  Various types of structures that enable a group of individuals to pool their
mortality risk have been proposed in the literature. Collectively, the
structures are called pooled annuity funds. Since the pooled annuity funds
propose different methods of pooling mortality risk, we investigate the
connections between them and find that they are genuinely different for a
finite heterogeneous membership profile.
  We discuss the importance of actuarial fairness, defined as the expected
benefits equalling the contributions for each member, in the context of pooling
mortality risk and comment on whether actuarial unfairness can be seen as
solidarity between members. We show that, with a finite number of members in
the fund, the group self-annuitization scheme is not actuarially fair: some
members subsidize the other members. The implication is that the members who
are subsidizing the others may obtain a higher expected benefit by joining a
fund with a more favourable membership profile. However, we find that the
subsidies are financially significant only for very small or highly
heterogeneous membership profiles.
