Dynamic Pricing of General Insurance in a Competitive Market

Abstract

A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer's objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation.In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer's risk aversion which leads to reduced market exposure and lower overall wealth generation.

Pricing general insurance with constraints

“…If one allows loss-leading strategies, then Emms (2006) found optimal strategies that lead to negative premiums for certain parameter sets. This is the ultimate lossleader and reflects the idea that the optimal strategy is to generate as much demand as possible initially in order to subsequently raise prices to generate a large profit.…”

“…Emms & Haberman (2005) formulated an accrued premium model, which although tractable did not yield an analytical expression for the optimal control. We use the model introduced in Emms (2006) because that model is easier to analyse with contraints.…”

“…The model in the body of Emms (2006) used the ratio of the breakeven premium to the market average premium to model claims and was unconstrained. Thus, the outstanding claims made by policyholders were not modelled directly.…”

“…If we wish to constrain the model then it is more appropriate to consider a model for which one can explicitly calculate the liabilities of the insurer. Therefore, we adopt the model described in the Appendix of Emms (2006), which uses a mean claim size rate to model claims. In this model the objective of the insurer is to maximise its net wealth, that is the wealth accumulated by selling insurance and settling claims including those claims, arising from in-force policies, that have yet to be settled.…”

“…Without constraints the optimal premium strategy is stochastic, and can be determined by solving the Bellman equation for the value function in a similar manner to Emms (2006). However, this is a difficult problem if there are constraints since the value function must be calculated in a state space of at least three dimensions and it is not smooth.…”