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Improved Analytical Bounds for Some Risk Quantities

Published online by Cambridge University Press:  29 August 2014

Werner Hürlimann
Werner Hürlimann*
Affiliation:
Allgemeine Mathematik, Paulstr. 9, Winterthur-Leben
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Abstract

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Simple analytical lower and upper bounds are obtained for stop-loss premiums and ruin probabilities of compound Poisson risks in case the mean, variance and range of the claim size distribution are known. They are based on stop-loss extremal distributions and improve the bounds derived earlier from dangerous extremal distributions. The special bounds obtained in case the relative variance of the claim size is unknown, but its maximal value is known, are related to other actuarial results.

Keywords

analytical boundsstop-lossruin probabilitystochastic orderingsrelative varianceatomic distributions
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 26 , Issue 2 , November 1996 , pp. 185 - 199
Copyright
Copyright © International Actuarial Association 1996

References

Benktander, G. (1977). On the rating of a special stop-loss cover. ASTIN Bulletin 9, 33–41.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J. (1986). Actuarial Mathematics. Society of Actuaries.Google Scholar
Brockett, P., Goovaerts, M.J., Taylor, G. (1991). The Schmitter problem. ASTIN Bulletin 21, 129–32.CrossRefGoogle Scholar
Bühlmann, H., Gaoliardi, B., Gerber, H.U., Straub, E. (1977). Some inequalities for stop-loss premiums. ASTIN Bulletin 9, 75–83.CrossRefGoogle Scholar
Daykin, C.D., Pentikäinen, T., Pesonen, M. (1994). Practical risk theory for actuaries. Monographs on Statistics and Applied Probability 53, Chapman and Hall.Google Scholar
De Vylder, F., Goovaerts, M.J. (1982). Upper and lower bounds on stop-loss premiums in case of known expectation and variance of the risk variable. Bulletin of the Swiss Association of Actuaries, 149–64.Google Scholar
Gerber, H.U. (1979). An introduction to mathematical risk theory. Huebner Foundation, Irwin.Google Scholar
Goovaerts, M.J., De Vylder, F., Haezendonck, J. (1984). Insurance premiums. North-Holland.Google Scholar
Goovaerts, M.J., Kaas, R. (1986). Best bounds for positive distributions with fixed moments. Insurance: Mathematics and Economics, 87–92.Google Scholar
Hürlimann, W. (1993). Solvabilité et réassurance. Bulletin of the Swiss Association of Actuaries, 229–49.Google Scholar
Hürlimann, W. (1995). A stop-loss ordered extremal distribution and some of its applications. XXVI-th ASTIN Colloquium, Leuven, September 1995.Google Scholar
Kaas, R. (1991). The Schmitter problem and a related problem: a partial solution. ASTIN Bulletin 21, 133–46.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M.J. (1986). Bounds on stop-loss premiums for compound distributions. ASTIN Bulletin 16, 13–17.CrossRefGoogle Scholar
Kaas, R., Heerwaarden, A.E. Van, Goovaerts, M.J. (1994). Ordering of actuarial risks. Caire Education Series 1, Brussels.Google Scholar
Steenackers, A., Goovaerts, M.J. (1991). Bounds on stop-loss premiums and ruin probabilities. Insurance: Mathematics and Economics 10, 153–59.Google Scholar
Stoyan, D. (1973). Bounds for the extrema of the expected value of a convex function of independent random variables. Studia Scientiarum Mathematicarum Hungarica 8, 153–59.Google Scholar