Multivariate location–scale mixtures of normals and mean–variance–skewness portfolio allocation

We show that the distribution of any portfolio whose components jointly follow a location–scale mixture of normals can be characterised solely by its mean, variance and skewness. Under this distributional assumption, we derive the mean–variance–skewness frontier in closed form, and show that it can be spanned by three funds. For practical purposes, we derive a standardised distribution, provide analytical expressions for the log-likelihood score and explain how to evaluate the information matrix. Finally, we present an empirical application in which we obtain the mean–variance–skewness frontier generated by the ten Datastream US sectoral indices, and conduct spanning tests.