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Calculating Ruin Probabilities via Product Integration

Published online by Cambridge University Press:  29 August 2014

Colin M. Ramsay  and
Miguel A. Usabel
Colin M. Ramsay
Affiliation:
Actuarial Science Program, University of Nebraska-Lincoln, Lincoln NE 68588-0426, USA
Miguel A. Usabel
Affiliation:
Dto. Economia Financiera y Actuarial, UCM Campus de Somosaguas, Ftad. CC Economicas, Madrid, Spain
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Abstract

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When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate methods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

Keywords

Integral equationconvergenceheavy-tailed distributions
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 27 , Issue 2 , November 1997 , pp. 263 - 271
Copyright
Copyright © International Actuarial Association 1997

References

REFERENCES

Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions. New York, N.Y.: Dover Publications.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986). Actuarial Mathematics. Ithasca, Ill.: Society of Actuaries.Google Scholar
Delves, L.M. and Mohamed, J.L. (1989) Computational Methods for Integral Equations. Cambridge, England: Cambridge University Press.Google Scholar
Dickson, D.C.M. (1989). “Recursive Calculation of the Probability and Severity of Ruin.” Insurance: Mathematics and Economics, 8, pp. 145–148.Google Scholar
Dickson, D.C.M., Egidio dos Reis, A.D. and Waters, H.R. (1995). “Some Stable Algorithms in Ruin Theory and Their Application.” Astin Bulletin 25, pp. 153–175.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (1991) “Recursive Calculation of Survival Probabilities.” Astin Bulletin 21, pp. 199–221.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8. Philadelphia, Pa.: University of Pennsylvania. (Distributed by Irwin, Homewood, IL.)Google Scholar
Goovaerts, M. and de Vylder, F. (1984) “A Stable Recursive Algorithm for Evaluation of Ultimate Ruin Probabilities.” Astin Bulletin, 14, pp. 53–59.CrossRefGoogle Scholar
Grandell, J. (1990). Aspects of Risk Theory. Springer Verlag, New York.Google Scholar
Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations. Philadelphia, Pa.: SIAM Studies in Applied Mathematics.CrossRefGoogle Scholar
Panjer, H.H. (1986) “Direct Calculation of Ruin Probabilities.” Journal of Risk and Insurance, 53, pp. 521–529.CrossRefGoogle Scholar
Panjer, H.H. and Wang, S (1993) “On the Stability of Recursive Formulas.” ASTIN Bulletin, 23, pp. 227–258.CrossRefGoogle Scholar
Ramsay, C.M. (1992a) “A Practical Algorithm for Approximating the Probability of Ruin.” Transactions of the Society of Actuaries, XLIV, 443–459.Google Scholar
Ramsay, C.M. (1992b) “Improving Goovaerts' and de Vylder's Stable Recursive Algorithm.” Astin Bulletin, 22, pp. 51–59.CrossRefGoogle Scholar
Thorjn, O. and Wikstad, N. (1977) “Calculation of Ruin Probabilities When the Claim Distribution is Lognormal.” Astin Bulletin, 9, pp. 231–246.CrossRefGoogle Scholar
Young, A. (1954) “Approximate Product Integration.” Proc. Royal Soc. London, Ser. A, 224, pp. 561–573.Google Scholar