[Paper Review] Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution

Enrique Calderín–Ojeda, Yuyu Chen+1 more|arXiv (Cornell University)|Jan 2, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper introduces a tail central moments (TCM) based capital allocation method and derives recursive analytical expressions for TCM and TCM-based capital allocation within the multivariate normal mean-variance mixture (NMVM) framework, including GH distributions.

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.

Motivation & Objective

• Motivate the need for improved tail-risk assessment beyond CTE-based capital allocation.
• Introduce tail central moments (TCM) as a generalisation of tail variance for capital allocation.
• Derive recursive analytical expressions for the TCM and TCM-based capital allocation in NMVM distributions.
• Show how NMVM and GH distributions model asymmetric and heavy-tailed financial data.
• Demonstrate via numerical analysis how TCM-based allocation reveals tail patterns not captured by CTE.

Proposed method

• Define k-th order tail moments (TM) and tail central moments (TCM) at level alpha.
• Propose a TCM-based capital allocation where K = TCM_alpha,k(S) and K_i = Cov[X_i, (S - CTE_alpha(S))^{k-1} | S > s_alpha].
• Develop recursive formulas for TM and TCM of the univariate NMVM distribution (Theorem 1).
• Extend TM/TCM results to multivariate NMVM to obtain explicit capital allocations for each component (Section 4).
• Specialise results to the GH/NMVM subclass when Theta follows a GIG distribution (Remark 4).
• Provide a numerical example using multivariate GH to illustrate the approach (Section 5).

Experimental results

Research questions

• RQ1 How can tail central moments be used to improve capital allocation beyond CTE-based methods in NMVM models?
• RQ2 What are the recursive formulas for TM and TCM for univariate NMVM distributions?
• RQ3 How can TCM-based capital allocation be computed and interpreted for multivariate NMVM distributions?
• RQ4 In what way does the NMVM/GH structure influence tail-risk contributions across components?

Key findings

• TCM-based capital allocation satisfies the full allocation property (Proposition 1).
• Explicit recursive expressions are derived for the TM and TCM of the NMVM aggregate loss S (Theorem 1).
• A closed-form or recursively computable TM/TCM-based allocation is obtained for the multivariate NMVM setting (Section 4).
• Corollaries provide CTE, TM_2, and TV results as special cases of the NMVM framework (Corollary 1).
• The approach accommodates GH distributions as a special NMVM case, enabling broader applicability (Remark 4).
• Numerical illustration with multivariate GH demonstrates that TCM-based allocation can reveal tail-risk patterns not detected by CTE alone (Section 5).