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**The expected utility applied to reinsurance.**
*(English)*
Zbl 0658.62123

Risk, decision and rationality, Sel. Pap. 3rd Int. Conf. Found. Appl. Utility, Risk Dec. Theories, FUR-3, Theory Decis. Libr., Ser. B 9, 679-689 (1988).

[For the entire collection see Zbl 0658.00021.]

Two of the most important questions to be answered by an insurer in relation to reinsurance for part of his portfolio are:

a) What form of reinsurance is optimal? b) How much reinsurance is optimal?

In this paper we attempt to provide some answers to these questions. We consider n risks (each risk is either a single policy or a group of policies) and we assume that the insurance company has the opportunity to reinsure each of these risks through a quota-share treaty or an excess of loss treaty or any combination of the two. By a combination of quota- share with excess of loss we mean a quota-share treaty topped by an excess of loss treaty. Combinations of quota-share with excess of loss are often used in motor insurance.

We will prove that the insurance company maximizes its expected utility of wealth with respect to the n risks, for an exponential utility function, if and only if the insurer’s expected utility of wealth with respect to each of the risks is maximized. It will be maximized reinsuring each of the risks by a pure excess of loss treaty or by a combination of quota-share with excess of loss but never by a pure quota- share treaty. Furthermore if for a risk quota-share, reinsurance is, in the obvious sense, more expensive than excess of loss, then excess of loss is optimal.

Two of the most important questions to be answered by an insurer in relation to reinsurance for part of his portfolio are:

a) What form of reinsurance is optimal? b) How much reinsurance is optimal?

In this paper we attempt to provide some answers to these questions. We consider n risks (each risk is either a single policy or a group of policies) and we assume that the insurance company has the opportunity to reinsure each of these risks through a quota-share treaty or an excess of loss treaty or any combination of the two. By a combination of quota- share with excess of loss we mean a quota-share treaty topped by an excess of loss treaty. Combinations of quota-share with excess of loss are often used in motor insurance.

We will prove that the insurance company maximizes its expected utility of wealth with respect to the n risks, for an exponential utility function, if and only if the insurer’s expected utility of wealth with respect to each of the risks is maximized. It will be maximized reinsuring each of the risks by a pure excess of loss treaty or by a combination of quota-share with excess of loss but never by a pure quota- share treaty. Furthermore if for a risk quota-share, reinsurance is, in the obvious sense, more expensive than excess of loss, then excess of loss is optimal.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

91B16 | Utility theory |