# What are the advantages of $EVaR$ over $CVaR$?

$CVaR$, which is short for Conditional Value-at-Risk, has long been accepted by both academe and practice as a good coherent risk measure. Entropic value-at-risk ($EVaR$) is a comparative new coherent risk measure compared to $CVaR$.

What is the advantages of $EVaR$ over $CVaR$? Or in what situation, $EVaR$ would perform better than $CVaR$?

Here $EVaR$ is defined as, $$\text{EVaR}_{1-\alpha}(X) := \inf_{z>0}\{z^{-1}ln(M_{X}(z))/\alpha\}= \inf_{z>0}\{z^{-1}ln(e^{-z*X}/\alpha\}.$$

Ahmadi-Javid, Amir. "Entropic value-at-risk: A new coherent risk measure." Journal of Optimization Theory and Applications 155.3 (2012): 1105-1123.

• Welcome! Could you give us the definition of EVaR? The link to a page that charges to get the info is not optimal. What do you already know about this risk measure? Commented Jul 9, 2018 at 10:17
• Isn't the answer to your question precisely in the research paper you linked ? Commented Jul 9, 2018 at 14:35
• Thank you for your interest. Entropic Value-at-Risk (EVaR) was first introduced by A. Ahmadi-Javid. The entropic value-at-risk of $X \in L_{M^{+}}$ with confidence level $1-\alpha$ is defined as: $$\text{EVaR}_{1-\alpha}(X) := \inf_{z>0}\{z^{-1}ln(M_{X}(z))/\alpha\}= \inf_{z>0}\{z^{-1}ln(e^{-z*X}/\alpha\}.$$ Ahmadi-Javid, Amir. "Entropic value-at-risk: A new coherent risk measure." Journal of Optimization Theory and Applications 155.3 (2012): 1105-1123. Commented Jul 9, 2018 at 15:20
• $CVaR$ is not generally accepted by practitioners as a good (coherent) measure of risk. In general, coherency is an irrelevant property for effective risk management. Commented Dec 26, 2021 at 11:14

The entropic value at risk (EVaR) is a coherent risk measure, developed to tackle some computational inefficiencies of the CVaR. It is the tightest possible upper bound for traditional VaR and CVaR, obtained from the Chernoff inequality.

EVaR can also be represented by using the concept of relative entropy, better known in statistics as the Kullback-Leibler (KL) divergence, which measures by how much one probability distribution is different from a reference distribution. Relative entropy is based on Shannon's entropy, or information entropy, which measures the unpredictability in a random variable's distribution. Statistical measures that include entropy within their formula arose from the field of information theory, and are called information theoretic measures.

For the normal distribution, the image shows what is meant by EVaR being an upper bound on both VaR and CVaR.

Two other important properties:

1. $$EVaR$$ is strictly monotone over continuous distributions, and strongly monotone over all distributions.

2. Formulations involving $$EVaR$$ are diffrentiable without need of adding many new variables or constraints.

$$VaR$$ and $$CVaR$$ both lack these important properties 1 and 2.