Entropy-Adjusted Value at Risk (EVaR)

• Entropy-adjusted VaR is a risk measure that integrates entropy concepts (e.g., KL divergence, Tsallis, and Rényi entropies) to overcome limitations of classical VaR.
• The framework employs coherent optimization techniques, providing closed-form and efficient solutions, such as those using the Lambert W function for common distributions.
• Practical applications include portfolio optimization, risk model calibration, and loss reconstruction, especially under non-Gaussian, heavy-tailed risk scenarios.

Entropy-adjusted Value at Risk (VaR) refers to a class of risk measures and methodologies that augment, replace, or meaningfully constrain classical Value at Risk using principles from entropy—primarily relative entropy (Kullback–Leibler divergence), generalized entropy such as Tsallis or Rényi, and the maximum-entropy principle. These constructions address common shortcomings of standard VaR, including lack of coherence, sensitivity to model misspecification, and inability to account robustly for non-Gaussian, heavy-tailed, or otherwise non-standard risk distributions. The most prominent form of entropy-adjusted VaR is the Entropic Value-at-Risk (EVaR), but more general entropy-based risk measures and entropy-calibrated probabilistic models are also in use. This article presents the main definitions, theoretical properties, solution methodologies, and applications of entropy-adjusted VaR, with attention to both conceptual underpinnings and concrete optimization frameworks.

1. Core Definitions and Dual Representations

1.1 Entropic Value at Risk (EVaR)

Let $X$ be a real-valued loss random variable on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and fix a risk level $\alpha \in (0,1)$. Assume the exponential moment exists, i.e., $\mathbb{E}[e^{t X}] < \infty$ for some $t>0$. The entropic value-at-risk at level $\alpha$ is

$$\mathrm{EVaR}_{\alpha}(X) = \inf_{t > 0} \frac{\ln \mathbb{E}[e^{tX}] - \ln(1-\alpha)}{t}.$$

This admits a robust dual representation as the supremum of expected loss under alternative probability measures $Q$ absolutely continuous with respect to $\mathbb{P}$, with a relative entropy (Kullback–Leibler) constraint: $$\mathrm{EVaR}_{\alpha}(X) = \sup_{Q \ll \mathbb{P},~D_{\mathrm{KL}}(Q\|\mathbb{P}) \leq -\ln(1-\alpha)} \mathbb{E}^Q[X].$$ This formulation makes explicit the connection between tail risk and an entropy-budgeted set of plausible alternative distributions (Ahmadi-Javid et al., 2017, Mishura et al., 3 Mar 2024, Pichler et al., 2018).

1.2 Generalized Entropic VaR: Tsallis and Rényi Extensions

• Tsallis Value at Risk (TsVaR): For $q > 0$, define generalized exponential functions $\exp_q$, and the TsVaR at level $\alpha$ and $q$ is

$$\mathrm{TsVaR}^+_{\alpha,q}(X) = \inf_{t>0} \left\{ \frac{1}{t} \left[ \frac{\mathbb{E}[\exp_q(t X)] - 1}{1-q} - \ln_q(1-\alpha) \right] \right\}.$$

TsVaR interpolates between VaR and EVaR; as $q \rightarrow 1$, TsVaR converges to EVaR (Yoshioka et al., 2023, Hajihasani et al., 2020).

• Rényi Entropic VaR: For $p \neq 1$, the Rényi-EVaR imposes constraints in terms of $p'$-order Rényi entropy and admits explicit representations in terms of $L^p$-norms and Hölder conjugates (Pichler et al., 2018).

2. Coherence, Law Invariance, and Theoretical Comparison

EVaR is a coherent risk measure, satisfying:

• Monotonicity: $X \leq Y$ a.s. implies $$\mathrm{EVaR}_\alpha(X) \leq \mathrm{EVaR}_\alpha(Y)$$.
• Translation-invariance: $$\mathrm{EVaR}_\alpha(X + c) = \mathrm{EVaR}_\alpha(X) + c$$.
• Positive homogeneity: $$\mathrm{EVaR}_\alpha(\lambda X) = \lambda\,\mathrm{EVaR}_\alpha(X)$$ for $\lambda > 0$.
• Subadditivity (Convexity): $$\mathrm{EVaR}_\alpha(X + Y) \leq \mathrm{EVaR}_\alpha(X) + \mathrm{EVaR}_\alpha(Y)$$.

Classical Value at Risk fails subadditivity and thus is not coherent. Conditional Value at Risk (CVaR) is coherent but, unlike EVaR, is not strongly monotone. EVaR also constitutes a tight law-invariant coherent upper bound for VaR and CVaR: $$\mathrm{VaR}_{\alpha}(X) \leq \mathrm{CVaR}_{\alpha}(X) \leq \mathrm{EVaR}_{\alpha}(X).$$ Strong and strict monotonicity of EVaR is notable: if $X \geq Y$ a.s., $\mathbb{P}\{X>Y\}>0$, and $\mathrm{ess\,sup}\, X > \mathrm{ess\,sup}\, Y$, then $$\mathrm{EVaR}_{\alpha}(X) > \mathrm{EVaR}_{\alpha}(Y)$$, which does not hold for classical VaR or CVaR (Ahmadi-Javid et al., 2017, Mishura et al., 3 Mar 2024, Pichler et al., 2018).

3. Entropy-Adjusted VaR via Maximum-Entropy and KL Calibration

3.1 KL-Minimization under Tail Constraints

An alternative entropy-adjusted approach reformulates the loss density under a relative-entropy penalty with tail constraints. Given a base measure $P_0$ (with density $p_0$), and a target $\mathrm{VaR}_\alpha = v_\alpha$, the calibrated measure $Q$ is obtained as the minimizer of KL divergence subject to $Q(L > v_\alpha) = 1-\alpha$: $$\min_{Q \ll P_0} \; D_{\mathrm{KL}}(Q \| P_0) \quad \text{subject to} \quad Q(L>v_\alpha) = 1-\alpha.$$ The solution is an exponential tilt on the tail: $$q(\ell) \propto p_0(\ell) \exp\{\theta \cdot 1_{\ell > v_\alpha}\},$$ with $\theta$ and the normalizer uniquely determined by the tail constraint (Dey et al., 2014).

3.2 Maximum-Entropy Under VaR and CVaR Constraints

The maximum-entropy portfolio return distribution constrained by given VaR $\epsilon$ and CVaR $\nu_-$ is a two-piece mixture of exponential densities (the "barbell" distribution): $$f_{MEE}(x) = \epsilon\,f_-(x) + (1-\epsilon)\,f_+(x),$$ where $f_-(x)$ is supported on $x \leq K$ and $f_+(x)$ on $x > K$, with scales fixed by the constraints (Geman et al., 2014). This construction provides the least-informative extension of tail constraints, yielding a fully specified density—and thus a robust entropy-adjusted VaR—that matches only the information input by the risk manager.

4. Computational Aspects and Closed-Form Results

4.1 Explicit Optimization for Jump-Diffusion and Sample-Based Models

In models with known Laplace or moment-generating function (MGF), the EVaR objective is closed-form and admits efficient convex optimization: $$\mathrm{EVaR}_{\alpha}(X) = \inf_{t > 0} \frac{\ln \mathbb{E}[e^{t X}] - \ln(1-\alpha)}{t}.$$ In portfolio optimization, EVaR leads to a differentiable convex objective with variable size independent of sample count, unlike sample-based CVaR formulations that scale linearly with the sample set (Ahmadi-Javid et al., 2017, Firouzi et al., 2014). For jump-diffusion processes or compound Poisson models, the entire risk functional becomes an explicit, low-dimensional convex program (Firouzi et al., 2014).

4.2 Closed-Form Solutions via Special Functions

Recent results provide closed-form expressions for EVaR in several classical distributions using the Lambert $W$ function, e.g.,

• Poisson: $$\mathrm{EVaR}_\alpha(\mathrm{Poisson}(\lambda)) = \frac{\beta}{W_0(\frac{\beta}{e\lambda})}$$, where $\beta = -\ln(1-\alpha) - \lambda$.
• Gamma: $-\theta k\, W_{-1}(-e^{-1}(1-\alpha)^{1/k})$ for $G(k,\theta)$.
• Laplace, Inverse Gaussian, and NIG also have explicit W-formulas (Mishura et al., 3 Mar 2024).

Such forms avoid the need for iterative root-finding in classical approaches and dramatically reduce computational burden.

4.3 Generalized Entropic VaR and Heavy-Tails

Tsallis-VaR (q-VaR) addresses cases where the exponential moment may not exist (e.g., long memory or heavy-tail processes), by using divergence balls based on Tsallis entropy. The computation is carried out by convex minimization over $t$ of a Tsallis-exponential-based objective, and leverages semi-implicit gradient descent for numerical efficiency (Yoshioka et al., 2023, Hajihasani et al., 2020).

5. Applications: Portfolio Optimization, Model Calibration, and Loss Reconstruction

5.1 Portfolio Optimization with Entropy-Adjusted VaR

Portfolio selection with EVaR leads to a convex optimization problem, even with non-elliptical (e.g., jump-diffusion) asset returns. The solution yields an efficient frontier analogous to but fundamentally different from the classical mean-variance Markowitz setup. EVaR-based portfolios outperform CVaR-optimized portfolios both in realized tail-risk and expected return at high confidence levels, especially in empirical studies with large asset universes (Ahmadi-Javid et al., 2017, Firouzi et al., 2014).

5.2 Risk Model Calibration and Stress Testing

Entropy-based calibration methods allow for incorporating expert views on portfolio loss quantiles or tails by minimally tilting the reference model, yielding new risk measures and probability distributions consistent with these views. This paradigm unifies and extends classical calibration, stress testing, and scenario analysis, as with the exponential tilt construction or barbell distribution (Dey et al., 2014, Geman et al., 2014).

5.3 Historical and Compound Losses: Maximum Entropy Reconstructions

For compound risk losses, entropy-based density reconstruction using finite fractional moments (Standard Maximum Entropy, SME, and Maximum Entropy in the Mean, MEM) enables estimation of VaR and TVaR closely tracking empirical or simulated statistics, even when only aggregate/historical loss data are available (Gomes-Gonçalves et al., 2014).

6. Practical Considerations and Limitations

• Moment Existence: EVaR requires existence of exponential moments, making it inapplicable for certain heavy-tailed or long-memory processes. TsVaR and Rényi-EVaR relax this requirement at the cost of coherence or positive homogeneity (Yoshioka et al., 2023, Pichler et al., 2018).
• Numerical Stability: Closed-form solutions using the Lambert $W$ function or maximum-entropy formulations are available for many common distributions, but edge cases (near branch points) may pose numerical challenges (Mishura et al., 3 Mar 2024).
• Model Uncertainty: Entropy-adjusted VaR is robust to model misspecification within the class of relative-entropy or generalized-entropy neighborhoods, but cannot compensate for structural model errors outside these ambiguity sets.

7. Comparative Summary Table

Approach	Risk Measure	Coherence	Moment Requirement
Classical VaR	Quantile	✗	None
CVaR	AVaR/ES	✓	1st moment finite
EVaR	Entropic	✓	Exponential
TsVaR (q-VaR)	Generalized	Convex	Polynomial
MaxEnt/KL-tilted	Calibrated	Model-based	Flexible

EVaR and entropy-adjusted VaR generalize and strengthen quantile-based risk assessment, providing rigorous, coherent, and tractable frameworks for risk management under distributional uncertainty, especially when non-Gaussian features or expert-driven tail constraints are material. The principle of entropy-penalized or maximum-entropy inference underlies both theoretical developments and computationally efficient tools for advanced risk management (Ahmadi-Javid et al., 2017, Firouzi et al., 2014, Geman et al., 2014, Pichler et al., 2018, Mishura et al., 3 Mar 2024, Yoshioka et al., 2023, Hajihasani et al., 2020, Dey et al., 2014, Gomes-Gonçalves et al., 2014).