Mirror Descent Algorithms for Risk Budgeting Portfolios

ArXiv ID: 2411.12323 “View on arXiv”

Authors: Unknown

Abstract

This paper introduces and examines numerical approximation schemes for computing risk budgeting portfolios associated to positive homogeneous and sub-additive risk measures. We employ Mirror Descent algorithms to determine the optimal risk budgeting weights in both deterministic and stochastic settings, establishing convergence along with an explicit non-asymptotic quantitative rate for the averaged algorithm. A comprehensive numerical analysis follows, illustrating our theoretical findings across various risk measures – including standard deviation, Expected Shortfall, deviation measures, and Variantiles – and comparing the performance with that of the standard stochastic gradient descent method recently proposed in the literature.

Keywords: Risk Budgeting, Mirror Descent, Expected Shortfall, Risk Measures, Portfolio Optimization, Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper involves advanced mathematical concepts such as Mirror Descent algorithms, convex optimization, and convergence rate analysis, indicating high math complexity; it also includes a comprehensive numerical analysis comparing methods across various risk measures, demonstrating strong empirical rigor.

  flowchart TD
    A["Research Goal:<br>Compute Risk Budgeting Portfolios<br>via Numerical Approximation"] --> B{"Methodology:<br>Mirror Descent Algorithms"}
    B --> C["Data/Inputs:<br>Positive Homogeneous &<br>Sub-additive Risk Measures"]
    C --> D["Computational Processes:<br>Deterministic & Stochastic Settings"]
    D --> E["Averaged Algorithm<br>with Convergence Rate"]
    E --> F["Outcomes:<br>1. Explicit Non-Asymptotic Rate<br>2. Numerical Analysis Across<br>Standard Deviation, ES, Deviation, Variantiles<br>3. Comparison vs. Stochastic Gradient Descent"]