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

摘要

Equations (2a), (2b), (3c), (3d), (4a), (4b), and (5a) have clear economic interpretations in terms of stock and flows. In each equation, the left side is the stock of 1, and the right side is the flow generated by the stock. We have focused on three decompositions of the stock of 1: a term annuity, including a tontine; whole life annuity; and a temporary life annuity, term insurance, and a pure endowment. The latter is a generalization of the decomposition featured in our previous paper. A term annuity, with or without residual payment, has smaller annual payments than a tontine's payments to survivors, but the expected present value of all payments to survivors and deceased members of the tontine are the same as a term annuity. In the example in Table 2, we saw that the whole life annuity had annual payments worth approximately 53 percent and insurance worth 47 percent of the decomposition of 1. These percentages will vary with i because smaller i reduces the relative value of annual payments and increases the value of residual insurance. For example, at i = .03, annuity payments comprise 38 percent and insurance 62 percent of 1. When 1 was decomposed into a temporary life annuity, term insurance, and pure endowment insurance, the term annuity was 49 percent of the value of 1, 35 percent was term insurance, and 16 percent was pure endowment. The value of the temporary annuity declines and insurances increase in value as i declines. For example, if i= .03, the relative proportions are 35 percent, 42 percent, and 23 percent, respectively. Finally, except for (4a) and (4b), there is an obvious dependence on n, the period over which the yield is spread.