Interval CVaR: A Unified Risk Measure

• Interval Conditional Value-at-Risk (In-CVaR) is a risk measure that averages quantiles over a specified interval to improve robustness against contamination and uncertainty.
• It unifies traditional estimators such as trimmed mean, least trimmed squares, and classical CVaR, extending their application in robust regression and risk management.
• The framework is applicable in portfolio optimization with interval-valued returns, providing practical, computationally tractable solutions validated by empirical studies.

Interval Conditional Value-at-Risk (In-CVaR) is a statistical risk measure that generalizes the classical Conditional Value-at-Risk (CVaR) by considering both lower and upper quantiles of a loss distribution or random return, as well as incorporating extensions to interval-valued uncertainties. The In-CVaR framework unifies traditional robust estimators such as trimmed mean, least trimmed squares (LTS), least median of squares (LMS), and classical CVaR by varying the interval over which the quantiles are considered. Recent developments demonstrate the advantages of In-CVaR in both regression modeling and financial risk management, especially in contexts requiring robustness against contamination, outliers, and data imprecision (You et al., 16 Jan 2026, Zhang et al., 2022).

1. Mathematical Foundations of In-CVaR

Let $D$ denote a probability distribution over a sample space $\mathcal X \times \mathcal Y$. For a regression model $f(x,\theta)$ parameterized by $\theta$, let $L(\theta) = \mathcal L(f(X_D, \theta) - Y_D)$ represent the induced random loss, where $\mathcal L:[0,\infty)\to[0,\infty)$ is a nondecreasing loss function.

• Value-at-Risk (VaR):

$$\mathrm{VaR}_\gamma(L(\theta)) = \inf\{ t : P(L(\theta) \le t) \ge \gamma \}$$

• Conditional Value-at-Risk (CVaR):

$$\mathrm{CVaR}_\alpha(L(\theta)) = \frac{1}{1-\alpha} \int_{\alpha}^1 \mathrm{VaR}_u(L(\theta))\,du$$

• Interval CVaR (In-CVaR) on $[\alpha, \beta]$:

$$\mathrm{In\text{-}CVaR}_\alpha^\beta(L(\theta)) = \frac{1}{\beta-\alpha} \int_\alpha^\beta \mathrm{VaR}_u(L(\theta))\,du$$

By adjusting $\alpha$ and $\beta$, In-CVaR interpolates between the expectation ($\alpha=0$, $\beta=1$), trimmed mean ($\alpha=0$, $\beta<1$), and median-based methods ($\alpha, \beta \to 1/2$). For interval-valued random returns $\tilde{X} = [X^L, X^U]$, the interval-valued In-CVaR (termed "ICVaR" in some literature) is defined componentwise: $\ICVaR_\alpha(\tilde X) = \Bigl[ -E[X^L | X^L \le -\VaR_\alpha(X^L)],\, -E[X^U | X^U \le -\VaR_\alpha(X^U)] \Bigr]$ which recovers scalar CVaR when $\tilde{X}$ is degenerate (Zhang et al., 2022).

2. Embedding In-CVaR in Statistical and Financial Models

In statistical estimation, In-CVaR is employed to construct estimators robust to extreme events: $$\hat S_\alpha^\beta(D, f) = \arg\min_{\theta\in\Theta} \mathrm{In\text{-}CVaR}_\alpha^\beta(\mathcal L(f(X_D,\theta) - Y_D))$$ This general framework includes:

• Trimmed least squares for $\ell_2$-loss and linear models.
• Direct analogs for $\ell_1$ and Huber losses.
• Nonlinear models such as piecewise-affine regressors or neural networks via the same minimization principle (You et al., 16 Jan 2026).

In finance, ICVaR has been used as a risk measure for interval-valued portfolio optimization. For $n$ assets with interval-valued returns $\tilde R_{ij}$, models include maximizing expected return subject to an ICVaR-budget and minimizing ICVaR subject to achieving a target return. These are transformed via the "midpoint–width" ($y$-index) method to real-valued linear programs for computational tractability (Zhang et al., 2022).

3. Robustness Properties: Breakdown Point and Contamination

The robustness of In-CVaR-based estimators is quantified via the distributional breakdown point (BP), defined relative to an $\varepsilon$-contaminated distribution $D = (1-\varepsilon)D_0 + \varepsilon G$. The value-based BP is: $$\varepsilon' = \sup\left\{ \varepsilon : \sup_{D=(1-\varepsilon)D_0+\varepsilon G}\sup_{\theta \in \hat S_\alpha^\beta(D, f)} |f(x, \theta)| < \infty \forall x \right\}$$ For models where $f(x, a\theta) = a f(x, \theta)$ and under mild support and loss growth conditions, the lower and upper bounds for the BP are established: $$\varepsilon'(\hat S_\alpha^\beta) = \min\{\beta, 1-\beta\}$$ In particular, setting $\beta = 1/2$ yields the optimal 50% BP, matching classical robust estimators (LMS/LTS). Expectation and classical CVaR ($\alpha=0, \beta=1$ or narrow tail trimming) have zero BP, indicating a lack of robustness to any contamination (You et al., 16 Jan 2026).

4. Qualitative Robustness under Perturbation

Robustness to weak perturbations in the data-generating distribution is characterized using the Prokhorov metric $d_\mathrm{P}$. An estimator is qualitatively robust at $D_0$ if small $d_\mathrm{P}$-perturbations produce small Hausdorff deviations in the solution set and empirical solutions converge uniformly to the population solution.

• Consistency: For any compact parameter set $C$, $\mathrm{In\text{-}CVaR}_\alpha^\beta(L(\theta))$ is continuous in $\theta$, and the empirical version converges uniformly.
• Stability: Under assumptions including local Lipschitz continuity and $\beta < 1$, the functional is uniformly continuous in $D$ and $\theta$.
• Necessity: If $\beta = 1$, In-CVaR loses qualitative robustness under arbitrarily small perturbations; hence, trimming the largest portion ($\beta<1$) is necessary (You et al., 16 Jan 2026).

5. Comparison with Expectation and Classical CVaR

Risk Measure	Breakdown Point	Qualitative Robustness
Expectation ($\alpha=0,\beta=1$)	0	Fails
CVaR	0 (if $\gamma < 1$)	Fails
In-CVaR$_\alpha^\beta$	$\min\{\beta,1-\beta\}$	Robust iff $\beta<1$

Expectation is degenerate with respect to contamination; the classical CVaR (trimming only a single tail) remains highly sensitive to outliers or contamination elsewhere in the distribution. In-CVaR, by trimming both lower $\alpha$ and upper $(1-\beta)$ fractions, achieves nontrivial BP and qualitative robustness, illustrated both theoretically and numerically (You et al., 16 Jan 2026).

6. Extensions to Interval-Valued Uncertainty and Portfolio Optimization

In financial settings with interval-valued returns, In-CVaR (ICVaR) is defined for intervals and inherits the properties of coherence, subadditivity, and convex dual representations. For random interval returns $\tilde X = [X^L, X^U]$:

• IVaR and ICVaR are computed on both interval endpoints.
• ICVaR is convex, coherent, and recover scalar CVaR for degenerate cases.
• Portfolio models under ICVaR are efficiently solved as LPs after conversion via midpoint–width ranking, providing tractable and interpretable risk management tools (Zhang et al., 2022).

Empirical analyses on Chinese stock market data confirm that ICVaR-based portfolio selections yield allocations consonant with investors’ risk tolerance, explicitly accounting for both randomness and imprecision in return data.

7. Assumptions, Computational Aspects, and Numerical Illustration

The theoretical analysis of In-CVaR relies on key assumptions regarding loss behavior (monotonicity, unboundedness), data support, and model regularity (positive homogeneity, parameter growth). Convex loss functions with polynomial growth, decomposability, and separation in parameter space underpin the robustness results.

Numerical experiments with piecewise affine regression and $\ell_1$ loss under contamination and outlier injection demonstrate that In-CVaR estimators remain stable up to their predicted breakdown point and exhibit negligible deviation under small perturbations, in stark contrast to expectation and classical CVaR estimators (You et al., 16 Jan 2026).

Computationally, ICVaR-based portfolio optimization is reduced to standard linear programming (via historical simulation and ranking transformations), and empirical results are consistent with risk–return trade-off intuition under uncertainty (Zhang et al., 2022).

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In summary, Interval Conditional Value-at-Risk generalizes classical tail risk measures, providing a framework that is both mathematically robust and adaptable to nonlinear estimation and financial risk management in environments characterized by contamination and interval uncertainty.