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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.

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  • $\begingroup$ Thanks for your answer. Can you give me an example for a particular quantile? I mean, for which distribution and quantile this should be possible? One case is sufficient, i just want to understand the reasoning behind that. $\endgroup$
    – partfin
    Commented Apr 30, 2021 at 9:48

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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?)

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  • $\begingroup$ You have been very clear about discrete distributions. For continuous ones, I can only think of possible limiting cases (when $\alpha\rightarrow1^-$ for instance) but I am not sure if I am right, I will definitely look into this further. Thank you very much for the help. $\endgroup$
    – partfin
    Commented Apr 30, 2021 at 16:06
  • $\begingroup$ @partfin, you are welcome! I do not have good intuition about the trickier distributions (but I also do not think very weird distributions would be all that relevant for actual applications of VaR and ES in the financial world). Should you discover something counter to what I wrote, it would be nice if you come back and tell me. $\endgroup$ Commented Apr 30, 2021 at 16:36

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