Efficient frontiers for portfolios under SSD and law-invariant risk measures with hyperbolic return distributions

Hasanjan Sayit

Papers from arXiv.org

Abstract: In the classical Markowitz mean variance framework, risk is measured by variance, and the portfolios on the efficient frontier can be derived in closed form using standard optimization methods. For broader mean risk formulations, however, obtaining closed form optimal portfolios is typically difficult. In this work, we derive explicit expressions for frontier portfolios corresponding to arbitrary law-invariant convex risk measures, assuming that the return vector follows a normal mean variance mixture distribution. Our approach first establishes stochastic dominance relations within the family of normal mean variance mixture models, and then leverages these relations to derive closed-form representations of frontier portfolios. The central finding demonstrates that when asset returns are described by a normal mean variance mixture models the associated mean risk efficient frontier can be obtained by solving a Markowitz mean variance problem for a suitably transformed return vector. The paper also extends the CAPM framework to the context of a normal mean variance mixture distribution.

Date: 2022-02, Revised 2026-08
New Economics Papers: this item is included in nep-cwa
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