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Optimal Risk Exchanges*

Published online by Cambridge University Press:  29 August 2014

Hans Bühlmann  and
William S. Jewell
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Abstract

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The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves.

To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 10 , Issue 3 , December 1979 , pp. 243 - 262
Copyright
Copyright © International Actuarial Association 1979

Footnotes

*

Presented at the 14th ASTIN Colloquium, Taormina, October 1978.

References

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