Pricing in a competitive stochastic insurance market

Abstract

This paper studies a one-period stochastic game to determine the optimal premium strategies of non-life insurers in a competitive market. Specifically, the optimal premium strategy is determined by the Nash equilibrium of an n-player game, in which each player is assumed to maximise the expected utility of terminal wealth. The terminal wealth is stochastic, since the number of policies and the size of claims are considered to be random variables. The total loss of each insurer is described by the collective risk model. The expected number of policies is affected by all the premiums in the market and further investigated by two distinct demand functions. Both models have an exponential functional form, that is characterised by market and price sensitivity parameters. The demand in the first model is zero for premiums above a given threshold, whereas the second model does not include such restriction. The pure strategy Nash equilibrium premiums are given as solutions to constrained optimisation problems. For the first model we prove the existence and uniqueness of a pure strategy Nash equilibrium, whereas for the second model we provide a formula when it exists. Two numerical examples are provided to illustrate the applicability of our findings.