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A Note on Subadditivity of Value at Risks (VaRs): A New Connection to Comonotonicity

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  • Yuri Imamura
  • Takashi Kato

Abstract

In this paper, we provide a new property of value at risk (VaR), which is a standard risk measure that is widely used in quantitative financial risk management. We show that the subadditivity of VaR for given loss random variables holds for any confidence level if and only if those are comonotonic. This result also gives a new equivalent condition for the comonotonicity of random vectors.

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  • Yuri Imamura & Takashi Kato, 2025. "A Note on Subadditivity of Value at Risks (VaRs): A New Connection to Comonotonicity," Papers 2509.12558, arXiv.org, revised Oct 2025.
    • Handle: RePEc:arx:papers:2509.12558
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    References listed on IDEAS

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    1. Cuestaalbertos, J. A. & Ruschendorf, L. & Tuerodiaz, A., 1993. "Optimal Coupling of Multivariate Distributions and Stochastic Processes," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 335-361, August.
    2. Denuit, Michel & Dhaene, Jan & Ghossoub, Mario & Robert, Christian Y., 2025. "Comonotonicity and Pareto optimality, with application to collaborative insurance," Insurance: Mathematics and Economics, Elsevier, vol. 120(C), pages 1-16.
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