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

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Author Info

  • 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''.

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File URL: http://arxiv.org/pdf/cond-mat/0103020
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number cond-mat/0103020.

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Date of creation: Mar 2001
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Publication status: Published in 18th International Conference in Finance, Namur - Belgium, 26, 27 & 28 JUNE 2001
Handle: RePEc:arx:papers:cond-mat/0103020

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Web page: http://arxiv.org/

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References

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  1. 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.
  2. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
  3. 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, vol. 3(2), pages 139-140, July.
  4. Didier Sornette, 1998. "Large deviations and portfolio optimization," Papers cond-mat/9802059, arXiv.org, revised Jun 1998.
  5. Sornette, Didier, 1998. "Large deviations and portfolio optimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 256(1), pages 251-283.
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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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