Paper: arXiv 2512.07787

Authors: Nawaf Mohammed

Abstract

We investigate the extremal aggregation behavior of Value-at-Risk (VaR) – that is, its additivity properties across all probability levels – for sums of one-sided random variables. For risks supported on ([0,\infty)), we show that VaR sub-additivity is impossible except in the degenerate case of exact additivity, which holds only under co-monotonicity. To characterize when VaR is instead fully super-additive, we introduce two structural conditions: negative simplex dependence (NSD) for the joint distribution and simplex dominance (SD) for a margin-dependent functional. Together, these conditions provide a unified and easily verifiable framework that accommodates non-identical margins, heavy-tailed laws, and a wide spectrum of negative dependence structures. All results extend to random variables with arbitrary finite lower or upper endpoints, yielding sharp constraints on when strict sub- or super-additivity can occur.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs expanding Value-at-Risk additivity theory to one-sided random variables without integrability assumptions. While highly theoretical with strong math complexity and clear exposition, it lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Analyze VaR Extremal Aggregation<br>for One-Sided Risks on [0, ∞)"] --> B["Inputs:<br>Joint Distributions & Margins<br>(Non-identical, Heavy-tailed)"]
    
    B --> C["Theoretical Analysis:<br>Evaluate Sub-additivity vs.<br>Super-additivity Conditions"]
    
    C --> D["Structural Framework:<br>Apply Negative Simplex Dependence (NSD)<br>& Simplex Dominance (SD)"]
    
    D --> E1["Outcome 1: Impossibility Result<br>Sub-additivity impossible except for<br>exact additivity (Co-monotonicity)"]
    
    D --> E2["Outcome 2: Super-additivity Characterization<br>Unified conditions for full super-additivity<br>extended to arbitrary finite endpoints"]