Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



On excess-of-loss reinsurance

Authors: Hansjörg Albrecher and Jozef L. Teugels
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 7-22
MSC (2000): Primary 62P05, 62H20
Published electronically: December 30, 2009
MathSciNet review: 2494532
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. H. Albrecher and G. Pirsic, Recursive Evaluation of Compound Distributions Revisited, Preprint, Radon Institute, Austrian Academy of Sciences, 2008.
  • 2. 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, 10.1007/s007800050075
  • 3. 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.
  • 4. Baruch Berliner, Correlations between excess of loss reinsurance covers and reinsurance of the 𝑛 largest claims, Astin Bull. 6 (1971/72), 260–275. MR 0314220
  • 5. Hans Bühlmann, Mathematical methods in risk theory, Die Grundlehren der mathematischen Wissenschaften, Band 172, Springer-Verlag, New York-Berlin, 1970. MR 0278448
  • 6. 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.
  • 7. Jan Grandell, Mixed Poisson processes, Monographs on Statistics and Applied Probability, vol. 77, Chapman & Hall, London, 1997. MR 1463943
  • 8. 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, 10.2143/AST.32.2.1030
  • 9. R. Kestemont and J. Paris, Sur l'ajustement du nombre des sinistres, Mitt. Ver. Schweiz. Versich. Math. (1985), 157-164.
  • 10. 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
  • 11. J. Kupper, Some aspects of cumulative risk, Astin Bulletin 3 (1963), 85-103.
  • 12. 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).
  • 13. T. Mack, Schadensversicherungsmathematik, Verlag Versicherungswirtschaft e.V., Karlsruhe, 1997.
  • 14. Ana J. Mata, Pricing excess of loss reinsurance with reinstatements, Astin Bull. 30 (2000), no. 2, 349–368. MR 1963403, 10.2143/AST.30.2.504640
  • 15. Harry H. Panjer, Recursive evaluation of a family of compound distributions, Astin Bull. 12 (1981), no. 1, 22–26. MR 632572
  • 16. L. Rȧde, Limit theorems for thinning of renewal point processes, J. Appl. Probability 9 (1972), 847–851. MR 0359052
  • 17. 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
  • 18. D. E. A. Sanders, When the wind blows: an introduction to catastrophe excess-of-loss reinsurance, CAS Forum (1995), 157-228.
  • 19. 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, 10.1080/03461238.1990.10413879
  • 20. 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.
  • 21. Bjørn Sundt and William S. Jewell, Further results on recursive evaluation of compound distributions, Astin Bull. 12 (1981), no. 1, 27–39. MR 632573
  • 22. 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
  • 23. B. Sundt, On allocation of excess-of-loss premiums, Astin Bulletin 22 (1992), 167-176.
  • 24. Paul Thyrion, Extension of the collective risk theory, Skand. Aktuarietidskr. suppl. 3-4 (1969), 84–98. Filip Lundberg Symposium on Risk Theory (Stockholm, 1968). MR 0350919
  • 25. H. G. Verbeek, An approach to the analysis of claims experience in motor liability excess-of-loss reinsurance, Astin Bulletin 6 (1972), 195-202.
  • 26. S. Wang and M. Sobrero, Further results on Hesselager's recursive procedure for calculation of some compound distributions, Astin Bulletin 24 (1994), 161-166.
  • 27. 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, 10.1080/03461238.1987.10413823
  • 28. G. E. Willmot, Sundt and Jewell's family of discrete distributions, Astin Bulletin 18 (1988), 17-29.
  • 29. Gordon E. Willmot, On recursive evaluation of mixed Poisson probabilities and related quantities, Scand. Actuar. J. 2 (1993), 114–133. MR 1272853
  • 30. 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, 10.1016/0167-6687(87)90006-0

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 62P05, 62H20

Retrieve articles in all journals with MSC (2000): 62P05, 62H20

Additional Information

Hansjörg Albrecher
Affiliation: Radon Institute, Austrian Academy of Sciences, Linz, Austria, and University of Linz, Altenbergerstrasse 69, A-4040 Linz, Austria

Jozef L. Teugels
Affiliation: EURANDOM, Technische Universiteit Eindhoven, The Netherlands, and Katholieke Universiteit Leuven, Leuven Center for Statistics, Celestijnenlaan 200B, B-3001 Heverlee, Belgium

Keywords: Reinsurance, point processes, thinning, Laplace--Stieltjes transform
Received by editor(s): August 20, 2008
Published electronically: December 30, 2009
Additional Notes: Supported by the Austrian Science Fund Project P18392
Article copyright: © Copyright 2009 American Mathematical Society