statistique, théorie et gestion de portefeuille - Docs at ISFA

452 14. Gestion de Portefeuilles multimoments et équilibre de marché
assessment and optimization. We have shown that these risk measures, which contain centered moments
(and cumulants with some restriction) as particular examples, generalize them significantly. We have presented
a generalization of previous generalizations of the efficient frontiers and of the CAPM based on these
risk measures in the cases of homogeneous and heterogeneous agents. We have then proposed a simple but
powerful specific von Mises representation of multivariate distribution of returns that allowed us to obtain
new analytical results on and empirical tests of a general framework for a portfolio theory of non-Gaussian
risks with non-linear correlations. Quantitative tests have been presented on a basket of seventeen stocks
among the largest capitalization on the NYSE.
This work opens several novel interesting avenues for research. One consists in extending the Gaussian copula
assumption, for instance by using the maximum-entropy principle with non-extensive Tsallis entropies,
known to be the correct mathematical information-theoretical representation of power laws. A second line
of research would be to extend the present framework to encompass simultaneously different time scales τ
in the spirit of (Muzy et al. 2001) in the case of a cascade model of volatilities.
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