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General framework for a portfolio theory with non-Gaussian risks and non-linear correlations

Author

Listed:
  • Y. Malevergne

    (Univ. Nice/CNRS)

  • D. Sornette

    (Univ. Nice/CNRS and UCLA)

Abstract

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 give exact formulas for the tails and for the moments and cumulants of the distribution of returns of a portfolio make of arbitrary compositions 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. We treat in details the problem of risk minimization using two different measures of risks (cumulants and value-at-risk) for a portfolio made of two assets and compare the theoretical predictions with direct empirical data. While good agreement is found, the remaining discrepancy between theory and data stems from the deviations from the Weibull parameterization for small returns. 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, 2001. "General framework for a portfolio theory with non-Gaussian risks and non-linear correlations," Papers cond-mat/0103020, arXiv.org.
  • Handle: RePEc:arx:papers:cond-mat/0103020
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    File URL: http://arxiv.org/pdf/cond-mat/0103020
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    References listed on IDEAS

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    1. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    2. P. Gopikrishnan & M. Meyer & L.A.N. Amaral & H.E. Stanley, 1998. "Inverse cubic law for the distribution of stock price variations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 3(2), pages 139-140, July.
    3. J-F. Muzy & D. Sornette & J. delour & A. Arneodo, 2001. "Multifractal returns and hierarchical portfolio theory," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 131-148.
    4. Sornette, Didier, 1998. "Large deviations and portfolio optimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 256(1), pages 251-283.
    5. Didier Sornette, 1998. "Large deviations and portfolio optimization," Papers cond-mat/9802059, arXiv.org, revised Jun 1998.
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    Cited by:

    1. Davies, G.B. & Satchell, S.E., 2004. "Continuous Cumulative Prospect Theory and Individual Asset Allocation," Cambridge Working Papers in Economics 0467, Faculty of Economics, University of Cambridge.
    2. L. Ingber, 2006. "Statistical mechanics of neocortical interactions: Portfolio of physiological indicators," Lester Ingber Papers 06pp, Lester Ingber.

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