Utility spanning and the Ordered Mean Difference Envelope

It is shown that the space of optimising portfolios for increasing risk averse utility functions forms a one dimensional manifold, which is the envelope of the ordered mean difference utility generators. The manifold also yields the set of second order stochastic dominant portfolios. The optimising portfolio for any utility function can be obtained by solving the simpler problem for a representative utility generator, which has just two linear segments. This can be done by using linear programming, which in turn can be iterated to trace out the entire efficient set, giving a computationally undemanding way of obtaining the stochastic dominance efficient set. The general efficiency frontier shares the one dimensional property with the mean variance efficient frontier, but unlike the latter, the associated portfolios do not form a convex set, so the two fund theorem of mean variance portfolio anaylsis does not hold in general.