Intra-group Risk Sharing Under Financial Fairness

We examine intra-group risk sharing under the assumption that prices are given
exogenously. Two situations are considered: one in which the entities that constitute the group choose to trade only among each other, and one in which the entities have limited access to the external market. In the first situation we prove that, if agents are expected utility maximizers, there is a unique risk-sharing rule that preserves value according to the exogenous rule and that is Pareto optimal among feasible allocations. This provides an extension of a result by Balasko (1979). In the second situation we give necessary and sufficient conditions for the internal market to be completable in the sense that all agents can reach all positions, subject only to the budget constraint, by combining their own partial access to the external market with internal trades under the given pricing rule.