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

2026-01-02

AI Summary

This study addresses the limitations of traditional capital allocation methods based on Conditional Tail Expectation (CTE), which often fail to accurately capture the contributions of individual risk components in the tail of loss distributions. For the first time, Tail Central Moments (TCM) are introduced into the capital allocation framework, and closed-form expressions for both TCM and its associated capital allocations are derived under multivariate normal mean-variance mixture distributions. This approach generalizes tail covariance allocation and effectively accommodates the skewness and heavy-tailedness commonly observed in financial and insurance data. Numerical analyses demonstrate that TCM reveals tail structures of equity losses that CTE cannot detect, substantially improving the precision with which the tail contributions of individual risk factors are characterized.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Decision/Utility Theory
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• capital allocation
• tail risk
• conditional tail expectation
• tail central moments
• risk contribution

Innovation

• Tail Central Moments
• Capital Allocation
• Normal Mean-Variance Mixture
• Tail Risk
• Conditional Tail Expectation