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On Wasserstein Distributionally Robust Mean Semi-Absolute Deviation Portfolio Model: Robust Selection and Efficient Computation

Weimi Zhou, Yong-Jin Liu

TL;DR

A robust Wasserstein profile inference approach to determine the size of the Wasserstein radius for DR-MLSAD model and an efficient proximal point dual semismooth Newton (PpdSsn) algorithm for the reformulated equivalent model are designed.

Abstract

This paper focuses on the Wasserstein distributionally robust mean-lower semi-absolute deviation (DR-MLSAD) model, where the ambiguity set is a Wasserstein ball centered on the empirical distribution of the training sample. This model can be equivalently transformed into a convex problem. We develop a robust Wasserstein profile inference (RWPI) approach to determine the size of the Wasserstein radius for DR-MLSAD model. We also design an efficient proximal point dual semismooth Newton (PpdSsn) algorithm for the reformulated equivalent model. In numerical experiments, we compare the DR-MLSAD model with the radius selected by the RWPI approach to the DR-MLSAD model with the radius selected by cross-validation, the sample average approximation (SAA) of the MLSAD model, and the 1/N strategy on the real market datasets. Numerical results show that our model has better out-of-sample performance in most cases. Furthermore, we compare PpdSsn algorithm with first-order algorithms and Gurobi solver on random data. Numerical results verify the effectiveness of PpdSsn in solving large-scale DR-MLSAD problems.

On Wasserstein Distributionally Robust Mean Semi-Absolute Deviation Portfolio Model: Robust Selection and Efficient Computation

TL;DR

A robust Wasserstein profile inference approach to determine the size of the Wasserstein radius for DR-MLSAD model and an efficient proximal point dual semismooth Newton (PpdSsn) algorithm for the reformulated equivalent model are designed.

Abstract

This paper focuses on the Wasserstein distributionally robust mean-lower semi-absolute deviation (DR-MLSAD) model, where the ambiguity set is a Wasserstein ball centered on the empirical distribution of the training sample. This model can be equivalently transformed into a convex problem. We develop a robust Wasserstein profile inference (RWPI) approach to determine the size of the Wasserstein radius for DR-MLSAD model. We also design an efficient proximal point dual semismooth Newton (PpdSsn) algorithm for the reformulated equivalent model. In numerical experiments, we compare the DR-MLSAD model with the radius selected by the RWPI approach to the DR-MLSAD model with the radius selected by cross-validation, the sample average approximation (SAA) of the MLSAD model, and the 1/N strategy on the real market datasets. Numerical results show that our model has better out-of-sample performance in most cases. Furthermore, we compare PpdSsn algorithm with first-order algorithms and Gurobi solver on random data. Numerical results verify the effectiveness of PpdSsn in solving large-scale DR-MLSAD problems.
Paper Structure (21 sections, 9 theorems, 137 equations, 1 figure, 1 table, 5 algorithms)

This paper contains 21 sections, 9 theorems, 137 equations, 1 figure, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. DRO formulation and computational tractability
  3. Choice of ambiguity radius $\epsilon$
  4. A preconditioned PpdSsn algorithm for DR-MLSAD model
  5. The PpdSsn algorithm and its convergence
  6. A semismooth Newton method for the subproblem
  7. The generalized HS-Jacobian of the projection $\Pi_{\mathcal{C}}(\cdot)$
  8. Constraint nondegeneracy condition for problem \ref{['ppadual']}
  9. A semismooth Newton algorithm and its convergence
  10. Efficient implementations of the linear system \ref{['newton']}
  11. Numerical experiments
  12. Comparison of out-of-sample performance
  13. Methodology
  14. Numerical results of model performance
  15. Comparison of algorithms
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose $\|\cdot\|_p=\|\cdot\|_{\infty}$. Considering the uncertain set $\Xi=\mathbb{R}^m$. Then for any $\epsilon\geq0$, DR-MLSAD model is equivalent to the following problem: where $\hat{\mu}=\frac{1}{N}\sum_{i=1}^{N}\hat{\xi}_i$.

Figures (1)

  • Figure 1: Portfolio out-of-sample cumulative wealth of different portfolio selection models for six datasets

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 1
  • proof
  • ...and 7 more