An Annuitization Problem in the Tax-Deferred Annuity Model

Abstract

This paper examines the optimal annuitization, investment, and consumption strategies of an individual facing a time-dependent mortality rate in the tax-deferred annuity model and considers both the case when the rate of buying annuities is unrestricted and the case when it is restricted. At the beginning, by using the dynamic programming principle, we obtain the corresponding HJB equation. Since the existence of the tax and the time-dependence of the value function make the corresponding HJB equation hard to solve, firstly, we analyze the problem in a simpler case and use some numerical methods to get the solution and some of its useful properties. Then, by using the obtained properties and Kuhn–Tucker conditions, we discuss the problem in general cases and get the value functions and the optimal annuitization strategies, respectively.