  Risk-sharing is one way to pool risks without the need for a third party. To
ensure the attractiveness of such a system, the rule should be accepted and
understood by all participants. A desirable risk-sharing rule should fulfill
actuarial fairness and Pareto optimality while being easy to compute. This
paper establishes a one-to-one correspondence between an actuarially fair
Pareto optimal (AFPO) risk-sharing rule and a fixed point of a specific
function. A fast numerical method for computing these risk-sharing rules is
also derived. As a result, we are able to compute AFPO risk-sharing rules for a
large number of heterogeneous participants in this framework.
