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Asymptotic Value-at-Risk Estimates for Sums of Dependent Random Variables

Published online by Cambridge University Press:  17 April 2015

Mario V. Wüthrich
Mario V. Wüthrich*
Affiliation:
Winterthur Insurance, Römerstrasse 17, P.O. Box 357, CH-8401 Winterthur, Switzerland, mario.wuethrich@winterthur.ch
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Abstract

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We estimate Value-at-Risk for sums of dependent random variables. We model multivariate dependent random variables using archimedean copulas. This structure allows one to calculate the asymptotic behaviour of extremal events. An important application of such results are Value-at-Risk estimates for sums of dependent random variables.

Keywords

Archimedean CopulaDependent RisksValue-at-RiskExtreme Value TheoryPickands-Balkema-de Haan TheoremMultivariate Extremes
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 1 , May 2003 , pp. 75 - 92
Copyright
Copyright © ASTIN Bulletin 2003

References

[1] Bäuerle, N. and Müller, A. (1998) Modelling and comparing dependencies in multivariate risk portfolios. Astin Bulletin 28, 1, 59–76.CrossRefGoogle Scholar
[2] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[3] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling extremal events for insurance and finance. Springer, Berlin.Google Scholar
[4] Embrechts, P., Mcneil, A. and Straumann, D. (2002) Correlation and Dependency in Risk Management: Properties and Pitfalls. In Risk Management: Value at Risk and Beyond, Dempster, M. (Ed.), Cambridge University Press, Cambridge, 176–223.CrossRefGoogle Scholar
[5] Frees, W.E. and Valdez, E.A. (1998) Understanding Relationships Using Copulas. North American Actuarial Journal 2, 1–25.CrossRefGoogle Scholar
[6] Hürlimann, W. (2002) Fitting bivariate cumulative returns with copulas. Proceedings of the 33rd International Astin Colloquium, Cancun, Mexico.Google Scholar
[7] Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
[8] Juri, A. and Wüthrich, M.V. (2002) Copula convergence theorems for tail events. Insurance: Math. Econom. 30, 405–420.Google Scholar
[9] Juri, A. and WÜthrich, M.V. (2002) Tail dependence from a distributional point of view. Preprint.Google Scholar
[10] Kimberling, C. (1974) A probabilistic interpretation of complete monotonicity. Aequationes Mathematicae 10, 152–164.CrossRefGoogle Scholar
[11] Klugman, S.A. and Prasa, R. (1999) Fitting bivariate loss distributions with copulas. Insurance: Math. Econom. 24, 139–148.Google Scholar
[12] Kremer, E. (1998) Largest claims reinsurance premiums under possible claims dependence. Astin Bulletin 28/2, 257–267.CrossRefGoogle Scholar
[13] Frey, R. and Mcneil, A.J. (2001) Modelling dependent defaults, Preprint.Google Scholar
[14] Nelsen, R.B. (1999) An introduction to copulas. Springer, New York.CrossRefGoogle Scholar
[15] Schweizer, B. and Sklar, A. (1983) Probabilistic Metric Spaces, North-Holland, New York.Google Scholar
[16] Sklar, A. (1959) Fonctions de répartition aux dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8, 229–231.Google Scholar