Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mean-Covariance Robust Risk Measurement

Viet Anh Nguyen, Soroosh Shafiee, Damir Filipović, Daniel Kuhn

Abstract

We introduce a universal framework for mean-covariance robust risk measurement and portfolio optimization. We model uncertainty in terms of the Gelbrich distance on the mean-covariance space, along with prior structural information about the population distribution. Our approach is related to the theory of optimal transport and exhibits superior statistical and computational properties than existing models. We find that, for a large class of risk measures, mean-covariance robust portfolio optimization boils down to the Markowitz model, subject to a regularization term given in closed form. This includes the finance standards, value-at-risk and conditional value-at-risk, and can be solved highly efficiently.

Mean-Covariance Robust Risk Measurement

Abstract

We introduce a universal framework for mean-covariance robust risk measurement and portfolio optimization. We model uncertainty in terms of the Gelbrich distance on the mean-covariance space, along with prior structural information about the population distribution. Our approach is related to the theory of optimal transport and exhibits superior statistical and computational properties than existing models. We find that, for a large class of risk measures, mean-covariance robust portfolio optimization boils down to the Markowitz model, subject to a regularization term given in closed form. This includes the finance standards, value-at-risk and conditional value-at-risk, and can be solved highly efficiently.
Paper Structure (26 sections, 39 theorems, 150 equations, 2 figures, 3 tables)

This paper contains 26 sections, 39 theorems, 150 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Structural Information
  4. Statistical Information
  5. Properties of the Gelbrich Ambiguity Set
  6. Relation between the Gelbrich and Wasserstein Ambiguity Sets
  7. Statistical Properties of the Gelbrich Ambiguity Set
  8. Computational Properties of the Gelbrich Ambiguity Set
  9. Gelbrich Risk of Linear Portfolio Loss Functions
  10. Basic Properties of Risk Measures
  11. Law Invariant, Translation Invariant and Positive Homogeneous Risk Measures
  12. Calculation of the Standard Risk Coefficient
  13. Mahalanobis-Gelbrich Risk
  14. Gelbrich Risk of Nonlinear Portfolio Loss Functions
  15. Gelbrich VaR of Piecewise Linear Loss Functions
  16. ...and 11 more sections

Key Result

Theorem 1

For any distributions $\mathbb Q_1,\mathbb Q_2\in\mathcal{M}_2$ with mean vectors $\mu_1$, $\mu_2 \in \mathbb{R}^n$ and covariance matrices $\Sigma_1$, $\Sigma_2 \in \mathbb{S}_{+}^n$, respectively, we have $\mathds{W}(\mathbb Q_1, \mathbb Q_2) \geq \mathds{G} ((\mu_1, \Sigma_1), (\mu_2, \Sigma_2))$

Figures (2)

  • Figure 1: Out-of-sample CVaR (top row) and mean-standard deviation return (bottom row) of distributionally robust portfolios as a function of $\rho$ for different training sample sizes: $N = 20$ (left), $N = 100$ (middle), and $N = 1{,}000$ (right).
  • Figure 2: Out-of-sample tracking error of distributionally robust tracking portfolios averaged over the backtesting interval as a function of $\rho$ (for better readability, only the best-performing Gelbrich and Delage-Ye portfolios are shown).

Theorems & Definitions (93)

  • Definition 1: Structural ambiguity set
  • Definition 2: Symmetric distribution
  • Definition 3: Linear unimodal distribution
  • Definition 4: Log-concave distribution
  • Definition 5: Elliptical distribution
  • Definition 6: Structural ambiguity set generated by $\widehat{\mathbb P}$
  • Definition 7: Gelbrich distance
  • Definition 8: Gelbrich ambiguity set
  • Definition 9: Wasserstein distance
  • Theorem 1: Gelbrich bound ref:gelbrich1990formula
  • ...and 83 more