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

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.

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

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.

Paper Structure

This paper contains 10 sections, 12 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. VaR Sub-additivity
  3. VaR super-additivity
  4. Further Generalizations and Remarks
  5. Conclusions
  6. Proofs of Section \ref{['sec:Generalizations']}
  7. Proof of Proposition \ref{['prop:XaXbImpossibility']}
  8. Proof of Proposition \ref{['prop:VaR_super-extension']}
  9. Proof of Proposition \ref{['prop:VaR_super-extension-sufficient']}
  10. Proof of Proposition \ref{['prop:XaXb_Xi_convex']}

Key Result

Theorem 2.2

$\boldsymbol{X}$ is VaR sub-additive if and only if $\boldsymbol{X}$ is VaR additive. In addition, $\boldsymbol{X}$ must be a co-monotonic vector.

Figures (2)

  • Figure 2: Plots of the functions $\phi_i$.
  • Figure 3: Comparison of ${\mathrm{VaR}}_p[S]$ and ${\mathrm{VaR}}_p[X_1]+{\mathrm{VaR}}_p[X_2]+{\mathrm{VaR}}_p[X_3]$.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • ...and 27 more