Optimal exposure strategies in insurance

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Abstract

Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'{e}r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'{e}vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

Keyword(s)

Dependent thinning; Levy Processes; Refracted Levy processes; Risk Theory; Stochastic processes; exit problem; markov chains; optimal control; point processes; ruin probability; stochastic control; underwriting competitive market; verification theorem

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