Abstract
We present new developments of a portfolio theory based on a characterization of the non-Gaussian properties of the multivariate distribution of the asset returns in terms of the moments and cumulants of the distribution of returns of a portfolio made of an arbitrary composition of these assets. We show how the concept of efficient frontiers generalizes naturally and how it depends on the chosen risk measure corresponding to different orders and/or combinations of centered moments or cumulants. Our extended formulas enable us to determine analytically the conditions under which it is possible to “have your cake and eat it too”, i.e., to construct a portfolio with both larger return and smaller “large risks.”