On certain geometric aspects of portfolio optimisation with higher moments

We discuss geometric properties related to the minimisation of a portfolio kurtosis given its first two odd moments, considering a risk-less asset and allowing for short sales. The findings are generalised for the minimisation of any given even portfolio moment with fixed excess return and skewness, and then for the case in which only excess return is constrained. An example with two risky assets provides a better insight on the problems related to the solutions. The importance of the geometric properties and their use in the higher moments portfolio choice context is highlighted.