Skip to content
All library documents

Negative GPD Shape Parameters and Bounded Tail Risk

Article Quant Q&A · Author: Marco Aurélio Guerra
Read the original

Summary

The document describes an attempt to estimate value at risk (VaR) and expected shortfall (ES) with a peaks-over-threshold method using a generalized Pareto distribution (GPD). In a rolling sample of Petrobras returns, the author encounters a software error when the fitted GPD shape parameter is negative and asks whether extreme value theory requires a positive shape.

The answer links a negative shape parameter to a distribution with a finite upper endpoint, which explains why some extrapolated risk measures may be unavailable or inconsistent with the software’s implementation. The response concludes that this is unsuitable for the author’s intended tail model. That conclusion is too broad as a general statement: negative shape is a valid GPD case for bounded tails. The document offers no further diagnosis of the software error or evidence that the fitted endpoint is implausible for the data, so users should distinguish model support from package-specific restrictions.

Key ideas

  • The question arises from a rolling peaks-over-threshold VaR and ES estimation using a GPD.
  • A negative GPD shape parameter corresponds to a distribution with a finite endpoint.
  • The software reports that a risk measure is not implemented for the fitted negative shape case.
  • Negative shape is a valid GPD parameter regime for bounded tails, despite the response’s broader claim.
  • The document does not diagnose the package behavior or assess whether the endpoint fits the return data.

Tags

StatisticsRisk managementEquities
Full text
# VaR and Expected Shorfall estimations with negative shape parameter of a GPD (Extreme Value Theory )


# VaR and Expected Shorfall estimations with negative shape parameter of a GPD (Extreme Value Theory )












So im trying to replicate an code from the Quantative Risk Management Book (https://github.com/qrmtutorial/qrm/blob/master/code/09_Market_Risk/09_Standard_methods_for_market_risk.R). But when i try a rolling window estimation -last 1000 observations- using the 'POT' method in my data ( log loss of "PETR4.SA" , Petrobras ticker in BOVESPA , from 2006 to end of 2020 ), i got an error ( around "2006-01-16/ + 1000 days" ) message telling that the Risk measure is not implemented for negative shape parameter of the GPD. I've read the book (QRM) and other papers about GPD for tail estimation but i could not find any reference to the restriction in the shape must being positive. The code for the VaR and ES estimation are below:

VaR_95 <- u + (beta/shape)(((1-alpha_95)/length(excess) / length(Losses))^(-shape)-1) ES_95 <- (VaR_95 + beta-shapeu) / (1-xi)

## Answer by Marco Aurélio Guerra (score 2, accepted)

https://quant.stackexchange.com/a/61541

it's me again... So i find out what a negative shape parameter in Generalized Pareto Distribuition means and why it's not possible to calculate EVT with it. negative shape parameter means that the distribuition has a limit, not quite what you are looking for when fitting an extreme value theory model.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.