[Paper Review] Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications & Puzzles

Jean-Charles Richard, Thierry Roncalli|arXiv (Cornell University)|Jan 1, 2019

Risk and Portfolio Optimization24 references4 citations

TL;DR

This paper develops a theoretically sound and computationally efficient framework for constructing constrained risk budgeting portfolios, where risk contributions are allocated according to predefined budgets despite portfolio weight constraints. It proposes a novel optimization formulation using logarithmic barrier functions and advanced numerical algorithms—such as cyclical coordinate descent, ADMM, and Dykstra’s algorithm—to solve the non-convex, constrained problem, while highlighting the critical role of risk measure homogeneity and the 'scaling puzzle' that arises when constraints conflict with this property.

ABSTRACT

This article develops the theory of risk budgeting portfolios, when we would like to impose weight constraints. It appears that the mathematical problem is more complex than the traditional risk budgeting problem. The formulation of the optimization program is particularly critical in order to determine the right risk budgeting portfolio. We also show that numerical solutions can be found using methods that are used in large-scale machine learning problems. Indeed, we develop an algorithm that mixes the method of cyclical coordinate descent (CCD), alternating direction method of multipliers (ADMM), proximal operators and Dykstra's algorithm. This theoretical body is then applied to some investment problems. In particular, we show how to dynamically control the turnover of a risk parity portfolio and how to build smart beta portfolios based on the ERC approach by improving the liquidity of the portfolio or reducing the small cap bias. Finally, we highlight the importance of the homogeneity property of risk measures and discuss the related scaling puzzle.

Motivation & Objective

• To formalize the mathematical structure of constrained risk budgeting portfolios, where risk budgets are no longer exactly achieved due to weight constraints.
• To develop a numerically stable and scalable algorithmic framework that combines CCD, ADMM, proximal operators, and Dykstra’s algorithm for large-scale portfolio optimization.
• To address the 'scaling compatibility puzzle' arising when constraints violate the homogeneity property of coherent risk measures.
• To demonstrate practical applications in turnover control, liquidity enhancement, and small-cap bias reduction in risk-based portfolios.
• To clarify the theoretical and numerical challenges in extending the ERC (Equal Risk Contribution) approach under constraints, ensuring uniqueness and stability.

Proposed method

• Formulates the constrained risk budgeting problem as a logarithmic barrier optimization: minimize risk measure R(x) subject to ∑bi ln xi ≥ κ and x ≥ 0.
• Uses the method of Lagrange multipliers with a barrier function to handle inequality constraints, embedding the risk budgeting condition into the optimization landscape.
• Employs cyclical coordinate descent (CCD) to iteratively optimize over one variable at a time, improving convergence and scalability.
• Integrates the alternating direction method of multipliers (ADMM) to decompose the problem into subproblems that are easier to solve, especially under complex constraints.
• Applies proximal operators and Dykstra’s algorithm to handle non-smooth or non-convex constraints, particularly in the presence of box or pointwise bounds.
• Calibrates the Lagrange multiplier λ via bisection to ensure feasibility and to minimize the Lagrangian function, selecting the optimal solution among potential local minima.

Experimental results

Research questions

• RQ1 How can risk budgeting be meaningfully extended to portfolios with explicit weight constraints, given that ex-ante risk budgets no longer match ex-post risk contributions?
• RQ2 What is the correct mathematical formulation of a constrained risk budgeting problem that ensures uniqueness and numerical stability?
• RQ3 How can large-scale, non-convex risk budgeting problems with complex constraints be solved efficiently using modern optimization techniques?
• RQ4 Why do numerical solutions to constrained risk budgeting problems depend on initialization, and what causes convergence to local minima?
• RQ5 What is the impact of the homogeneity property of risk measures on the compatibility of constraints, and how does this lead to the 'scaling puzzle'?

Key findings

• The standard formulation of risk budgeting as minimizing the sum of squared deviations between risk contributions and budgets is ill-posed and non-convex, leading to numerical instability and non-unique solutions.
• The correct formulation uses a logarithmic barrier function to embed risk budget constraints into the optimization problem, ensuring a unique and stable solution.
• The proposed algorithm—combining CCD, ADMM, proximal operators, and Dykstra’s algorithm—achieves high scalability and robustness, even for large portfolios with complex constraints.
• When constraints are incompatible with the homogeneity of the risk measure, multiple local minima may exist; the optimal solution is selected by minimizing the Lagrangian function value.
• In numerical experiments, the first and third portfolios in Table 12 were optimal, achieving the lowest volatility (5.20% and 5.98%) and Lagrangian value (13.29% and 10.68%), respectively.
• The ERC property is preserved for unconstrained assets, but constrained assets deviate from equal risk contribution, highlighting the need for careful constraint design in risk-based investing.