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PATHS AND INDICES OF MAXIMAL TAIL DEPENDENCE

Published online by Cambridge University Press:  19 June 2015

Edward Furman ,
Jianxi Su  and
Ričardas Zitikis
Edward Furman*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
Jianxi Su
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada E-Mail: gavinsox@mathstat.yorku.ca
Ričardas Zitikis
Affiliation:
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada E-Mail: zitikis@stats.uwo.ca
*
E-Mail: efurman@mathstat.yorku.ca
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Abstract

We demonstrate both analytically and numerically that the existing methods for measuring tail dependence in copulas may sometimes underestimate the extent of extreme co-movements of dependent risks and, therefore, may not always comply with the new paradigm of prudent risk management. This phenomenon holds in the context of both symmetric and asymmetric copulas with and without singularities. As a remedy, we introduce a notion of paths of maximal (tail) dependence and utilize the notion to propose several new indices of tail dependence. The suggested new indices are conservative, conform with the basic concepts of modern quantitative risk management, and are capable of differentiating between distinct risky positions in situations when the existing indices fail to do so.

Keywords

Multivariate distributioncopulatail dependencemaximal dependencefatal shockmultivariate Paretoenterprise risk management
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 661 - 678
Copyright
Copyright © Astin Bulletin 2015 

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References

Asimit, A.V. and Badescu, A.L. (2010) Extremes on the discounted aggregate claims in a time dependent risk model. Scandinavian Actuarial Journal, 2010, 93–104.CrossRefGoogle Scholar
Asimit, A.V., Furman, E., Tang, Q. and Vernic, R. (2011) Asymptotics for risk capital allocations based on conditional tail expectation. Insurance: Mathematics and Economics, 49, 310–324.Google Scholar
Asimit, A.V., Furman, E. and Vernic, R. (2010) On a multivariate Pareto distribution. Insurance: Mathematics and Economics, 46 (2), 308–316.Google Scholar
Asimit, A.V. and Jones, B.L. (2008) Dependence and the asymptotic behavior of large claims reinsurance. Insurance: Mathematics and Economics, 43, 407–411.Google Scholar
Balakrishnan, N. and Lai, C.D. (2009) Continuous Bivariate Distributions, 2nd ed.New York: Springer.Google Scholar
Biener, C. and Eling, M. (2012) Insurability in microinsurance markets: an analysis of problems and potential solutions. The Geneva Papers, 37, 77–107.Google Scholar
Churchill, C. and Matul, M. (2012) Protecting the Poor: A Microinsurance Compendium, vol. II. Geneva, Switzerland: International Labour Organization.Google Scholar
Coles, S., Heffernan, J. and Tawn, J. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339–365.CrossRefGoogle Scholar
Dharmadhikari, S. and Joag-dev, K. (1987) Unimodality, Convexity and Applications. New York: Academic Press.Google Scholar
Durante, F., Fernández-Sánchez, J. and Pappadà, R. (2015) Copulas, diagonals, and tail dependence. Fuzzy Sets and Systems, 264, 22–41.CrossRefGoogle Scholar
Durante, F., Pappadà, R. and Torelli, N. (2014) Clustering of financial time series in risky scenarios. Advances in Data Analysis and Classification, 8, 359–376.CrossRefGoogle Scholar
Embrechts, P. (2009) Copulas: A personal view. Journal of Risk and Insurance, 76, 639–650.CrossRefGoogle Scholar
Embrechts, P., Lindskog, F. and McNeil, A. (2003) Modeling dependence with copulas and applications to risk management. In Handbook of Heavy Tailed Distributions in Finance (ed. Rachev, S.) Chapter 8, pp. 329–384. Amsterdam: Elsevier.CrossRefGoogle Scholar
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013) Model uncertainty and VaR aggregation. Journal of Banking and Finance, 37, 2750–2764.CrossRefGoogle Scholar
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014) An academic response to Basel 3.5. Risks, 2, 25–48.CrossRefGoogle Scholar
Fischer, M.J. and Klein, I. (2007) Some results on weak and strong tail dependence coefficients for means of copulas. Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie, No. 78/2007.Google Scholar
Frees, E.W. (2010) Regression Modeling with Actuarial and Financial Applications. New York: Cambridge University Press.Google Scholar
Frees, E.W., Carriere, J. and Valdez, E. (1996) Annuity valuation with dependent mortality. Journal of Risk and Insurance, 63, 229–261.CrossRefGoogle Scholar
Frees, E.W. and Wang, P. (2005) Credibility using copulas. North American Actuarial Journal, 9 (2), 31–48.CrossRefGoogle Scholar
Furman, E., Kuznetsov, A., Su, J. and Zitikis, R. (2015) The Maximal Tail Dependence Path of the Gaussian Copula is Diagonal. Technical report available at SSRN: http://ssrn.com/abstract=2558675. Accessed on June 10, 2015.Google Scholar
Furman, E. and Zitikis, R. (2010) General Stein-type covariance decompositions with applications to insurance and finance. ASTIN Bulletin, 40, 369–375.CrossRefGoogle Scholar
Heffernan, J.E. (2000) A directory of coefficients of tail dependence. Extremes, 3, 279–290.CrossRefGoogle Scholar
Hua, L. and Joe, H. (2014) Strength of tail dependence based on conditional tail expectation. Journal of Multivariate Analysis, 123, 143–159.CrossRefGoogle Scholar
Jaworski, P., Durante, F. and Härdle, W. (2013) Copulae in Mathematical and Quantitative Finance. Berlin: Springer.CrossRefGoogle Scholar
Jaworski, P., Durante, F., Härdle, W. and Rychlik, T. (2010) Copulas Theory and its Applications. Berlin: Springer.CrossRefGoogle Scholar
Joe, H., Li, H. and Nikoloulopoulos, A.K. (2010) Tail dependence functions and vine copulas. Journal of Multivariate Analysis, 101, 252–270.CrossRefGoogle Scholar
Ledford, A.W. and Tawn, J.A. (1996) Statistics for near independence in multivariate extreme values. Biometrika, 83, 169–187.CrossRefGoogle Scholar
Li, H. and Wu, P. (2013) Extremal dependence of copulas: A tail density approach. Journal of Multivariate Analysis, 144, 99–111.CrossRefGoogle Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management. Princeton: Princeton University Press.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas, 2nd ed.New York: Springer.Google Scholar
Nikoloulopoulos, A.K. and Karlis, D. (2008) Multivariate logit copula model with an application to dental data. Statistics in Medicine, 27, 6393–6406.CrossRefGoogle ScholarPubMed
ORSA (2014). Own Risk and Solvency Assessment. Office of the Superintendent of Financial Institutions, Government of Canada, Ottawa. http://www.osfi-bsif.gc.ca/eng/fi-if/rg-ro/gdn-ort/gl-ld/Pages/e19.aspx. Accessed on March 17, 2015.Google Scholar
Patton, A.J. (2012) A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18.CrossRefGoogle Scholar
Puccetti, G. and Rüschendorf, L. (2014) Asymptotic equivalence of conservative value-at-risk- and expected shortfall-based capital charges. Journal of Risk, 16, 3–22.CrossRefGoogle Scholar
van de Geer, S.A. (2009) Empirical Processes in M-Estimation. Cambridge: Cambridge University Press.Google Scholar
van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. With Applications to Statistics. New York: Springer.CrossRefGoogle Scholar
Weng, C. and Zhang, Y. (2012) Characterization of multivariate heavy-tailed distribution families via copula. Journal of Multivariate Analysis, 106, 178–186.CrossRefGoogle Scholar