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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

REFERENCES

Beard, R. E., Pentikainen, T. and Pesonen, E. (1969). Risk Theory, Wiley, New York.Google Scholar
Borch, K. (1960). Reciprocal Reinsurance Treaties, Astin Bulletin, 1, 170–191.CrossRefGoogle Scholar
Borch, K. (1960). The Safety Loading of Insurance Premiums, Skandinavisk Aktuarietidskrift, 43, 163–184.Google Scholar
Borch, K. (1962). Equilibrium in a Reinsurance Market, Econometrica, 30, 424–444.CrossRefGoogle Scholar
Borch, K. (1968). General Equilibrium in the Economics of Uncertainty, in Risk and Uncertainty, Borch, K. and Mossin, J. (Eds.), Macmillan, London, 247–264.Google Scholar
Borch, K. (1974). The Mathematical Theory of Insurance, Heath, Lexington.Google Scholar
Bühlmann, H. and Jewell, W. S.Unicity of Fair Pareto-optimal Risk Exchanges, Astin Colloquium 78.Google Scholar
DuMouchel, W. (1968). The Pareto Optimality of an n-Company Reinsurance Treaty, Skandinavisk Aktuarietidskrift, 51, 165–170.Google Scholar
Gale, D. (1977). Fair Division of a Random Harvest, ORC 77-21, Operations Research Center, University of California, Berkeley.Google Scholar
Gerber, H. U. (1978). Pareto-Optimal Risk Exchanges and Related Decision Problems, Astin Bulletin, 10, 25–33.CrossRefGoogle Scholar
Lemaire, J. (1977). Echange de Risques Entre Assureurs et Théorie des Jeux, Astin Bulletin, 9, 155–179.CrossRefGoogle Scholar
Seal, H. (1969). Stochastic Theory of a Risk Business, Wiley, New York.Google Scholar
Wilson, R. (1968). The Theory of Syndicates, Econometrica, 36, 119–132.CrossRefGoogle Scholar