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Three Methods to Calculate the Probability of Ruin

Published online by Cambridge University Press:  29 August 2014

François Dufresne  and
Hans U. Gerber
François Dufresne*
Affiliation:
University of Lausanne, Switzerland
Hans U. Gerber*
Affiliation:
University of Lausanne, Switzerland
*
École des H.E.C., Université de Lausanne, CH-1015 Lausanne, Switzerland.
École des H.E.C., Université de Lausanne, CH-1015 Lausanne, Switzerland.
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Abstract

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The first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

Keywords

Probability of ruindiscretizationcombination of exponentialssimulation
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 19 , Issue 1 , April 1989 , pp. 71 - 90
Copyright
Copyright © International Actuarial Association 1989

References

Beekman, J. A. (1974) Two Stochastic Processes, Almqvist & Wiksell, Stockholm.Google Scholar
Bohman, H. (1971) “The ruin probability in a special case”, ASTIN Bulletin 7, 66–68.CrossRefGoogle Scholar
Bowers, N.L. Jr, Gerber, H.U., Hickman, J.C., Jones, D. A. and Nesbitt, C.J. (1986) Actuarial Mathematics, Society of Actuaries, Itasca, Illinois.Google Scholar
Cramér, H. (1955) “Collective Risk Theory – A survey of the theory from the point of view of the theory of stochastic processes”, The Jubilee Volume of Skandia.Google Scholar
Dubourdieu, J. (1952) Théorie mathématique du risque dans les assurances de répartition, Gauthier-Villars, Paris.Google Scholar
Dufresne, F. and Gerber, H. U. (1988) “The Probability and Severity of Ruin for Combinations of Exponential Claim Amount Distributions and their Translations”, Insurance: Mathematics & Economics 7, 75–80.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Volume 2, Wiley, New York.Google Scholar
Gerber, H.U., Goovaerts, M.J., and Kaas, R. (1987) “On the Probability and Severity of Ruin”, ASTIN Bulletin, 17, 151–163.CrossRefGoogle Scholar
Goovaerts, M.J. and De Vylder, F. (1984) “A Stable Algorithm for Evaluation of Ultimate Ruin Probabilities”, ASTIN Bulletin, 14, 53–59.CrossRefGoogle Scholar
Hoel, P. G., Port, S. C. and Stone, C. J. (1972) Introduction to Stochastic Processes, Boston: Houghton Mifflin.Google Scholar
Meyers, G., and Beekman, J.A. (1988) “An improvement to the convolution method of calculating ψ(u)”, Insurance: Mathematics and Economics, 6, 267–274.Google Scholar
Panjer, H. H. (1986) “Direct Calculation of Ruin Probabilities”, The Journal of Risk and Insurance Vol. 53. Nr. 3, 521–529.CrossRefGoogle Scholar
Panjer, H. H. (1981) “Recursive Evaluation of a Family of Compound Distributions”, ASTIN Bulletin, 12, 22–26.CrossRefGoogle Scholar
Seal, H. L. (1972) “Risk Theory and the Single Server Queue”, Journal of the Swiss Association of Actuaries, 171–178.Google Scholar
Shiu, E.S.W. (1984) Discussion of “Practical Applications of the Ruin Function”, Transactions of the Society of Actuaries. 36, 480–486.Google Scholar
Shiu, E.S.W. (1988) “Calculation of the Probability of Eventual Ruin by Beekman's Convolution Series”, Insurance: Mathematics and Economics, 7, 41–47.Google Scholar
Ströter, B. (1985) “The Numerical Evaluation of the Aggregate Claim Density Function via Integral Equation”, Bulletin of the German Association of Actuaries 17, 1–14.Google Scholar
Taylor, G.C. (1985) “A Heuristic Review of Some Ruin Theory Results”, ASTIN Bulletin, 15, 73–88.CrossRefGoogle Scholar
Thorin, O., and Wikstad, N. (1976) “Calculation and use of ruin probabilities”, Transactions of the 20th International Congress of Actuaries, vol, III, 773–781.Google Scholar
Thorin, O., and Wikstad, N. (1977) “Calculation of ruin probabilities when the claim distribution is lognormal”, ASTIN Bulletin, 9, 231–246.CrossRefGoogle Scholar