Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T22:49:50.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximations of Ruin Probability by Di-Atomic or Di-Exponential Claims

Published online by Cambridge University Press:  29 August 2014

Joshua Babier  and
Beda Chan
Joshua Babier*
Affiliation:
University of Toronto, Canada
Beda Chan*
Affiliation:
University of Toronto, Canada
*
Department of Statistics, 100 St George Street, University of Toronto, Toronto, Ontario, CanadaM5S 1A1.
Department of Statistics, 100 St George Street, University of Toronto, Toronto, Ontario, CanadaM5S 1A1.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The sensitivity of the ruin probability depending on the claim size distribution has been the topic of several discussion papers in recent ASTIN Bulletins. This discussion was initiated by a question raised by Schmitter at the ASTIN Colloquium 1990 and attempts to make further contributions to this problem. We find the necessary and sufficient conditions for fitting three given moments by diatomic and diexponential distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with diatomics and diexponentials whenever the necessary and sufficient conditions are met, and compute the ruin probabilities using well known formulas for discrete and for combination of exponentials claim amounts. We then compare our approximations with the exact values that appeared in the literature. Finally we propose using diatomic and diexponential claim distributions as tools to study the Schmitter problem.

Keywords

Ruin probabilitydiatomic distributiondiexponential distribution
Type
Discussion Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 22 , Issue 2 , November 1992 , pp. 235 - 246
Copyright
Copyright © International Actuarial Association 1992

References

Beekman, J.A. (1968) Collective risk results. TSA 20, 182–199.Google Scholar
Beekman, J.A. (1969) A ruin function approximation. TSA 21, 41–48, 275–282.Google Scholar
Bowers, N. L. Jr., Gerber, H.U., Hickman, J.C., Jones, D. A. and Nesbitt, C. J. (1986) Actuarial Mathematics. Society of Actuaries, Schaumburg, Illinois.Google Scholar
Brockett, P., Goovaerts, M. and Taylor, G. (1991) The Schmitter problem. ASTIN Bulletin 21, 129–132.CrossRefGoogle Scholar
Chan, B. (1990) Ruin probability for translated combination of exponential claims. ASTIN Bulletin 20, 113–114.CrossRefGoogle Scholar
Cramér, H. (1955) Collective Risk Theory. Skandia Jubilee Volume, Nordiska Bokhandeln, Stockholm.Google Scholar
Dufresne, F. and Gerber, H.U. (1988) The probability and severity of ruin for combination of exponential claim amount distribution and their translations. Insurance: Mathematics and Economics 7, 75–80.Google Scholar
Dufresne, F. and Gerber, H. U. (1989) Three methods to calculate the probability of ruin. ASTIN Bulletin 19, 71–90.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H.U. (1991) Rational ruin problems — a note for the teacher. Insurance: Mathematics and Economics 10, 21–29.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. 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
Kaas, R. (1991) The Schmitter problem and a related problem: a partial solution. ASTIN Bulletin 21, 133–146.CrossRefGoogle Scholar
Mereu, J.A. (1972) An algorithm for computing expected stop-loss claims under a group life contract. TSA 24, 311–320.Google Scholar
Reckin, G.E., Schwark, D.J. and Snyder, J.B. II (1984) Practical applications of the ruin function. TSA 36, 453–477.Google Scholar
Schmitter, H. (1990) The ruin probability of a discrete claims distribution with a finite number of steps. XXII ASTIN Colloquium, Montreux, Switzerland.Google Scholar
Seah, E. S. (1990) Computing the probability of eventual ruin. TSA 42, 421–446.Google Scholar
Shiu, E.S.W. (1984) Discussion on “Practical applications of the ruin function”. TSA 36, 480–486.Google Scholar
Shiu, E.S.W. (1987) Convolution of uniform distribution and ruin probability. Scandinavian Actuarial Journal 70, 191–197.CrossRefGoogle Scholar
Shiu, E.S.W. (1989) Ruin probability by operational calculus. Insurance: Mathematics and Economics 8, 243–250.Google Scholar
Täcklind, S. (1942) Sur le risque de ruine dans des jeux inéquitables. Scandinavian Actuarial Journal, 1–42.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar