Optimal Life Insurance Purchase Problems in a Life-cycle Model

Abstract

We first study optimal consumption/investment of a retiree who has luxury bequest motives and faces the nonnegative bequest constraint. His lifetime is uncertain but actuarially fair life insurance-annuity policies are available. We obtain a closed form solution by using a dynamic programming method, and investigate the effects of lifetime uncertainty and the presence of life insurance-annuity on the consumption/investment policies.

We also consider a consumption-investment problem where the relevant economic conditions change depending on whether the wealth exceeds a critical level or not. We propose a dynamic programming method to solve the problem by dividing the problem into subproblems by wealth levels and imposing freeze condition at the boundaries. We then join the solutions of the subproblems so that the resulting value function is piecewise $C^2$. The methodology is illustrated through an application to a problem with nonnegative life insurance constraint.