Optimal Management of an Insurer's Exposure in a Competitive General Insurance Market

The qualitative behaviour of the optimal premium strategy is determined for an insurer in a finite and an infinite market using a deterministic general insurance model. The optimisation problem leads to a system of forward-backward differential equations obtained from Pontryagin's Maximum Principle. The focus of the modelling is on how this optimisation problem can be simplified by the choice of demand function and the insurer's objective. Phase diagrams are used to characterise the optimal control. When the demand is linear in the relative premium the structure of the phase diagram can be determined analytically. Two types of premium strategy are identified for insurer in an infinite market, and which is optimal depends on the existence of equilibrium points in the phase diagram. In a finite market there are four more types of premium strategy, and optimality depends on the initial exposure of the insurer and the position of a saddle point in the phase diagram. The effect of a nonlinear demand function is examined by perturbing the linear price function. An analytical optimal premium strategy is also found using inverse methods when the price function is nonlinear