The Utility Value of Longevity Risk Pooling: Analytic Insights

The consensus among researchers is that (some) longevity risk pooling is the optimal strategy for drawing down wealth in retirement, and a robust literature has developed around its measurement via annuity equivalent wealth. However, most of the published work is conducted numerically, and authors usually report only a handful of limited values. In this article we derive closed-form expressions for the value of longevity risk pooling with fixed life annuities under constant relative risk aversion preferences. We show, for example, that this value converges to e−1≈65% when the interest rate happens to be the inverse of life expectancy, remaining lifetimes are exponentially distributed, and utility is logarithmic. In general the various formula we derive match previously published numerical results when properly calibrated to discrete time and tables. More importantly, we focus attention on the incremental utility from annuitization when the retiree is already endowed with preexisting pension income such as Social Security benefits. Indeed, because of the difficulty in working with the so-called wealth depletion time in lifecycle models, we believe this is an area that hasn’t received proper attention in the literature. Overall, our article offers an assortment of tools to help explain the value of longevity risk pooling.