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An insurance company offers an insurance contract (p, K), consisting of a premium p and a deductible K. In this paper, we consider the problem of choosing the premium optimally as a function of the deductible. The insurance company is facing a market of N customers, each characterized by their personal claim frequency, α, and risk aversion, β. When a customer is offered an insurance contract, she/he will, based on these characteristics, choose whether or not to insure. The decision process of the customer is analyzed in detail. Since the customer characteristics are unknown to the company, it models them as i.i.d. random variables; A 1 , . . . , A N for the claim frequencies and B 1 , . . . , B N for the risk aversions. Depending on the distributions of A i and B i , expressions for the portfolio size n(p; K) ∈ [0, N] and average claim frequency α(p; K) in the portfolio are obtained. Knowing these, the company can choose the premium optimally, mainly by minimizing the ruin probability.

Two insurance companies I 1 , I 2 with reserves R 1 ( t ) , R 2 ( t ) compete for customers, such that in a suitable differential game the smaller company I 2 with R 2 ( 0 ) < R 1 ( 0 ) aims at minimizing R 1 ( t ) − R 2 ( t ) by using the premium p 2 as control and the larger I 1 at maximizing by using p 1 . Deductibles K 1 , K 2 are fixed but may be different. If K 1 > K 2 and I 2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed rates of claim arrivals, explicit equilibrium premiums are obtained, and shown to depend on the running reserve difference. The analysis is based on the diffusion approximation to a standard Cramér-Lundberg risk process extended to allow investment in a risk-free asset.

We view the retail non-life insurance decision from the perspective of the insured. We formalize different consumption–insurance problems depending on the flexibility of the insurance contract. For exponential utility and power utility we find the optimal flexible insurance decision or insurance contract. For exponential utility we also find the optimal position in standard contracts that are less flexible and therefore, for certain nonlinear pricing rules, lead to a welfare loss for the individual insuree compared to the optimal flexible insurance decision. For the exponential loss distribution, we quantify a significant welfare loss. This calls for product development in the retail insurance business.

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