On excess-of-loss reinsurance

Albrecher, Hansjörg; Teugels, Jozef

doi:10.1090/S0094-9000-09-00787-X

On excess-of-loss reinsurance

Authors: Hansjörg Albrecher and Jozef L. Teugels
Journal: Theor. Probability and Math. Statist. 79 (2009), 7-22
MSC (2000): Primary 62P05, 62H20
DOI: https://doi.org/10.1090/S0094-9000-09-00787-X
Published electronically: December 30, 2009
MathSciNet review: 2494532
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Abstract: We discuss a unified framework to analyze the distribution of the number of claims and the aggregate claim sizes in an excess-of-loss reinsurance contract based upon the use of point processes and work out several examples explicitly. We first deal with a single excess-of-loss situation with an extra upper bound on the coverage of individual claims. Subsequently the results are extended to a reinsurance chain with $k$ partners.

References

H. Albrecher and G. Pirsic, Recursive Evaluation of Compound Distributions Revisited, Preprint, Radon Institute, Austrian Academy of Sciences, 2008.

Søren Asmussen, Bjarne Højgaard, and Michael Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance Stoch. 4 (2000), no. 3, 299–324. MR 1779581, DOI https://doi.org/10.1007/s007800050075

G. Benktander and C. O. Segerdahl, On the analytical representation of claim distributions with special reference to excess of loss reinsurance, Trans. 16-th Intern. Congress Actuaries, 1960, pp. 626–646.

Baruch Berliner, Correlations between excess of loss reinsurance covers and reinsurance of the $n$ largest claims, Astin Bull. 6 (1971/72), 260–275. MR 314220, DOI https://doi.org/10.1017/S0515036100011041

Hans Bühlmann, Mathematical methods in risk theory, Die Grundlehren der mathematischen Wissenschaften, Band 172, Springer-Verlag, New York-Berlin, 1970. MR 0278448

L. Centeno and O. Simões, Combining quota-share and excess-of-loss treaties on the reinsurance on $n$ independent risks, Astin Bulletin 21 (1991), 41–55.

Jan Grandell, Mixed Poisson processes, Monographs on Statistics and Applied Probability, vol. 77, Chapman & Hall, London, 1997. MR 1463943

Klaus Th. Hess, Anett Liewald, and Klaus D. Schmidt, An extension of Panjer’s recursion, Astin Bull. 32 (2002), no. 2, 283–297. MR 1942940, DOI https://doi.org/10.2143/AST.32.2.1030

R. Kestemont and J. Paris, Sur l’ajustement du nombre des sinistres, Mitt. Ver. Schweiz. Versich. Math. (1985), 157–164.

Stuart A. Klugman, Harry H. Panjer, and Gordon E. Willmot, Loss models, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1998. From data to decisions; With the assistance of Gary G. Venter; A Wiley-Interscience Publication. MR 1490300

J. Kupper, Some aspects of cumulative risk, Astin Bulletin 3 (1963), 85–103.

S. A. Ladoucette and J. L. Teugels, Exact and asymptotic properties for a generic reinsurance layer based on an ordered sample size, Scand. Actuar. J. (to appear).

T. Mack, Schadensversicherungsmathematik, Verlag Versicherungswirtschaft e.V., Karlsruhe, 1997.

Ana J. Mata, Pricing excess of loss reinsurance with reinstatements, Astin Bull. 30 (2000), no. 2, 349–368. MR 1963403, DOI https://doi.org/10.2143/AST.30.2.504640

Harry H. Panjer, Recursive evaluation of a family of compound distributions, Astin Bull. 12 (1981), no. 1, 22–26. MR 632572, DOI https://doi.org/10.1017/S0515036100006796

L. Rȧde, Limit theorems for thinning of renewal point processes, J. Appl. Probability 9 (1972), 847–851. MR 359052, DOI https://doi.org/10.2307/3212621

Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt, and Jozef Teugels, Stochastic processes for insurance and finance, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1999. MR 1680267

D. E. A. Sanders, When the wind blows: an introduction to catastrophe excess-of-loss reinsurance, CAS Forum (1995), 157–228.

Klaus J. Schröter, On a family of counting distributions and recursions for related compound distributions, Scand. Actuar. J. 3-4 (1990), 161–175. MR 1157783, DOI https://doi.org/10.1080/03461238.1990.10413879

H. Sichel, On a family of discrete distributions particularly suited to represent long tailed frequency data, Proc. 3-rd Symp. Math. Statistics, Pretoria, CSIR, 1971.

Bjørn Sundt and William S. Jewell, Further results on recursive evaluation of compound distributions, Astin Bull. 12 (1981), no. 1, 27–39. MR 632573, DOI https://doi.org/10.1017/S0515036100006802

Bjørn Sundt, On excess of loss reinsurance with reinstatements, Schweiz. Verein. Versicherungsmath. Mitt. 1 (1991), 51–66 (English, with French and German summaries). MR 1116983

B. Sundt, On allocation of excess-of-loss premiums, Astin Bulletin 22 (1992), 167–176.

Paul Thyrion, Extension of the collective risk theory, Skand. Aktuarietidskr. suppl, suppl. 3- (1971), 84–98. Filip Lundberg Symposium on Risk Theory (Stockholm, 1968). MR 350919, DOI https://doi.org/10.1080/03461238.1969.10404611

H. G. Verbeek, An approach to the analysis of claims experience in motor liability excess-of-loss reinsurance, Astin Bulletin 6 (1972), 195–202.

S. Wang and M. Sobrero, Further results on Hesselager’s recursive procedure for calculation of some compound distributions, Astin Bulletin 24 (1994), 161–166.

Gordon E. Willmot, The Poisson-inverse Gaussian distribution as an alternative to the negative binomial, Scand. Actuar. J. 3-4 (1987), 113–127. MR 943576, DOI https://doi.org/10.1080/03461238.1987.10413823

G. E. Willmot, Sundt and Jewell’s family of discrete distributions, Astin Bulletin 18 (1988), 17–29.

Gordon E. Willmot, On recursive evaluation of mixed Poisson probabilities and related quantities, Scand. Actuar. J. 2 (1993), 114–133. MR 1272853

G. E. Willmot and H. H. Panjer, Difference equation approaches in evaluation of compound distributions, Insurance Math. Econom. 6 (1987), no. 1, 43–56. MR 904968, DOI https://doi.org/10.1016/0167-6687%2887%2990006-0