# VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

## Author Info

• Y. Malevergne

(Univ. Nice and Univ. Lyon)

• D. Sornette

(CNRS-Univ. Nice 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 multivariate generalizations with Gaussian copulas, we offer exact formulas for the tails of the distribution $P(S)$ of returns $S$ of a portfolio of arbitrary composition of these assets. We find that the tail of $P(S)$ is also asymptotically a modified Weibull distribution with a characteristic scale $\chi$ function of the asset weights with different functional forms depending on the super- or sub-exponential behavior of the marginals and on the strength of the dependence between the assets. We then treat in details the problem of risk minimization using the Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically) equivalent in this framework.

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File URL: http://arxiv.org/pdf/physics/0301009

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number physics/0301009.

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Publication status: Published in Quantitative Finance 4 (1), 17-36 (2003)
Handle: RePEc:arx:papers:physics/0301009

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Web page: http://arxiv.org/

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## References

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1. Y. Malevergne & D. Sornette, 2001. "Testing the Gaussian Copula Hypothesis for Financial Assets Dependences," Finance 0111003, EconWPA.
2. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
3. Bekaert, Geert & Wu, Guojun, 2000. "Asymmetric Volatility and Risk in Equity Markets," Review of Financial Studies, Society for Financial Studies, vol. 13(1), pages 1-42.
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.
6. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
7. A. Johansen & D. Sornette, 1998. "Stock market crashes are outliers," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 1(2), pages 141-143, January.
8. 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.
9. Con Keating & Hyun Song Shin & Charles Goodhart & Jon Danielsson, 2001. "An Academic Response to Basel II," FMG Special Papers sp130, Financial Markets Group.
10. 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.
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## Citations

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Cited by:
1. Saralees Nadarajah & Samuel Kotz, 2006. "The modified Weibull distribution for asset returns," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 449-449.
2. C. James Hueng & Ruey Yau, 2006. "Investor preferences and portfolio selection: is diversification an appropriate strategy?," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 255-271.
3. Ibragimov, Rustam & Walden, Johan, 2008. "Portfolio diversification under local and moderate deviations from power laws," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 594-599, April.
4. Vicente Medina Martínez & Ángel Pardo Tornero, 2012. "Stylized facts of CO2 returns," Working Papers. Serie AD 2012-14, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
5. Y. Malevergne & V. Pisarenko & D. Sornette, 2006. "The modified weibull distribution for asset returns: reply," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 451-451.
6. Walden, Johan & Ibragimov, Rustam, 2008. "Portfolio Diversification under Local and Moderate Deviations from Power Laws," Scholarly Articles 2640586, Harvard University Department of Economics.
7. Roger Bowden, 2006. "The generalized value at risk admissible set: constraint consistency and portfolio outcomes," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 159-171.

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