Efficient Portfolios and Extreme Risks: An Extended Dirichlet Approach

Abstract

This paper solves the mean-variance-skewness-kurtosis (MVSK) portfolio optimization
problem using a new approach based on the Dirichlet distribution. To obtain efficient portfolios,
we generalize the Dirichlet distribution using the student copula to produce an increased
proportion of portfolios concentrated on a subset of the index. We illustrate our approach
using the forty stocks of the French index. Our study also compares MVSK efficient portfolios
with the efficient portfolios that are obtained in a reduced three-dimensional setting, such as
a mean-variance-skewness or mean-variance-kurtosis setting. To incorporate the preferences
of decision-makers and to determine an optimal portfolio within the efficient set, we introduce
and illustrate a new indicator called the generalized Sharpe ratio that groups variance,
skewness, and kurtosis into a single quantity.

Keywords: Portfolio Choice, Efficient Frontier, Extreme Risk, Dirichlet Distribution

JEL Classification: G11, D81, C63