The paper considers systems of payments which are not fully funded, i.e partially funded or fully unfunded. Generally, the objective is to be able to establish a premium formula which is consistent with long term planning as to e.g. a target rate of funding, limited variation in premiums from year to year, etc. The premium formulas considered are those which relate premiums to prior years' premiums, claims experience and accumulated funds. These questions are reviewed in Section 1 which suggests the use of control theory. Section 2 formulates and provides a formal solution to the problem.

Section 3 provides a couple of results which relate premiums to targeting of accumulated funds.

Subsequent sections consider the construction of premium formulas. It is emphasized that the intention is not to supply a definitive set of instructions as to how the premium formula might be constructed, but rather to illustrate some of the principles relevant to it.

In particular, two numerical examples are provided in § 6.2. Premium formulas are constructed which appear to respond reasonably satisfactorily to simulated claims experience.

It is found that accurate prediction of future claims escalation is crucial to the operation of formulas of the linear control theory type (§ 6.2.1). Brief comment on possible further research aimed at dealing with this aspect of the question is given in Section 7.