Modeling Insurance Claim Distribution via Mixture Distribution and Copula

Document Type : Research Paper


1 Ph.D. Candidate of Finance, University of Tehran, Tehran, Iran

2 Prof. of Finance, University of Tehran, Tehran, Iran

3 Associate Prof. of Finance, University of Tehran, Tehran, Iran


This paper analyses whether joint probability distribution function of losses due to different exposures covered under the same policy could be modeled in an appropriate manner via mixture distribution proposed and copula concept.
Special type of distribution which is a mixture of Generalized Hyperbolic Skew t distribution and Extreme Value theory (EVT) has been used for modeling marginal distributions of claims and copula function has been considered as a means of modeling dependency structure among claims. Most important copula including; Gaussian, t, Frank, Gumbel and Clayton was tested from goodness of fit point of view.
The data used in this study are the amount of property damage and bodily injury covered under automobile liability insurance.
Results reveal that joint probability distribution of claims could be effectively modeled by Clayton copula and proposed mixture distribution.


Main Subjects

Aas, K., Haff, I. (2006). The Generalized Hyperbolic Skew Student’s t-Distribution. Journal of Financial Econometrics 4(2), 275–309.
Anderson, T.W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley & Sons, New York.
Balkema, A. A., de Haan, L. (1974). Residual Life Time at Great Age. Annals of Probability, 2(5), 792-804.
Bassi, F., Embrechts, P., Kafetzaki, M. (1998). Risk management and Quantile Estimation. In A Practical Guide to Heavy Tails, Adler, R. J., Feldman, F., and Taqqu, M. (eds), 111–130. Birkhäuser.
Beirlant, J., Joossens, E., Segers, J. (2004). Generalized Pareto Fit to the Society of Actuaries’ Large Claims Database. North American Actuarial Journal 8(2), 108–111.
Belguise, O., Levy, C. (2003). Tempêtes : Etude des dépendances entre les branches Automobile et Incendie à l’aide de la théorie des copulas Topic 1 Risk evaluation. 2003 ASTIN Colloquium, Berlin, Germany. Available in:
Bouyè, E. (2002). Multivariate Extremes at Work for Portfolio Risk Measurement. Warwick Business School Working Paper Series, WP01-10.
Chava, S., Stefanescu, C., Turnbull, S. (2008). Modeling the Loss Distribution. Working Paper. Available in: Stefanescu_Turnbull.pdf.
Cherubini, U., Lucianco, E., Vecchiato, W. (2004). Copula Methods in Finance. John Wiley & Sons, Sussex, England.
Debbie, J, D., Jones, B, L. (2006). Multivariate Extreme Value Theory and Its Usefulness in Understanding Risk. North American Acutarial Jouranl, 10(4), 1-27.
Eling, E. (2012). Fitting Insurance Claims to Skewed Distributions: Are the Skew-Normal and Skew-Student Good Models? Insurance: Mathematics and Economics, 51(2), 239-248.
Embrechts, P., Resnick, S. I., Samorodnitsky, G. (1999). Extreme Value Theory as a Risk Management Tool. North American Actuarial Journal 3(2), 30–41.
Frees, E.W., Carrière, J.F., Valdez, E. (1996). Annuity Valuation with Dependent Mortality. Journal of Risk and Insurance, 63(2), 229–261.
Frees, E.W., Valdez, E. A. (1998). Understanding Relationships Using Copulas. North American Actuarial Journal 2(1), 1–25.
Johnson, N. L., Kotz, S., Balakrishnan, N. (1997). Discrete Multivariate Distributions. John Wiley & Sons, New York.
Johnson, R. A., Wichern, D. W. (1988). Applied Multivariate Statistical Analysis (2 ed). Prentice-Hall Inc, New Jersey.
Krzanowski, W. J. (1988). Principles of Multivariate Analysis: a user’s perspective. Oxford University Press, Oxford.
Lane, M.N. (2000). Pricing Risk Transfer Transactions. ASTIN Bulletin, 30 (2), 259-293.
Lee, W. C., Fang, C. J. (2010). The Measurement of Capital for Operational |Risk of Taiwanese Commercial Banks. The Journal of Operational Risk 5(2), 79-102.
McNeil, A.J., Frey, R., Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton Series in Finance, Princeton University Press. New Jersey.
McNeil, A. J., Saladin, T. (1997).The Peaks Over Thresholds Method for Estimating High Quantiles of Loss Distributions. Preprint, Department Mathematik, ETH Zentrum, Zurich.
Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.
Pickands, J. I. (1975). Statistical Inference Using Extreme Value Order Statistics. Annals of Statististics, 3,119-131.
Venter, G.G. (2002). Tails of Copulas. Proceedings of the Casualty Actuarial Society, 89(1), 68–113.
Vernic, R. (2006). Multivariate Skew-Normal Distributions with Applications in Insurance. Insurance: Mathematics and Economics, 38(2), 413-426.