Building Loss Models

Burnecki, Krzysztof and Janczura, Joanna and Weron, Rafal (2010): Building Loss Models.

Abstract

This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.

• Item Type: MPRA Paper
• Language: English
• Keywords: Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing
• Subjects: G - Financial Economics > G2 - Financial Institutions and Services > G22 - Insurance ; Insurance Companies ; Actuarial Studies; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; G - Financial Economics > G3 - Corporate Finance and Governance > G32 - Financing Policy ; Financial Risk and Risk Management ; Capital and Ownership Structure ; Value of Firms ; Goodwill

References

Ahrens, J. H. and Dieter, U. (1982). Computer generation of Poisson deviates from modified normal distributions, ACM Trans. Math. Soft. 8: 163–179.

Albrecht, P. (1982). On some statistical methods connected with the mixed Poisson process, Scandinavian Actuarial Journal: 1–14.

Bratley, P., Fox, B. L., and Schrage, L. E. (1987). A Guide to Simulation, Springer-Verlag, New York.

Burnecki, K., Hardle, W., and Weron, R. (2004). Simulation of Risk Processes, in J. Teugels, B. Sundt (eds.) Encyclopedia of Actuarial Science, Wiley, Chichester, 1564–1570.

Burnecki, K., Kukla, G., and Weron, R. (2000). Property insurance loss distributions, Physica A 287: 269-278.

Burnecki, K., and Weron R. (2005). Modeling of the Risk Process, in P. Cizek, W. Hardle, R. Weron (eds.) Statistical Tools for Finance and Insurance, Springer-Verlag, Berlin, 319–339.

Chernobai, A., Burnecki, K., Rachev, S. T., Trueck, S. and Weron, R. (2006). Modelling catastrophe claims with left-truncated severity distributions, Computational Statistics 21: 537-555.

Chernobai, A.S., Rachev, S.T., and Fabozzi, F.J. (2007). Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis, Wiley.

D’Agostino, R. B. and Stephens, M. A. (1986). Goodness-of-Fit Techniques, Marcel Dekker, New York.

Daykin, C.D., Pentikainen, T., and Pesonen, M. (1994). Practical Risk Theory for Actuaries, Chapman, London.

Devroye, L. (1986). Non-Uniform Random Variate Generation, Springer-Verlag, New York.

Embrechts, P. and Kluppelberg, C. (1993). Some aspects of insurance mathematics, Theory Probab. Appl. 38: 262–295.

Grandell, J. (1991). Aspects of Risk Theory, Springer, New York.

Hogg, R. and Klugman, S. A. (1984). Loss Distributions, Wiley, New York.

Hormann, W. (1993). The transformed rejection method for generating Poisson random variables, Insurance: Mathematics and Economics 12: 39–45.

Huber, P. J. (2004). Robust statistics, Wiley, Hoboken.

Johnk, M. D. (1964). Erzeugung von Betaverteilten und Gammaverteilten Zufallszahlen, Metrika 8: 5-15.

Kaas, R., Goovaerts, M., Dhaene, J., and Denuit, M. (2008). Modern Actuarial Risk Theory: Using R, Springer.

Klugman, S. A., Panjer, H.H., and Willmot, G.E. (2008). Loss Models: From Data to Decisions (3rd ed.), Wiley, New York.

Panjer, H.H. (2006). Operational Risk : Modeling Analytics, Wiley.

Panjer, H.H. and Willmot, G.E. (1992). Insurance Risk Models, Society of Actuaries, Chicago.

Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance, Wiley, Chichester.

Ross, S. (2002). Simulation, Academic Press, San Diego.

Stadlober, E. (1989). Sampling from Poisson, binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Math. Statist. Sektion 303, Forschungsgesellschaft Joanneum Graz.

Stephens, M. A. (1978). On the half-sample method for goodness-of-fit, Journal of the Royal Statistical Society B 40: 64-70.

Teugels, J.L. and Vynckier, P. (1996). The structure distribution in a mixed Poisson process, J. Appl. Math. & Stoch. Anal. 9: 489–496.

Tse, Y.-K. (2009). Nonlife Actuarial Models: Theory, Methods and Evaluation, Cambridge University Press, Cambridge.

Weron, R. (2004). Computationally Intensive Value at Risk Calculations, in J. E. Gentle, W. Hardle, Y. Mori (eds.) Handbook of Computational Statistics, Springer, Berlin, 911–950.

Willmot, G. E. (2001). The nature of modelling insurance losses, The Munich Re Inaugural Lecture, December 5, 2001, Toronto.