Note on Random Survivorship Group Benefits

Consider a group of n independent lives age x where each life puts § 1 in a fund at time 0. The fund earns interest at rate t, and at the end of t years the accumulated value of the fund is divided equally among the survivors. The traditional approach to calculating the expected lump sum benefit per survivor from the initial group of n lives is based on the concept of a deterministic survivorship group. This approach ignores the stochastic nature of the survivorship process. In reality, the benefit per survivor is actually a random variable with an expected value which depends on the first inverse moment of a positive binomial random variable. Using GRAB'S and SAVAGE'S (1954) recursive formula for the first reverse moment, it is shown that the traditional approach yields a fairly accurate approximation to the solution even when one assumes a random number of survivors. KEYWORDS inverse moments; positive binomial distribution, actuarial accumulated value.