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arxiv: 2512.07787 · v4 · submitted 2025-12-08 · 💱 q-fin.RM · math.PR

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

Nawaf Mohammed This is my paper

Pith reviewed 2026-05-17 00:43 UTC · model grok-4.3

classification 💱 q-fin.RM math.PR
keywords value-at-risksub-additivitysuper-additivityone-sided random variablesnegative simplex dependenceco-monotonicityrisk aggregationdependence structures
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The pith

For risks supported on [0,∞), Value-at-Risk sub-additivity is impossible except in the degenerate case of exact additivity under co-monotonicity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the additivity properties of Value-at-Risk for sums of one-sided random variables across all probability levels. For risks supported on the non-negative reals, sub-additivity fails except when the variables are co-monotonic, in which case VaR is exactly additive. The authors introduce negative simplex dependence for the joint distribution and simplex dominance for a margin-dependent functional to identify when full super-additivity instead holds. These conditions accommodate non-identical margins, heavy tails, and a range of negative dependence structures while remaining easy to check. The same results apply to random variables with arbitrary finite lower or upper endpoints, giving sharp limits on strict sub- or super-additivity.

Core claim

For risks supported on [0,∞), 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, the paper introduces 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

What carries the argument

Negative simplex dependence (NSD) for the joint distribution together with simplex dominance (SD) for a margin-dependent functional, which together guarantee full super-additivity of VaR.

If this is right

  • VaR of the sum equals the sum of VaRs precisely when the variables are co-monotonic.
  • Full super-additivity of VaR holds whenever the joint distribution meets negative simplex dependence and the functional meets simplex dominance.
  • The same additivity constraints apply to risks with any finite lower or upper endpoint.
  • The framework covers non-identical margins and heavy-tailed distributions without requiring identical laws.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Portfolio managers could test the NSD and SD conditions on loss data to determine whether aggregated VaR is conservative.
  • The structural conditions may generalize to other quantile-based risk measures in multi-period settings.
  • Regulatory capital calculations for positive loss distributions might incorporate these dependence checks to avoid understating aggregate risk.

Load-bearing premise

The joint distribution must satisfy negative simplex dependence and the functional must satisfy simplex dominance at the specific probability levels and for the distributions considered.

What would settle it

A pair of non-co-monotonic non-negative random variables for which VaR of the sum is smaller than the sum of the individual VaRs at some probability level would falsify the impossibility of sub-additivity.

Figures

Figures reproduced from arXiv: 2512.07787 by Nawaf Mohammed.

Figure 1
Figure 1. Figure 1: Support of the Ordinal Sum copula C(u, v) on the unit square [0, 1]2 . Since both marginals are Pareto (II) with unit shape then their expectations are infinite. Their common VaR is VaRp[X1] = VaRp[X2] = p 1 − p . For the sum S = X1 + X2, the VaR is piecewise and given by VaRp[S] =    6 + 8p 2 9 − 4p 2 , 0 < p ≤ 1 2 , 2 − 2p(1 − p) p(1 − p) , 1 2 < p < 1. A direct comparison between VaRp[S] and Va… view at source ↗
Figure 2
Figure 2. Figure 2: The marginal ϕi functions. By Proposition 3.6, this implies that Φ is not globally non-increasing. Nevertheless, we now verify that the SD condition for Φ(x1, x2, x3) = ϕ1(x1) + ϕ2(x2) + ϕ3(x3) 28 [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: compares VaRp[S] with the sum of marginal VaRs [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript investigates the extremal aggregation properties of Value-at-Risk (VaR) for sums of one-sided random variables. For risks supported on [0, ∞), it establishes that sub-additivity is impossible except in the degenerate case of exact additivity, which occurs only under comonotonicity. To characterize full super-additivity, the authors introduce two new structural conditions: negative simplex dependence (NSD) for the joint distribution and simplex dominance (SD) for a margin-dependent functional. These are presented as sufficient to guarantee the property across probability levels, accommodating non-identical margins, heavy-tailed distributions, and various negative dependence structures. All results are extended to random variables with arbitrary finite lower or upper endpoints.

Significance. If the central claims hold, the work delivers sharp theoretical constraints on VaR additivity for positive risks, which is relevant to risk management and regulatory practice where sub-additivity is often invoked for diversification. The introduction of NSD and SD supplies a unified, verifiable framework that handles non-identical margins and heavy tails without reducing to fitted parameters. Credit is due for the clean separation of the impossibility result from the sufficient conditions and for the explicit extension to bounded supports.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: The impossibility of strict sub-additivity is derived from the one-sided support constraint; the argument appears to use the fact that P(X+Y > q) ≤ P(X > q) + P(Y > q) cannot hold strictly for all q when supports are [0,∞). Clarify whether the same contradiction arises immediately for the finite-endpoint extension stated in the abstract, or whether additional boundary terms must be controlled.
  2. [§4.3, Theorem 4.4] §4.3, Theorem 4.4: The sufficiency claim for full super-additivity under NSD and SD is load-bearing for the characterization result. The derivation of the inequality chain for non-identical margins relies on the margin-dependent functional satisfying SD at the specific levels α and β; an explicit verification step or counter-example for Pareto margins with different shape parameters would strengthen the generality statement.
minor comments (3)
  1. [§2] The notation for the probability levels (α, β, etc.) is introduced in §2 but used inconsistently in the statements of Theorems 4.1 and 5.1; a single global convention would improve readability.
  2. [Definition 4.1] Definition 4.1 of negative simplex dependence would benefit from an immediate example with a concrete copula (e.g., Clayton with negative parameter) to illustrate how the simplex condition is checked in practice.
  3. [§4] The abstract claims the conditions are 'easily verifiable'; the main text should include a short algorithmic checklist or pseudocode for verifying NSD and SD for a given joint distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive evaluation of the manuscript. We address each major comment below, indicating where revisions will be made to improve clarity and strengthen the presentation.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: The impossibility of strict sub-additivity is derived from the one-sided support constraint; the argument appears to use the fact that P(X+Y > q) ≤ P(X > q) + P(Y > q) cannot hold strictly for all q when supports are [0,∞). Clarify whether the same contradiction arises immediately for the finite-endpoint extension stated in the abstract, or whether additional boundary terms must be controlled.

    Authors: The core contradiction in Theorem 3.2 indeed stems from the one-sided support preventing strict sub-additivity for all q. For the finite-endpoint extensions, the same logical contradiction arises, but the proof requires explicit control of boundary probability masses at the lower or upper endpoints. We adjust the quantile definitions by subtracting endpoint contributions and show that the inequality P(X+Y > q) > P(X > q) + P(Y > q) still cannot hold strictly unless the variables are comonotonic. We will add a clarifying paragraph in §3 detailing this boundary adjustment to make the extension fully transparent. revision: yes

  2. Referee: [§4.3, Theorem 4.4] §4.3, Theorem 4.4: The sufficiency claim for full super-additivity under NSD and SD is load-bearing for the characterization result. The derivation of the inequality chain for non-identical margins relies on the margin-dependent functional satisfying SD at the specific levels α and β; an explicit verification step or counter-example for Pareto margins with different shape parameters would strengthen the generality statement.

    Authors: The proof of Theorem 4.4 is formulated for arbitrary non-identical margins and holds whenever the SD condition is satisfied at the relevant α and β levels, which encompasses heterogeneous Pareto margins under suitable NSD. To strengthen the exposition as suggested, we will insert a brief explicit verification example in §4.3 for two Pareto distributions with distinct shape parameters, confirming that the SD inequality chain applies directly and yields full super-additivity. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained with no circular reductions

full rationale

The paper derives the impossibility of VaR sub-additivity for risks on [0,∞) except under exact additivity and comonotonicity directly from the definition of VaR and the one-sided support constraint, without any reduction to fitted inputs or self-referential equations. The new structural conditions NSD for the joint law and SD for the margin-dependent functional are introduced as independent, verifiable assumptions that suffice for full super-additivity; these are not outputs of the same model but external characterizations that accommodate non-identical margins and heavy tails. All extensions to arbitrary finite endpoints follow from the same support-based arguments, yielding a self-contained mathematical framework with no load-bearing self-citations or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on standard properties of VaR as a quantile functional and on the newly introduced notions of negative simplex dependence and simplex dominance. No free parameters are fitted to data. The two new conditions function as invented structural entities whose independent verification would require checking the joint distribution against the stated definitions.

axioms (2)
  • standard math VaR is defined as the left-continuous quantile function at level p for each marginal and for the sum.
    Invoked throughout the abstract as the object whose additivity properties are studied.
  • domain assumption Random variables are supported on [0, ∞) or have arbitrary finite lower or upper endpoints.
    Explicitly stated as the setting in which the impossibility and characterization results hold.
invented entities (2)
  • Negative simplex dependence (NSD) no independent evidence
    purpose: A joint-distribution condition that, together with simplex dominance, guarantees full super-additivity of VaR.
    Newly introduced structural condition for the joint law; independent evidence would be verification on concrete copulas or distributions outside the paper.
  • Simplex dominance (SD) no independent evidence
    purpose: A margin-dependent functional condition that, together with NSD, guarantees full super-additivity of VaR.
    Newly introduced condition relating the individual distributions; independent evidence would be checking the functional inequality on given margins.

pith-pipeline@v0.9.0 · 5449 in / 1585 out tokens · 34353 ms · 2026-05-17T00:43:33.449992+00:00 · methodology

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Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    https://doi.org/10.1007/s10687-008-0071-5 Embrechts, P., Neˇ slehov´ a, J., & W¨ uthrich, M. V. (2009). Additivity properties for value-at- risk under archimedean dependence and heavy-tailedness.Insurance: Mathematics and Economics,44(2), 164–169. https://doi.org/10.1016/j.insmatheco.2008.08.001 48 Gautschi, W. (1959). Some elementary inequalities relatin...

    work page doi:10.1007/s10687-008-0071-5 2009

  2. [2]

    https://doi.org/10.1016/s0378-4266(02)00272-8 Zhu, W., Li, L., Yang, J., Xie, J., & Sun, L. (2023). Asymptotic subadditivity/superadditivity of value-at-risk under tail dependence.Mathematical Finance,33(4), 1314–1369. https: //doi.org/10.1111/mafi.12393 49

    work page doi:10.1016/s0378-4266(02)00272-8 2023