Maximizing the utility of consumption with commutable life annuities

Abstract

The purpose of this paper is to reveal the relation between commutability of life annuities and retirees’ willingness to annuitize. To this end, we assume the existence of commutable life annuities, whose surrender charge is a proportion of their actuarial 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 commutable life 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 exclude the actuarial value of her annuity income in calculating wealth; therefore, we do not allow the individual to borrow from her future annuity income because this income is contingent on her being alive.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, it is optimal for the retiree to behave as if annuities are not commutable. We provide numerical examples to illustrate our results.