We minimize the probability of lifetime ruin in a deterministic financial and insurance model, although the investor's time of death is random, with an age-dependent force of mortality. By contrast with the traditional anything-anytime annuitization model (that is, individuals can annuitize any fraction of their wealth at anytime), the individual only purchases life annuity income gradually, using a bounded, absolutely continuous rate. As in the anything-anytime annuitization case, we find that it is optimal for the individual not to purchase additional annuity income when her wealth is less than a specific linear function of her existing annuity income, which we call the buy boundary. Interestingly, we find the buy boundary in our model is identical to the one in the anything-anytime annuitization model. However, there is a separate threshold, which we call the safe level. (This threshold degenerates to the buy boundary in the anything-anytime annuitization model.) When wealth is greater than the safe level, the minimum probability of lifetime ruin is zero; when wealth lies between the buy boundary and the safe level, the individual's best choice is to purchase annuity income at the maximum allowable rate.

尽管投资者的死亡时间是随机的，并且死亡率与年龄有关，但我们在确定性金融和保险模型中将终生破产的可能性降至最低。与传统的随时年金化模式（即个人可以随时将其财富的任何部分进行年金化）相比，个人仅使用有界的、绝对连续的利率逐步购买终身年金收入。与随时年金化的情况一样，我们发现，当个人的财富小于现有年金收入的特定线性函数（我们称之为购买边界）时，不购买额外年金收入是最佳选择。。有趣的是，我们发现模型中的购​​买边界与随时年金模型中的购​​买边界相同。然而，有一个单独的阈值，我们称之为安全级别。（这个阈值退化为随时年金模型中的购​​买边界。）当财富大于安全水平时，一生破产的最小概率为零；当财富大于安全水平时，一生破产的最小概率为零；当财富处于购买边界和安全水平之间时，个人的最佳选择是以最高允许利率购买年金收入。