Closed-form Solutions for an Explicit Modern Ideal Tontine with Bequest Motive

Abstract: In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way of providing income under a specified bequest motive, from a defined contribution pension pot. A key feature of the present paper is that it relaxes the assumption of fixed proportions invested in tontine and bequest accounts. In making the bequest proportion an additional control function I obtain, hitherto unavailable, closed-form solutions for the fractional consumption rate, wealth, bequest amount, and bequest proportion under a constant relative risk averse utility. I show that the optimal bequest proportion is the product of the optimum fractional consumption rate and an exponentiated bequest parameter. I show that under certain circumstances, such as a very high bequest motive, a life-cycle utility maximisation strategy will necessitate negative mortality credits analogous to a member paying life insurance premiums. Typical scenarios are explored using UK Office of National Statistics life tables.