Equilibrium in a Reinsurance Syndicate; Existence, Uniqueness and Characterization

This paper attempts to give an overview of the pricing of risks in a pure exchange economy, where trade takes place at time zero and where uncertainty is revealed at time one. An economic equilibrium model under uncertainty is formulated, where conditions characterizing a Pareto optimal exchange equilibrium are derived. We present two sets of sufficient conditions for the existence of an equilibrium, and demonstrate how equilibria can be characterized through several examples. Uniqueness of equilibrium is also discussed. Special attention is given to the principal components that the premiums in a reinsurance market must depend upon. We also apply the general theory to the risk exchange problem between a policyholder and an insurer, and in particular we compute market premiums of the resulting optimal contracts. It is emphasized throughout how the formulation of a competitive equilibrium, rather than merely a general risk exchange formulation, is of particular interest in deriving a well-defined and unique set of equilibrium premiums in an insurance market. The theory is put into a framework which is fruitful for extensions beyond the one-period case.