Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-23T22:46:15.142Z Has data issue: false hasContentIssue false
  • English
  • Français

On Pareto Conjugate Priors and Their Application to Large Claims Reinsurance Premium Calculation

Published online by Cambridge University Press:  17 April 2015

José L. Vilar-Zanón  and
Cristina Lozano-Colomer
José L. Vilar-Zanón
Affiliation:
Dept. Economía Financiera I., Facultad Ciencias Económicas y Empresariales. Universidad Complutense de Madrid. Campus de Somosaguas. Pozuelo de Alarcón, 28223, España. E-mail: jlvilarz@ccee.ucm.es
Cristina Lozano-Colomer
Affiliation:
Dept. Métodos Cuantitativos, Universidad Pontificia de Comillas (ICADE). Madrid. E-mail: clozano@cee.upcomillas.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to premium calculation of large claims excess of loss (XL) reinsurance contracts. It studies the use of the generalized inverse Gaussian (GIG) as a Pareto prior conjugate, a family that contains as a particular case the gamma distribution. An exact credibility formula is deduced allowing the calculation of individual reinsurance premiums. These are premiums suited to the excesses history of a sole portfolio. A family of predictive distributions for the excesses is derived. We apply our exact credibility model to a sample of excesses arisen in ten Spanish portfolios of liability motor insurance from year 1992 to year 2001.

Keywords

ReinsuranceXL premiumslarge claimsextreme value modelingBayesexact credibilityPareto distributiongeneralized inverse Gaussian familyAMS subject classification62P0560G7062F15
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 37 , Issue 2 , November 2007 , pp. 405 - 428
Copyright
Copyright © ASTIN Bulletin 2007

References

Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487–1503.CrossRefGoogle Scholar
D’Agostino, R.B. and Stephens, M.A. (1986) Goodness-of-fit techniques. Statistics: textbooks and monographs Vol. 68. Marcel Dekker Inc. New York and Bassel.Google Scholar
Arnold, B.C. and Press, S.J. (1989) Bayesian estimation and prediction for Pareto data. Journal of the American Statistical Association. Theory and Methods, 84(408), 1079–1084.CrossRefGoogle Scholar
Beirlant, J., Teugels, J.L. and Vynckier, P. (1996) Practical analysis of extreme values. Leuven University Press.Google Scholar
Berger, J.O. (1980) Statistical decision theory and Bayesian analysis. Springer-Verlag. London.CrossRefGoogle Scholar
Bühlmann, H. and Gisler, A. (2005) A course in credibility theory and its applications. Springer.Google Scholar
Coles, S. and Powell, E.A. (1996) Bayesian methods in extreme value modeling: a review and new developments. International Statistical Review, 64(1), 119–136.CrossRefGoogle Scholar
Coles, S. (2001) An introduction to statistical modeling of extreme values. Springer-Verlag. London.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modeling extremal events for insurance and finance. Springer.CrossRefGoogle Scholar
Fürst, D. (1964) Formulation Bayesienne du problème des valeurs extrêmes en relation à la réassurance en excédent des sinistres. Astin Bulletin, 3(2), 153–162.CrossRefGoogle Scholar
Hesselager, O. (1993) A class of conjugate priors with applications to excess-of-loss reinsurance. Astin Bulletin, 23(1), 77–93.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5), 1163–1174.CrossRefGoogle Scholar
Jorgensen, (1982) Statistical properties of the generalized inverse Gaussian distribution. Springer-Verlag. New York.CrossRefGoogle Scholar
Lemaire, J. (1995) Bonus-malus systems in automobile insurance. Kluwer Academic Publishers.CrossRefGoogle Scholar
McNeil, A.J. (1997) Estimating the tails of loss severity distributions using extreme value theory. Astin Bulletin, 27(1), 117–137.CrossRefGoogle Scholar
Panjer, H.H. and Willmot, G.E. (1992) Insurance risk models. Society of Actuaries.Google Scholar
Patrick, G. and Mashitz, I. (1989) Credibility for reinsurance excess pricing. 21st Astin Colloquium. New York.Google Scholar
Pin-Hung, Hsieh (2004) A data record analytic method for forecasting next record catastrophe loss. Journal of Risk and Insurance, 71(2), 309–322.Google Scholar
Reiss, R.D. and Thomas, M. (1999) A new class of Bayesian estimators in paretian excess-of-loss reinsurance. Astin Bulletin, 29(2), 339–349.CrossRefGoogle Scholar
Reiss, R.D. and Thomas, M. (2001) Statistical analysis of extreme values. 2nd edition. Birkhäuser.Google Scholar
Rytgaard, M. (1990) Estimation in the Pareto distribution. Astin Bulletin, 20(2), 201–216.CrossRefGoogle Scholar
Siegel, S. and Castellan, N.J. (1988) Nonparametric statistics for the behavioural sciences. 2nd edition. New York. McGraw-Hill.Google Scholar
Straub, E. (1971) Estimation of the number of excess claims by means of the credibility theory. Astin Bulletin, 5(3), 388–392.CrossRefGoogle Scholar
Tremblay, L. (1992) Using the Poisson inverse Gaussian in bonus-malus systems. Astin Bulletin, 22(1), 97–106.CrossRefGoogle Scholar