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Optimal risk allocation in reinsurance networks. (English) Zbl 1416.91155

Summary: In this paper we consider reinsurance or risk sharing from a macroeconomic point of view. Our aim is to find socially optimal reinsurance treaties. In our setting we assume that there are \(n\) insurance companies, each bearing a certain risk, and one representative reinsurer. The optimization problem is to minimize the sum of all capital requirements of the insurers where we assume that all insurance companies use a form of range-value-at-risk. We show that in case all insurers use value-at-risk and the reinsurer’s premium principle satisfies monotonicity, then layer reinsurance treaties are socially optimal. For this result we do not need any dependence structure between the risks. In the general setting with range-value-at-risk we obtain again the optimality of layer reinsurance treaties under further assumptions, in particular under the assumption that the individual risks are positively dependent through the stochastic ordering. Our results include the findings in [Y. Chi and K. S. Tan, ibid. 52, No. 2, 180–189 (2013; Zbl 1284.91216)] in the special case \(n=1\). At the end, we discuss the difference between socially optimal reinsurance treaties and individually optimal ones by looking at a number of special cases.

MSC:

91B30 Risk theory, insurance (MSC2010)
60E15 Inequalities; stochastic orderings

Citations:

Zbl 1284.91216
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References:

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