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Pareto-Optimal Risk Exchanges and Related Decision Problems

Published online by Cambridge University Press:  29 August 2014

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In various branches of applied mathematics the problem arises of making decisions to reconcile conflicting criteria. One example is the classical statistical problem, where a type 1 error cannot be arbitrarily reduced without increasing the probability for a type 2 error. Another example, quite familiar to actuaries, is graduation, where a compromise between smoothness and fit has to be reached. This motivates the concept of Pareto-optimal decisions, which is discussed in section 2. There is a simple method, maximizing a weighted average of the scores, to obtain certain Pareto-optimal decisions. In section 3 a condition is given, which is satisfied in most applications, that guarantees that all the Pareto-optimal decisions can be found by this method. This is applied in section 4, where the problem of risk exchange between n insurance companies is considered. The original model of Borch is generalized: it is assumed that some of the companies are not willing to contribute more than a certain fixed amount towards the aggregate loss of the other companies. The theorem in section 4 gives a characterization of all the Pareto-optimal risk exchanges. Because of the restrictions, these risk exchanges do not just depend on the combined surplus (which would amount to pooling) in general, and can be found by an algorithm. One benefit of this generalization of Borch's Theorem is that two seemingly unrelated results (optimality of a stop loss contract, and optimality of certain dividend formulas in group insurance) follow from it as special cases.

Type
Research Article
Copyright
Copyright © International Actuarial Association 1978

References

REFERENCES

[1] Arrow, K., (1963), “Uncertainty and the Welfare Economics of Medical Care”, American Economic Review, LIII, 941–73.Google Scholar
[2] Borch, K., (1962), “Equilibrium in a reinsurance market”, Econometrica 30, 424–44.CrossRefGoogle Scholar
[3] Bühlmann, H., 1970, Mathematical Methods in Risk Theory. New York: Springer.Google Scholar
[4] DuMouchel, W. H., (1968), “The Pareto-optimality of a n-company reinsurance treaty”, Skandinavisk Aktuarietidskrift 51, 165–70.Google Scholar
[5] DeGroot, M. H., 1975. Probability and Statistics, Reading: Addison-Wesley.Google Scholar
[6] Greville, T. N. E., (1973), Part Five Study Notes on Graduation, Chicago: Society of Actuaries.Google Scholar
[7] Jones, D. A. and Gerber, H. U., (1974), “Dividend Formulas in Group Insurance”, Transactions of the Society of Actuaries 26, 7793.Google Scholar