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.

在本文中，我扩展了 Bernhardt 和 Donnelly (2019) 处理现代显性联合养老保险的工作，作为在特定遗赠动机下从固定缴款养老金罐中提供收入的一种方式。本文件的一个主要特点是它放宽了投资于联合养老保险和遗赠账户的固定比例的假设。在使遗赠比例成为一个额外的控制函数时，我获得了在恒定的相对风险规避效用下的部分消费率、财富、遗赠金额和遗赠比例的封闭形式解，迄今为止是不可用的。我证明了最优遗产比例是最优部分消费率和一个取幂的遗产参数的乘积。我表明在某些情况下，例如非常高的遗赠动机，生命周期效用最大化策略将需要负死亡率信用，类似于支付人寿保险费的成员。使用英国国家统计局生命表探索典型场景。