The Utility Value of Longevity Risk Pooling: Analytic Insights

Abstract

The consensus among scholars 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 (AEW). However, most of the published work is conducted numerically and authors usually report only a handful of limited values. In this paper we derive some 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 the square root of Euler's e -1 (=65%), when the interest rate is the inverse of life expectancy, lifetimes are exponentially distributed and utility is logarithmic. In general the various formulae 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 pre-existing pension income such as Social Security benefits. Indeed, due to the difficulty in working with the so-called wealth depletion time (WDT) in lifecycle models, we believe this is an area that hasn't received proper attention from actuarial researchers. Our paper offers tools to explain the value of longevity risk pooling.