Decompositions of 1 That Incorporate Growth

摘要

We have developed generalizations of formulae presented in our two previous papers in this series. Generalizations allow for payments to grow in both additive and multiplicative decompositions of 1 using the NDR. We also generalized the decomposition of 1 consisting of whole life annuity and insurance components by allowing whole life annuity payments to grow. Finally, we considered a tontine that provided higher payments in early retirement and imposed negative growth that was offset by positive mortality subsidies from dead to living tontine members. This type of tontine might be of interest to people who want high incomes in early retirement (when relatively young and healthy), settle for somewhat lower income in middle retirement years (when more likely to be more sedentary but not dependent on others), and have much higher incomes if they live to their 90s and beyond (when more likely to need services of care givers and others in daily life).