What is entropic VaR, and how does it provide a tighter bound on tail risk than traditional VaR?

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AcadiFi · Accepted Answer

Entropic Value at Risk (EVaR) is a coherent risk measure derived from the Chernoff bound on tail probabilities. It provides a tighter upper bound on tail losses than VaR and connects to information theory through the concept of relative entropy (Kullback-Leibler divergence). **Definition:** EVaR_alpha(X) = inf over z > 0 of { (1/z) x ln(M_X(z) / (1-alpha)) } where M_X(z) = E[e^{z*X}] is the moment-generating function (MGF) of the loss X, and the infimum is taken over all positive z. **Intuition:** The Chernoff inequality states: P(X >= x) <= e^{-zx} x M_X(z) for all z > 0 Setting the right side equal to (1-alpha) and solving for x gives the tightest exponential bound on the alpha-quantile. This bound is always at least as tight as VaR and no looser than ES. **Risk Measure Ordering:** VaR_alpha <= EVaR_alpha <= ES_alpha (for the same alpha) This ordering means EVaR sits between VaR and ES, providing a compromise: more conservative than VaR but less extreme than ES. **Worked Example (Normal Distribution):** For X ~ N(mu, sigma^2), the MGF is M_X(z) = exp(mu*z + 0.5*sigma^2*z^2). EVaR_alpha = inf over z > 0 of { mu + 0.5*sigma^2*z + (1/z)*ln(1/(1-alpha)) } Taking the derivative and setting to zero: 0.5*sigma^2 - (1/z^2)*ln(1/(1-alpha)) = 0 z* = sqrt(2*ln(1/(1-alpha))) / sigma Substituting back: EVaR_alpha = mu + sigma x sqrt(2*ln(1/(1-alpha))) For alpha = 0.99: EVaR = mu + sigma x sqrt(2 x ln(100)) = mu + sigma x sqrt(9.21) = mu + **3.035 x sigma** Compare: VaR(99%) = mu + 2.326*sigma, ES(99%) = mu + 2.665*sigma. So for a portfolio with mu = 0 and sigma = $2M daily: - VaR(99%) = **$4.65M** - ES(99%) = **$5.33M** - EVaR(99%) = **$6.07M** EVaR is the most conservative, reflecting its origin as an exponential tail bound. **Properties:** - Coherent (satisfies all four Artzner axioms) - Connects to exponential premium principle in insurance - Robust to model misspecification in the tails - Computationally tractable when the MGF has a closed form Study advanced risk measures in our FRM Market Risk course.

What is entropic VaR, and how does it provide a tighter bound on tail risk than traditional VaR?

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