Optimal timing of an annuity purchase: a free boundary analysis

Locating the optimal age (time) at which to purchase an irreversible life annuity is a problem that has received considerable attention in the literature over the past decade. As Hainaut and Deelstra noticed (J. Econ. Dyn. Control 44 (2014) 124-146), one might think that individuals should annuitize their wealth in case their financial investments performs poorly, due to the fear that the performance might become even worst. An alternative reasoning suggest that, in order to have an acceptable annuity payment, individuals should switch to annuities if the financial performance are high enough. The annuitization problem is formulated as a continuous time optimal stopping problem. The optimal time of the annuity purchases depends both on the individual's wealth and life expectancy. It is characterized as the first time the individual's wealth crosses an unknown boundary, that divides the time-wealth plane into the so called continuation and stopping regions (where it is optimal respectively to postpone or immediately purchase an annuity). The problem of finding the optimal stopping boundary is converted into a parabolic free boundary problem. From this free boundary set-up we deduce an integral equation for the boundary, which can be used to compute its values numerically. A variety of numerical examples are presented in case of Gompertz-Makeham mortality and proportional hazard rate models.