  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. At the end, we discuss the difference between socially
optimal reinsurance treaties and individually optimal ones by looking at a
number of special cases.
