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I have a time series with monthly data from which I compute the expected shortfall empirically, following the classical definition which can be found, for example, in wikipedia's definition.

That is, assuming I have 200 monthly returns and I am looking to compute the 10% expected shortfall, I take the worst 20 returns and I compute their mean to get my $ES_{10}$.

The thing is, I have a "monthly" expected shortfall, and I would like to annualize this result.

I wonder whether I should consider it as a "return" and do $(1+ES_{10})^{12}-1$ or maybe use the volatility annualization $ES_{10} \cdot \sqrt{12}$? Or is it something different?

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If you are willing to hypothesize a normal distribution of returns for these purposes, then you scale by the square root. However, normal tails are fairly skinny, so a lot of people like to fit Student-t or Pareto distributions to the tails. In this case, you have to convolve 12 copies of the fitted distribution together, and in the general case there is no simple formula to help you.

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  • $\begingroup$ Thanks. Do you know any reference that would allow me to find an example of how they did it assuming pareto or student? $\endgroup$ – SRKX Apr 7 '11 at 7:09
  • $\begingroup$ Not for this exact problem, but Goldstein at Barra research has published some whitepapers on fat-tailed distributions for expected shortfall. $\endgroup$ – Brian B Apr 7 '11 at 13:53
  • $\begingroup$ couldn't find the paper, do you have a link by any chance? $\endgroup$ – SRKX Apr 8 '11 at 6:58
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    $\begingroup$ Oops that is Goldberg...here is an SSRN link papers.ssrn.com/sol3/papers.cfm?abstract_id=1341363 $\endgroup$ – Brian B Apr 26 '11 at 18:52
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I believe the ES is a linear measure as opposed to VaR and should be annualized using a the same approach that is used in the return space.

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    $\begingroup$ Hmmm. According to the post below, for an $N(0,\sigma)$ variable the ES is given by $ES(\alpha)=-\sigma\frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha}$. So it scales like $\sigma$ i.e. it scales like $\sqrt T$. See blog.smaga.ch/… $\endgroup$ – Alex C Jun 20 '18 at 17:13
  • $\begingroup$ This answer is not correct. Like VaR is a quantile, ES is kind of an average of quantiles. If you assume a gaussian distribution for example, the scaling is the same. $\endgroup$ – byouness Jun 22 '18 at 15:07

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