A theoretical generalization of the Markowitz model incorporating skewness and kurtosis

Abstract

This paper proposes a generalization of Markowitz model that incorporates skewness and kurtosis into the classical mean–variance allocation framework. The principal appeal of the present approach is that it provides the closed-form solution of the optimization problem. The four moments optimal portfolio is then decomposed into the sum of three portfolios: the mean–variance optimal portfolio plus two self-financing portfolios, respectively, accounting for skewness and kurtosis. Theoretical properties of the optimal solution are discussed together with the economic interpretation. Finally, an empirical exercise on real financial data shows the contribution of the two portfolios accounting for skewness and kurtosis when financial returns depart from Normal distribution.

Keywords:

• Portfolio optimization
• Higher moments
• Skewness
• Kurtosis
• Mathematical programming

Acknowledgments

I am grateful to the Editor and the anonymous reviewers for their insightful comments on my paper. I would also like to thank my beloved dad Roberto.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 A non-Normal distribution of assets' returns is the necessary condition to apply the proposed four moments optimization model obtaining a solution that differs from the standard mean variance solution. Considering that monthly returns calculated on equity indexes can show limited deviations from the Normal distribution, it has been chosen to use daily returns to have data characterized by a more significant deviation from the Normal distribution.