Net Premiums in Stochastic Life Contingencies

A fundamental problem in actuarial science is the determination of the level of a premium payment plan necessary to meet future contingent obligations. In the modern stochastic version of life contingencies, this net level is determined by the principle that expected value of a certain loss random variable is set equal to zero. This principle is desirable in that it yields the same net premium level as under the traditional deterministic life contingencies. To achieve this reduction to the traditional framework, in stochastic contingencies the actuary must use care in selecting random variables to be used in calculating premiums. In this paper we use expectations of different random variables to define the notions of retrospective, prospective, individual, and aggregate net premiums. Unlike deterministic contingencies, these premiums are not equivalent. Their respective relationships are discussed in order to provide a deeper understanding of the notion of a net premium. The individual premium turns out to have desirable properties, and therefore this approach is extended from premium to reserving considerations.