statistique, théorie et gestion de portefeuille - Docs at ISFA

Multi-Moments Method for Portfolio Management:
Generalized Capital Asset Pricing Model
in Homogeneous and Heterogeneous markets ∗
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée
CNRS UMR6622 and Université de Nice-Sophia Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095
e-mails: Yannick.Malevergne@unice.fr and sornette@unice.fr
Abstract
We use a new set of consistent measures of risks, in terms of the semi-invariants of pdf’s, such that the
centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on
the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of
risks and present the generalized CAPM, both in the cases of homogeneous and heterogeneous markets.
Then, using a family of modified Weibull distributions, encompassing both sub-exponentials and superexponentials,
to parameterize the marginal distributions of asset returns and their natural multivariate
generalizations, we offer exact formulas for the moments and cumulants of the distribution of returns of
a portfolio made of an arbitrary composition of these assets. Using combinatorial and hypergeometric
functions, we are in particular able to extend previous results to the case where the exponents of the
Weibull distributions are different from asset to asset and in the presence of dependence between assets.
In this parameterization, we treat in details the problem of risk minimization using the cumulants as
measures of risks for a portfolio made of two assets and compare the theoretical predictions with direct
empirical data. Our extended formulas enable us to determine analytically the conditions under which
it is possible to “have your cake and eat it too”, i.e., to construct a portfolio with both larger return and
smaller “large risks”.
1 Introduction
The Capital Asset Pricing Model (CAPM) is still the most widely used approach to relative asset evaluation,
although its empirical roots are been found weaker and weaker in recent years. This asset valuation model
describing the relationship between expected risk and expected return for marketable assets is strongly
∗ We acknowledge helpful discussions and exchanges with J.V. Andersen, J.P. Laurent and V. Pisarenko. We are grateful to
participants of the workshop on “Multi-moment Capital Asset Pricing Models and Related Topics”, ESCP-EAP European School
of Management, Paris, April,19, 2002, and in particular to Philippe Spieser, for their comments. This work was partially supported
by the James S. Mc Donnell Foundation 21st century scientist award/studying complex system.
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