Annualizing Expected Shortfall Under Different Return Distributions

Summary

The document asks how to convert an empirically estimated monthly expected shortfall into an annual figure. Its example estimates a lower-tail expected shortfall by averaging the worst observations in a monthly return sample, then considers two possible scaling rules: compounding it like a return or multiplying by the square root of the number of months in a year.

The accepted response says square-root scaling is appropriate if returns are assumed to be normally distributed for this purpose. It warns that normal distributions have relatively thin tails, so analysts may instead fit heavier-tailed distributions such as Student-t or Pareto. Under those models, annual tail risk requires combining the monthly distributions across the year; there is generally no simple scaling formula. A second answer asserts that expected shortfall is linear and should use a return-style annualization, but offers no derivation. The replies therefore do not provide a universal rule, and the competing suggestion underscores that scaling depends on assumptions about return aggregation and the distribution’s tail behavior.

Key ideas

• The question concerns annualizing an empirical monthly expected shortfall estimated from lower-tail returns.
• Square-root-of-time scaling is proposed when returns are assumed to be normal.
• Heavy-tailed return models can invalidate simple square-root scaling.
• For fitted tail distributions, annual risk may require aggregating monthly distributions across the year.
• The responses offer differing views and do not establish a general scaling rule.

Tags

# How to annualize Expected Shortfall?


# How to annualize Expected Shortfall?












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 by Brian B (score 5, accepted)

https://quant.stackexchange.com/a/930

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.

## Answer by Sofiger (score -3)

https://quant.stackexchange.com/a/40410

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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