Universal Value-at-Risk superadditivity

2026-06-22

AI Summary

This study investigates whether Value-at-Risk (VaR) exhibits universal superadditivity (UVS)—i.e., superadditivity at all confidence levels—under heavy-tailed losses. It introduces, for the first time, UVS and its weighted variant (WUVS) as global properties of random vectors, proposing strict notions of UVS/WUVS and revealing their profound implications: any distortion risk measure becomes superadditive, and optimal portfolio allocation collapses to a single asset. Using tools such as convex increasing transformations, weak convergence, and distributional mixtures, the paper establishes preservation results and sufficient conditions for UVS/WUVS within risk families characterized by complete subscalability, super-Cauchy behavior, or reverse subadditivity. The findings demonstrate that various classes of non-identically distributed heavy-tailed risks satisfy UVS/WUVS, thereby invalidating diversification benefits in these settings and offering rigorous theoretical support for risk management practices.

Abstract

Value-at-Risk (VaR) is a standard regulatory risk measure, and its failure of subadditivity is well known. Much less appreciated is that for sufficiently heavy-tailed losses, VaR can be superadditive uniformly across all probability levels, a phenomenon strictly stronger than the asymptotic superadditivity studied in extreme value theory. We call this property universal VaR superadditivity (UVS). We study UVS and its stronger weighted version (WUVS) as properties of random vectors rather than of marginal distributions. This perspective unifies and extends a recent line of work on iid infinite-mean models. UVS, except for trivial cases, imposes an infinite-mean structure. We establish preservation properties of UVS and WUVS under increasing and convex transformations, weak convergence, and certain distributional mixtures, and use these tools to prove UVS and WUVS for non-identically distributed risks in several large families including completely subscalable, super-Cauchy, and inverted subadditive risks, extending results previously available only in the iid case. In many results, we also establish strict versions of UVS and WUVS, which lead to stronger decision-theoretic implications. As a consequence, for any portfolio satisfying WUVS, every distortion risk measure is superadditive, so an optimal allocation concentrates on a single asset, and diversification is never beneficial.

Problem

• Value-at-Risk
• superadditivity
• heavy-tailed distributions
• risk diversification
• distortion risk measures

Innovation

• Universal VaR Superadditivity
• Weighted UVS
• Infinite-mean distributions
• Distortion risk measures
• Non-iid risks