A Perpetuity, Life Annuity, and Life Insurance Related to a Decomposition of 1

摘要

The well-known actuarial equation 1 - ia_x = {1+ i)A_x has its roots in work by de Moivre and Simpson approximately 275 years ago. They clearly understood the left side of the equation as illustrated by the problems in Section Ⅱ and gave meaning to (1 - ia_x)as a residual without explicitly mentioning insurance. The answer to the question posed by Price in Section Ⅲ draws heavily on de Moivre and Simpson and shows the insurance component in the equation was understood as well. The equation continues to be of contemporary importance. For example, Table S in Internal Revenue Service's Actuarial Tables (Publication 1457, 2014) contains entries with column headings labeled Annuity, Life Estate, and Remainder, the concepts we denote by a_x, ia_x, and 1 - ia_x, respectively, in this note. At age x = 50 and i = .02, Table S has a_(50) = 21.5904, .02a_(50) = .43181, and 1 - .02a_(50) = .56819. In words, when the unit 1 denotes $1 and i= .02, then one dollar supports a life annuity of $.02 per year for a 50-year-old person; and the expected present value of this annuity is ia_(50) = .02(21.5904) = $.43181. The remainder has an expected present value of 1 - ia_x = 1 - .43181 = $.56819. From the equation 1 - ia_x = (1 + i)A_x, we know that the latter expected present value is equivalent to an insurance payment of $1.02 at the end of the year of death of the 50-year-old annuitant. That is, $1 sustains a life annuity of $.02 per year and an insurance payment of $1.02; the expected present value of the former is $.43181 and the latter $.56819.