Higher Order Moments and Beyond

As we skip beyond the realms of the normal world, many desirable properties fall short. Specifically, the central moment of linear combinations of random variables do not preserve the features of the addenda. For example, not even null-correlated returns preserve the signs of odd central moments as the returns are combined into a portfolio. A mathematical explanation of this counter-intuitive phenomenon is provided. However, using one-sided higher-order moments instead of higher-order moments may be a way to overcome these drawbacks. Thanks to the fact that they are coherent risk measures, a number of desirable marginal ordering properties are preserved.