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I want to calculate the expected shortfall of a return series with the block maxima (BM, link) method in Extreme Value Theory, however I can't seem to find out how this should be done. All papers I've found only consider the peak over threshold (POT) method where it can be calculated via

$$ ES_q = \frac{VaR_q}{1-\gamma} + \frac{\beta - \gamma u}{1-\gamma}$$ Where $\gamma$ is the GPD shape parameter, $\beta$ the scale parameter and $u$ the threshold. I reckon that it must be possible to use this with the BM method as well, as the GPD (Generalized Pareto Distribution) is a special case of the limit to the GEV (Generalized Extreme Value Distribution) (Am I right?). However, I can't seem to find any paper which calculated ES with the BM method so I'm a bit stuck right now. Is it possible to use this expression for the BM method as well? If yes, what do I need to fill in for the threshold $u$?

I found the expression for the POT ES in McNeil (1999): Extreme Value Theory for Risk Managers, but it's stated in many other papers and books as well.

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We can define the algorithm of Expected Shortfall (ES) based on Block Maxima Method as follows:

Where α is the scale, ξ is the shape parameter and β is the location. All of them, parameters of the GEV distribution. Where n is the block size and p the probability.

You can find the complete proof in Ou &Yi (2009): “Robustness Analysis and Algorithm of Expected Shortfall Based on Extreme-Value Block Minimum Model”.

I hope it helps.

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