Maximizing the Utility of Consumption with Reversible Annuities

Abstract

The purpose of this paper is to reveal the relation between the reversibility of annuities and retirees' reluctance to annuitize. To this end, we assume the existence of reversible annuities, whose surrender charge is a proportion of their purchase value. We model a retiree as a utility-maximizing economic agent who can invest in a financial market with a risky and a riskless asset and who can purchase or surrender reversible annuities. We define the wealth of an individual as the total value of her risky and riskless assets, which is required to be non-negative during her lifetime. We solve this incomplete market utility maximization problem via duality arguments and obtain semi-analytical solutions. We find that the optimal annuitization strategy depends on the size of proportional surrender charge, with lower proportional surrender charges leading to more annuitization. We also find that full annuitization is optimal when there is no surrender charge or when the retiree is very risk averse. Surprisingly, we find that in the case for which the proportional surrender charge is larger than a critical value, an individual behaves as if annuities are not reversible at all. Numerical examples are given to illustrate our results.

Keywords: Reversible Annuities, Utility Maximization, Retirement, Optimal Investment, Stochastic Control, Free-Boundary Problem