Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-30T05:41:13.597Z Has data issue: false hasContentIssue false
  • English
  • Français

On the use of Archimedean copulas for insurance modelling

Published online by Cambridge University Press:  17 June 2020

Thilini Dulanjali Kularatne ,
Jackie Li  and
David Pitt
Thilini Dulanjali Kularatne
Affiliation:
Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, New South Wales, Australia
Jackie Li*
Affiliation:
Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, New South Wales, Australia
David Pitt
Affiliation:
Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, New South Wales, Australia
*
*Corresponding author. E-mail: jackie.li@mq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we explore the use of an extensive list of Archimedean copulas in general and life insurance modelling. We consider not only the usual choices like the Clayton, Gumbel–Hougaard, and Frank copulas but also several others which have not drawn much attention in previous applications. First, we apply different copula functions to two general insurance data sets, co-modelling losses and allocated loss adjustment expenses, and also losses to building and contents. Second, we adopt these copulas for modelling the mortality trends of two neighbouring countries and calculate the market price of a mortality bond. Our results clearly show that the diversity of Archimedean copula structures gives much flexibility for modelling different kinds of data sets and that the copula and tail dependence assumption can have a significant impact on pricing and valuation. Moreover, we conduct a large simulation exercise to investigate further the caveats in copula selection. Finally, we examine a number of other estimation methods which have not been tested in previous insurance applications.

Keywords

Archimedean copulasTail dependenceLosses and allocated loss adjustment expensesMulti-population mortality modellingMortality bondEstimation methodsEmpirical copulas

Information

Type
Paper
Information
Annals of Actuarial Science , Volume 15 , Issue 1 , March 2021 , pp. 57 - 81
Copyright
© Institute and Faculty of Actuaries 2020

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.)

Article purchase

Temporarily unavailable

References

Abdallah, A., Boucher, J.P. & Cossette, H. (2015). Modeling dependence between loss triangles with hierarchical Archimedean copulas. ASTIN Bulletin, 45(3), 577–599.CrossRefGoogle Scholar
Brechmann, E.C. & Schepsmeier, U. (2013). Modeling dependence with C- and D-vine copulas: the R package CDVine. Journal of Statistical Software, 52(3), 1–27.CrossRefGoogle Scholar
Carriere, J.F. (2000). Bivariate survival models for coupled lives. Scandinavian Actuarial Journal, 2000(1), 17–32.CrossRefGoogle Scholar
Chen, H., MacMinn, R. & Sun, T. (2015). Multi-population mortality models: a factor copula approach. Insurance: Mathematics and Economics, 63, 135–146.Google Scholar
Chen, H., MacMinn, R.D. & Sun, T. (2017). Mortality dependence and longevity bond pricing: a dynamic factor copula mortality model with the GAS structure. Journal of Risk and Insurance, 84(S1), 393–415.CrossRefGoogle Scholar
Czado, C. (2019). Analyzing Dependent Data with Vine Copulas – A Practical Guide with R. Springer International Publishing, Springer Nature Switzerland AG.CrossRefGoogle Scholar
Denuit, M., Purcaru, O. & Van Keilegom, I. (2006). Bivariate Archimedean copula models for censored data in non-life insurance. Journal of Actuarial Practice, 13, 5–32.Google Scholar
Durante, F. & Sempi, C. (2015). Principles of Copula Theory. Chapman and Hall/CRC, Boca Raton.CrossRefGoogle Scholar
Eling, M. & Jung, K. (2018). Copula approaches for modeling cross-sectional dependence of data breach losses. Insurance: Mathematics and Economics, 82, 1676–180.Google Scholar
Embrechts, P., Lindskog, F. & McNeil, A. (2001). Modelling Dependence with Copulas and Applications to Risk Management. Department of Mathematics, ETHZ. Available online at the address https://people.math.ethz.ch/~embrecht/ftp/copchapter.pdfGoogle Scholar
Erhardt, V. & Czado, C. (2012). Modeling dependent yearly claim totals including zero claims in private health insurance. Scandinavian Actuarial Journal, 2012(2), 106–129.CrossRefGoogle Scholar
Fernandez, C. & Steel, M.F.J. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93(441), 359–371.Google Scholar
Frees, E.W. & Valdez, E.A. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), 1–25.CrossRefGoogle Scholar
Frees, E.W., Carriere, J. & Valdez, E. (1996). Annuity valuation with dependent mortality. Journal of Risk and Insurance, 63(2), 229–261.CrossRefGoogle Scholar
Gaillardetz, P. & Lin, X.S. (2006). Valuation of equity-linked insurance and annuity products with binomial models. North American Actuarial Journal, 10(4), 117–144.CrossRefGoogle Scholar
Genest, C., Nešlehová, J. & Quessy, J.F. (2012). Tests of symmetry for bivariate copulas. Annals of the Institute of Statistical Mathematics, 64, 811–834.CrossRefGoogle Scholar
Genest, C., Rémillard, B. & Beaudoin, D. (2009). Goodness-of-fit tests for copulas: a review and a power study. Insurance: Mathematics and Economics, 44, 199–213.Google Scholar
Genest, C. & Rivest, L.P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88(423), 1034–1043.CrossRefGoogle Scholar
Gourieroux, C. & Lu, Y. (2015). Love and death: a freund model with frailty. Insurance: Mathematics and Economics, 63, 191–203.Google Scholar
Guégan, D. & Ladoucette, S.A. (2004). Dependence modelling of the joint extremes in a portfolio using Archimedean copulas: application to MSCI indices. Institutions et Dynamiques Historiques de l’Economie (IDHE). Available online at the address https://halshs.archives-ouvertes.fr/halshs-00189214/documentGoogle Scholar
Haug, S., Klüppelberg, C. & Peng, L. (2011). Statistical models and methods for dependence in insurance data. Journal of the Korean Statistical Society, 40, 125–139.CrossRefGoogle Scholar
Human Mortality Database (HMD). (2018). University of California, Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). Available online at the address www.mortality.orgGoogle Scholar
Joe, H. (2014). Dependence Modeling with Copulas. Chapman and Hall/CRC, New York.CrossRefGoogle Scholar
Kaishev, V.K., Dimitrova, D.S. & Haberman, S. (2007). Modelling the joint distribution of competing risks survival times using copula functions. Insurance: Mathematics and Economics, 41, 339–361.Google Scholar
Klugman, S.A. & Parsa, R. (1999). Fitting bivariate loss distributions with copulas. Insurance: Mathematics and Economics, 24, 139–148.Google Scholar
Kupiec, P.H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73–84.CrossRefGoogle Scholar
Li, H. & Lu, Y. (2019). Modeling cause-of-death mortality using hierarchical Archimedean copula. Scandinavian Actuarial Journal, 2019(3), 247–272.CrossRefGoogle Scholar
Li, J., Li, J.S.H., Tan, C.I. & Tickle, L. (2018). Assessing basis risk in index-based longevity swap transactions. Annals of Actuarial Science, 13(1), 166–197.CrossRefGoogle Scholar
Luciano, E., Spreeuw, J. & Vigna, E. (2008). Modelling stochastic mortality for dependent lives. Insurance: Mathematics and Economics, 43, 234–244.Google Scholar
Nelsen, R.B. (1999). An Introduction to Copulas. Springer-Verlag, New York.CrossRefGoogle Scholar
Okhrin, O. & Ristig, A. (2014). Hierarchical Archimedean copulae: the HAC package. Journal of Statistical Software, 58(4), 1–20.CrossRefGoogle Scholar
Pešta, M. & Okhrin, O. (2014). Conditional least squares and copulae in claims reserving for a single line of business. Insurance: Mathematics and Economics, 56, 28–37.Google Scholar
Peters, G.W., Dong, A.X.D. & Kohn, R. (2014). A copula based Bayesian approach for paid-incurred claims models for non-life insurance reserving. Insurance: Mathematics and Economics, 59, 258–278.Google Scholar
Pfeifer, D., Mändle, A., Ragulina, O. & Girschig, C. (2019). New copulas based on general partitions-of-unity (part III) – the continuous case. Dependence Modeling, 7(1), 181–201.CrossRefGoogle Scholar
Savelli, N. & Clemente, G.P. (2011). Hierarchical structures in the aggregation of premium risk for insurance underwriting. Scandinavian Actuarial Journal, 2011(3), 193–213.CrossRefGoogle Scholar
Segers, J., Sibuya, M. & Tsukahara, H. (2017). The empirical beta copula. Journal of Multivariate Analysis, 155(C), 35–51.CrossRefGoogle Scholar
Shi, P., Feng, X. & Ivantsova, A. (2015). Dependent frequency-severity modeling of insurance claims. Insurance: Mathematics and Economics, 64, 417–428.Google Scholar
Shi, P. & Frees, E.W. (2010). Long-tail longitudinal modeling of insurance company expenses. Insurance: Mathematics and Economics, 47, 303–314.Google Scholar
Shi, P. & Valdez, E.A. (2014). Multivariate negative binomial models for insurance claim counts. Insurance: Mathematics and Economics, 55, 18–29.Google Scholar
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Institut Statistique de l’Université de Paris, 8, 229–231.Google Scholar
Spreeuw, J. (2006). Types of dependence and time-dependent association between two lifetimes in single parameter copula models. Scandinavian Actuarial Journal, 2006(5), 286–309.CrossRefGoogle Scholar
Staudt, A. (2010). Tail Risk, Systemic Risk, and Copulas. Casualty Actuarial Society E-Forum, Volume 2, Fall 2010.Google Scholar
Sun, J., Frees, E.W. & Rosenberg, M.A. (2008). Heavy-tailed longitudinal data modeling using copulas. Insurance: Mathematics and Economics, 42, 817–830.Google Scholar
Tsay, R.S. (2002). Analysis of Financial Time Series. John Wiley & Sons, New York.CrossRefGoogle Scholar
Venter, G.G. (2002). Tails of Copulas. Casualty Actuarial Society Proceedings, Volume LXXXIX, Numbers 170 & 171.Google Scholar
Villegas, A.M., Haberman, S., Kaishev, V.K. & Millossovich, P. (2017). A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bulletin, 47(3), 631–679.CrossRefGoogle Scholar
Wang, C.W., Yang, S.S. & Huang, H.C. (2015). Modeling multi-country mortality experience and its application in pricing survivor index swaps – a dynamic copula approach. Insurance: Mathematics and Economics, 63, 30–39.Google Scholar
Wang, S.S. (2000). A class of distortion operations for pricing financial and insurance risks. Journal of Risk and Insurance, 67(1), 15–36.CrossRefGoogle Scholar
Wang, S.S. (2004). Cat bond pricing using probability transforms. Geneva Papers on Risk and Insurance – Issues and Practice (Special Issue).Google Scholar
Zhang, Y. & Dukic, V. (2013). Predicting multivariate insurance loss payments under the Bayesian copula framework. Journal of Risk and Insurance, 80(4), 891–919.CrossRefGoogle Scholar
Zhao, X. & Zhou, X. (2010). Applying copula models to individual claim loss reserving methods. Insurance: Mathematics and Economics, 46, 290–299.Google Scholar
Zhao, X. & Zhou, X. (2012a). Copula models for insurance claim numbers with excess zeros and time-dependence. Insurance: Mathematics and Economics, 50, 191–199.Google Scholar
Zhao, X. & Zhou, X. (2012b). Estimation of medical costs by copula models with dynamic change of health status. Insurance: Mathematics and Economics, 51, 480–491.Google Scholar