Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution

ArXiv ID: 2601.00568 “View on arXiv”

Authors: Enrique Calderín-Ojeda, Yuyu Chen, Soon Wei Tan

Abstract

Capital allocation is a procedure used to assess the risk contributions of individual risk components to the total risk of a portfolio. While the conditional tail expectation (CTE)-based capital allocation is arguably the most popular capital allocation method, its inability to reflect important tail behaviour of losses necessitates a more accurate approach. In this paper, we introduce a new capital allocation method based on the tail central moments (TCM), generalising the tail covariance allocation informed by the tail variance. We develop analytical expressions of the TCM as well as the TCM-based capital allocation for the class of normal mean-variance mixture distributions, which is widely used to model asymmetric and heavy-tailed data in finance and insurance. As demonstrated by a numerical analysis, the TCM-based capital allocation captures several significant patterns in the tail region of equity losses that remain undetected by the CTE, enhancing the understanding of the tail risk contributions of risk components.

Keywords: Capital Allocation, Tail Central Moments (TCM), Conditional Tail Expectation (CTE), Tail Risk Contributions, Normal Mean-Variance Mixture, Equities

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper focuses on deriving analytical expressions (recursive theorems) for tail central moments within the complex normal mean-variance mixture distribution class, indicating very high mathematical density. While it mentions numerical analysis, the description is generic without specific backtesting pipelines or data sources, suggesting theoretical development rather than immediate empirical implementation.

  flowchart TD
    A["Research Goal<br>Develop new capital allocation method<br>capturing tail behavior"] --> B["Methodology<br>Introduce Tail Central Moments<br>TCM-based allocation"]
    B --> C["Target Model<br>Multivariate Normal Mean-Variance Mixture<br>Asymmetric, heavy-tailed data"]
    C --> D["Input Data<br>Equity loss returns<br>Risk component data"]
    D --> E["Computation<br>Analytical expressions for TCM &<br>TCM-based capital allocation"]
    E --> F["Comparison<br>CTE-based allocation vs<br>TCM-based allocation"]
    F --> G["Key Findings<br>TCM captures tail patterns<br>missed by CTE"]
    G --> H["Outcome<br>Enhanced understanding of<br>tail risk contributions"]