Risk exchange with distorted probabilities

We study the equilibrium in a risk exchange, where agents' preferences are characterised by generalised (rank-dependent) expected utility, i.e. by a concave utility and a convex probability distortion (Quiggin, 1993). We obtain explicit results for the equilibrium price density, thus generalising Büuhlmann's (1980, 1984) formulas. For linear utility functions, we show that the agents' preference maximisation problem is equivalent to minimisation of portfolio risk and reformulate it in an insurance context, as premium maximisation under risk capital constraints induced by a coherent risk measure. We find that equilibrium is only reached if the same risk measure is applied throughout the market. Finally, we discuss the analogy of the exchange to a pooling arrangement and show that equilibrium prices can be obtained as marginal cost prices for an agent representing the collective (pool) of market players. From that perspective we discuss the links between equilibrium pricing, cooperative games and capital allocation.

Keywords: competitive equilibrium, risk exchange, generalised expected utility, cooperative games, coherent risk measures