Decompositions of 1 That Incorporate Growth

This paper is the third, and final, paper of a three-part series. The purpose of these papers was to develop several decompositions of 1. We began with the historical development of the formula 1 ¼ iax þ (1 þ i)Ax that has its origin in work by Abraham de Moivre, Thomas Simpson, and Richard Price in the 18th and 19th centuries. The equation decomposes 1 into life annuity and life insurance components and remains the basis of life estate and remainder calculations in taxation matters. Our second paper decomposed three types of annuities: (1) an ordinary term annuity 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. Finally, the current paper adds growth to annuities and provides yet another generalization of 1¼iaxþ(1þi)Ax but with growth in annuity payments and insurance. The combination of all three papers provides a partial taxonomy of future payments that 1 can generate.