Hidden Dependence and Aggregate Tail Risk

Abstract: We study risk aggregation problems for arbitrary non-decreasing aggregation functions and tail risk measures under dependence uncertainty in a distributionally robust setting. To this end, we introduce the notion of hidden dependence for random vectors, which is built on the concepts of risk concentration and common tail events developed in Wang and Zitikis (2020). We show that, starting from a tail event $A$ of the aggregate loss for an arbitrary random vector $Y$, one can construct a random vector with hidden dependence that dominates $Y$ on the tail event $A$. We then focus on the case in which model uncertainty is described by small perturbations of the distribution of a random vector with respect to a suitable probability distance without changing the marginals. We show that these perturbations of the reference distribution are compatible with hidden dependence and thus lead to the same worst-case risk bounds as in the unconstrained case for arbitrary $γ$-tail risk measures with a suitable level $γ$. Finally, we apply our results in a credit risk context and quantify the potential underestimation of portfolio risk arising from uncertainty in the dependence structure. In particular, we show that even small deviations from a reference Gaussian dependence model can, in principle, justify dramatic increases in capital requirements.

• The paper introduces (c,γ)-hidden dependence and proves that worst-case γ-tail risk with fixed marginals can be characterized using models whose dependence matches a reference copula outside a common extreme-tail event.
• It shows that arbitrarily small Wasserstein, integral probability metric, or dependence-measure deviations from a reference model can preserve unconstrained worst-case tail risk, meaning constraints must directly govern far-tail dependence to be effective.
• A 500-loan Merton portfolio demonstrates the impact: models close to a Gaussian benchmark can produce worst-case VaR⁰·⁹⁹⁹ equal to total exposure, versus 41 under the benchmark, implying more than tenfold underestimation.

Overview and motivation

"Hidden Dependence and Aggregate Tail Risk" (De Vecchi, Nendel, and Vanduffel) studies worst-case values of tail risk measures for aggregated losses when marginal distributions are fixed but the dependence structure is only partially known. The paper's central object is a new notion, $(c,\gamma)$-hidden dependence, which formalizes situations in which a fitted copula $c$ correctly describes the joint behavior of a random vector except on a common $\gamma$-tail event — that is, except precisely when all components simultaneously realize their largest values. The authors show that such hidden dependence structures are compatible with arbitrarily small perturbations of a reference multivariate distribution measured by consistent probability distances (including $p$-Wasserstein distances and integral probability metrics), and consequently that distance- or dependence-measure constraints around a reference model may fail to reduce worst-case tail risk at high confidence levels. The practical upshot is stark: even small deviations from a reference Gaussian copula can justify worst-case VaR equal to total portfolio exposure.

The work sits in the literature on risk aggregation under dependence uncertainty (2606.30193), which ranges from fully unconstrained settings — where worst-case VaR of a sum can approach the sum of individual TVaRs — to partial-information settings using variance constraints, factor models, or correlation bounds. Compared with Eckstein, Kupper, and Pohl's numerical approach to robust risk aggregation, this paper provides structural conditions; compared with De Vecchi, Nendel, and Streicher's result that arbitrarily small positive dependence can hide co-monotonic tails, it shows that adding explicit dependence information does not remedy the problem.

Hidden dependence and reduction of the optimization problem

Building on Wang and Zitikis' concepts of $\gamma$-tail events and common tail events, the paper defines $(c,\gamma)$-hidden dependence: a random vector $X$ has $(c,\gamma)$-hidden dependence if there exists a common $\gamma$-tail event $A$ (with $c$0) such that, conditionally on $c$1, the vector has copula $c$2. A characterization lemma gives four equivalent formulations, including a stochastic representation in which each component satisfies

$c$3

with $c$4 on $c$5 and $c$6 on $c$7, as well as an explicit patchwork formula for the full copula $c$8 combining $c$9 below level $\gamma$0 and an arbitrary tail copula above it.

The backbone result (Proposition 3.6) is a dominance construction: starting from any random vector $\gamma$1 and any event $\gamma$2 of probability $\gamma$3, one can build a vector $\gamma$4 with identical marginals and $\gamma$5-hidden dependence whose components dominate those of $\gamma$6 almost surely on $\gamma$7. Since common $\gamma$8-tail events are $\gamma$9-tail events for $p$0 under any non-decreasing aggregation function $p$1, and since $p$2-tail risk measures depend only on conditional distributions given such events, the consequence is a structural reduction: the supremum of any $p$3-tail risk measure over all couplings with fixed marginals equals the supremum restricted to vectors with $p$4-hidden dependence (Corollary 3.9). More generally, if the feasible set $p$5 contains all $p$6-hidden-dependence distributions, the constrained and unconstrained suprema coincide (Theorem 3.10). In the identically distributed, sublinear case with co-monotonic tail copula, the worst case equals $p$7.

An immediate implication is that dependence information does not improve tail risk bounds unless it constrains behavior specifically inside the far tail — a region where data are essentially never informative.

Distributional perturbations do not help

The second main result addresses ambiguity sets defined by a ball of radius $p$8 around a reference coupling, where the radius is measured by a consistent probability distance — a map dominating weak convergence along fixed-marginal sequences. Examples include $p$9-Wasserstein distances, integral probability metrics with bounded continuous generators (e.g., the Fortet–Mourier metric), Kolmogorov-type sup distances, and distances induced by regular dependence measures.

The key estimate shows that for a bounded continuous IPM generator $\gamma$0,

$\gamma$1

which vanishes as $\gamma$2 by dominated convergence. Hence, for every $\gamma$3 there exists $\gamma$4 such that every $\gamma$5-hidden-dependence vector with $\gamma$6 lies within the ball of radius $\gamma$7 around the reference model (Theorem 3.14). Combining this with the structural reduction yields the paper's central theorem: for any $\gamma$8 there is a confidence level $\gamma$9 at which the constrained worst-case value equals the unconstrained one (Theorem 3.16). Via the correspondence between consistent probability distances and regular dependence measures, the same conclusion holds when partial information is imposed through broad classes of dependence measures (Corollary 3.18), including Pearson correlation, Spearman's rho, Kendall's tau, and transport dependencies.

Beyond the class of $(c,\gamma)$0-tail risk measures, the paper extends parts of the analysis to left-continuous VaR at levels $(c,\gamma)$1 and to expectiles. For VaR, if the survival copula satisfies $(c,\gamma)$2, then $(c,\gamma)$3 for $(c,\gamma)$4 — i.e., the comonotonic value. For expectiles at levels $(c,\gamma)$5 with $(c,\gamma)$6, the worst-case expectile of a weighted sum over any feasible set containing hidden-dependence vectors equals $(c,\gamma)$7, again matching the unconstrained bound.

Quantitative calibration: Wasserstein radii and dependence measures

To make the qualitative results operational, the paper derives explicit relations between the tail level $(c,\gamma)$8 and the admissible radius $(c,\gamma)$9. For $X$0-Wasserstein distances ($X$1) with the normalized $X$2 norm, closed-form upper bounds on $X$3 for $X$4-hidden-dependence vectors are obtained; for $X$5 the bound involves $X$6 of the maximum of two independent copies of each marginal, and for $X$7 and non-negative marginals it reduces to $X$8. Explicit invertible conditions linking $X$9 and $(c,\gamma)$0 are worked out for uniform, exponential, and Bernoulli marginals; e.g., for uniform marginals on $(c,\gamma)$1, $(c,\gamma)$2 whenever $(c,\gamma)$3.

For dependence measures, sharp bounds are derived independently of the tail copula $(c,\gamma)$4: for Spearman's rho, $(c,\gamma)$5 whenever $(c,\gamma)$6 (under a mild condition on the reference copula), and analogously for Kendall's tau with a square-root threshold. For exchangeable Bernoulli vectors — the natural setting for default indicators — the exact identity

$(c,\gamma)$7

provides a principled way to calibrate the radius $(c,\gamma)$8: if a Student-$(c,\gamma)$9 alternative model is deemed plausible, all models no farther than its Wasserstein distance from the Gaussian benchmark are admissible.

Credit risk implications

The final section applies these results to a homogeneous Merton-model loan portfolio ($\gamma$0, default probability $\gamma$1, asset correlation $\gamma$2, unit exposures), taking a Gaussian copula as reference and calibrating $\gamma$3 via the exact exchangeable-Bernoulli Wasserstein distance to a Student-$\gamma$4 alternative. The numbers are striking:

$\gamma$5-copula dof	$\gamma$6 radius $\gamma$7	Compatible $\gamma$8	Underestimation TVaR$\gamma$9	Underestimation VaR$A$0
3	0.1063	0.9944	883%	1219%
5	0.0917	0.9958	740%	1219%
10	0.0733	0.9973	534%	1219%
20	0.0550	0.9985	362%	1219%

Because the compatible $A$1 falls below $A$2 in every row, the worst-case VaR$A$3 attains the unconstrained bound of 500 (total exposure), whereas the Gaussian reference model yields a VaR of only 41 — a more-than-tenfold underestimation even when the plausible alternative is a $A$4-copula that is statistically nearly indistinguishable from the Gaussian benchmark. For large portfolios ($A$5), using Vasicek's asymptotic formula — which also underlies the RWA calculation in Article 153 of the CRR — the implied VaR$A$6 underestimation ratios range from roughly 778% to 2703% across default probabilities between 0.75% and 1.25% and correlations between 5% and 15%, always with $A$7 so that the unconstrained bound applies. These figures indicate that capital requirements based on very high confidence levels are highly sensitive to dependence assumptions that available data cannot discipline.

Limitations and open questions

Several caveats should be noted. First, the compatibility result is existential: for a given $A$8 it guarantees some $A$9 below which constraints cease to bind, but whether the relevant regulatory confidence level (e.g., $c$00) actually exceeds the compatible $c$01 depends on the specific marginals, reference copula, and metric — the credit-risk examples confirm this for their parameterizations but do not establish universality. Second, the quantitative Wasserstein bounds are upper bounds on attainable distances, so the calibrated radii are conservative in the sense of possibly overstating the feasible set; conversely, the hidden-dependence model with co-monotonic tails is one admissible model rather than a certified worst case for TVaR$c$02 when $c$03. Third, the expectile result requires the condition $c$04 and identically distributed marginals, and the VaR extension requires positivity of the diagonal of the survival copula. Finally, the analysis presumes fixed, exactly known marginals; how marginal uncertainty interacts with hidden dependence remains outside the scope of the paper, as does the empirical question of which tail copulas are statistically identifiable from realistic default datasets.

Conclusion

The paper establishes that $c$05-hidden dependence characterizes worst-case aggregate tail risk for arbitrary non-decreasing aggregations and $c$06-tail risk measures, and that such structures survive arbitrarily tight distributional or dependence-measure constraints anchored at any reference model once the confidence level is pushed sufficiently close to one. The credit-risk application quantifies the consequence: deviations from a Gaussian copula small enough to be statistically undetectable can inflate worst-case VaR$c$07 by an order of magnitude, up to total exposure. The results caution against interpreting high-confidence tail risk estimates from fitted multivariate models as robust, and they identify precisely what kind of information — namely, constraints operative inside the far tail itself — would be required to improve upon the unconstrained bounds.