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Entropic Value at Risk (EVaR)

Entropic Value at Risk is an upper bound for both VaR and CVaR, derived from the Chernoff inequality and moment generating functions. It provides the tightest possible exponential upper bound on the tail probability and serves as a coherent risk measure with connections to information theory.

Overview

Entropic Value at Risk was introduced by Ahmadi-Javid (2012) as a new coherent risk measure that bridges the gap between CVaR and the worst-case risk measure. While CVaR averages losses in the tail, EVaR provides an exponentially-weighted bound that places greater emphasis on extreme outcomes.

The name "entropic" comes from its deep connection to the exponential (Esscher) premium principle in actuarial science and to the concept of relative entropy (Kullback-Leibler divergence) in information theory. EVaR can be interpreted as the tightest upper bound on the VaR obtained from the Chernoff inequality applied to the moment generating function of the loss distribution.

From a practical standpoint, EVaR is more conservative than CVaR and better captures the risk of heavy-tailed distributions. It maintains all the desirable mathematical properties of a coherent risk measure while providing additional sensitivity to extreme tail events that CVaR may underweight.

Mathematical Formulation

Definition

For a random variable representing portfolio loss and confidence level , the Entropic Value at Risk is defined as:

where the infimum is taken over all positive real numbers . The optimization over selects the tightest exponential bound on the tail probability at level .

Moment Generating Function

The key ingredient in EVaR is the moment generating function (MGF) of the loss distribution:

The MGF must exist (be finite) for all for EVaR to be well-defined. This requirement is satisfied by normal, log-normal, and most light-tailed distributions, but excludes some heavy-tailed distributions such as the Pareto or Cauchy distribution. For the normal distribution with mean and variance , the MGF is:

Derivation via Chernoff Bound

EVaR arises naturally from the Chernoff inequality. For any threshold and any :

Setting the right-hand side equal to and solving for gives the EVaR as the tightest such bound optimized over .

Risk Measure Ordering

EVaR fits into a strict hierarchy of risk measures. For any confidence level and any random variable :

This ordering shows that EVaR is the most conservative of the three measures. Furthermore, as , all three measures converge to the essential supremum of (the worst-case loss). The gap between EVaR and CVaR is largest for heavy-tailed distributions, where the exponential weighting in EVaR penalizes extreme outcomes more heavily.

Connection to Relative Entropy

EVaR has a dual representation in terms of relative entropy (Kullback-Leibler divergence):

where is the Kullback-Leibler divergence from the reference measure to the alternative measure . This shows that EVaR computes the worst-case expected loss over all probability distributions within a specified "information ball" around the reference distribution.

Advantages & Limitations

Advantages

  • Coherent risk measure: Satisfies subadditivity, monotonicity, positive homogeneity, and translation invariance.
  • Tighter tail bound: Provides the tightest possible exponential bound on tail probability via the Chernoff inequality.
  • More conservative than CVaR: Better captures extreme tail risk for heavy-tailed distributions.
  • Information-theoretic foundation: Dual representation via relative entropy provides robust optimization interpretations.

Limitations

  • MGF existence required: EVaR is not defined for distributions whose moment generating function does not exist (e.g., heavy-tailed Pareto).
  • Computational complexity: Requires numerical optimization over the parameter and estimation of the MGF.
  • Less intuitive: Harder to interpret and communicate compared to VaR or CVaR.
  • Potentially overconservative: May lead to excessively conservative risk estimates for near-normal distributions.
  • Limited industry adoption: Not yet widely used in practice or regulatory frameworks compared to VaR and CVaR.

References

  1. Ahmadi-Javid, A. (2012). "Entropic Value-at-Risk: A New Coherent Risk Measure." Journal of Optimization Theory and Applications, 155(3), 1105-1123.
  2. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228.
  3. Ahmadi-Javid, A., & Fallah-Tafti, M. (2019). "Portfolio Optimization with Entropic Value-at-Risk." European Journal of Operational Research, 279(1), 225-241.
  4. Breuer, T., & Csiszár, I. (2013). "Systematic Stress Tests with Entropic Plausibility Constraints." Journal of Banking & Finance, 37(5), 1552-1559.