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Ordering of multivariate probability distributions with respect to extreme portfolio losses

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  • Georg Mainik
  • Ludger Ruschendorf

Abstract

A new notion of stochastic ordering is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of ordering conditions on the spectral measures, which allows for analytical or numerical verification in practical applications. Additional comparison criteria in terms of further stochastic orderings are derived. The application examples include worst case and best case scenarios, elliptically contoured distributions, and multivariate regularly varying models with Gumbel, Archimedean, and Galambos copulas.

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  • Georg Mainik & Ludger Ruschendorf, 2010. "Ordering of multivariate probability distributions with respect to extreme portfolio losses," Papers 1010.5171, arXiv.org.
    • Handle: RePEc:arx:papers:1010.5171
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    References listed on IDEAS

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    1. Alink, Stan & Lowe, Matthias & V. Wuthrich, Mario, 2004. "Diversification of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 77-95, August.
    2. Georg Mainik & Ludger Rüschendorf, 2010. "On optimal portfolio diversification with respect to extreme risks," Finance and Stochastics, Springer, vol. 14(4), pages 593-623, December.
    3. Alink, Stan & Löwe, Matthias & Wüthrich, Mario V., 2005. "Analysis of the Expected Shortfall of Aggregate Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 25-43, May.
    4. Shaked, Moshe & Shanthikumar, J. George, 1997. "Supermodular Stochastic Orders and Positive Dependence of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 86-101, April.
    5. Wei, Gang & Hu, Taizhong, 2002. "Supermodular dependence ordering on a class of multivariate copulas," Statistics & Probability Letters, Elsevier, vol. 57(4), pages 375-385, May.
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