Policyholders often decide to buy, renew, or cancel insurance based on the premium charged by the insurer compared with what they expect their claims will be. It is important for actuaries to consider the persistency of policyholders because the financial well-being of the insurer depends on spreading its risk over a large book of business. We use statistical decision theory to develop premium formulas that account for the past experience of a given policyholder, the experience of the entire collection of policyholders, and the likelihood of the policyholder renewing with or buying from a given insurer, that is, persistency.

We assume that the persistency of policyholders depends on the arithmetic difference between the premium charged and their anticipated claims. We extend the work of Taylor (1975) in which he obtains linear credibility formulas by minimizing loss functions that incorporate the persistency of policyholders. We consider Taylor's loss functions and other objective functions, including those that account for the amount of business the insurer writes or renews.

When the number of candidates is unknown, the problem of selecting the best during a sequence of interviews has many reasonable solutions. A simple condition for admissibility is established and it is shown that the class of Bayes solutions obtained by treating the number of candidates as a random variable is by no means complete. On the other hand, there is a single improper prior distribution for which the extended Bayes solutions constitute the whole family of admissible procedures.