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.
Highlights
Insurer 4’s Nash equilibrium premium is larger than Insurer 2’s, since she has higher risk aversion, resulting to larger reduction in her exposure volume
For all cases that we study the pure strategy Nash equilibrium exists and is unique
The pure strategy Nash equilibrium premiums are given as solutions to constrained optimisation problems
Summary
There is a continuous and strong interest in the modelling of insurance premiums in a competitive market Insurers determine their premiums in response to the premiums that are being offered by the competitor companies. In non-life insurance, are almost exclusively devoted on calculating the distribution of the underlying risk, following the underlying trends in distribution instead of formulating the underwriting strategies (Emms et al, 2007). Such approaches ignore the actions made by the competitors which affect the movement of premium rates and, eventually, the exposure of the company into the corresponding market (Taylor, 1986). Due to the complex nature of price competition, several modelling and pricing challenges arise, and this is the motivation of our paper
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