Paper: arXiv 2601.00568

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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"]

Cited References (60)

  1. Coherent measures of risk (1999)
  2. A polynomial goal programming model for portfolio optimization based on entropy and higher moments (2018)
  3. Properties and comparison of risk capital allocation methods (2017)
  4. TVaR -based capital allocation for multivariate compound distributions with positive continuous claim amounts (2012)
  5. Multivariate P areto portfolios: TCE -based capital allocation and divided differences (2007)
  6. Coherent allocation of risk capital (2001)
  7. Optimal capital allocation principles (2012)
  8. Size-biased risk measures of compound sums (2020)
and 52 more
  1. Hyperbolic distributions in finance (1995)
  2. Tail conditional moment for generalized skew-elliptical distributions (2021)
  3. Mean-risk analysis with risk associated with below-target returns (1977)
  4. Tail variance premium with applications for elliptical portfolio of risks (2006)
  5. Multivariate T weedie distributions and some related capital-at-risk analyses (2010)
  6. The gradient allocation principle based on the higher moment risk measure (2022)
  7. Capital allocation techniques: review and comparison (2021)
  8. Portfolio selection with higher moments (2010)
  9. Extreme conditional tail moment estimation under serial dependence (2019)
  10. Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions (2015)
  11. Conditional tail risk measures for the skewed generalised hyperbolic family (2019)
  12. A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures (2021)
  13. Tail variance for generalised hyper-elliptical models (2025)
  14. An axiomatic approach to capital allocation (2005)
  15. Higher moment coherent risk measures (2007)
  16. Conditional tail moments of the exponential family and its related distributions (2010)
  17. Tail risk measures and risk allocation for the class of multivariate normal mean-variance mixture distributions (2019)
  18. Capital allocation for a sum of dependent compound mixed Poisson variables: a recursive algorithm approach (2019)
  19. A comparison of generalized hyperbolic distribution models for equity returns (2014)
  20. Tail variance premiums for log-elliptical distributions (2013)
  21. Multivariate tail conditional expectation for elliptical distributions (2016)
  22. A multivariate tail covariance measure for elliptical distributions (2018)
  23. Asymptotic results on tail moment and tail central moment for dependent risks (2023)
  24. An optimization approach to the dynamic allocation of economic capital (2004)
  25. Tail conditional expectations for elliptical distributions (2003)
  26. Tail conditional expectations for exponential dispersion models (2005)
  27. The tail Stein’s identity with applications to risk measures (2016)
  28. Quantitative Risk Management: Concepts, Techniques, and Tools - Revised Edition (2015)
  29. Modeling heavy-tailed stock index returns using the generalized hyperbolic distribution (2009)
  30. Allocation of Economic Capital in Loan Portfolios (2000)
  31. Measurement of Risk, Solvency Requirements and Allocation of Capital Within Financial Conglomerates (2002)
  32. Higher order measures of risk and stochastic dominance — → 2402.15387
  33. Generalized deviations in risk analysis (2006)
  34. Tail Moments of Compound Distributions (2022)
  35. Optimization of conditional Value-at-Risk (2000)
  36. Markowitz revisited: mean-variance models in financial portfolio analysis (2001)
  37. The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments (1970)
  38. Conditional expectation as quantile derivative (2001)
  39. Allocating portfolio economic capital to sub-portfolios (2004)
  40. Capital allocation to business units and sub-portfolios: the E uler principle (2008)
  41. The Multivariate Normal Distribution (2012)
  42. On tail conditional variance and tail covariances (2004)
  43. Tail conditional variance for elliptically contoured distributions (2005)
  44. Multivariate skew-normal distributions with applications in insurance (2006)
  45. The moments of the multivariate normal (1985)
  46. Capital allocation based on the tail covariance premium adjusted (2014)
  47. Optimal capital allocation based on the Tail Mean–Variance model (2013)
  48. Tail moments and tail joint moments for multivariate generalized hyperbolic distribution (2025)
  49. A generalized tail mean-variance model for optimal capital allocation (2025)
  50. Multivariate doubly truncated moments for generalized skew-elliptical distributions with applications (2024)
  51. Tail conditional expectations for generalized skew-elliptical distributions (2022)
  52. Statistical Properties of the Generalized Inverse Gaussian Distribution (1982)

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