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My goal is to test if ES (CVaR) empirically is a better risk measure than VaR for a set of given variables (assumed underlying distribution, confidence level, sample size) for different asset classes.

As VaR and ES are like apples and oranges, they can not be compared directly. Most backtesting frameworks that I have found test explicitly VaR or ES (I am aware of the difficulties of backtesting ES) and these backtests are not comparable and it is seems hard to infer if e.g ES is superior to VaR from given samples. I assume there must be involved some function that include a common success factor.

Based on this, I am wondering if there are any known approaches for statistically testing (backtests) and comparing VaR and ES?

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Var is a risk measure and ES is a risk measure. What do you mean by "test if ES ... is better than VaR"? Both measures have certain properties. Based on these properties you choose one. – Richard Oct 28 '13 at 14:35
Yes, they do have some different mathematical and statistical properties so neither risk measure trump the other based on that. However, these are only their theoretical properties, what really matters is how they do in application based on the tools that are developed, which was how I phrased my question. – Chris Nov 2 '13 at 12:08
In my mind the following is a fact: risk measures have theoretical properties. They can be derived from their definitons under specific assumptions on the probability space, given distribution functions and so forth. Any statistical estimator should have these properties too (e.g. ES is sudadditive, any statistical estimator of ES should be subadditive too). – Richard Nov 4 '13 at 8:27

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