Some general results on risk budgeting portfolios

1. Problem Statement

Risk Budgeting (RB) is a portfolio construction strategy where an investor assigns a predetermined risk contribution ($x^b_i$) to each asset $i$ such that the total portfolio risk is allocated according to these targets. When all targets are equal ($x^b_i = 1/N$), the strategy is known as Risk Parity.

The core challenge addressed in the paper is the numerical calculation of the RB portfolio.

• Current State: Existing literature (e.g., Maillard et al., 2010; Cetingoz et al., 2024) primarily relies on solving an auxiliary optimization problem to find the RB portfolio.
• Limitations of Current Methods:
  • Economic Interpretation: The auxiliary optimization problems often lack a direct economic interpretation, making it difficult to justify why an investor should solve them.
  • Numerical Instability: Optimization-based approaches can suffer from numerical instability, especially as the number of assets increases or when inputs are near the boundary of the simplex.
  • Computational Cost: Solving non-linear optimization problems is computationally expensive compared to simpler iterative methods.
  • Theoretical Constraints: Proofs of existence and uniqueness often rely heavily on the convexity of the auxiliary optimization problem rather than the intrinsic properties of the risk measure.

The authors propose a novel, non-optimization-based approach to calculate the RB portfolio for general risk measures, with a specific implementation for the standard deviation.

2. Methodology

The authors introduce a framework based on fixed-point theory and Cauchy sequences within the probability simplex.

A. General Framework (Arbitrary Risk Measure)

1. Definitions:
   • Let $S$ be the simplex of long-only portfolios ($x \in \mathbb{R}^N, \sum x_i = 1, x_i \geq 0$).
   • Let $\rho$ be a reference risk measure and $RC_\rho(x)$ be the vector of marginal risk contributions.
   • The goal is to find $x^* \in S$ such that the normalized risk contributions match the budget: $RC_\rho(x^*) / \mathbf{1}^T RC_\rho(x^*) = x^b$.
2. The Iterative Sequence:
   The authors define a function $\Delta(x)$ representing the deviation from the target risk budget:
   $$\Delta(x) = RC_\rho(x) - x^b \mathbf{1}^T RC_\rho(x)$$
   They construct a sequence $\{x_n\}$ starting from an initial point $x_1 \in S$:
   $$x_{n+1} = x_n + k(x_n)\Delta(x_n)$$
   where $k(x_n)$ is a step-size function chosen to ensure convergence.
3. Convergence Proof:
   • Theorem 1: If a continuous function $k(x)$ exists such that $\|\Delta(x_{n+1})\|_2 \leq L \|\Delta(x_n)\|_2$ (with $0 < L < 1$), the sequence $\{x_n\}$ is a Cauchy sequence.
   • Since the simplex with the Euclidean norm is a complete metric space, the sequence converges to a limit $\bar{x}$.
   • Theorem 2: The limit $\bar{x}$ satisfies $\Delta(\bar{x}) = 0$, meaning it is the risk budgeting portfolio.
4. Existence and Uniqueness:
   • The problem is reformulated as finding a fixed point of the function $G(x) = x - k(x)\Delta(x)$.
   • Theorem 3 (Kellogg's Theorem): Sufficient conditions for the existence and uniqueness of the fixed point in the simplex are provided, relying on the differentiability of the function and the absence of eigenvalue 1 in the Jacobian.

B. Specific Implementation: Standard Deviation

When the risk measure is the standard deviation ($\rho(x) = \sqrt{x^T V x}$, where $V$ is the covariance matrix):

1. Marginal Risk: $RC_\rho(x) = \text{diag}(x)Vx$.
2. Step Size Calculation ($k_n$):
   The authors derive a specific algorithm to find $k_n$ that satisfies the contraction condition $\|\Delta(x_{n+1})\|_2 < L \|\Delta(x_n)\|_2$.
   • This leads to a quartic polynomial $Q(k_n)$. If real roots exist, $k_n$ is chosen to satisfy $Q(k_n) < 0$.
   • If no real roots exist (which can happen if $L$ is too small), a cubic polynomial $C(k_n)$ is solved to ensure $\|\Delta(x_{n+1})\|_2 \leq \|\Delta(x_n)\|_2$.
3. Uniqueness in Simplex:
   • Theorem 7 & Corollary 1: By leveraging results from O'Leary (2003) on matrix scaling, the authors prove that for a positive definite covariance matrix $V$ and a budget $x^b \in S$, there exists a unique fixed point in the simplex.
   • Theorem 8: Provides an explicit bound on $|k_n|$ to ensure the sequence remains within the simplex $S$ (preventing negative weights).

3. Key Contributions

1. Novel Algorithmic Approach: The paper proposes a fixed-point iteration method that avoids solving auxiliary optimization problems. This removes the need for complex solvers (like fmincon) and their associated numerical instabilities.
2. Theoretical Rigor:
   • Establishes existence and uniqueness conditions for the RB portfolio based on the properties of the marginal risk contribution function, rather than just the convexity of an optimization objective.
   • Proves that the proposed iterative sequence is a Cauchy sequence, guaranteeing convergence to the unique solution in the simplex.
3. Generalizability: The framework is defined for general risk measures, provided the marginal risk contributions are continuous. The paper details the implementation specifically for standard deviation but outlines the path for other measures (e.g., VaR, Expected Shortfall).
4. Handling Negative Marginal Contributions: The paper addresses the counter-intuitive case where an asset has negative marginal risk (due to negative correlations) and demonstrates how the algorithm handles these scenarios robustly.

4. Empirical Results

The authors compared their Fixed-Point (FP) algorithm against three standard methods:

• OP1: Optimization minimizing squared deviation of risk contributions (Maillard et al., 2010).
• OP2: Optimization using log-transformed variables (Maillard et al., 2010).
• NLS: Solving the non-linear system directly using lsqnonlin.

Experimental Setup:

• Tested on portfolios with $N = 5, 10, 50, 100, 200$ assets.
• 1,000 random experiments per size with random covariance matrices and risk budgets.
• Metrics: Computational time (seconds) and accuracy (distance of realized risk budget to target).

Key Findings:

• Speed: The FP algorithm was significantly faster. For $N=200$, FP took 0.038s, while OP1 took 1.41s, OP2 took 0.27s, and NLS took 3.65s.
• Accuracy: FP achieved the highest accuracy (lowest error) across all portfolio sizes.
  • At $N=200$, FP error was 0.0040, compared to 0.0556 (OP1), 0.2877 (OP2), and 0.0458 (NLS).
• Scalability: The FP algorithm showed the best resilience to increasing dimensionality ($N$). Its computational time grew much slower than the optimization-based methods.
• Parameter Sensitivity: The parameter $L$ (convergence rate control) was tested. Values of $L$ close to 1 (e.g., 0.95 or 0.99) offered the best balance between step reliability and convergence speed, whereas very small $L$ values caused the algorithm to stall due to the lack of real roots in the quartic polynomial.

5. Significance and Conclusion

This paper offers a paradigm shift in calculating risk budgeting portfolios. By moving away from optimization-based formulations toward a fixed-point iteration derived from Cauchy sequences, the authors provide:

• Computational Efficiency: A method that is orders of magnitude faster and more stable, making it suitable for high-dimensional portfolios and real-time applications.
• Theoretical Clarity: A clearer economic and mathematical justification for the solution, decoupling the existence of the portfolio from the convexity of an auxiliary objective function.
• Practical Robustness: The algorithm handles edge cases (near-boundary portfolios, negative correlations) effectively without requiring complex regularization or log-transformations.

The authors conclude that this approach is superior to standard optimization techniques and suggest future work to extend these results to a broader class of risk measures (e.g., Expected Shortfall) and to refine the theoretical conditions for existence and uniqueness.