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, \(\alpha\), and risk aversion, \(\beta\). When a customer is offered an insurance contract, she will based on these characteristics choose whether or not to insure. The decision process of the customer is analyzed in details. Since the customer characteristics are unknown to the company, it models them as iid random variables; \(A_1, \ldots, A_N\) for the claim frequencies and \(B_1, \ldots, B_N\) for the risk aversions. Depending on the distributions of \(A_i\) and \(B_i\), expressions for the portfolio size \(n(p;K)\in [0,N]\) and average claim frequency \(\alpha(p;K)\) in the portfolio are obtained. Knowing these, the company can choose the premium optimally, mainly by minimizing the ruin probability.

Keywords: microeconomic insurance; customer characteristics; portfolio size; average claim frequency; ruin theory