Wealth heterogeneity in a closed pooled annuity fund

The income stability of a closed pooled annuity fund is studied. The focus is on quantifying the impact of inhomogeneous initial savings amounts on idiosyncratic longevity risk. Besides wealth inhomogeneity, the members of the pool are independent and identical copies of each other. We ignore systematic investment risk or mortality risk and define income stability as keeping the income within a specified tolerance of each member's initial income payment in a fixed proportion of future scenarios. For a given group of people with different savings amounts, we find an analytical expression that closely approximates the time for which the fund provides a stable income. Our main result uses this expression to determine if everyone benefits from pooling their funds with the whole group or how to split a cohort into different pooled annuity funds.