Mirror Descent Algorithms for Risk Budgeting Portfolios

TL;DR

This paper develops and analyzes Mirror Descent algorithms for efficiently computing risk budgeting portfolios under various risk measures, providing convergence guarantees and numerical comparisons.

Contribution

It introduces a novel application of Mirror Descent to risk budgeting, with convergence analysis and extensive numerical experiments across multiple risk measures.

Findings

Mirror Descent converges with explicit rates for risk budgeting.

The proposed algorithms outperform stochastic gradient descent in numerical tests.

Applicable to various risk measures including standard deviation and Expected Shortfall.

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.