Worst-Case Higher Moment Coherent Risk Based on Optimal Transport with Application to Distributionally Robust Portfolio Optimization

Abstract

The tail risk management is of great significance in the investment process. As an extension of the asymmetric tail risk measure—Conditional Value at Risk (CVaR), higher moment coherent risk (HMCR) is compatible with the higher moment information (skewness and kurtosis) of probability distribution of the asset returns as well as capturing distributional asymmetry. In order to overcome the difficulties arising from the asymmetry and ambiguity of the underlying distribution, we propose the Wasserstein distributionally robust mean-HMCR portfolio optimization model based on the kernel smoothing method and optimal transport, where the ambiguity set is defined as a Wasserstein “ball” around the empirical distribution in the weighted kernel density estimation (KDE) distribution function family. Leveraging Fenchel’s duality theory, we obtain the computationally tractable DCP (difference-of-convex programming) reformulations and show that the ambiguity version preserves the asymmetry of the HMCR measure. Primary empirical test results for portfolio selection demonstrate the efficiency of the proposed model.

Multi-Variate Risk Measures under Wasserstein Barycenter

“…In this case, the well-studied Wasserstein metric supports the method and provides fundamental connections for the rising concept of barycenter, in the sense of Agueh and Carlier (2011); this is considered seminal for a number of generalizations and applications (see, e.g., Bigot et al (2018), Álvarez Esteban et al (2018), Le Gouic and Loubes (2017), and the references therein). The Wasserstein metric has notably enriched the risk management literature (see, e.g., Feng and Erik (2018), Wang et al (2020), Pesenti (2022), and Liu and Liu (2022)).…”