This paper addresses the three-moment portfolio selection problem. We develop a comprehensive characterization of this problem through a duality perspective, meticulously analyzing the linear systems associated with the multipliers of three interconnected optimization formulations (variance minimization, skewness maximization, and return maximization, each with two moments fixed). Our methodological contribution extends and completes existing analyses, particularly that of Athayde and Flores (2004), by providing an alternative, geometrically-based approach that effectively navigates the highly non-linear nature of these problems without requiring a challenging study of multiplier signs, including in degenerate cases. Our results offer crucial conditions to precisely determine whether a portfolio does not belongs to the efficient frontier, thereby enabling a more robust identification of optimal asset allocations in practice. Furthermore, our framework clarifies when solutions deviate from the classical Markowitz mean-variance model, providing deeper insights into the role of higher moments in investment decisions. By relying on computationally efficient checks for linear dependence, our approach not only advances theoretical understanding but also presents a practical and accessible tool for portfolio managers seeking to integrate higher moments into their optimization strategies, ultimately leading to more informed and efficient portfolio construction.
Martins et al. (Thu,) studied this question.
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