The Analytics of a Single-Period Tontine

Abstract

A single-period tontine is an arrangement in which a group of members contribute to an investment pool, and after a fixed period of time, the pool is distributed to those members who are still alive. The distribution is made in unequal amounts based on member death probabilities and contribution amounts. In a companion article we apply the single-period tontine to propose products that we feel have commercial potential. Here, we focus on the analytics. The analysis applies not just to single-period tontines, but also to other arrangements that can be analyzed as a sequence of single-period tontines, such as pooled annuity funds.The core of the paper is the development of formulas for the mean and variance of the random amount that a surviving member receives in a single-period tontine. We use the formulas to resolve an open question about bias in pooled annuity funds and to show practical conditions under which it can be made negligible. We show how the provider can use the formulas to manage the subscription process, determining who is allowed to participate and how much they are allowed to contribute, so that the statistics of the tontine are favorably controlled. We use the formulas to compare mixing different cohorts within a single tontine versus creating a separate tontine for each cohort, finding that mixing cohorts is better because it reduces both idiosyncratic risk and systematic risk.