×

zbMATH — the first resource for mathematics

Optimal retention levels, given the joint survival of cedent and reinsurer. (English) Zbl 1143.91029
In the present paper the optimal reinsurance is considered from the point of view of both the interests of the primary insurer (cedent) and reinsurer, as two parties jointly liable for the risk they share. For this purpose two alternative joint risk measures and two optimality criteria for setting the excess of loss retention level are introduced. One such measure is the probability that both cedent and reinsurer survive up to finite time horizon. Then the retention level is said to be optimal if it maximizes the probability of joint survival. The alternative measure is the absolute value of the difference between the probability of survival to a finite moment of the cedent and probability of survival of the reinsurer, given the survival of the cedent up to that moment. The optimal retention level is the value which minimizes this difference. To calculate the optimal retention levels the authors derive explicit expressions for these two criteria, assuming that the originally occurring claims to cedent (claims before reinsurance) have a discrete joint distribution, their arrivals form a Poisson point process and the premium income function is a nonnegative increasing real function. Necessary calculations can be carried out using the corresponding Mathematica modules, developed for this purpose. The quota share contracts are also considered under the same model. It is shown that the probability of joint survival of the cedent and reinsurer coincides with the probability of survival of solely the insurer. A number of examples and numerical comparisons, illustrating the performance of the proposed reinsurance optimality criteria, are presented and discussed.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1080/03461230110106237 · Zbl 1015.62104 · doi:10.1080/03461230110106237
[2] Andersen KM, University of Copenhagen, Laboratory of Actuarial Mathematics (2000)
[3] DOI: 10.1007/s007800050075 · Zbl 0958.91026 · doi:10.1007/s007800050075
[4] Bäuerle N (2002) Approximation of optimal reinsurance and dividend pay-out policies Paper presented at: Scientific Conference on Insurance and Finance Weimar (April, 24, 2002)
[5] Borch K, ASTIN Bulletin pp 170– (1960)
[6] Borch K, ASTIN Bulletin pp 293– (1969)
[7] Bowers NL Gerber HU Hickman JC Jones DA Nesbitt CJ (1997) Actuarial Mathematics. The Society of Actuaries second edition
[8] Bühlmann H, Mathematical methods in risk theory (1970) · Zbl 0209.23302
[9] Bühlmann H, ASTINBulletin 11 pp 52– (1980)
[10] Bühlmann H, ASTIN Bulletin 14 pp 13– (1984) · doi:10.1017/S0515036100004773
[11] DOI: 10.1016/0167-6687(86)90043-0 · Zbl 0598.62141 · doi:10.1016/0167-6687(86)90043-0
[12] DOI: 10.2143/AST.27.1.542067 · doi:10.2143/AST.27.1.542067
[13] Centeno ML, Reidel Publishing Company pp pp. 679–689– (1988)
[14] Centeno ML, Scandinavian Actuarial J. pp 97– (1991)
[15] DOI: 10.2143/AST.21.1.2005400 · doi:10.2143/AST.21.1.2005400
[16] Clark J, Reinsurance (2002)
[17] de Finetti B, Giorn. Ist.Ital. Attuari I 1 pp 1– (1940)
[18] DOI: 10.1016/S0167-6687(96)00011-X · Zbl 0894.62110 · doi:10.1016/S0167-6687(96)00011-X
[19] DOI: 10.2143/AST.27.2.542048 · doi:10.2143/AST.27.2.542048
[20] DOI: 10.1016/S0167-6687(99)00063-3 · Zbl 0964.62099 · doi:10.1016/S0167-6687(99)00063-3
[21] Gerber HU, University of Pennsylvania (1979)
[22] Hesselager O, Scand. Actuarial J. 1 pp 80– (1990) · Zbl 0728.62100 · doi:10.1080/03461238.1990.10413873
[23] Hipp C, University of Karsruhe (2001)
[24] DOI: 10.1080/034612300750066728 · Zbl 0958.91030 · doi:10.1080/034612300750066728
[25] DOI: 10.1016/S0167-6687(01)00078-6 · Zbl 1074.62528 · doi:10.1016/S0167-6687(01)00078-6
[26] Ignatov ZG, Explicit finite time ruin probabilities for Discrete, Dependent Claims (2001)
[27] Kaluszka M, Insurance: Mathematics and Economics 24 pp 219– (2001)
[28] Krvavych Y (2001) On existence of insurer’s optimal excess of loss reinsurance strategy Paper presented at the 5-th International Congress onInsurance:Mathematics and Economics
[29] Mnif M Sulem A (2001) Optimal risk control under excess of loss reinsuranceINRIA Research Report No 4317
[30] von Neumann J, Theory of Games and Economic Behavior, Princeton (1944)
[31] Picard P, Scand. Actuarial J. 1 pp 58– (1997) · Zbl 0926.62103 · doi:10.1080/03461238.1997.10413978
[32] DOI: 10.1080/034612301750077338 · Zbl 0971.91039 · doi:10.1080/034612301750077338
[33] Schmidli H, On minimizing the ruin probability by investment and reinsurance, Preprint, University of Aarhus (2002) · Zbl 1021.60061
[34] Schmitter H, Setting optimal reinsurance retentions, Swiss Reinsurance Company (2001)
[35] Straub E, Non-life insurance mathematics, Springer (1988) · Zbl 0677.62098 · doi:10.1007/978-3-662-03364-7
[36] DOI: 10.1016/S0167-6687(97)00038-3 · Zbl 0922.62112 · doi:10.1016/S0167-6687(97)00038-3
[37] DOI: 10.1007/s007800200073 · Zbl 1066.91052 · doi:10.1007/s007800200073
[38] Taylor G, Scand. Actuarial J. 1 pp 40– (1992) · Zbl 0758.90015 · doi:10.1080/03461238.1992.10413896
[39] Verbeek HG, ASTIN Bulletin pp 29– (1966) · doi:10.1017/S0515036100008886
[40] Waters H, Scand. Actuarial J. pp 37– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.