How to annualize Expected Shortfall?

Question

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?

Answer 1

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.