statistique, théorie et gestion de portefeuille - Docs at ISFA

426 14. Gestion de Portefeuilles multimoments et équilibre de marché
entangled with the Mean-Variance Portfolio Model. Indeed both of them fundamentally rely on the description
of the probability distribution function (pdf) of asset returns in terms of Gaussian functions. The
Mean-Variance description is thus at the basis of Markovitz’s portfolio theory (Markovitz 1959) and of the
CAPM (see for instance (Merton 1990)).
Otherwise, the determination of the risks and returns associated with a given portfolio constituted of N
assets is completely embedded in the knowledge of their multivariate distribution of returns. Indeed, the
dependence between random variables is completely described by their joint distribution. This remark
entails the two major problems of portfolio theory: 1) determine the multivariate distribution function of
asset returns; 2) derive from it useful measures of portfolio risks and use them to analyze and optimize
portfolios.
The variance (or volatility) of portfolio returns provides the simplest way to quantify its fluctuations and
is at the fundation of the (Markovitz 1959)’s portfolio selection theory. Nonetheless, the variance of a
portfolio offers only a limited quantification of incurred risks (in terms of fluctuations), as the empirical
distributions of returns have “fat tails” (Lux 1996, Gopikrishnan et al. 1998, among many others) and the
dependences between assets are only imperfectly accounted for by the covariance matrix (Litterman and
Winkelmann 1998). It is thus essential to extend portfolio theory and the CAPM to tackle these empirical
facts.
The Value-at-Risk (Jorion 1997) and many other measures of risks (Artzner et al. 1997, Sornette 1998,
Artzner et al. 1999, Bouchaud et al. 1998, Sornette et al. 2000b) have then been developed to account for the
larger moves allowed by non-Gaussian distributions and non-linear correlations but they mainly allow for the
assessment of down-side risks. Here, we consider both-side risk and define general measures of fluctuations.
It is the first goal of this article. Indeed, considering the minimum set of properties a fluctuation measure
must fulfil, we characterize these measures. In particular, we show that any absolute central moments and
some cumulants satisfy these requirement as well as do any combination of these quantities. Moreover,
the weights involved in these combinations can be interpreted in terms of the portfolio manager’s aversion
against large fluctuations.
Once the definition of the fluctuation measures have been set, it is possible to classify the assets and portfolios
using for instance a risk adjustment method (Sharpe 1994, Dowd 2000) and to develop a portfolio
selection and optimization approach. It is the second goal of this article.
Then a new model of market equilibrium can be derived, which generalizes the usual Capital Asset Pricing
Model (CAPM). This is the third goal of our paper. This improvement is necessary since, although the use
of the CAPM is still widely spread, its empirical justification has been found less and less convincing in the
past years (Lim 1989, Harvey and Siddique 2000).
The last goal of this article is to present an efficient parametric method allowing for the estimation of the
centered moments and cumulants, based upon a maximum entropy principle. This parameterization of
the problem is necessary in order to obtain accurate estimates of the high order moment-based quantities
involved the portfolio optimization problem with our generalized measures of fluctuations.
The paper is organized as follows.
Section 2 presents a new set of consistent measures of risks, in terms of the semi-invariants of pdf’s, such as
the centered moments and the cumulants of the portfolio distribution of returns, for example.
Section 3 derives the generalized efficient frontiers, based on these novel measures of risks. Both cases with
and without risk-free asset are analyzed.
Section 4 offers a generalization of the Sharpe ratio and thus provides new tools to classify assets with
respect to their risk adjusted performance. In particular, we show that this classification may depend on the
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