Apart from the risk premium of equity over bonds, volatility of asset prices and trading volumes are two aspects of the already developed general equilibrium asset pricing theory that fail to show any resemblance with real data. The assumption of agent homogeneity has been relaxed in a number of studies where agents face uninsurable income shocks but fail to provide a consistent explanation that addresses all issues. This study assumes a dynamically complete asset market with two states and two assets, where the agents are allowed to be heterogeneous either in terms of their risk preferences, or in terms of their beliefs for the probabilities of the exogenous shock. The assumption that the (logarithm of the) aggregate dividend follows an autoregressive process with an iid state dependent shock, helps us construct a non-trivial equilibrium in which the endogenously determined wealth distribution across agents is part of the economy's state vector. Qualitative results show that this kind of setup works in the right direction in all issues addressed, while quantitative results are produced using a projection method as well as a variant of the Krusell and Smith method

Costas Xiouros

English

Research Papers in Economics

Asset Price Volatilities and Trading Volumes
in Dynamically Complete Markets with
Heterogeneous Agents
Costas Xiouros¤
February 28, 2006
Abstract
Apart from the risk premium of equity over bonds, volatility of asset prices and
trading volumes are two aspects of the already developed general equilibrium asset
pricing theory that fail to show any resemblance with real data. The assumption of
agent homogeneity has been relaxed in a number of studies where agents face unin-
surable income shocks but fail to provide a consistent explanation that addresses all
issues. This study assumes a dynamically complete asset market with two states and
two assets, where the agents are allowed to be heterogeneous either in terms of their
risk preferences, or in terms of their beliefs for the probabilities of the exogenous shock.
The assumption that the (logarithm of the) aggregate dividend follows a mean reverting
process with an iid state dependent shock, helps us construct a non-trivial equilibrium
in which the endogenously determined wealth distribution across agents is part of the
economy's state vector. Qualitative results show that this kind of setup works in the
right direction in all issues addressed, while quantitative results are produced using a
projection method as well as a variant of the Krusell and Smith (1997) method.
¤PhD student at the department of Finance and Business economics, Marshall School of Business, Uni-
versity of Southern California, Los Angeles, CA 90089-1427, USA (e-mail: xiouros@marshall.usc.edu)
11 Introduction
The standard representative agent model, as exempli¯ed by the Lucas (1978) tree model,
even though the paradigm for a framework to study asset prices, is known to fail in a number
of predictions. Apart from the Equity Premium Puzzle [see for example Mehra and Prescott
(1985) and Campbell (1999)], the agent homogeneity assumption stands as an obstacle in ex-
plaining asset price volatilities as well as trading volumes. As Hansen and Jagannathan (1991)
show, the discrepancy between the volatilities of the asset prices and that of the aggregate
consumption growth, is impossible to explain with a sensible set of preference parameters.
As trading is concerned, things are even worse as the homogeneous agent models are not
the only ones that predict no-trade equilibria. In particular, Judd, Kubler and Schmedders
(2003a) in an extended Lucas (1978) model, that incorporates agent heterogeneity in terms
of preferences, show that general equilibrium restrictions rule out trading in long lived as-
sets. Further, Constantinides and Du±e (1996) construct an incomplete market setting with
uninsurable income shocks that results also into a no-trade equilibrium.
While common intuition dictates that the agent heterogeneity is most probably the element
required to explain endogenously, volatility and volume, there are two issues that remain
widely unresolved. Firstly, what type of heterogeneity is mostly responsible in each case
and secondly, through what avenues? Further, given the evidence that the cross-sectional
variation in consumption growth goes a long way in explaining the risk premia observed,
as shown by Brav, Constantinides and Geczy (2002), can the agent heterogeneity produce
similar variation in marginal rates of intertemporal substitutions across agents?
The heterogeneous agent literature has mostly concentrated on ex-post agent heterogeneity
coming from completely or partially uninsurable income shocks, like in Constantinides and
Du±e (1996), Mankiw (1986), Heaton and Lucas (1996) and Krusell and Smith (1997). In
their paper Constantinides and Du±e need to assume persistent income shocks so that they
avoid close to complete risk sharing by the agents. However, they go to the other extreme
allowing for no risk sharing with a no-trade equilibrium. Building on the same framework
Storesletten, Telmer and Yaron (2001) present an overlapping generations model with idio-
syncratic income shocks where there is some trade, and therefore partial risk sharing, where
the interesting pricing implications come from the way agents accumulate or deccumulate
assets as they age. The main problem of this framework is that it is not able to address
all issues at the same time. On the one end of the spectrum is the risk premium and the
volatility puzzles while on the other end is the issue of trading, with the deciding factor being,
how much risk sharing is allowed.
This study concentrates on ex-ante agent heterogeneity, in a dynamically complete market
setting, either in terms of risk preferences, as in Dumas (1989), Wang (1996) and Chan and
Kogan (2002), or in terms of beliefs as in Abel (1989) and Jouini and Napp (2004). In
particular we use standard time and state separable preferences with power utility, where the
agents are allowed to have di®erent risk aversion parameters. Alternatively, the agents are
allowed to have their own beliefs as to the probabilities of the exogenous shock, which they
do not update.
2The model is constructed within the Lucas (1978) framework, while we try to built weakly
stationary recursive equilibria that requires the wealth distribution in order to predict the
evolution of the system. This is achieved in our complete asset markets, by assuming that
the (logarithm of the) dividend follows a mean reverting process that is driven by a time-
independent and state-dependent exogenous shock. For the case of heterogeneity in beliefs,
an interesting evolution of the wealth distribution, that a®ects trading and prices, is achieved
even with the standard Markovian process for the dividends. However, for the case of risk
preference heterogeneity, the results of Judd, Kubler and Schmedders (2003a) apply. The
mechanism through which the wealth distribution evolves in a non-trivial way (meaning being
dependent only on the exogenous shock) is di®erent in the two cases considered. In general,
the hedging demands of the agents depend on their risk aversion parameter, their beliefs
about the probabilities of the exogenous shock, their relative wealth and the aggregate risk
that they face. When the agents have di®erent beliefs for the probabilities of the states, in
equilibrium they disagree on their consumption growth rates. This results in a non-trivial
evolution of their relative wealth levels which has signi¯cant qualitative as well as quantitative
implications both in terms of trading as well as asset prices. When agents have di®erent risk
aversion parameters obviously make di®erent investment decisions. However, the essential
element here is that the risk that they face each period changes with the current level of the
dividend, which in turn implies continuous adjustments in their portfolios.
A major issue in general equilibrium models with heterogeneous agents is the way the equi-
librium is approximated, when the wealth distribution (or alternatively the distribution of
asset holdings) plays a role in the evolution of the system [see Wang (1994), Du±e et al.
(1994) and Kubler and Polemarchakis (2004)]. Since closed form solutions are not available,
many methods have been developed to approximate closely the evolution of the system. See
for example Judd, Kubler and Schmedders (2003b) for projection and perturbation methods,
Rios-Rull (1997), Krusell and Smith (1997, 1998) or den Haan (1994) for methods that use
bounded rationality and Heaton and Lucas (1996) for a discretized space method, just to
mention a few.
An endogenous state space normally arises in an incomplete market setting, but as we show
here it is not necessary. Our complete market setting as well as the assumption of power
utility helps us in solving the agent's value function, as well the decision rules, in a semi-closed
form and expressing the unknown part of the solution in terms of an assumed equilibrium
pricing kernel and the law of motion of the state vector. A method, therefore is developed
where projection methods are used in order to approximate the pricing kernel and the law
of motion. The advantages of our method are mainly three. First, the number of unknown
functions that are required to approximate, decrease considerably from the case where the
individual Euler equations are used together with the market clearing conditions. Secondly,
with the semi-closed form solution of the value function, its shape is ensured to be preserved.
Last but not least, with the way the individual decision rules are constructed, they are smooth
in the state vector, and they can be used when assuming frequent trading.
A second method is also used to approximate our equilibrium. It is a variant of the Krusell
and Smith (1997) method, which is extended to incorporate heterogeneous agents and long
lived assets. This method however cannot be used for many assets of many di®erent types of
3agents. With two assets and two agents (or types of agents) the state vector is the wealth of
the ¯rst agent (or the wealth distribution of the ¯rst type), and the current dividend level.
The functions that need to be approximated are hence, the two price functions as well as the
law of motion.
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5

Aggregation of Heterogeneous Beliefs.

and A.A.Smith

and D.Du±e

and H.Polemarchakis (2004). Stationary Markov equilibrium for overlapping generations. Economic Theory,

and L.Kogan (2002). Catching Up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices.

Asset prices in an exchange economy.

Asset Prices under Heterogeneous Beliefs: Implications for the Equity Premium. working paper,

Asset prices, consumption, and the business cycle.

Asset Pricing with Heterogeneous Consumers and Limited Participation: Empirical Evidence.

C.I.Telmer and A.Yaron (2001). Asset pricing with idiosyncratic risk and overlapping generations. Working Paper.

Computation of Equilibria in Heterogeneous Agent Models. working paper.

Evaluating the e®ects of incomplete markets on risk sharing and asset pricing.

Heterogeneity, Aggregate Uncertainty and the Short Term Interest Rate: A Case Study of two Solution Techniques. working paper.

Restrictions on intertemporal marginal rates of substitution implied by asset returns.

Schmedders (2003a). Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents.

Schmedders (2003b). Computational Methods for Dynamic Equilibria with Heterogeneous Agents.

Stationary Markov equilibria in an OLG model with correlated production shocks.

The equity premium and the concentration of aggregate shocks.

The Equity Premium.

The term structure of interest rates in a pure exchange economy with heterogeneous investors.

Two person dynamic equilibrium in the capital market.

Asset price volatilities and trading volumes in heterogeneous agent economies

http://repec.org/sce2006/up.2985.1141173552.pdf

Asset price volatilities and trading volumes in heterogeneous agent economies

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