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Actuarial Applications of a Hierarchical Insurance Claims Model

Published online by Cambridge University Press:  09 August 2013

Edward W. Frees ,
Peng Shi  and
Emiliano A. Valdez
Edward W. Frees
Affiliation:
School of Business, University of Wisconsin, Madison, Wisconsin 53706USA, E-mail: jfrees@bus.wisc.edu
Peng Shi
Affiliation:
School of Business, University of Wisconsin, Madison, Wisconsin 53706USA, E-mail: pshi@bus.wisc.edu
Emiliano A. Valdez
Affiliation:
Department of Mathematics, College of Liberal Arts and Sciences, University of Connecticut, Storrs, Connecticut 06269-3009USA, E-mail: valdez@math.uconn.edu
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Abstract

This paper demonstrates actuarial applications of modern statistical methods that are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy and claims files from a major Singaporean insurance company. A hierarchical statistical model, developed in prior work (Frees and Valdez (2008)), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders.

Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to “unbundle” contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described.

Keywords

Long-tail regressioncopulasrisk measuresreinsurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 1 , May 2009 , pp. 165 - 197
Copyright
Copyright © International Actuarial Association 2009

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References

Angers, J.-F., Desjardins, D., Dionne, G. and Guertin, F. (2006) Vehicle and fleet random effects in a model of insurance rating for fleets of vehicles. ASTIN Bulletin 36(1), 25–77.CrossRefGoogle Scholar
Antonio, K., Beirlant, J., Hoedemakers, T. and Verlaak, R. (2006) Lognormal mixed models for reported claims reserves. North American Actuarial Journal 10(1), 30–48.Google Scholar
Boucher, J.-Ph. and Denuit, M. (2006) Fixed versus random effects in Poisson regression models for claim counts: A case study with motor insurance. ASTIN Bulletin 36(1), 285–301.Google Scholar
Brockman, M.J. and Wright, T.S. (1992) Statistical motor rating: making effective use of your data. Journal of the Institute of Actuaries 119, 457–543.Google Scholar
Cairns, A.J.G. (2000) A discussion of parameter and model uncertainty in insurance. Insurance: Mathematics and Economics 27(3), 313–330.Google Scholar
Coutts, S.M. (1984) Motor insurance rating, an actuarial approach. Journal of the Institute of Actuaries 111, 87–148.Google Scholar
Desjardins, D., Dionne, G. and Pinquet, J. (2001) Experience rating schemes for fleets of vehicles. ASTIN Bulletin 31(1), 81–105.Google Scholar
Frangos, N.E. and Vrontos, S.D. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin 31(1), 1–22.Google Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal 2(1), 1–25.Google Scholar
Frees, E.W. and Valdez, E.A. (2008) Hierarchical insurance claims modeling. Journal of the American Statistical Association 103(484), 1457–1469.CrossRefGoogle Scholar
Gourieroux, Ch. and Jasiak, J. (2007) The Econometrics of Individual Risk: Credit, Insurance, and Marketing. Princeton University Press, Princeton, NJ.Google Scholar
Hardy, M. (2003) Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance. John Wiley & Sons, New York.Google Scholar
Hsiao, C., Kim, C. and Taylor, G. (1990) A statistical perspective on insurance rate-making. Journal of Econometrics 44, 5–24.Google Scholar
Kahane, Y. and Haim Levy, H. (1975) Regulation in the insurance industry: determination of premiums in automobile insurance. Journal of Risk and Insurance 42, 117–132.Google Scholar
Klugman, S., Panjer, H. and Willmot, G. (2004) Loss Models: From Data to Decisions (Second Edition), Wiley, New York.Google Scholar
McDonald, J.B. and Xu, Y.J. (1995) A generalization of the beta distribution with applications. Journal of Econometrics 66, 133–152.Google Scholar
Pinquet, J. (1997) Allowance for cost of claims in bonus-malus systems. ASTIN Bulletin 27(1), 33–57.Google Scholar
Pinquet, J. (1998) Designing optimal bonus-malus systems from different types of claims. ASTIN Bulletin 28(2), 205–229.CrossRefGoogle Scholar
Renshaw, A.E. (1994) Modeling the claims process in the presence of covariates. ASTIN Bulletin 24(2), 265–285.Google Scholar
Sun, J., Frees, E.W. and Rosenberg, M.A. (2008) Heavy-tailed longitudinal data modeling using copulas. Insurance: Mathematics and Economics, 42(2), 817–830.Google Scholar
Terza, J.V. and Wilson, P.W. (1990) Analyzing frequencies of several types of events: A mixed multinomial-Poisson approach. The Review of Economics and Statistics 108–115.Google Scholar
Weisberg, H.I. and Tomberlin, T.J. (1982) A statistical perspective on actuarial methods for estimating pure premiums from cross-classified data. Journal of Risk and Insurance 49, 539–563.Google Scholar
Weisberg, H.I., Tomberlin, T.J. and Chatterjee, S. (1984) Predicting insurance losses under cross-classification: A comparison of alternative approaches. Journal of Business & Economic Statistics 2(2), 170–178.Google Scholar