Before revision:  We determine the relationship between some of these structures and we show thatthey are not all actuarially fair .
After revision:  We discuss the importance of actuarial fairness, defined as the expected benefits equalling the contributions for each member, and 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 subsidise the other members. The implication is that the members who are subsidising the others may obtain a higher expected benefit by joining a fund with a more favourable membership profile. 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 .