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Credibility for the Chain Ladder Reserving Method

Published online by Cambridge University Press:  17 April 2015

Alois Gisler
Affiliation:
AXA-Winterthur Insurance Company, P.O. Box 357, Email: [email protected]
Mario V. Wüthrich
Affiliation:
Departement Mathematik, ETH Zurich, Rämistrasse 101, CH 8401 Winterthur, Email: [email protected]
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Abstract

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We consider the chain ladder reserving method in a Bayesian set up, which allows for combining the information from a specific claims development triangle with the information from a collective. That is, for instance, to consider simultaneously own company specific data and industry-wide data to estimate the own company's claims reserves. We derive Bayesian estimators and credibility estimators within this Bayesian framework. We show that the credibility estimators are exact Bayesian in the case of the exponential dispersion family with its natural conjugate priors. Finally, we make the link to the classical chain ladder method and we show that using non-informative priors we arrive at the classical chain ladder forecasts. However, the estimates for the mean square error of prediction differ in our Bayesian set up from the ones found in the literature. Hence, the paper also throws a new light upon the estimator of the mean square error of prediction of the classical chain ladder forecasts and suggests a new estimator in the chain ladder method.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2008

References

[1] Barnett, G. and Zehnwirth, B. (2000) Best estimates for reserves. Proc. CAS LXXXII, 245321.Google Scholar
[2] Buchwalder, M., Bühlmann, H., Merz, M. and Wüthrich, M.V. (2006) The mean square error of prediction in the chain ladder reserving method (Mack and Murphy revisited). ASTIN Bulletin 36/2, 521542.CrossRefGoogle Scholar
[3] Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Universitext, Springer Verlag.Google Scholar
[4] Dobson, A.J. (1990) An Introduction to Generalized Linear Models. Chapman and Hall, London.CrossRefGoogle Scholar
[5] England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial J. 8/3, 443518.CrossRefGoogle Scholar
[6] Gisler, A. (2006) The estimation error in the chain-ladder reserving method: a Bayesian approach. ASTIN Bulletin 36/2, 554565.CrossRefGoogle Scholar
[7] Hess, K. and Schmidt, K.D. (2000) A comparison of models for the chain-ladder method. Dresdner Schriften zur Versicherungsmathematik, 3/2000, Technische Universität Dresden.Google Scholar
[8] Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. ASTIN Bulletin 8, 7790.CrossRefGoogle Scholar
[9] Mack, T. (1991) A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin 21/1, 93109.CrossRefGoogle Scholar
[10] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23/2, 213225.CrossRefGoogle Scholar
[11] Mack, T. (1994) Which stochastic model is underlying the chain ladder method? Insurance: Mathematics and Economics 15, 133138.Google Scholar
[12] Mack, T. and Venter, G. (2000) A comparison of stochastic models that reproduce chain ladder reserve estimates. Insurance: Mathematics and Economics 26, 101107.Google Scholar
[13] Mack, T., Quarg, G. and Braun, C. (2006) The mean square error of prediction in the chain ladder reserving method – a comment. ASTIN Bulletin 36/2, 543552.CrossRefGoogle Scholar
[14] Mccullagh, P. and Nelder, J.A. (1989) Generalized Linear Models. Chapman and Hall, Cambridge, 2nd edition.CrossRefGoogle Scholar
[15] Murphy, D.M. (1994) Unbiased loss development factors. Proc. CAS LXXXI, 154222.Google Scholar
[16] Ohlsson, E. and Johansson, B. (2006) Exact credibility and Tweedie models. ASTIN Bulletin 36/1, 121133.CrossRefGoogle Scholar
[17] Venter, G. (2006) Discussion of the mean square error of prediction in the chain ladder reserving method. ASTIN Bulletin 36/2, 566572.CrossRefGoogle Scholar
[18] Verrall, R.J. (2000) An investigation into stochastic claims reserving models and chain-ladder technique. Insurance: Mathematics and Economics 26, 9199.Google Scholar
[19] Verrall, R.J. (2004) A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Act. J. 813, 6789.CrossRefGoogle Scholar
[20] Wüthrich, M.V. (2003) Claims reserving using Tweedie’s compound Poisson model. ASTIN Bulletin 33/2, 331346.CrossRefGoogle Scholar
[21] Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Wiley Finance.Google Scholar