Abstract
In this paper, we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times being generalized Erlang(n)-distributed, we obtain the integro-differential equation for the Gerber–Shiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albrecher H, Claramunt MM, Mármol M (2005) On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times. Insur Math Econ 37(2):324–334
Bowers NL, Gerber HU, Hickman JC, Jones DA, Nesbitt CJ (1997) Actuarial mathematics. Society of Actuaries, Schaumburg
Bühlmann H (1996) Mathematical methods in risk theory. Springer, Berlin
Centeno L (1986) Measuring the effects of reinsurance by the adjustment coefficient. Insur Math Econ 5(2):169–182
Centeno L (2002) Measuring the effects of reinsurance by the adjustment coefficient in the Sparre Anderson model. Insur Math Econ 30(1):37–49
Centeno L (2005) Dependent risks and excess of loss reinsurance. Insur Math Econ 37(2):229–238
Dickson DCM (2005) Insurance risk and ruin. Cambridge University Press, Cambridge
Dickson DCM, Drekic S (2004) The joint distribution of the surplus prior to ruin and the deficit at ruin in some Sparre Andersen models. Insur Math Econ 34(1):97–107
Dickson DCM, Waters HR (1996) Reinsurance and ruin. Insur Math Econ 19(1):61–80
Dickson DCM, Waters HR (2006) Optimal dynamic reinsurance. ASTIN Bull 36(2):415–432
Gerber HU (1979) An introduction to mathematical risk theory. S.S. Huebner Foundation monograph. University of Pennsylvania, Philadelphia
Gerber HU, Shiu ES (1998) On the time value of ruin. N Am Actuar J 2(1):48–78
Gerber HU, Shiu ES (2005) The time value of ruin in a Sparre Andersen model. N Am Actuar J 9(2):49–84
Goovaerts MJ, Van Heervaarden AE, Kaas R (1989) Optimal reinsurance in relation to ordering of risk. Insur Math Econ 8(1):11–17
Hesselager O (1990) Some results on optimal reinsurance in terms of the adjustment coefficient. Scand Actuar J 1990(1–2):80–95
Hipp C (2006) Speedy convolution algorithms and Panjer recursions for phase-type distributions. Insur Math Econ 38(1):176–188
Hipp C, Vogt M (2003) Optimal dynamic XL reinsurance. ASTIN Bull 33(2):193–207
Hojgaard B, Taksar M (1998) Optimal proportional reinsurance policies for diffusion models with transaction costs. Insur Math Econ 22(1):41–51
Landriault D, Willmot GE (2008) On the Gerber–Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. Insur Math Econ 42(2):600–608
Li S, Garrido J (2004) On ruin for the Erlang(n) risk process. Insur Math Econ 34(3):391–408
Lin XS, Pavlova K (2006) The compound Poisson risk model with a threshold dividend strategy. Insur Math Econ 38(1):57–80
Lin XS, Willmot GE (1999) Analysis of a defective renewal equation arising in ruin theory. Insur Math Econ 25(1):63–84
Lin XS, Willmot GE (2000) The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insur Math Econ 27(1):19–44
Schmidli H (2001) Optimal proportional reinsurance policies in a dynamic setting. Scand Actuar J 2001(1):55–68
Schmidli H (2002) On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab 12(3):890–907
Schmidli H (2006) Optimisation in non-life insurance. Stoch Models 22(4):689–722
Taksar M, Markussen C (2003) Optimal dynamic reinsurance policies for large insurance portfolios. Finance Stoch 7(1):97–121
Verlaak R, Beirlant J (2003) Optimal reinsurance programs: an optimal combination of several reinsurance protections on a heterogeneous insurance portfolio. Insur Math Econ 33(2):381–403
Waters HR (1979) Excess of loss reinsurance limits. Scand Actuar J 1979(1):37–43
Waters HR (1983) Some mathematical aspects of reinsurance. Insur Math Econ 2(1):17–26
Willmot GE (2007) On the discounted penalty function in the renewal risk model with general interclaim times. Insur Math Econ 41(1):17–31
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Castañer, A., Claramunt, M.M. & Mármol, M. Ruin probability and time of ruin with a proportional reinsurance threshold strategy. TOP 20, 614–638 (2012). https://doi.org/10.1007/s11750-010-0165-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-010-0165-5