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

Paper: arXiv 2509.12558

Authors: 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.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This short theoretical note provides a clean, elegant mathematical proof establishing a necessary and sufficient condition for VaR subadditivity across all confidence levels via comonotonicity. However, it is purely theoretical with no empirical backtesting or real-world financial data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Investigate VaR Subadditivity & Comonotonicity Connection] --> B[Inputs: Loss Random Variables & VaR Confidence Levels]
    B --> C[Mathematical Analysis: Evaluate VaR Sums vs. Combined VaR]
    C --> D[Theoretical Proof: Connect Subadditivity to Joint Distribution Properties]
    D --> E[Outcome 1: VaR is Subadditive for All Levels ⇔ Variables are Comonotonic]
    D --> F[Outcome 2: New Equivalent Condition for Comonotonicity of Random Vectors]