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Multi-Moments Method for Portfolio Management: Generalized Capital Asset Pricing Model in Homogeneous and Heterogeneous markets

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  • Y. Malevergne

    (Univ. Nice and Univ. Lyon)

  • D. Sornette

    (CNRS-Univ. Nice and UCLA)

Abstract

We introduce 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 super-exponentials, 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''.

Suggested Citation

  • Y. Malevergne & D. Sornette, 2002. "Multi-Moments Method for Portfolio Management: Generalized Capital Asset Pricing Model in Homogeneous and Heterogeneous markets," Papers cond-mat/0207475, arXiv.org.
    • Handle: RePEc:arx:papers:cond-mat/0207475
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    References listed on IDEAS

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    Cited by:

    1. Malevergne, Y. & Sornette, D., 2007. "Self-consistent asset pricing models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 149-171.
    2. Y. Malevergne & V. F. Pisarenko & D. Sornette, 2003. "Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?," Papers physics/0305089, arXiv.org.
    3. Y. Malevergne & D. Sornette, 2003. "VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions," Papers physics/0301009, arXiv.org.
    4. Rockafellar, R. Tyrrell & Uryasev, Stan & Zabarankin, Michael, 2006. "Master funds in portfolio analysis with general deviation measures," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 743-778, February.
    5. Ba Chu, 2012. "Approximation of Asymmetric Multivariate Return Distributions," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 19(3), pages 293-318, September.

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