Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:32:09.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Calculating and communicating tail association and the risk of extreme loss

Published online by Cambridge University Press:  06 December 2012

Paul Sweeting  and
Fotis Fotiou
Paul Sweeting*
Affiliation:
University of Kent and J.P. Morgan Asset Management
Fotis Fotiou
Affiliation:
University of Kent
*
*Correspondence to: Paul Sweeting, J.P. Morgan, Asset Management, Finsbury Dials, 20 Finsbury Street, London, EC2Y 9AQ. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we examine two aspects of extreme events: their calculation and their communication. In relation to calculation, there are two types of extreme events that are considered.

The first is the extent to which extreme events in two or more variables occur together. This can be gauged by using measures of tail association. Higher levels of tail association are useful for highlighting the extent to which there are concentrations of risk. We investigate the range of approaches used to measure tail association and propose a pragmatic alternative, the coefficient of finite tail dependence.

The second type of extreme event arises from combinations of losses from a series of risks that together result in total losses exceeding a particular level. This is measured using ruin lines or, in higher dimensions, planes and hyperplanes. The probability of ruin and the economic cost of ruin are considered here. In this context, it is important to consider what the term “loss” actually means, and whether it is in relation to a current set of exposures or a potential strategy.

The communication of extreme events is discussed not just in terms of the numbers that can be used, but in terms of the graphical methods that can be used to aggregate information on a range of risk combinations. This involves communicating not just the level of risk but also the importance of the risk considered.

Keywords

CopulasCorrelationDependenceTail associationExtreme Value TheoryProbability of Ruin
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 18 , Issue 1 , March 2013 , pp. 13 - 72
Copyright
Copyright © Institute and Faculty of Actuaries 2012 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Behboodian, J., Dolati, A., Úbeda-Flores, M. (2007). A Multivariate Version of Gini's Rank Association Coefficient. Statistical Papers, 48, 295–304.Google Scholar
Blomqvist, N. (1950). On a measure of dependence between two random variables. Annals of Mathematical Statistics, 21, 593–600.CrossRefGoogle Scholar
Boyer, B.H., Gibson, M.S., Lauretan, M. (1997). Pitfalls in tests for changes in correlations. International Finance Discussion Paper 597 from the Board of the Governors of the Federal Reserve System.Google Scholar
Capéraà, P., Fougères, A.-L., Genest, C. (1997). A Nonparametric Estimation Procedure for Bivariate Extreme Value Copulas. Biometrika, 84(3), 567–577.Google Scholar
Charpentier, A. (2003). Tail Distribution and Dependence Measures. Proceedings of the 43 rdASTIN Conference. Available at: http://www.actuaries.org/ASTIN/Colloquia/Berlin/Charpentier.pdfGoogle Scholar
Coles, S., Heffernan, J., Tawn, J. (1999). Dependence Measures for Extreme Value Analyses. Extremes, 2(4), 339–365.Google Scholar
Deheuvels, P. (1991). On the Limiting Behavior of the Pickands Estimator for Bivariate Extreme-Value Distributions. Statistics and Probability Letters, 12, 429–439.Google Scholar
Dolati, A., Úbeda-Flores, M. (2006). On measures of multivariate concordance. Journal of Probability and Statistical Sciences, 4(2), 147–164.Google Scholar
Embrechts, P., Lindskog, F., McNeil, A.J. (2003). Modelling Dependence with Copulas and Applications to Risk Management, in The Handbook of Heavy Tailed Distributions in Finance (ed. S.T. Rachev), Elsevier Science, Amsterdam.Google Scholar
Embrechts, P., McNeil, A.J., Strauman, D. (2002). Correlation and Dependency in Risk Management: Properties and Pitfalls, in Risk Management: Value at Risk and Beyond (ed. M. Dempster), Cambridge University Press, Cambridge.Google Scholar
Fischer, M., Dörflinger, M. (2006). A Note on a Non-Parametric Tail Dependence Estimator, discussion paper 76/2006, Friedrich-Alexander-University Erlangen-Nuremberg, downloaded from econpapers.repec.org.Google Scholar
Frahm, G., Junker, M., Schmidt, R. (2005). Estimating the Tail-Dependence Coefficient: Properties and Pitfalls. Insurance: Mathematics and Economics, 37(1), 80–100.Google Scholar
Haug, S., Klüppelberg, C., Kuhn, G. (2009). Dimension reduction based on extreme dependence, draft downloaded from tum.de.Google Scholar
Hult, H., Lindskog, F. (2002). Multivariate Extremes, Aggregation and Dependence in Elliptical Distributions. Advances in Applied Probability, 34, 587–608.Google Scholar
Joe, H. (1990). Multivariate concordance. Journal of Multivariate Analysis, 35(1), 12–30.Google Scholar
Juri, A., Wüthrich, M.V. (2003). Tail Dependence from a Distributional Point of View. Extremes, 6, 213–246.CrossRefGoogle Scholar
Khoudraji, A. (1995). Contributions à la Théorie des Copules et à la Modélisation de Données aux Valeurs Extrêmes”, unpublished doctoral dissertation, Université Laval.Google Scholar
Küppelberg, C., Kuhn, G., Peng, L. (2008). Semi-Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case. Scandinavian Journal of Statistics, 35(4), 701–718.Google Scholar
Malevergne, Y., Sornette, D. (2006). Extreme Financial Risks: From Dependence to Risk Management. Springer-Verlag, Berlin Heidelberg.Google Scholar
McNeil, A.J., Frey, R., Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, New Jersey.Google Scholar
Picklands, J. (1981). Multivariate extreme value distributions. Proceedings of the 43 rdSession of the International Statistical Institute, 859–878.Google Scholar
Ruymgaart, F.H., van Zuijlen, M.C.A. (1978). Asymptotic Normality of Multivariate Linear Rank Statistics in the Non-i.i.d. Case. Annals of Statistics, 6(3), 588–602.Google Scholar
Scarsini, M. (1984). On Measures of Concordance. Stochastica, 8, 201–218.Google Scholar
Schmid, F., Schmidt, R. (2007). Multivariate Conditional Versions of Spearman's Rho and Related Measures of Tail Dependence. Journal of Multivariate Analysis, 98, 1123–1140.CrossRefGoogle Scholar
Schmidt, R., Stadtmüller, U. (2006). Nonparametric Estimation of Tail Dependence. Scandinavian Journal of Statistics, 33, 307–335.Google Scholar
Shaw, R., Smith, A.D., Spivak, G. (2010). Calibration and Communication of Dependencies with a Case Study based on Market Returns, paper presented to the 2010 UK Actuarial Profession GIRO Conference. Available at: http://www.actuaries.org.uk/research-and-resources/documents/a04-calibrating-dependenciesGoogle Scholar
Sibuya, M. (1960). Bivariate Extreme Statistics. Annals of the Institute of Statistical Mathematics, 11, 195–210.Google Scholar
Sklar, A. (1959). Fonctions de Répartition à n Dimensions et Leurs Marges. Publications de l'Institut de Statistique de l'Université de Paris, 8, 229–231.Google Scholar
Taylor, M.D. (2007). Multivariate Measures of Concordance. Annals of the Institute of Statistical Mathematics, 59(4), 789–806.CrossRefGoogle Scholar
Úbeda Flores, M. (2005). Multivariate Versions of Blomqvist's beta and Spearman's footrule. Annals of the Institute of Statistical Mathematics, 57, 781–788.Google Scholar
Venter, G.G. (2002). Tails of Copulas. Proceedings of the Casualty Actuarial Society, 89, 68–113.Google Scholar
Wolff, E.F. (1980). N-Dimensional Measures of Dependence. Stochastica, 4(3), 175–188.Google Scholar