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EXISTENCE AND UNIQUENESS OF CHAIN LADDER SOLUTIONS

Published online by Cambridge University Press:  12 August 2016

Greg Taylor
Greg Taylor*
Affiliation:
UNSW Business School, Level 6, West Lobby, UNSW Business School Building E12, UNSW Sydney 2052 Australia E-Mail: [email protected]
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Abstract

The cross-classified chain ladder has a number of versions, depending on the distribution to which observations are subject. The simplest case is that of Poisson distributed observations, and then maximum likelihood estimates of parameters are explicit. Most other cases, however, including Bayesian chain ladder models, lead to implicit MAP (Bayesian) or MLE (non-Bayesian) solutions for these parameter estimates, raising questions as to their existence and uniqueness. The present paper investigates these questions in the case where observations are distributed according to some member of the exponential dispersion family.

Keywords

Bayesian chain laddercross-classified chain ladderEDF chain ladderexistenceloss reservingMAP estimatormaximum likelihood estimateTweedie chain ladderuniqueness
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 1 , January 2017 , pp. 1 - 41
Copyright
Copyright © Astin Bulletin 2016 

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References

England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal, 8 (iii), 443–518.CrossRefGoogle Scholar
England, P.D., Verrall, R.J. and Wüthrich, M.V. (2012) Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method. Annals of Actuarial Science, 6 (2), 258–281.CrossRefGoogle Scholar
Gisler, A. and Müller, P. (2007) Credibility for additive and multiplicative models. Paper presented to the 37th ASTIN Colloquium, Orlando FL, USA. See http://www.actuaries.org/ASTIN/Colloquia/Orlando/Presentations/Gisler2.pdf.Google Scholar
Gisler, A. and Wüthrich, M.V. (2008) Credibility for the chain ladder reserving method. Astin Bulletin, 38 (2), 565–597.Google Scholar
Hachemeister, C. A. and Stanard, J.N. (1975) IBNR claims count estimation with static lag functions. Spring meeting of the Casualty Actuarial Society.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J.P. (2008) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika, 95, 979–986.Google Scholar
Kuang, D., Nielsen, B. and Nielsen, J.P. (2009) Chain-ladder as maximum likelihood revisited. Annals of Actuarial Science, 4 (1), 105–121.Google Scholar
Mack, T. (1993) Distribution-free calculation of the standard error of chain ladder reserve estimates. Astin Bulletin, 23 (2), 213–225.CrossRefGoogle Scholar
McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models, 2nd ed. London, UK: Chapman & Hall.CrossRefGoogle Scholar
Merz, M., Wüthrich, M.V. and Hashorva, E. (2013) Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science, 7 (1), 3–25.CrossRefGoogle Scholar
Renshaw, A.E. and Verrall, R.J. (1998) A stochastic model underlying the chain-ladder technique. British Actuarial Journal, 4 (iv), 903–923.Google Scholar
Shi, P., Basu, S. and Meyers, P.P. (2012) A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal, 16 (1), 29–51.Google Scholar
Taylor, G. (2000) Loss Reserving: An Actuarial Perspective. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
Taylor, G. (2009) The chain ladder and Tweedie distributed claims data. Variance, 3, 96–104.Google Scholar
Taylor, G. (2011) Maximum likelihood and estimation efficiency of the chain ladder. Astin Bulletin, 41 (1), 131–155.Google Scholar
Taylor, G. (2015) Bayesian chain ladder models. Astin Bulletin, 45 (1), 75–99.Google Scholar
Tweedie, M. C. K. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions, Proceedings of the Indian Statistical Golden Jubilee International Conference (ed. Ghosh, J.K. and Roy, J.), pp. 579–604. Indian Statistical Institute.Google Scholar
Verrall, R.J. (2000) An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: Mathematics and Economics, 26 (1), 91–99.Google Scholar
Verrall, R.J. (2004) A Bayesian generalised linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8 (3), 67–89.Google Scholar
Wüthrich, M.V. (2003) Wüthrich M V (2007). Claims reserving using Tweedie's compound Poisson model. Astin Bulletin, 33 (2), 331–346.Google Scholar
Wüthrich, M.V. (2007) Using a Bayesian approach for claims reserving. Variance, 1 (2), 292–301.Google Scholar
Wüthrich, M.V. (2012) Discussion of “A Bayesian log-normal model for multivariate loss reserving” by Shi-Basu-Meyers. North American Actuarial Journal, 16 (3), 398–401.Google Scholar
Wüthrich, M.V. and Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance, Chichester, UK: John Wiley & Sons Ltd.Google Scholar