VaR at Its Extremes: Impossibilities and Conditions for One-Sided Random Variables

Abstract

Value-at-Risk (VaR) may reward diversification at some probability levels and penalize it at others. We study the two extremal regimes in which one effect prevails at every level simultaneously, for sums of one-sided risks. For risks supported on $[0,\infty )$, VaR sub-additivity at all levels, exact VaR additivity and co-monotonicity are equivalent without any integrability assumption; for such risks this removes the finite-mean hypothesis of Imamura and Kato (2026), which cannot be dropped for two-sided risks. Global super-additivity, in turn, is equivalent to first-order stochastic dominance of the aggregate over its co-monotonic counterpart. We factorize this condition through the maximum of independent copies of the margins into negative simplex dependence (NSD), a condition on the joint law that is strictly weaker than negative lower orthant dependence, and simplex dominance (SD) of the aggregator $\Phi (x_1,\dots ,x_n)={\sum }_{i=1}^n{x}_i\log F_{X_i}(x_i)$, which for continuous margins holds whenever each reciprocal risk $1/X_i$ has a decreasing failure rate on average. The criterion admits heterogeneous margins, yields a lower bound on the diversification penalty that depends on the margins only, is exact for unit Fréchet margins, and accommodates positively and tail-dependent risks such as common-shock models. Independently of any factorization, a vector that is VaR super-additive but not VaR additive has at least two infinite-mean components, and this bound is sharp. A reflection principle extends all results to finite endpoints: lower-bounded risks are never strictly sub-additive, upper-bounded risks are never strictly super-additive, and compactly supported risks are neither.