Can Value at Risk and Expected Shortfall be equal?

Question

I am struggling with the study of value at risk (VaR) and expected shortfall (ES). In particular, I am looking for some cases in which these two measures can be equal. I know that, given a loss cdf $F_L(x)$ VaR is defined as $\inf\{x\in\mathbb{R}:F_L(x)\geq\alpha\}$, for $\alpha\in(0,1)$. ES instead is $(1-\alpha)^{-1}\int_{\alpha}^{1}\text{VaR}_L(u)du$ for $\alpha\in0$.

Is there a distribution for which VaR and ES are equal? If yes, how can I prove it mathematically? I was wondering about a function with jumps but i'm not sure about that. Thanks for your help.

Answer 1

Let us call the random variable of interest $X$. To make VaR and ES equal for a quantile level $\alpha$, we need $q_\alpha=q_{\alpha'}$ for all $\alpha'<\alpha$. (Otherwise, $\text{ES}_\alpha<\text{VaR}_\alpha$.) That implies the distribution of the random variable must have a point mass $m\geq\alpha$ at the minimum value of $X$: $m:=P(\min(X))\geq\alpha$. An example would be a Bernoulli distribution with parameter $p$ and any $\alpha\leq1-p$.

To make VaR and ES equal for all quantile levels $\alpha$, the point mass must be unity, turning the random variable into a constant.

_{(This seems kind of obvious for discrete distributions with a finite number of possible values. I wonder if I have missed any tricky or degenerate case among discrete distributions with an infinite number of possible values or among continuous distributions?)}