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On Optimal Cancellation of Policies

Published online by Cambridge University Press:  29 August 2014

Hans U. Gerber
Hans U. Gerber*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan
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One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.

In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?

Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.

Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 9 , Issue 1-2 , January 1977 , pp. 125 - 138
Copyright
Copyright © International Actuarial Association 1977

References

[1]Blackwell, D., “Discounted dynamic programming,” Ann. Math. Statist. 36, 226–235 (1965).CrossRefGoogle Scholar
[2]Breiman, L., “Stopping-rule problems,” Applied Combinatorial Mathematics (ed. by Beckenbach, ), 284–319, Wiley, New York (1964).Google Scholar
[3]Breiman, L., Probability, Addison-Wesley, Reading (1968).Google Scholar
[4]Chow, Y. S., Robbins, H. and Siegmund, D., Great expectations: The theory of optimal stopping, Houghton Mifflin, Boston (1971).Google Scholar
[5]Cox, D. R. and Miller, H. D., The theory of stochastic processes, Methuen, London (1965).Google Scholar
[6]Cozzolino, J. M. and Freifelder, L., “Underwriting individual drivers: A sequential approach,” forthcoming in the Transactions of the Casualty-Actuarial Society.Google Scholar
[7]DeGroot, M. H., Optimal statistical decisions, McGraw-Hill, New York (1970).Google Scholar
[8]Sirjaev, A. N., Statistical Sequential Analysis (optimal stopping rules) Transactions of Mathematical Monographs, Volume 38, American Mathematical Society, Providence, Rhode Island (1973).Google Scholar
[9]Feller, W., “An Introduction To Probability Theory And Its Applications”, vol. 2. Wiley, New York (1966).Google Scholar