  A collectivised fund is a proposed form of pension investment, in which all
investors agree that any funds associated with deceased members should be split
among survivors. For this to be a viable financial product, it is necessary to
know how to manage the fund even when it is heterogeneous: that is when
different investors have different preferences, wealth and mortality. There is
no obvious way to define a single objective for a heterogeneous fund, so this
is not an optimal control problem. In lieu of an objective function, we take an
axiomatic approach. Subject to our axioms on the management of the fund, we
find an upper bound on the utility that can be achieved for each investor,
assuming a complete markets and the absence of systematic longevity risk. We
give a strategy for the management of such heterogeneous funds which achieves
this bound asymptotically as the number of investors tends to infinity.
