So, does ES capture tail risk, but VaR not? Therefore the Basel committee is correct, and we all should use ES. Is that true?
Not quite. As we can see here, ES is just a low multiple of VaR, and if we are considering 99% VaR and 97.5% ES, they are just about the same. Therefore,
ES says just about as much about the tails as VaR. Not much as all.
Still, there are good reasons to pick ES over VaR, but they are not purely statistical.
Artzner et. al (1999) show that ES is subadditive and VaR not. But is VaR really not subadditive?
We could use some fancy math to show that, and I do indeed have a paper showing exactly that but there is a much easier way. where is the tail index, so they are related by constant, and hence if ES is subadditive, so must VaR.