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

TL;DR

The paper addresses the problem of subadditivity for Value at Risk $VaR_\alpha$ and shows that the inequality $VaR_\alpha(\sum_{i=1}^n X_i) \le \sum_{i=1}^n VaR_\alpha(X_i)$ holding for all $\alpha\in(0,1)$ is equivalent to the random vector $(X_1,\ldots,X_n)$ being comonotonic. It then collects equivalent characterizations of comonotonicity, including a minimal copula $M(u)=\min\{u_i\}$, a uniform random variable representation $(F^{-1}_{X_1}(U),\ldots,F^{-1}_{X_n}(U))$, and a Fréchet-class ordering condition. The results extend to elliptic distributions, where subadditivity holds for $\alpha\in[1/2,1)$ and equality implies comonotonicity, clarifying when VaR behaves coherently. The analysis relies on integrability and atomless probability spaces, and highlights the necessity of these conditions by noting counterexamples when integrability fails.

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.

Theorems & Definitions (6)

• Definition 1
• Theorem 1
• Remark 1: Equivalent conditions for comonotonicity
• Remark 2: Elliptic distributions
• Remark 3: Integrability condition
• proof : Proof of Theorem \ref{['th_main']}