The aim of this paper is to show, within the mean-variance framework, how the market belief can be constructed as the result of the aggregation of heterogeneous beliefs and how the market equilibrium prices of risky assets can thus be determined. The heterogeneous beliefs are defined in terms of not only the means but also variances and covariances. By constructing the mean and variance of the market belief, we analyze the impact of the heterogeneous beliefs on the market equilibrium asset pricing relation. In particular, we extend the standard CAPM under homogenous beliefs to the one under the heterogeneous beliefs.Mean variance analysis, heterogeneous beliefs, aggregation, asset pricing

Carl Chiarella

Roberto Dieci

Tony He

English

Research Papers in Economics

AGGREGATION OF HETEROGENEOUS BELIEFS AND ASSET PRICING:
A MEAN-VARIANCE ANALYSIS
CARL CHIARELLA*, ROBERTO DIECI** AND XUE-ZHONG HE*
*School of Finance and Economics
University of Technology, Sydney
PO Box 123 Broadway
NSW 2007, Australia
**Universita’ degli Studi di Bologna
Facolta’ di Economia del Polo di Rimini
Via D. Anghera’ 22
I-47900 Rimini, Italy
The Sharpe-Lintner-Mossin Capital Asset Pricing Model (CAPM) plays a central
role in modern ﬁnance theory. It is founded on the paradigm of homogeneous beliefs
and rational representative agent. However, from the theoretical perspective this para-
digm has been criticised on a number of grounds (e.g. Williams (1977), Varian (1985),
Detemple and Murthy (1994), Cecchetti et al. (2000), Abel (1989, 2002), Calvet et
al. (2004), Jouini and Napp (2006)), in particular concerning its extreme assumptions
about information of the economic environment and computational ability on the part
of the rational representative economic agent. It is found from the empirical literature
that (see Harvey (1989)) the fundamental CAPM equation that relates expected returns
on assets to their portfolio risk is not well supported. For instance the security market
line does not have the predicted slope and the measure of portfolio risk, the so-called
beta coefﬁcient, is time varying in ways that are not consistent with the theory.
12 CHIARELLA, DIECI AND HE
There has been some work on extending the mean-variance model to allow for dif-
ferences in beliefs across agents (see Williams (1977), Varian (1985), Detemple and
Murthy (1994) Cecchetti et al. (2000), Abel (1989, 2002), Calvet et al. (2004), Jouini
and Napp (2006)). The impact of heterogeneous beliefs among agents on the market
equilibrium price has been focused. In particular, we are more interested in how the
market equilibrium price is determined when agents are bounded rational and hetero-
geneous in beliefs.
Recent research has witnessed a shift towards a paradigm of boundedly rational and
heterogeneous agents (see Frankel and Froot (1987), Day and Huang (1990), Chiarella
(1992), Kirman (1993), Lux (1995), Brock and Hommes (1998), Lux and Maechesi
(1999), ChiarellaandHe(2001, 2002, 2003), ChiarellaandBohm(2004)andChiarella
et. al (2002, 2006)). This strand of research has essentially taken the framework of
the standard paradigm of ﬁnance theory (the rational representative agent choosing a
portfolio of risky assets so as to maximise expected utility) and added to it bounded
rationality, agent heterogeneity, agents seeking to learn about the economic environ-
ment and agents using various heuristics to change ﬁnancial decision strategies. The
result is a model of ﬁnancial markets as nonlinear adaptive evolutionary systems that
is able to reproduce at least some of the stylised facts of ﬁnancial markets that are
not well-explained by the standard paradigm, we refer Hommes (2006) and LeBaron
(2006)) for two excellent surveys of these developments.
There remain a number of key issues with which this literature has still to come
to grips, these include: the implications for portfolio diversiﬁcation strategies; the
implications for the relationship between expected risk adjusted returns on different
assets (in other words how does the traditional Capital Asset Pricing Model - CAPM
- look in this new framework?); whether this framework can provide a theory of a
time-varying beta coefﬁcient.AGGREGATION OF HETEROGENEOUS BELIEFS AND ASSET PRICING 3
In most of the literature, a portfolio of one risk-free asset and one-risky asset is con-
sidered and agents are heterogeneous in the mean, rather than the variance. In this
paper, we consider a portfolio of one risk-free asset and many risky assets and extend
the mean-variance model in the discrete time setting to allow for heterogeneity in not
only the mean but also the variance across agents. By assuming a Walrasian auction-
eer scenario, we show that the market aggregate expected payoffs of the risky assets
are weighted averages of the heterogeneous expected payoffs of the risky assets across
the agents, in which the weights are given by the heterogeneous covariance matrices
adjusted by the risk aversion coefﬁcients of the agents. We also show that the mar-
ket equilibrium prices of risky assets are determined jointly by the aggregate mean
and variance of the market portfolio. In particular, we derive an equilibrium relation
between the market aggregate expected payoff of the risky assets and the market port-
folio expected payoff, leading to a CAPM-like relation with heterogeneous beliefs. An
exact formula for the so-called ¯ coefﬁcient is derived. When the agents are heteroge-
neous in means but homogeneous in covariances of the risky assets, we show that, in
equilibrium, the market aggregate expected payoffs of the risky assets and the market
portfolio are the risk aversion weighted averages of the heterogeneous expected pay-
offs of the risk assets and the market portfolio, respectively. A further assumption of
the homogeneous beliefs in the means then leads to the standard CAPM.
Some numerical examples are used to illustrate the impact of the heterogeneous
beliefs on the aggregate returns of the market portfolio. In particular we examine
the inﬂuence of the heterogeneous beliefs on the ¯ coefﬁcient in the standard CAPM.
Furtheranalysisontheexplanatorypoweroftheresultwithrespecttotheriskpremium
puzzle will be conducted.4 CHIARELLA, DIECI AND HE
REFERENCES
Abel, A. (1989), Asset prices under heterogeneous beliefs: implications for the equity premium, work-
ing paper 09-89, Rodney L. White Center for Financial Research.
Abel, A. (2002), ‘An exploration of the effects of pessimism and doubt on asset returns’, Journal of
Economic Dynamics and Control 26, 1075–1092.
Bohm, V. and Chiarella, C. (2004), ‘Mean variance preferences, expectations formation, and the dy-
namics of random asset prices’, Mathematical Finance 15, 61–97.
Brock, W. and Hommes, C. (1998), ‘Heterogeneous beliefs and routes to chaos in a simple asset pricing
model’, Journal of Economic Dynamics and Control 22, 1235–1274.
Calvet, L., Grandmont, J.-M. and Lemaire, I. (2004), Aggregation of heterogeneous beliefs and asset
pricing in complete ﬁnancial markets, working paper 2004-12, CREST.
Cecchetti, S., Lam, P. and Mark, N. (2000), ‘Asset pricing with distorted beliefs: are equity returns too
good to be true?’, American Economic Review 90, 787–805.
Chiarella, C. (1992), ‘The dynamics of speculative behaviour’, Annals of Operations Research 37, 101–
123.
Chiarella, C., Dieci, R. and Gardini, L. (2002), ‘Speculative behaviour and complex asset price dynam-
ics’, Journal of Economic Behavior and Organization 49, 173–197.
Chiarella, C., Dieci, R. and He, X. (2006), ‘Heterogeneous expectations and speculative behaviour in a
dynamic multi-asset framework’, Journal of Economic Behavior and Organization . in press.
Chiarella, C. and He, X. (2001), ‘Asset price and wealth dynamics under heterogeneous expectations’,
Quantitative Finance 1, 509–526.
Chiarella, C. and He, X. (2002), ‘Heterogeneous beliefs, risk and learning in a simple asset pricing
model’, Computational Economics 19, 95–132.
Chiarella, C. and He, X. (2003), ‘Heterogeneous beliefs, risk and learning in a simple asset pricing
model with a market maker’, Macroeconomic Dynamics 7, 503–536.
Day, R. and Huang, W. (1990), ‘Bulls, bears and market sheep’, Journal of Economic Behavior and
Organization 14, 299–329.
Detemple, J. and Murthy, S. (1994), ‘Intertemporal asset pricing with heterogeneous beliefs’, Journal
of Economic Theory 62, 294–320.AGGREGATION OF HETEROGENEOUS BELIEFS AND ASSET PRICING 5
Frankel, F. and Froot, K. (1987), ‘Using survey data to test propositions regarding exchange rate expec-
tations’, American Economic Review 77, 133–153.
Harvey, C. R. (1989), ‘Time-varying conditional covariances in tests of asset pricing models’, Journal
of Financial Economics 24, 289–317.
Hommes, C. (2006), Handbook of Computational Economics, Vol. 2:Agent-based Computational Eco-
nomics, North-Holland, chapter Heterogeneous Agent Models in Economics and Finance. to ap-
pear.
Jouini, E. and Napp, C. (2006), ‘Heterogeneous beliefs and asset pricing in discrete time: An analysis
of pessimism and doubt’, Journal of Economic Dynamics and Control .
Kirman, A. (1993), ‘Ants, rationality, and recruitment’, Quarterly Journal of Economics 108, 137–156.
LeBaron, B. (2006), Handbook of Computational Economics, Vol. 2: Agent-based Computational Eco-
nomics, North-Holland, chapter Agent-based Computational Finance. to appear.
Lux, T. (1995), ‘Herd behaviour, bubbles and crashes’, Economic Journal 105, 881–896.
Lux, T. and Marchesi, M. (1999), ‘Scaling and criticality in a stochastic multi-agent model of a ﬁnancial
markets’, Nature 397(11), 498–500.
Varian, H. (1985), ‘Divergence of opinion in complete markets’, Journal of Finance 40, 309–317.
Williams, J. (1977), ‘Capital asset prices with heterogeneous beliefs’, Journal of Financial Economics
5, 219–239.

Aggregation of heterogeneous beliefs and asset pricing in complete ﬁnancial markets, working paper 2004-12,

An exploration of the effects of pessimism and doubt on asset returns’,

Ants, rationality, and recruitment’,

Asset price and wealth dynamics under heterogeneous expectations’,

Asset prices under heterogeneous beliefs: implications for the equity premium, working paper 09-89, Rodney L. White Center for Financial Research.

Asset pricing with distorted beliefs: are equity returns too good to be true?’,

Bulls, bears and market sheep’,

Capital asset prices with heterogeneous beliefs’,

Divergence of opinion in complete markets’,

Herd behaviour, bubbles and crashes’,

Heterogeneous beliefs and asset pricing in discrete time: An analysis of pessimism and doubt’,

Heterogeneous beliefs and routes to chaos in a simple asset pricing model’,

Heterogeneous beliefs, risk and learning in a simple asset pricing model with a market maker’,

Heterogeneous beliefs, risk and learning in a simple asset pricing model’,

Heterogeneous expectations and speculative behaviour in a dynamic multi-asset framework’,

Intertemporal asset pricing with heterogeneous beliefs’,

Mean variance preferences, expectations formation, and the dynamics of random asset prices’,

Scaling and criticality in a stochastic multi-agent model of a ﬁnancial markets’,

Speculative behaviour and complex asset price dynamics’,

The dynamics of speculative behaviour’,

Time-varying conditional covariances in tests of asset pricing models’,

Aggregation of Heterogeneous Beliefs and Asset Pricing: A Mean-Variance Analysis

http://repec.org/sce2006/up.20076.1139813031.pdf

Aggregation of Heterogeneous Beliefs and Asset Pricing: A Mean-Variance Analysis

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