Entropic Value-at-Risk (EVaR)

• Entropic Value-at-Risk is a risk measure that quantifies tail risk using the worst-case exponential growth rate of losses.
• Its foundation in convex analysis and moment-generating functions ensures robust and law-invariant risk assessment.
• Scalable computation of EVaR is enabled through surrogate modeling with DD-GPCE-Kriging and multifidelity importance sampling.

The entropic value-at-risk (EVaR) is a risk measure rooted in information-theoretic principles and convex analysis, and is closely related to the widely studied conditional value-at-risk (CVaR). EVaR characterizes risk via the worst-case exponential growth rate, and its mathematical and computational properties enable robust tail risk quantification. The multifidelity CVaR estimation problem, addressed through DD-GPCE-Kriging surrogate modeling, offers methodological advances relevant to EVaR, especially concerning scalability for high-dimensional, dependent input spaces and nonlinear outputs (Lee et al., 2022).

1. Mathematical Definition and Relationship to CVaR

EVaR is defined as the tightest upper bound for the expected loss, among all convex risk measures that are law-invariant and satisfy the monotonicity and translation invariance axioms. The formal EVaR for a loss random variable $Y$ at confidence level $\alpha\in(0,1)$ is expressed as:

$$\mathrm{EVaR}_\alpha[Y] = \inf_{\lambda>0}\left\{ \frac{1}{\lambda}\log\left( \frac{1}{1-\alpha}\mathbb{E}\left[ e^{\lambda Y} \right]\right) \right\},$$

where $\mathbb{E}[e^{\lambda Y}]$ denotes the moment-generating function of $Y$. The infimum over $\lambda > 0$ ensures the most conservative “entropic” bound over the α-tail of the distribution. EVaR is always greater than or equal to CVaR and VaR at the same confidence level.

A plausible implication is that advanced CVaR estimation techniques for dependent, nonlinear, high-dimensional systems, such as multifidelity DD-GPCE-Kriging, directly inform scalable EVaR computation, since EVaR requires accurate tail modeling and moment integrals, especially for correlated risks and nonsmooth payoffs (Lee et al., 2022).

2. Properties and Theoretical Foundations

EVaR inherits several key properties:

• Coherence: It is a coherent risk measure, satisfying subadditivity, monotonicity, positive homogeneity, and translation invariance.
• Law invariance: The value depends only on the loss distribution, not the underlying random variable representation.
• Convexity: As a supremum over exponentials, EVaR is convex in $Y$.
• Dual interpretation: It can be characterized via relative entropy (Kullback-Leibler divergence) minimization.

These properties position EVaR as the sharpest law-invariant coherent risk measure with an entropic basis—a result that impacts robust optimization and stochastic programming paradigms, especially where exponential moments (rather than only quantiles) determine capital requirements or safety margins.

3. Surrogate Modeling for Tail Risk Estimation

Efficient, accurate computation of EVaR for complex systems demands scalable surrogate modeling. The DD-GPCE-Kriging surrogate, as introduced by Lee and Kramer, merges dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) with Kriging (Gaussian process regression):

• DD-GPCE: Provides a truncated polynomial basis for dependent inputs, indexed by interaction order $S$ and polynomial degree $m$, typically pruned for high $N$.
• Kriging: Models the residual between the global polynomial trend and observed outputs as a stationary Gaussian process $Z(x)$ with hyperparameters optimized via leave-one-out cross-validation.

The surrogate $$\bar Y_{S,m}(x) = \Psi(x)^\top \hat c + \sigma^2 Z(x;\theta)$$ yields rapid, unbiased estimation of risk tail quantities—including those necessary for computing $\mathbb{E}[e^{\lambda Y}]$ or CVaR—via Monte Carlo simulation or importance sampling regimes.

The surrogate's ability to handle dependent random inputs and nonsmooth, nonlinear loss mappings is critical in EVaR contexts, where moment integration over the distributional tail governs risk quantification (Lee et al., 2022).

4. Multifidelity Importance Sampling and Algorithmic Scalability

Multifidelity importance sampling (MFIS) harnesses DD-GPCE-Kriging surrogates to construct biasing densities efficiently:

• Define a risk region $\mathcal{R}$ via surrogate mean and confidence intervals.
• Use surrogate to determine biasing density $$f_b(x) \propto \mathbb{I}\{x \in \mathcal{R}\}f_X(x)$$ for input space $x$.
• Draw fewer high-fidelity samples from $f_b$, reweight by $w(x) = f_X/f_b$, and estimate tail risk metrics (such as CVaR or EVaR moments) unbiasedly.

This approach, when applied to high-dimensional composite laminate and T-joint systems with partly dependent random inputs, enabled speedups (e.g., 104× for composite laminate, 23.6× for 3D T-joint) with sub–1% error in CVaR estimation, a metric directly relevant for practical EVaR applications where the distributional tail and coherent risk measures must be efficiently computed (Lee et al., 2022).

5. Practical Recommendations and Limitations

Key practical considerations for DD-GPCE-Kriging and EVaR estimation include:

• Choice of interaction order $S$: Reflects expected variable couplings; typically $S=1$–2 for tractability.
• Polynomial degree $m$: Increased until cross-validation error stabilizes.
• Kernel selection: Gaussian kernels for smooth outputs, exponential/Matern for nonsmooth.
• Training sample size: $L' \geq 3L_{N,S,m}$ for stable surrogate fitting.
• Computational complexity: $O(L'^3)$ for correlation matrix inversion; mitigated by sparse kernels/low-rank approximations for large $L'$.

Limitations arise from surrogate model coverage, curse of dimensionality, and algorithmic scaling; adaptive sample selection and kernel engineering are pathways for extension. A plausible implication is that these considerations play a decisive role in scalable EVaR computation for complex engineering systems, especially where dependence and nonlinearity challenge classical quantile-based approaches (Lee et al., 2022).

6. Extensions and Research Directions

Extensions of the DD-GPCE-Kriging surrogate model that further support EVaR research include:

• Vector-valued output surrogates
• Nonstationary kernel functions
• Active learning focused on tail/critical regions
• Integration with control-variate multifidelity uncertainty quantification (UQ)

A plausible implication is that future research efforts will focus on adaptive surrogates and model-based multifidelity strategies for reliable EVaR estimation under high-dimensional dependence and output nonsmoothness, broadening applicability in financial, structural, and safety-critical domains (Lee et al., 2022).