A life insurance policy is a contract between the insurer and another party known as the policyholder. In return for a payment of premiums, the insurer will pay a predetermined amount of money, known as a death benefit, upon the death of the policyholder. The amount of the benefit can vary with the time of death. In practice, this money will be paid immediately upon death (or more realistically, a short time after, to allow for processing the claim) but for mathematical convenience we assume in this chapter that it will be paid at the end of the year of death. For example, if the policyholder purchases a policy on January 1 and dies a week later, our assumption means that the death benefit will not be paid until December 31. We will consider the more realistic situation of payment at the moment of death in Chapter 8.
The reader should distinguish carefully between life annuity and life insurance contracts. The life annuity provides a sequence of periodic benefit payments. The typical life insurance contract provides only a single benefit payment, paid on the occasion of death.
Consider a policy on (x). Let bk be the amount that will be paid at time k + 1 for death between time k and time k + 1. We will refer to the vector
as the death benefit vector.
Notation The reader is cautioned that many authors ...