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.
Highlights
Portfolio selection is a typical optimization problem under parameter uncertainty when facing uncertain asset returns
To overcome the difficulties arise from the asymmetry and ambiguity of the underlying distribution, we propose a Wasserstein distributionally robust portfolio model based on kernel smoothing method and mean-higher-moment coherent risk (HMCR), where the ambiguity set is a Wasserstein “ball” in the finite-dimensional probability distribution space spanned by the weighted kernel density estimation (KDE); Leveraging Optimal Transport theory, we define KDE–Wasserstein distance by incorporating KDE into Wasserstein distance and prove it enjoys a metric property; Leveraging Fenchel’s duality theory to obtain a finite-dimensional dual problem of the inner maximization problem, we overcome the difficulty arise from the nonliear functional with respect to probability distribution associated with HMCR, and derive the computationally tractability result of the Wasserstein distributionally robust portfolio model based on KDE and mean-HMCR
We extend φ-divergence ambiguity set in [40] to Wasserstein ambiguity set by integrating the weight KDE with Optimal Transport theory, and we discover that the tractable reformulations of our model involve some difference-of-convex programming (DCP) constraints different from those of [40], which deeply reflects the insights arise from incorporating KDE into Wasserstein distance by optimal transport
Summary
Portfolio selection is a typical optimization problem under parameter uncertainty when facing uncertain asset returns. [33] study distributionally robust optimization problems with the Wasserstein ambiguity set centering at the empirical distribution and the objective of the inner maximization is an expectation of certain loss function, under certain mild assumptions, derive a finite convex programming problem. We prove the tractability result of the Wasserstein distributionally robust portfolio p optimization based on the KDE and mean-HMCRγ model (12), starting from giving the following Lemma 1 in [30]. We prove the tractable reformulation of the Wasserstein distributionally robust p portfolio optimization based on KDE and mean-HMCRγ model (12) with given ξ i , i =. In order to obtain the computationally tractable reformulation of problem (12), we need to derive the closed expressions of the support function of the uncertainty set Λτd in (9) and the concave conjugate function of FK (w, α; ·) in (19).
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