On the Structure of Risk Contribution: A Leave-One-Out Decomposition into Inherent and Correlation Risk

Paper: arXiv 2604.10375

Authors: Nolan Alexander, Frank Fabozzi

Abstract

This paper develops a decomposition of standard Risk Contribution (RC) into two economically interpretable components: inherent risk and correlation risk. Using a leave-one-out representation, each position’s RC separates into a term reflecting its own volatility contribution independent of the portfolio and a term capturing its covariance with the remainder of the portfolio. The inherent component is always positive, arising from the intrinsic volatility of the position, while the correlation component may amplify or mitigate total portfolio risk depending on how the position moves relative to other holdings. Because the decomposition operates within standard RC, it preserves the property of strict additivity. This separation provides diagnostic insight not visible from aggregate risk contributions alone. It distinguishes whether a position contributes risk because it is volatile in isolation or because it is highly correlated with the rest of the portfolio, and it clarifies when a negatively correlated position functions as an effective hedge. Two approaches to time-series analysis are presented to track how inherent and correlation risk evolve across market regimes, revealing whether changes in portfolio risk during stress periods are driven by volatility shocks, correlation shifts, or both. Empirical illustrations suggest that the decomposition provides stable, transparent, and easily implementable risk diagnostics that can support portfolio risk reporting, stress testing, and performance attribution.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a mathematically clean and intuitive leave-one-out decomposition of Euler risk contributions into inherent volatility and correlation components. While the empirical illustrations are mostly introductory diagnostic applications rather than exhaustive backtests, the theoretical contribution is elegant, additive, and immediately actionable for risk reporting.

Research Flowchart

  flowchart TD
    A["Research Question: How to decompose Risk Contribution into interpretable inherent vs. correlation components?"] --> B["Methodology: Leave-One-Out (LOO) Representation of Portfolio Variance"]
    C["Inputs: Multi-Asset Class Time-Series Data & Portfolio Weights"] --> D["Computational Process: Calculate LOO Volatility (Inherent) & Remainder Covariance (Correlation)"]
    B --> D
    D --> E["Time-Series Analysis: Track Evolution Across Market Regimes & Stress Periods"]
    E --> F["Outcomes: Additive RC Decomposition, Highlighting Volatility Shocks vs. Correlation Shifts for Risk Diagnostics"]

Cited References (17)

1. Does Gold Glitter in the Long-Run? Gold as a Hedge and Safe Haven across Time and Investment Horizon — link
2. The Effect of Errors in Means , Variances , and Covariances on Optimal Portfolio Choice — link
3. Hedges and Safe Havens: An Examination of Stocks, Bonds, Gold, Oil and Exchange Rates — link
4. Beyond Value at Risk : The New Science of Risk Management — link
5. Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH — link
6. Active Portfolio Management ( PB ) — link
7. Portfolio Selection by Using Time Varying Covariance Matrices
8. Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection — link

0 of 17 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.