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Bayesian Optimization for CVaR-based portfolio optimization

Robert Millar, Jinglai Li

TL;DR

The paper tackles constrained portfolio optimization by minimizing $g( extbf{x})=\text{CVaR}_{\alpha}[f(\textbf{x},\textbf{Z})]$ subject to $R(\textbf{x})=\mathbb{E}_{\mathbf Z}[f(\textbf{x},\mathbf Z)]\ge r^{\min}$, where CVaR evaluations are computationally expensive. It advances Bayesian Optimization by introducing an active-constraint acquisition, a two-stage sampling strategy that only fully evaluates CVaR for near-feasible cases, and batchable implementations that exploit parallelism. The core innovations are the ACW-EI acquisition and its two-stage extension (2S-ACW-EI), plus a theoretical justification that the optimum lies on the constraint boundary, which motivates near-boundary sampling. Empirical results on three portfolio scenarios demonstrate faster convergence and lower CVaR compared to baseline CW-EI/ACW-EI methods, with batch variants offering substantial speedups. The work enables efficient, scalable risk-aware asset allocation when CVaR is costly to compute and feasibility is determined by a cheap expected-return constraint.

Abstract

Optimal portfolio allocation is often formulated as a constrained risk problem, where one aims to minimize a risk measure subject to some performance constraints. This paper presents new Bayesian Optimization algorithms for such constrained minimization problems, seeking to minimize the conditional value-at-risk (a computationally intensive risk measure) under a minimum expected return constraint. The proposed algorithms utilize a new acquisition function, which drives sampling towards the optimal region. Additionally, a new two-stage procedure is developed, which significantly reduces the number of evaluations of the expensive-to-evaluate objective function. The proposed algorithm's competitive performance is demonstrated through practical examples.

Bayesian Optimization for CVaR-based portfolio optimization

TL;DR

The paper tackles constrained portfolio optimization by minimizing subject to , where CVaR evaluations are computationally expensive. It advances Bayesian Optimization by introducing an active-constraint acquisition, a two-stage sampling strategy that only fully evaluates CVaR for near-feasible cases, and batchable implementations that exploit parallelism. The core innovations are the ACW-EI acquisition and its two-stage extension (2S-ACW-EI), plus a theoretical justification that the optimum lies on the constraint boundary, which motivates near-boundary sampling. Empirical results on three portfolio scenarios demonstrate faster convergence and lower CVaR compared to baseline CW-EI/ACW-EI methods, with batch variants offering substantial speedups. The work enables efficient, scalable risk-aware asset allocation when CVaR is costly to compute and feasibility is determined by a cheap expected-return constraint.

Abstract

Optimal portfolio allocation is often formulated as a constrained risk problem, where one aims to minimize a risk measure subject to some performance constraints. This paper presents new Bayesian Optimization algorithms for such constrained minimization problems, seeking to minimize the conditional value-at-risk (a computationally intensive risk measure) under a minimum expected return constraint. The proposed algorithms utilize a new acquisition function, which drives sampling towards the optimal region. Additionally, a new two-stage procedure is developed, which significantly reduces the number of evaluations of the expensive-to-evaluate objective function. The proposed algorithm's competitive performance is demonstrated through practical examples.

Paper Structure

This paper contains 20 sections, 1 theorem, 18 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Optimal portfolio allocation
  3. Risk Measures
  4. Problem Set-up
  5. Bayesian Optimization
  6. Gaussian Process
  7. Unconstrained Bayesian Optimization
  8. Bayesian Optimization with Constraints
  9. Proposed methodology
  10. Activeness of the Constraint
  11. Two-Stage Weight Selection
  12. New Acquisition Function
  13. The complete algorithm
  14. Batch Implementation
  15. Numerical Experiments
  16. ...and 5 more sections

Key Result

Theorem 1

If function $f(\mathbf{x},\mathbf{Z})$ and distribution $p_{\mathbf z}(\cdot)$ satisify assumptions 1-4, $\alpha$ is chosen such that $v_f(\mathbf{x},\alpha) \geq 0$$\forall$$w \in \mathbb{W}$, and solutions to the constrained optimization problem exist, then there must exist a solution, denoted as

Figures (4)

  • Figure 1: Plots showing the optimal solution (green-x) for numerical example one and the design points generated by each of the three methods. The figures include both the fully evaluated points (red) and those for which only the constraint was evaluated (blue). The feasible region is dark grey, the active region is light grey and the infeasible region is white. The objective function contours are shown too.
  • Figure 2: The best objective value obtained after each iteration for the portfolio allocation problems across the existing method (CW-EI BO) and the four new proposed methods.
  • Figure 3: The best objective value obtained after each iteration for problems 1a - 3a, with $r^{\max} = 105\% r^{\min}$.
  • Figure 4: Schematic illustration of the connection between $B_i$ and $D_i$.

Theorems & Definitions (1)

  • Theorem 1