Heterogeneity among new entrants to a pooled annuity fund

Pooled annuity funds pay an income for life to their members, by pooling directly the members’ longevity risk together. It is well known that the more members in the fund, the less volatile are the income payments to the members. However, this is true only as long as the members are independent and identical copies of each other.
A group of heterogeneous members, who are of different ages and have different amounts of money in the fund, have a higher income volatility than a same-sized group of homogeneous members. A measure of heterogeneity is proposed and studied in conjunction with a measure of income volatility.
Despite heterogeneity increasing income volatility, the effects are in general small.
The general rule that pooling more participants together is better, regardless of their characteristics, is still broadly supported. It is only in a fund with very heterogeneous participants that restrictions on the membership should be considered, such as capping the amount of money that an individual can bring to the fund.