Simply put, Expected Shortfall is the average of all VaRs above a certain level of $\alpha$

Expected Shortfall
$$ES_{\alpha}(L) = \dfrac{1}{1 – \alpha} \int_{\alpha}^1 \text{VaR}_u (L) du $$

One of the major stipulations of Basell III is that the Capital Requirements should be based on Expected Shortfall. The requirement for back testing is still based on VaR$_{\alpha}$

1. Can we prove that Expected Shortfall is an average of VaR$_{\alpha}$?

Start by with an assumption about the distribution of the loss random variable $L$. Assume the loss $L$ has a distribution denoted by $F$. Then, by definition, the Value at Risk at confidence level $\alpha \in (0, 1)$ is the smallest realised loss $l$ of the random loss variable $L \leq l$, that will only be exceeded $( 1 – \alpha )\times 100$ percent of the time during the $N-day$ trading window. Mathematically, VaR is the unique inverse of the distribution at the confidence level $\alpha$

$$\text{VaR}_{\alpha} = F^{-1}(\alpha)$$

The definition of VaR$_{\alpha}$ highlights one the main problems and criticisms of it as a risk measure. It is a lower bound on the loss or, the smallest loss. Yes, there is a slightly comforting confidence level associated with the low frequency of that loss being attained or exceeded. But no sense of by how much the threshold is breached if or when it is breached. In response to this limitation, Expected Shortfall is the average of all VaRs ablove the level of $\alpha$.