Range Value-at-Risk (RVaR): Robust Risk Measure

• Range Value-at-Risk (RVaR) is a robust risk measure defined as the average of VaR levels over a probability band, bridging VaR and Expected Shortfall.
• It offers bounded sensitivity and enhanced stability in risk estimation, mitigating issues of model perturbations and tail misspecification.
• Multivariate extensions of RVaR facilitate efficient portfolio optimization, regulatory capital assessment, and risk sharing under uncertainty.

Range Value-at-Risk (RVaR) is a risk measure defined as the average of Value-at-Risk (VaR) levels across a probability range, designed to capture tail risk robustly while maintaining statistical and practical advantages over classical VaR and Expected Shortfall (ES). RVaR generalizes the notion of a single quantile-based risk cutoff by integrating over a specified probability band, providing a flexible tool that interpolates between VaR and ES, and admitting robust extensions to multivariate and model-uncertainty frameworks.

1. Definition and Mathematical Foundations

Let $X$ be a real-valued loss random variable with cumulative distribution function (cdf) $F_X$. The Value-at-Risk at probability level $\alpha$ is defined as: $$\mathrm{VaR}_\alpha(X) = \inf\{x \in \mathbb{R} : F_X(x) \ge \alpha\}, \quad 0 < \alpha < 1.$$ Range Value-at-Risk at levels $\alpha_1 < \alpha_2$ is given by: $$\mathrm{RVaR}_{\alpha_1,\alpha_2}(X) = \frac{1}{\alpha_2-\alpha_1} \int_{\alpha_1}^{\alpha_2} \mathrm{VaR}_u(X) \, \mathrm{d}u = E[X | \mathrm{VaR}_{\alpha_1}(X) \leq X \leq \mathrm{VaR}_{\alpha_2}(X)]$$ (Bairakdar et al., 2020, Rehman et al., 22 Jul 2024, Fissler et al., 2019, Zuo et al., 2023, Liu et al., 26 Nov 2025).

Special cases recover VaR for $\alpha_1 = \alpha_2$ and ES (Tail VaR) as $\alpha_1 \to 0, \alpha_2 \to \alpha$: $$\mathrm{RVaR}_{0,\alpha}(X) = ES_\alpha(X) = \frac{1}{\alpha} \int_0^\alpha \mathrm{VaR}_u(X) \mathrm{d}u.$$

RVaR is law-invariant and inherits key properties from VaR: translation invariance, positive homogeneity, and monotonicity. Subadditivity may fail in general, but holds for comonotonic risks (Bairakdar et al., 2020, Zuo et al., 2023).

2. Robustness and Sensitivity Characteristics

RVaR was introduced by Cont et al. (2010) to provide robustness with respect to model perturbations. The influence function of RVaR is bounded, preventing the measure from "blowing up" under small data contamination. In contrast, ES has unbounded sensitivity to tail misspecification, increasing the risk of regulatory arbitrage in capital estimation (Bairakdar et al., 2020, Rehman et al., 22 Jul 2024).

Empirical and simulation studies demonstrate negligible bias in RVaR estimation even with moderate sample sizes, and lower dispersion in capital requirements compared to ES across admissible models. This robustness is especially relevant in heavy-tailed settings (Rehman et al., 22 Jul 2024, Bairakdar et al., 2020).

3. Multivariate Extensions and Dependence Structure

Multivariate Range Value-at-Risk (MRVaR) generalizes RVaR to risk vectors $X = (X_1, \ldots, X_n)$ via: $$\mathrm{MRVaR}_{p, q}(X) = E\left[X \mid \mathrm{VaR}_{p_k}(X_k) \leq X_k \leq \mathrm{VaR}_{q_k}(X_k) \; \forall k \right]$$ (Bairakdar et al., 2020, Zuo et al., 2023).

MRVaR accounts for dependence via lower-orthant and upper-orthant definitions. In the lower-orthant case, conditioning is based on joint events where each component falls within its truncation interval, with other coordinates below certain cutoffs. For elliptical and log-elliptical distributions (normal, Student-$t$, Laplace, etc.), MRVaR admits closed-form expressions for practical computation (Zuo et al., 2023). The associated multivariate range covariance and "truncated" correlation matrices reveal dependence restricted to the specified quantile intervals.

Robustness and bounded sensitivity also hold in the multivariate setting. MRVaR remains additivity-preserving under comonotonic aggregations, though general subadditivity does not always hold (Bairakdar et al., 2020, Liu et al., 26 Nov 2025).

4. Aggregation, Bounds, and Risk Sharing under Uncertainty

RVaR is central in the theory of robust risk aggregation under model uncertainty and dependence ambiguity. For a collection of marginal distributions, sharp bounds on the aggregate RVaR are obtained using Fréchet-type convolution inequalities, extreme-value information (partial maxima/minima), or copula restrictions and proximity measures (Lux et al., 2016, Liu et al., 26 Nov 2025).

For model uncertainty, worst-case and best-case RVaR bounds are derived by considering the maximum and minimum VaR across candidate distributions. In risk-sharing among agents with average-quantile risk measures, the inf-convolution problem admits a closed-form minimizer under quantile band restrictions, leading to comonotonic or counter-comonotonic allocations depending on the loss regime (Liu et al., 26 Nov 2025, Bairakdar et al., 2020).

Table: Summary of Aggregation and Risk Sharing Bounds (from (Liu et al., 26 Nov 2025))

Risk Measure	Aggregation Bound	Allocation Scheme
RVaR	Extended convolution bounds	Comonotonic/counter-
Inter-RVaR Difference	Duality with quantile bounds	comonotonic by region
Averaged Quantile Risk	Inf-convolution formula	Truncated allocation

5. Estimation Techniques and Numerical Performance

Estimation of RVaR is implemented via quantile integration over observed or simulated data. For parametric models (GEV, GPD, elliptical), explicit formulas facilitate direct calculation. Empirical estimators use empirical cdfs, Riemann sums, and simulation techniques (e.g., Monte Carlo or copula-based sampling for dependent asset returns) (Bairakdar et al., 2020, Rehman et al., 22 Jul 2024, Zuo et al., 2023).

Quantum algorithms, leveraging amplitude estimation, offer quadratic speedup for RVaR computation in simulation-heavy regimes, although denominator noise can inflate variance for small probability bands. Error mitigation strategies (iterative amplitude estimation, polynomial circuit depth reduction) improve stability for practical use on contemporary quantum hardware (Laudagé et al., 2022).

Monte Carlo and backtesting frameworks employ strictly consistent scoring functions for the triplet $(\VaR_\alpha, \RVaR_{\alpha,\beta}, \VaR_\beta)$, as RVaR alone is not elicitable. Murphy diagrams are used for diagnostic assessment of estimator performance under a family of loss functions (Fissler et al., 2019, Rehman et al., 22 Jul 2024).

6. Applications in Portfolio Optimization, Regulatory Capital, and Model Ambiguity

RVaR and MRVaR are used for optimal portfolio selection, substituting for mean and covariance in classical Markowitz frameworks. Efficient frontiers are computed for MRVaR and variance, reflecting different tail-risk appetites (Zuo et al., 2023).

In regulatory capital, RVaR provides coherent and robust quantification, decreasing susceptibility to regulatory arbitrage seen with ES. Legal Robustness indices, computed as relative dispersion across approved models, show RVaR’s resistance to model misspecification (Rehman et al., 22 Jul 2024).

Under distributional ambiguity with constraints (mean, variance, increasing failure rate), extreme-case RVaR formulæ yield sharp capital allocations for insurance and risk management. Explicit parametric distributions (truncated exponential families) form the basis for stress-testing under such uncertainty (Su et al., 29 Jun 2025).

7. Controversies, Limitations, and Open Problems

The non-elicitability of RVaR as a univariate functional hinders direct use in forecast verification and backtesting, but joint elicitability with bounding quantiles resolves practical issues. Subadditivity is not guaranteed in all cases—careful selection of the probability band and aggregation regime is necessary for coherence (Fissler et al., 2019, Bairakdar et al., 2020).

RVaR’s sensitivity to denominator estimation in slicing-based algorithms (quantum or classical) presents numerical challenges, particularly for small tail bands. This suggests balancing band width against desired tail sensitivity in practical implementation (Laudagé et al., 2022).

The extension of RVaR to more general aggregation functionals and to distributions outside classical parametric families remains an active area for research, with computational tractability and robustness properties as primary considerations.

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In sum, Range Value-at-Risk advances the risk measurement toolkit by interpolating between VaR and ES, maintaining robustness to both statistical and model perturbations, supporting multivariate, aggregation, and risk-sharing applications, and offering practical estimation and optimization pathways for modern financial, insurance, and regulatory environments (Bairakdar et al., 2020, Zuo et al., 2023, Liu et al., 26 Nov 2025, Rehman et al., 22 Jul 2024, Lux et al., 2016, Fissler et al., 2019, Laudagé et al., 2022, Su et al., 29 Jun 2025, Hu, 2019).