How to set the premium size is obviously one of the main decisions to be made by an insurance company, but quantitative studies of the effect of lowering or increasing the premium are few in the literature compared to other control problems, say for dividend pay-out or reinsurance.

Intuitively one expects that lowering the premium p per customer will increase the portfolio size n(p) (the number of people insured) but it is less clear what is the effect on the net income pn(p), on the gain, the ruin probability etc. One difficulty is to quantify the dependence of n(p) on p and of the phenomenon of adverse selection (a higher premium will bias the portfolio towards customers prone to many claims and thereby less attractive to the company).

We formulate a general criterion for a risk-averse customer to insure, based on calculations of his present values of the alternative strategies of insuring or not insuring and further parameters such as his discount rate d and the risk-free interest rate r<d. Implications for the classical empirical Bayes model with the rate of claims of a customer being a random variable are derived and extensions given to situations with customers having only partial information on their claim arrival rate or/and a stochastic d.

As example of a control problem, minimizing the ruin probability as function of p is studied. Joint work with Michael Taksar.