Decompositions of 1 Related to Term Annuities, Whole Life Annuities, and Temporary Life Annuities

In the previous issue of the Journal, we gave a historical development of the equation 1 ¼ iax þ (1 þ i)Ax (Skoog and Ciecka, 2014), showing that 1 consists of whole life annuity ax and insurance Ax components. We used the symbol ‘‘1’’ to denote the natural number 1, but ‘‘1’’ also could be interpreted as one unit, such as one dollar or one million dollars. In this paper, we provide additive and multiplicative decompositions of 1. Additive decompositions have
remainder payments made to survivors or heirs. Multiplicative decompositions do not have residual payments. We decompose three types of annuities: (1) an ordinary term annuity and a fixed-term tontine with, and without, residual payments; (2) whole life annuities with, and without, residual payments; and (3) a generalization of 1 ¼ iax þ (1 þ i)Ax into a temporary life annuity, term insurance, and a pure endowment. The basic idea for this paper may be understood by asking the question: What streams of future payments can 1 generate?