Portfolio optimization using modified herfindahl constraint

Modern portfolio theory started with Markowitz (J Financ 7(1):77–91, 1952; Portfolio selection efficient diversification of investments. Wiley, New York, 1959). Early works developed necessary conditions on utility function that would result in mean-variance theory being optimal, see Tobin (Rev Econ Stud 25(2):65–86, 1958). Recently, considering the stylized facts of asset returns, mean-variance model has been extended to higher moments. Despite all, empirical evidence has shown that mean-variance model and its variants often yield overly concentrated portfolios. Portfolio diversification is still an open question. To avoid this problem different constraints have been introduced in the portfolio optimization procedure. In this paper we study from an empirical point of view the impact of imposing a constraint on the Modified Herfindahl index of the portfolio, in case of mean-variance and mean-variance-skewness optimization. We find that imposing a constraint on the level of the portfolio diversification leads to better out of sample performance and significant gains, despite the use of shrinkage estimators for moments and comoments, in particular when long estimation periods are considered.