Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-11T22:12:41.632Z Has data issue: false hasContentIssue false
  • English
  • Français

The Schmitter Problem and a Related Problem: A Partial Solution

Published online by Cambridge University Press:  29 August 2014

R. Kaas
R. Kaas*
Affiliation:
University of Amsterdam, the Netherlands
*
Institute for Actuarial Science and Econometrics, University of Amsterdam, Jodenbreestraat 23, NL-1011 NH Amsterdam, the Netherlands.
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.

At the 1990 ASTIN-colloquium, Schmitter posed the problem of finding the extreme values of the ultimate ruin probability ψ(u) in a risk process with initial capital u, fixed safety margin θ, and mean μ and variance σ2 of the individual claims. This note aims to give some more insight into this problem. Schmitter's conjecture that the maximizing individual claims distribution is always diatomic is disproved by a counterexample. It is shown that if one uses the distribution maximizing the upper bound e−Ru to find a ‘large’ ruin probability among risks with range [0, b], incorrect results are found if b is large or u small.

The related problem of finding extreme values of stop-loss premiums for a compound Poisson (λ) distribution with identical restrictions on the individual claims is analyzed by the same methods. The results obtained are very similar.

Type
Discussion Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 21 , Issue 1 , April 1991 , pp. 133 - 146
Copyright
Copyright © International Actuarial Association 1991

References

REFERENCES

Bowers, N. L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itasca, Illinois.Google Scholar
Brockett, P.L., Goovaerts, M.J. and Taylor, G.C. (1991) The Schmitter problem. ASTIN Bulletin 21, 129–132.CrossRefGoogle Scholar
Gerber, H.U. (1990) From the convolution of uniform distributions to the probability of ruin. Mitteilungen der VSVM Heft 2/1989, 283–292.Google Scholar
Goovaerts, M.J., De Vylder, F. and Haezendonck, J. (1984) Insurance premiums. North-Holland, Amsterdam.Google Scholar
Goovaerts, M.J., Kaas, R., Van Heerwaarden, A.E. and Bauwelinckx, T. (1990) Effective Actuarial Methods. North-Holland, Amsterdam.Google Scholar
Kaas, R. and Goovaerts, M.J. (1986) Bounds on stop-loss premiums for compound distributions. ASTIN Bulletin XVI, 1, 13–17.CrossRefGoogle Scholar
Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.A. (1986) Numerical recipes; the art of scientific computing. Cambridge University Press.Google Scholar
Schmitter, H. (1990) The ruin probability of a discrete claims distribution with a finite number of steps. Paper presented at the XXII ASTIN-colloquium, Montreux, Switzerland.Google Scholar
Shiu, E.S.W. (1989) Ruin probability by operational calculus. Insurance: Mathematics and Economics 8, 243–249.Google Scholar
Steenackers, A. and Goovaerts, M.J. (1990) Bounds on slop-loss premiums and ruin probabilities for given values of the mean, variance and maximal value of the claim size. Paper presented at the XXII ASTIN-colloquium, Montreux, Switzerland.Google Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York, Reprinted by Krieger, Huntington, NY (1977).Google Scholar