Optimal Purchasing of Deferred Income Annuities When Payout Yields are Mean-Reverting*

Abstract

We determine the optimal lifecycle purchasing strategy for deferred income annuities (DIAs)—which are distinct from single-premium income annuities (SPIAs)—for an individual who wishes to maximize the expected utility of his/her annuity income at a fixed time in the future. In contrast to the vast portfolio-choice literature for SPIAs, we focus on the stochasticity of the DIA’s payout yield and address concerns that rates are currently “too low” to justify irreversible annuitization. We assume a mean-reverting model for payout yields and show that a risk-neutral consumer who wishes to maximize his/her expected retirement income should wait until yields reach a threshold—which lies above historical averages—and then purchase the DIA in one lump sum. In contrast, a risk-averse consumer who is concerned the payout yield will remain below average for an extended period and worries about losing mortality credits while waiting, should employ a barrier purchasing strategy, as in the portfolio choice problem under transaction costs. We illustrate how this insight is applied in the context of annuitization. In fact, the optimal behavior of a risk-averse consumer resembles an asymmetric dollar-cost averaging strategy, with a portion of the DIA-budget spent even while payout rates are below historical averages. As part of our analysis we offer an easy-to-use asymptotic approximation for the optimal purchasing strategy (threshold) and provide some numerical examples to illustrate the concept.