Paper: arXiv 2411.12323

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: 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.

Research Flowchart

  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"]