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Optimal Insurance Arrangements

Published online by Cambridge University Press:  29 August 2014

Karl Borch
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In a recent paper on the theory of demand for insurance Arrow [1] has proved that the optimal policy for an insurance buyer is one which gives complete coverage, beyond a fixed deductible. The result is proved under very general assumptions, but its content can be illustrated by the following simple example.

Assume that a person is exposed to a risk which can cause him a loss x, represented by a stochastic variable with the distribution F(x). Assume further that he by paying the premium P(y) can obtain an insurance contract which will guarantee him a compensation y(x), if his loss amounts to x. The problem of our person is to find the optimal insurance contract, i.e. the optimal function y(x), when the price is given by the functional P(y).

In order to give an operational formulation to the problem we have outlined, we shall assume that the person's attitude to risk can be represented by a Bernoulli utility function u(x), and we shall write S for his “initial wealth”. His problem will then be to maximize

when the functional P(y) is given, and y(x) є Y. The set Y can be interpreted as the set of insurance policies available in the market. It is, natural to assume that o ≤ y(x) ≤ x, but beyond this there is no need for assuming additional restrictions on the set Y.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 8 , Issue 3 , September 1975 , pp. 284 - 290
Copyright
Copyright © International Actuarial Association 1975

References

[1]Arrow, K. J., “Optimal Insurance and Generalized Deductibles”, Skandinavian Actuarial Journal, 1974, pp. 1–42.Google Scholar
[2]Borch, K., “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, Transactions of the 16th International Congress of Actuaries, Vol.2, pp. 579–610.Google Scholar
[3]Borch, K., “The Safety Loading of Reinsurance Premiums”, Skandinavisk Aktuarietidskrift, 1960, pp. 163–184.Google Scholar
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[5]Bühlmann, H., Mathematical Methods in Risk Theory, Springer Verlag, 1970.Google Scholar
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