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 her annuity income at a fixed time in the future. In contrast to the portfolio-choice literature for SPIAs, we focus on the "stochasticity" of the DIA's payout yield and address concerns that payout yields are currently too low to justify irreversible annuitization. After formulating and solving the optimal control problem analytically, our main qualitative result is that when payout yields are mean-reverting, a risk-neutral consumer who wishes to maximize 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. 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. We conclude with numerical examples calibrated to a time series of DIA prices and provide some estimates of the expected purchasing time.

Keywords: deferred income annuities, stochastic interest rates, optimal stopping, instantaneous control, retirement, pensions

JEL Classification: D14, G11, G22, J26