A note on separation in mean-lower-partial-moment portfolio optimization with fixed and moving targets

As measures of portfolio risk, Lower Partial Moments (LPMs) have several advantages over variance, the traditional measure of risk. A separation theorem can be proven in the context of mean-LPM portfolio optimization, when the target is equal to the risk-free interest rate. The question of which other targets admit separation has remained open, however. We attempt to answer this question and clear up some confusion in the literature caused by previous attempts. We distinguish between a fixed and a moving target, that is, a target that depends on the (distribution of the) random return whose risk measure is being evaluated. We show that the risk-free interest rate is the only fixed target that guarantees linear separation. Among moving targets, we show that linear separation holds if the target is set equal to the mean return of the portfolio under consideration.