pμ |μ
t
,ν
2
t
qdμ
γ
It is well known from Bayesian analysis that if y |θ N pθ,σ
2
1
q and θ N px,σ
2
2
q then
y N px,σ
2
1
σ
2
2
q. Using this result I get,
˜
f
t
pg
t 1
q a
2πγ pσ
2
t
u
2
t
q
γ
a
2π pσ
2
t
u
2
t
q
ϕ
g
t 1
|μ
t
,γ pσ
2
t
u
2
t
q ν
2
t
γ
d
1 ν
2
t
γ pσ
2
t
u
2
t
q
1 γ
ϕ
g
t 1
μ
t
,σ
2
t
u
2
t
ν
2
t
γ
It is clear from the final expression that the first part is the belief compensator exp p b
t
q
and the second term is the consensus beliefsf
r
.
Proof of Corollary 3.6. The continuously compounded risk-free rate is given by r
f
t
logE
t
pM
t,t 1
q. Since the SDF is log-normally distributed by lemma 3.2 then,
r
f
t
E
t
rlogM
t,t 1
s 1
2
V
t
rlogM
t,t 1
s
where
E
t
rlogM
t,t 1
s logδ γμ
t
p1 λ
x
qpγ η qω
t
b
t
V
t
rlogM
t,t 1
s γ
2
σ
2
t
u
2
t
ν
2
t
γ
,
and this completes the proof.
Example 3.1. In this example every agent has dogmatic beliefs, μ
i
t
μ
1
for all t, with-
out uncertainty, u
t
0. Let us suppose that the consumption distribution over beliefs,
175
˜ α, at some period t is N pμ
t
,ν
2
t
q. Re-expressing the law of motion for the consumption
distribution (B.1) in terms of the convolution ˜ α I get:
˜ α
t 1
pdμ q ˜ α
t
pdμ q
ϕ pg
t 1
|μ,σ
2
t
q
˜
f
t
pg
t 1
q
1 {γ
Using the results obtained in lemma 3.2 I get:
˜ α
t 1
pdμ q ?
2πσ
t
1 {γ
?
2πγσ
t
?
2πγσ
t
?
2πσ
t
1 {γ
ϕ pg
t 1
|μ,γσ
2
t
q
ϕ pg
t 1
|μ
t
,γσ
2
t
ν
2
t
q
ϕ pμ |μ
t
,ν
2
t
qdμ
1
ϕ pg
t 1
|μ
t
,γσ
2
t
ν
2
t
q
ϕ pg
t 1
|μ
t
,γσ
2
t
ν
2
t
qϕ pμ |μ
t 1
,ν
2
t 1
qdμ
where
μ
t 1
ν
2
t 1
ν
2
t
μ
t
ν
2
t 1
γσ
2
t
g
t 1
1
ν
2
t 1
1
ν
2
t
1
γσ
2
t
which completes the example.
Computational Approach
The complete computational approach that I employ to approximate the function of the
price-dividend ratio in equilibrium is quite involved. Here I present an outline of the algo-
rithm. In both homogeneous and heterogeneous beliefs economies the state vector is given
byz pμ,σ,ω q and the exogenous shock is the aggregate endowment growth whose natu-
ral logarithm isg. What differs in these two economies is the law of motion which I denote
in general withz
1
L pz,g q and the equilibrium stochastic discount factor (SDF) denoted
withM under the consensus beliefs. The price-dividend ratio under the consensus beliefs
is given by
P pz q E rM pz,g qe
g
pP pz
1
q 1 q |z s.
and letJ pz,g q M pz,g qe
g
. I first discretize the variableg by using32 Hermite quadrature
points of a standard normal variable and appropriately adjusting for the conditional mean
and conditional standard deviation. LetG pz q be the set of32 points and pw
j
,j 1...32 q
be the Hermite quadrature weights.
The price dividend function is approximated by a complete product of Chebyshev poly-
nomials of order d 20 denoted with T
i
. Since the Chebyshev polynomials are over the
176
interval r 1,1 s I set upper and lower bounds for all the state variables that are sufficiently
broad. The approximate price dividend function is expressed as follows:
˜
P pz q ‚
0 ⁄i
1
i
2
i
3
⁄d
β pi
1
,i
2
,i
3
qT
i
1
2
μ μ
μ μ
1
T
i
2
2
σ σ
σ σ
1
T
i
3
2
ω ω
ω ω
1
βT pz q.
To determine the unknown coefficients β I use the Galerkin projection method. With n
state variables an approximation of order d implies a number m pd n q!
n!d!
of unknown β
coefficients. I define the residual to be the Euler equation error,
R pz |β q ˜
P pz q ‚
g PG pz q
J pz,g q
˜
P pz
1
qw pg q ‚
g PG pz q
J pz,g qw pg q,
β
T pz q ‚
g PG pz q
J pz,g qT pz
1
qw pg q
‚
g PG pz q
J pz,g qw pg q,
and then use the Galerkin projections,
P
i
pβ q ‚
z PZ
R pz |β qT
i
pz q.
for all the Chebyshev polynomials used in the approximation. Z is the set of grid points
constructed from the zeros of the n Chebyshev polynomials of order p1 d q. Since the
number of projections is the same as the number of unknown coefficients, the coefficients
are determined by setting all the projections to zero and solving the resulting system of
equations. The residual function is linear in the coefficients and therefore the system of
equations is also linear. The approximation error is defined to be,
err pβ q sup
z PZ R
n
|R pz |β q|
˜
P pz q
,
which is solved numerically.
177
C Appendix to Chapter 4
Proofs
In this appendix I provide the proofs of all the results in chapter 4. Time t is discrete
and infinite with intermediate periods denoted either t
or t
. I have a complete market
economy with two fundamental shocks: (i) aggregate consumption growthg P tg
h
,g
l
u and
(ii) innovations to the probability of the high state,ǫ
π
P t0,1 u, and the process is given by
equation (4.1). ǫ
π
is unobservable but the two agents in the economy i P t1,2 u observe
signals tǫ
1
,ǫ
2
u that believe are informative according to (4.3). Agents eat every whole
period and have time and state separable preferences with external habit as given by (4.11)
and (4.12).
Proof of Lemma 4.1. For the first result I only need to note that,
P
i
pǫ
i
|ǫ
π
q ‚
ǫ
j
P
i
pǫ
i
,ǫ
j
|ǫ
π
q
and use assumption (4.3). Next because of the fact thatP
i
pǫ
π
q 1 {2 ǫ
π
P t0,1 u I derive
thatP
i
pǫ
j
q 1 {2 pi,j,ǫ
j
q P t1,2 u t1,2 u t0,1 u through:
P
i
pǫ
j
q ‚
ǫ
π
P
i
pǫ
j
|ǫ
π
qP
i
pǫ
π
q.
Finally fori j,
P
i
pǫ
i
|ǫ
j
q ǫ
π
P
i
pǫ
i
,ǫ
j
|ǫ
π
qP
i
pǫ
π
q
P
i
pǫ
j
q
1
2
,
which completes the proof.
Proof of Lemma 4.2. I only need to derive the probabilities P
i
pǫ
π
|ǫ
i
,ǫ
j
q which is done
through the general formula:
P
i
pǫ
π
|ǫ
i
,ǫ
j
q P
i
pǫ
i
,ǫ
j
|ǫ
π
qP
i
pǫ
π
q
P
i
pǫ
i
,ǫ
j
q
and the second result of lemma 4.1.
Proof of Lemma 4.3. This result is a simple application of the tower property of conditional
expectations for the specific case where the vector of probabilities which I denote here with
P pP pz q,z PZ q is unknown:
E
i
rf pz qs E
i
E
i
rf pz q|P s
E
i
‚
z
f pz qP pz q
‚
z
f pz qE
i
rP pz qs.
178
In forming their expectations about the probabilities agents only disagree on the probability
of the growth state since from lemma 4.1 they agree on the unconditional probabilities of
the two signals and agree on the conditional probability ofǫ
ρ
. Finally,ǫ
ρ
is independent of
the two signals.
Proof of Lemma 4.4. The optimization problem of an agent that invests in state contingent
claims every periodp
t
pz
t 1
q that pay the aggregate wealth next periodW
t 1
if shockz
t 1
realizes is given by
max
pθ
i
t
,t 0,1,... q
E
i
#
8
‚
t 0
δ
t
u pc
i
t
,y
t
q
F
a
0
+
s.t. c
i
t
θ
i
t 1
pz
t
qW
t
θ
i
t
p
t
, t 0,1,...
The first order conditions are:
δ
t
u
1
pc
i
t
,y
t
qp
t
pz
t 1
q δ
t 1
u
1
pc
i
t 1
,y
t 1
qW
t 1
P
i
pz
t 1
|F
a
t
q,
whereu
1
pc,y q denotes the first derivative with respect to the first argument. c
i
t 1
andW
t 1
refer to the state ifz
t 1
realizes. Letα c
1
{y. Rearranging I get the following expression
for agent1:
p
t
pz
t 1
q δg
γ
2
t 1
α
t 1
α
t
γ
1
W
t 1
P
1
pz
t 1
|F
a
t
q.
Similar expression is derived for the second agent where his consumption proportion is
1 α. Therefore, for each exogenous state z
t 1
I have two equations and hence I can
express p
t
pz
t 1
q and α
t 1
in terms of the rest of the variables which gives us the results.
The initial consumption proportion α
0
is trivially equal to 1 {2 since agents start with the
same wealth and the same beliefs.
Proof of Lemma 4.5. From lemma 4.4 I see that the consumption proportion next period
α ps
t
,z
t 1
q α
t 1
depends only on the previous consumption α α
t
and the ratio of
probabilities for the statez
t 1
. In lemma 4.3 I have seen that the beliefs differ only on the
probability of the growth state that depends on the beliefs at states and the two signals.
Proof of Lemma 4.6. The first result is a straightforward application of Blackwell’s suffi-
cient conditions and the Contraction Mapping Theorem. Let s
1
L ps,z q be the law of
motion of the states that takes values in the setS and define
T pw qps q α δ
‚
z PZ
g
1 γ
2
q pz |s qw ps
1
q.
Then T :S S is clearly a mapping whereS is the space of bounded functions onS.
It is both monotone and it discounts iffδg
1 γ
2
⁄ 1. It is therefore a contraction. Hence by
the Contraction Mapping Theorem there exists a functionw PS such thatw Tw.
179
Functionw ps q has the formαA ps q{p1 δ q whereα outsideA ps q is the individual state
variable whereas inA ps q I have the aggregate state variableα for which I do not distinguish
for notational simplicity. Function w ps q is homogeneous of degree 1 wrt. the individual
state variableα because its derivative is homogeneous of degree0.
In the special case ofγ
1
γ
2
1 I conjecture thatA ps q 1 which is verified since
α
1 δ
α δ
‚
s
απ
1
ps,ǫ
1
,ǫ
2
q p1 α qπ
2
ps,ǫ
1
,ǫ
2
q
α
1
1 δ
and from the law of motion in lemma 4.5α
1
rαπ
1
ps,ǫ
1
,ǫ
2
q p1 α qπ
2
ps,ǫ
1
,ǫ
2
qs α.
Proof of Lemma 4.7. Within one time step I use the subscript0 to denote the beginning of
the time-step, 1 to denote the intermediate time point and 2 to denote the end of the time-
step. The state of the economy at 0 iss pα,π
1
,π
2
,ρ q and the state of the economy at 1
is described by ps,ǫ
1
,ǫ
2
q. The set of exogenous shocks isz pǫ
1
,ǫ
2
,g,ǫ
ρ
q andy denotes
the aggregate endowment at 0. θ
1
ps q denotes the vector of holdings at the beginning of 1
andθ
2
ps q the vector of holdings at the beginning of2. The first order conditions of the two
problems (P
0
) and (P
1
) are given by:
u
1
pc
i
0
,y qp ps,z qW pL ps,z qqyg BV
1
pθ
i
1
ps q|y,s,ǫ
1
,ǫ
2
q
Bθ
i
1
ps,z q
P pǫ
1
,ǫ
2
q, z PZ, (FOC0)
˜
λ
i
ps,ǫ
1
,ǫ
2
qp ps,ǫ
1
,ǫ
2
,z qW pL ps,z qqyg δ
BV
0
pθ
i
2
ps,z q|yg,L ps,z qq
Bθ
i
2
ps,z q
P
i
pz |s,ǫ
1
,ǫ
2
q, z PZ, (FOC1)
where
˜
λ
i
ps,ǫ
1
,ǫ
2
q is the Lagrange multiplier. I then derive the conditions from the Enve-
lope Theorem applied to the two optimization problems:
BV
0
pθ
i
0
|y,s q
Bθ
i
0
u
1
pc
i
0
,y qW ps qy, (ET0)
BV
1
pθ
i
1
ps q|y,s,ǫ
1
,ǫ
2
q
Bθ
i
0
˜
λ
i
ps,ǫ
1
,ǫ
2
qp ps,ǫ
1
,ǫ
2
,z qW pL ps,z qqyg. (ET1)
From (FOC0) and (ET1) I get:
u
1
pc
i
0
,y qp ps,z q ˜
λ
i
ps,ǫ
1
,ǫ
2
,z qP pǫ
1
,ǫ
2
q
and note thatu
1
pc
i
0
,y q{
˜
λ
i
ps,ǫ
1
,ǫ
2
,z q is agent independent. Therefore I can define:
λ ps,ǫ
1
,ǫ
2
q u
1
pc
i
0
,y q
˜
λ
i
ps,ǫ
1
,ǫ
2
,z q
.
180
From equation (FOC1), equation (ET0) for time point 2 and the definition ofλ ps,ǫ
1
,ǫ
2
q I
get:
p ps,ǫ
1
,ǫ
2
,z q δλ ps,ǫ
1
,ǫ
2
q
u
1
pc
i
2
,yg q
u
1
pc
i
0
,y q
π
i
ps,ǫ
1
,ǫ
2
q, z PZ. (C.1)
I know that in equilibrium c
1
0
αy and c
2
0
p1 α qy and likewise c
1
2
α ps,z qyg and
c
2
2
p1 α ps,z qqyg. For each z PZ I have two equations (C.1), one for each agent. In
these two equations I treatp ps,ǫ
1
,ǫ
2
q andα ps,z q as unknowns and solve forp ps,ǫ
1
,ǫ
2
q by
eliminatingα ps,z q to get to the final result.
Proof of Proposition 4.1. The prices can be derived from the wealth function at every time
point according to:
P ps q w pα,π
1
,π
2
,ρ q w p1 α,π
2
,π
1
,ρ q 1
for whole periods and
P ps,ǫ
1
,ǫ
2
q w pα,π
1
,π
2
,ρ,ǫ
1
,ǫ
2
q w p1 α,π
2
,π
1
,ρ,ǫ
1
,ǫ
2
q
for intermediate periods. For the special case ofγ
1
γ
2
1 I already know thatw ps q α {p1 δ q andw ps,ǫ
1
,ǫ
2
q αδ {p1 δ q which give the result.
Agents in every half period trade in the stock in unit net supply and three half period
Arrow-Debreu securities in zero net supply in order to finance their wealth next period.
The AD securities pay the current aggregate endowment for their corresponding state. In
every states the three AD securities are redundant since their wealth does not vary in the
intermediate period,w ps,ǫ
1
,ǫ
2
q αδ {p1 δ q. Therefore, the first agent only needs to hold
α of the stock. In the next period wealth varies only to the extend that their consumption
varies. From lemma 4.5 I know that their consumption only varies due to the realization
of g and is independent of ǫ
ρ
. In case h the wealth of the first agent in the next period
after the realization of pg
h
,0.5 q isα ps,ǫ
1
,ǫ
2
,g
h
q{p1 δ q and is only given by his holding
of the stock and the endowment share. Therefore his consumption share must also be his
holding of the stock at the beginning of the period. The AD security for the state pg
h
,0 q is
redundant since his consumption share is the same as in pg
h
,0.5 q, while the holdings of the
other two securities are the same and denoted withϑ. In state pg
l
,ǫ
ρ
q his wealth is given by
w pL ps,ǫ
1
,ǫ
2
,g
l
,ǫ
ρ
qq α ps,ǫ
1
,ǫ
2
,g
h
qpP 1 q ϑ.
From the equilibrium values of the price and wealth I get the result forϑ. Similarly I derive
the conditions for the second case.
Proof. Lets pα,π
1
,π
2
,ρ q and ˜ s p1 α,π
2
,π
2
,ρ q. In the same way that I derived that
the first agent’s wealth to be
w ps q α
A ps q
1 δ
181
I can show that in the intermediate period it is
w ps,ǫ
1
,ǫ
2
q αδ
A ps,ǫ
1
,ǫ
2
q
1 δ
.
The functions A ps q and A ps,ǫ
1
,ǫ
2
q are non-linear in the states. From the relation of the
price of the stock in an intermediate period with the individual wealth functions I have,
P ps,ǫ
1
,ǫ
2
q
w ps,ǫ
1
,ǫ
2
q
1 w p˜ s,ǫ
1
,ǫ
2
q
w ps,ǫ
1
,ǫ
2
q
.
However, the ratio of the two wealth functions is not constant unless in the special case of
proposition 4.1. Therefore, agents need to trade on the rest of the securities every whole
period. Similarly it is true for the intermediate periods.
Computational Approach
The computational method for solving for the equilibrium prices is the same as the one used
for solving the model of chapter 3 as described in appendix B. The advantage in this case
is that the exogenous shocks are discrete and the endogenous state variables are bounded.
This adds accuracy to the results.
182

Abstract (if available)

Abstract This thesis examines how and to what extend certain types of heterogeneity of agents in an economy with complete financial markets can explain the variation in aggregate prices and the volume of trade that we observe. There are a number of characteristics of the financial markets that have been particularly puzzling researchers

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Creator Xiouros, Costas (author)

Core Title Asset prices and trading in complete market economies with heterogeneous agents

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 04/28/2009

Defense Date 03/09/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag complete markets,financial prices,heterogeneous agents,OAI-PMH Harvest,trading volume

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Zapatero, Fernando (committee chair), Jones, Christopher S. (committee member), Magill, Michael (committee member), Sangvinatsos, Antonios (committee member)

Creator Email cxiouros@gmail.com,xiouros@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2140

Unique identifier UC1196181

Identifier etd-Xiouros-2871 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-222611 (legacy record id),usctheses-m2140 (legacy record id)

Legacy Identifier etd-Xiouros-2871.pdf

Dmrecord 222611

Document Type Dissertation

Rights Xiouros, Costas

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu