Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T11:28:36.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Modelling Dependence in Insurance Claims Processes with Lévy Copulas

Published online by Cambridge University Press:  09 August 2013

Benjamin Avanzi ,
Luke C. Cassar  and
Bernard Wong
Benjamin Avanzi
Affiliation:
School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney NSW 2052, Australia, E-mail: [email protected]
Luke C. Cassar
Affiliation:
School of Risk and Actuarial Studies, Australian School of Business, University of New South Wales, Sydney NSW 2052, Australia, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the potential of Lévy copulas as a tool for modelling dependence between compound Poisson processes and their applications in insurance. We analyse characteristics regarding the dependence in frequency and dependence in severity allowed by various Lévy copula models. Through the introduction of new Lévy copulas and comparison with the Clayton Lévy copula, we show that Lévy copulas allow for a great range of dependence structures.

Procedures for analysing the fit of Lévy copula models are illustrated by fitting a number of Lévy copulas to a set of real data from Swiss workers compensation insurance. How to assess the fit of these models with respect to the dependence structure exhibited by the dataset is also discussed.

Finally, we provide a decomposition of the trivariate compound Poisson process and discuss how trivariate Lévy copulas model dependence in this multivariate setting.

Keywords

Lévy copulaDependenceCompound Poisson processInsuranceReal data
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 2 , November 2011 , pp. 575 - 609
Copyright
Copyright © International Actuarial Association 2011

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

Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities. World Scientific, Singapore, 2nd edition.Google Scholar
Bargès, M., Cossette, H. and Marceau, É. (2009) TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics, 45(3), 348–361.Google Scholar
Barndorff-Nielsen, O.E. and Lindner, A.M. (2007) Lévy Copulas: Dynamics and Transforms of Upsilon Type. Scandinavian Journal of Statistics, 34, 298–316.Google Scholar
Bäuerle, N. and Blatter, A. (2011) Optimal control and dependence modeling of insurance portfolios with Lévy dynamics. Insurance: Mathematics and Economics, 48(3), 398–405.Google Scholar
Bäuerle, N. and Blatter, A. and Müller, A. (2008) Dependence properties and comparison results for Lévy processes. Mathematical Methods of Operations Research, 67, 161–186.Google Scholar
Bäuerle, N. and Grübel, R. (2005) Multivariate counting processes: copulas and beyond. ASTIN Bulletin, 35(2), 379–408.Google Scholar
Biagini, F. and Ulmer, S. (2009) Asymptotics for operational risk quantii ed with expected shortfall. ASTIN Bulletin, 39(2), 735–752.Google Scholar
Böcker, K. and Klüppelberg, C. (2008) Modelling and Measuring Multivariate Operational Risk with Lévy Copulas. Journal of Operational Risk, 3, 3–27.Google Scholar
Böcker, K. and Klüppelberg, C. (2010) Multivariate models for operational risk. Quantitative Finance, 1–15.Google Scholar
Bowers, N.L.J., Gerber, H.U., Hickman, J.C, Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics. The Society of Actuaries, Schaumburg, Illinois, 2nd edition.Google Scholar
Bregman, Y. and Klüppelberg, C. (2005) Ruin estimation in multivariate models with Clayton dependence structure. Scandinavian Actuarial Journal, 2005(6), 462–480.Google Scholar
Cont, R. and Tankov, P. (2004) Financial Modelling With Jump Processes. Chapman & Hall/CRC, London.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kass, R. (2005) Actuarial Theory for Dependent Risks. John Wiley & Sons, Inc., West Sussex.Google Scholar
Eder, I. and Klüppelberg, C. (2009) The quintuple law for sums of dependent Lévy processes. The Annals of Applied Probability, 19(6), 2047–2079.Google Scholar
Embrechts, P., McNeil, A.J. and Straumann, D. (2002) Correlation and Dependence in Risk Management: Properties and Pitfalls. In Risk management: value at risk and beyond (ed. Dempster, M.), 176–223. Cambridge University Press, Cambridge.Google Scholar
Esmaeili, H. and Klüppelberg, C. (2010a) Parameter estimation of a bivariate compound Poisson process. Insurance: Mathematics and Economics, 47(2), 224–233.Google Scholar
Esmaeili, H. and Klüppelberg, C. (2010b) Two-step estimation of a multivariate Lévy process. Available at http://www-m4.ma.tum.de/Papers/.Google Scholar
Fosker, P., Scanlon, M. and Simpson, E. (2010) Insurance ERM advances: Global leaders point way for regional players. URL: http://www.towerswatson.com/assets/pdf/3372/1210-ERM.pdf Google Scholar
Genest, C. and Nešlehová, J. (2007) A primer on copulas for count data. Astin Bulletin, 37(2), 475–515. ISSN 0515-0361.Google Scholar
Genest, C, Rémillard, B. and Beaudoin, D. (2009) Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199–213.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Kallsen, J. and Tankov, P. (2006) Characterisation of dependence of multidimensional Lévy processes using Lévy copulas. Journal of Multivariate Analysis, 97(7), 1551–1572.Google Scholar
Klugman, S. and Rioux, J. (2006) Toward a Unified Approach to Fitting Loss Models. North American Actuarial Journal, 10(1), 63–83.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss Models: From Data to Decisions. John Wiley and Sons, Hoboken, New Jersey.CrossRefGoogle Scholar
Lindskog, F. and McNeil, A.J. (2003) Common Poisson shock models: Applications to insurance and credit risk modelling. ASTIN Bulletin, 33(2), 209–238.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.Google Scholar
Mikosch, T. (2009) Non-Life Insurance Mathematics: An Introduction with the Poisson Process. Springer, 2nd edition.Google Scholar
Nelsen, R.B. (1999) An Introduction to Copulas. Springer, New York.Google Scholar
Sato, K.-I. (1999) Levy Processes and Inifinitely Divisible Distributions. Cambridge University Press.Google Scholar
Tankov, P. (2003) Dependence structure of spectrally positive multidimensional Lévy processes. Available at http://www.math.jussieu.fr/tankov/.Google Scholar
Yuen, K.C. and Wang, G. (2002) Comparing Two Models with Dependent Classes of Business. ARCH, Society of Actuaries, 22.Google Scholar