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Yuyu Chen, Liyuan Lin, Ruodu Wang · 2026-06-22
Value-at-Risk usually rewards diversification, but this paper shows that when losses are extremely heavy-tailed (so heavy their average is effectively infinite), VaR can behave the opposite way — combining risks makes the measured risk worse, not better, and this holds across all probability levels rather than just in extreme tails. The authors formalize this 'universal VaR superadditivity' and prove it applies to broad families of risks, including ones that aren't identically distributed.
Why it matters: It's a warning about diversification assumptions in the presence of catastrophic, infinite-mean-type risks (think certain insurance, operational, or tail-catastrophe exposures): for such portfolios, standard risk measures can penalize spreading across assets, and concentrating in a single asset may look optimal. Practitioners modeling extreme tail risk should recognize that the usual 'diversification reduces risk' logic can break down.
⚠ This is an abstract mathematical result that hinges on infinite-mean loss distributions rarely encountered in typical asset returns, so it has no direct bearing on ordinary portfolio diversification.
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.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.