Financial/Actuarial Mathematics Seminar

This paper examines the optimal annuitization, investment and consumption strategies of a utility-maximizing retiree who can invest in a market with a risky and a riskless asset and who can purchase a reversible life annuity. The surrender charge of a life annuity is a proportion of its purchasing value. We focus on the impact of a proportional surrender charge on an individualÕs optimal strategy. 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 find that, when the proportional surrender charge is smaller than some critical value, an individual keeps wealth to one side of a separating ray in wealth-annuity space by purchasing more annuities. The slope of this ray decrease as the the proportional surrender charge increases; that is, an individual is more willing to annuitize under a smaller surrender charge. When her wealth reaches zero, the individual continues to invest in the risky asset by borrowing from the riskless account and surrenders just enough annuity income to keep her wealth non-negative when needed. In contrast, when the proportional surrender charge is larger than this critical value, an individual does not invest in the risky asset when her wealth is zero. Instead, the individual reduces her consumption to a level lower than annuity income in order to accumulate wealth. More surprisingly, we find that in the case when surrender charge is larger than the critical value, the optimal annuitization, investment, consumption strategies as well as the maximized utility of an retiree do not depend on the size of surrender charge. An individual behaves as if the annuity is not reversible at all and does not surrender existing annuities under any circumstance. All these optimal solutions are established via duality of the primal problem. Numerical examples are also given to illustrate our results.
Joint work with Virginia Young.