Expected Shortfall Estimation with the Block Maxima Method

Summary

The document asks how to estimate Expected Shortfall (ES) from a return series using the block maxima approach in Extreme Value Theory. It contrasts that question with a peaks-over-threshold formula based on Generalized Pareto Distribution parameters and asks whether the same expression applies to block maxima, given the relationship between the GPD and the Generalized Extreme Value distribution. The stated goal is to identify the necessary parameters, including whether a threshold is needed.

The response says ES can be defined from a GEV distribution fitted to block maxima or minima and identifies scale, shape, location, block size, and probability as relevant quantities. It points to a cited paper for a proof, but the document itself omits the promised formula and derivation. Consequently, it does not show how to calculate ES from those parameters, clarify sign conventions for returns versus losses, or explain how block size and probability map to the desired risk quantile. The response is too incomplete to serve as a standalone procedure.

Key ideas

• The question distinguishes block maxima estimation from peaks-over-threshold estimation of Expected Shortfall.
• The response identifies the GEV location, scale, and shape parameters as relevant to a block maxima approach.
• Block size and probability are also named as inputs to the proposed definition.
• The actual ES expression and derivation are absent, so the calculation cannot be reproduced from this document alone.

Tags

# Block maxima estimation of Expected Shortfall


# Block maxima estimation of Expected Shortfall












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.

## Answer by Maria A. (score 1)

https://quant.stackexchange.com/a/55659

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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