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Entropic value at risk

EVaR minimises an exponential bound on loss risk. Derive its normal-loss formula, compare it with VaR and ES, and recognise its sensitivity to tail assumptions.

Entropic value at risk (EVaR) is a loss risk measure obtained by minimising an exponential probability bound over its tuning parameter. It gives a conservative alternative to value at risk and expected shortfall, with a form useful in portfolio optimisation. Its conservatism is relative to the specified loss distribution: a larger number does not establish that the model captures real crashes.

From a probability bound to a monetary threshold

Let LL be loss over a fixed horizon, with positive values denoting losses and negative values gains. Write α\alpha for confidence and p=1−αp=1-\alpha for the upper-tail probability. Thus 99% confidence means p=0.01p=0.01; confusing these conventions reverses the intended tail emphasis.

The moment-generating function is ML(θ)=EeθL]M_L(\theta)=\mathbb{E}e^{\theta L}]. For any θ>0\theta>0 where this expectation is finite, the [Chernoff bound gives

P(L≥x)≤e−θxML(θ).\mathbb{P}(L\ge x)\le e^{-\theta x}M_L(\theta).

Setting the right side equal to pp produces a loss threshold

bp(θ)=log⁡ML(θ)−log⁡pθ.b_p(\theta)=\frac{\log M_L(\theta)-\log p}{\theta}.

EVaR takes the smallest threshold available from this family:

EVaR⁡α(L)=inf⁡θ>0log⁡E[eθL]−log⁡(1−α)θ.\operatorname{EVaR}_{\alpha}(L) =\inf_{\theta>0}\frac{\log\mathbb{E}[e^{\theta L}]-\log(1-\alpha)}{\theta}.

Infinite exponential moments contribute an infinite objective. The infimum notation matters: a finite positive optimiser need not exist. For a certain loss L=cL=c, the objective is c−log⁡(p)/θc-\log(p)/\theta and approaches cc only as θ→∞\theta\to\infty.

If LL is in USD, θ\theta is in inverse USD, making the exponent dimensionless and EVaR a USD amount. Under the usual finite-domain conditions, the same-confidence ordering is VaR⁡α≤ES⁡α≤EVaR⁡α\operatorname{VaR}_{\alpha}\le\operatorname{ES}_{\alpha}\le\operatorname{EVaR}_{\alpha}. EVaR is an upper bound, not the average of the worst pp fraction of losses; that average is ES. Ahmadi-Javid's original paper introduces this construction.

A normal loss where the minimisation is explicit

Suppose a fictional one-day monetary loss follows a normal distribution with mean USD 1 and standard deviation USD 2. These are stipulated loss-model parameters, not estimates from market observations. In USD numerical units, L∼N(1,22)L\sim\mathcal{N}(1,2^2).

For a general normal loss with mean μ\mu and standard deviation σ>0\sigma>0,

log⁡ML(θ)=μθ+σ2θ22,bp(θ)=μ+σ2θ2−log⁡pθ.\log M_L(\theta)=\mu\theta+\frac{\sigma^2\theta^2}{2}, \qquad b_p(\theta)=\mu+\frac{\sigma^2\theta}{2}-\frac{\log p}{\theta}.

Differentiating gives bp′(θ)=σ2/2+log⁡(p)/θ2b_p'(\theta)=\sigma^2/2+\log(p)/\theta^2. Its zero yields

θ∗=−2log⁡pσ,EVaR⁡α(L)=μ+σ−2log⁡p.\theta^*=\frac{\sqrt{-2\log p}}{\sigma}, \qquad \operatorname{EVaR}_{\alpha}(L) =\mu+\sigma\sqrt{-2\log p}.
At 99% confidence, θ∗≈1.517427\theta^*\approx1.517427 per USD and EVaR is USD 7.069709. Choosing θ=1\theta=1 per USD would instead give a looser threshold of USD 7.605170.

For comparison, put z=Φ−1(0.99)z=\Phi^{-1}(0.99), where Φ\Phi and ϕ\phi are the standard normal CDF and density. VaR is μ+σz\mu+\sigma z and ES is μ+σϕ(z)/0.01\mu+\sigma\phi(z)/0.01.

Same one-day normal loss modelUSDMeaning
99% VaR5.652696Exact model quantile
99% ES6.330428Model average of the worst 1%
99% EVaR7.069709Optimised exponential-bound threshold
Fictional one-day normal USD loss with mean 1 and standard deviation 2. At 99% confidence the exponential bound is minimised at theta 1.517427 per USD, giving EVaR USD 7.069709. VaR is USD 5.652696 and expected shortfall is USD 6.330428.
Fictional one-day normal USD loss with mean 1 and standard deviation 2. At 99% confidence the exponential bound is minimised at theta 1.517427 per USD, giving EVaR USD 7.069709. VaR is USD 5.652696 and expected shortfall is USD 6.330428.
Original fictional normal-loss example. The curve plots the Chernoff loss threshold against its exponential parameter; its minimum defines EVaR. All three risk measures use the same one-day distribution and 99% confidence.
The normal model's actual probability of exceeding USD 7.069709 is about 0.120326%, below the 1% Chernoff bound. The gap is the price of using this bound. EVaR is neither an exact 99th percentile nor a maximum possible loss; this normal distribution has no finite upper endpoint.
For the standardisation behind that probability check, the Normal Distribution lesson develops normal tail calculations. Its public page is a preview; the full lesson requires a free signed-in account.

What the exponential emphasis changes

Large positive losses receive exponentially increasing weight inside ML(θ)M_L(\theta). A finite sample can therefore make EVaR computable while leaving substantial uncertainty about the population tail. A nondegenerate lognormal positive loss, for example, has finite ordinary moments but no finite positive exponential moment, so its population EVaR is infinite; the Virtual Laboratories notes, “Moments,” give this contrast. Every finite empirical sample of that distribution has finite EVaR. A finite software result does not settle the existence question for the underlying model.
EVaR is a coherent risk measure on its appropriate domain. It respects loss ordering and certain-loss shifts, scales proportionally with nonnegative exposure, and is subadditive. Subadditivity compares combined risk with the sum of standalone risks; it does not say that adding a position always reduces risk. These properties support portfolio aggregation, but they do not insert missing liquidity costs or dependence scenarios into LL.
The name “entropic” also has a precise interpretation. For bounded losses, EVaR equals the largest expected loss obtained by reweighting the reference probabilities within a relative-entropy budget of −log⁡p-\log p. Delbaen's analysis, §1, states this dual representation using the opposite, utility sign convention. The budget limits probability distortion; it does not protect against every imaginable loss model or create outcomes absent from the reference model.

Finally, the entropic risk measure at a fixed parameter is ρθ(L)=log⁡E[eθL]/θ\rho_\theta(L)=\log\mathbb{E}[e^{\theta L}]/\theta. EVaR adds −log⁡p/θ-\log p/\theta and then optimises over θ\theta. A chosen risk-aversion parameter and a chosen confidence level are different inputs. Neither calculation by itself establishes a regulatory capital requirement.

Sources

  • Amir Ahmadi-Javid, Entropic Value-at-Risk: A New Coherent Risk Measure, Journal of Optimization Theory and Applications 155, 1105–1123 (2012; online December 2011). The checked publisher abstract establishes the original construction and its relation to VaR, CVaR and entropy; full subscription text was not used.
  • Amir Ahmadi-Javid and Malihe Fallah-Tafti, Portfolio Optimization with Entropic Value-at-Risk, author manuscript, §2, equation (2-1), normal comparison and Remark 1; §3.2. These supply the defining formula, domain discussion and sample-based optimisation setting. The USD scenario, calculations and figure here are original.
  • Freddy Delbaen, Remark on the Paper “Entropic Value-at-Risk: A New Coherent Risk Measure”, version July 13, 2015, §1. This provides the bounded-variable entropy dual representation and its technical qualification; signs are translated to positive losses here.
  • Virtual Laboratories, The Lognormal Distribution, “Moments,” exercises 8–10, hosted by Universidad Nacional de Mar del Plata. This gives finite ordinary moments alongside divergence of every positive exponential moment.
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  • tail risk
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  • chernoff bound
  • risk measures
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