Using Generalized Pareto Models to Extrapolate Portfolio Tail Losses
Summary
The document discusses fitting a generalized Pareto distribution (GPD) to losses above a chosen threshold as part of an extreme value theory approach to portfolio tail risk. It addresses why the fitted distribution’s expected loss can exceed the observed average of threshold exceedances, and whether that difference alone signals a poor fit. The responses say a higher modeled expectation can be consistent with using the GPD to extrapolate beyond historical observations, including into more severe losses that have not yet occurred.
Threshold selection has no single definitive answer. The document recommends checking whether conclusions change substantially across threshold choices; stability is presented as an indication that the GPD may be appropriate, rather than a conclusive test. It also points to plots of the fitted distribution as a way to inspect fit. The discussion offers limited diagnostic detail and no data example or formal goodness-of-fit procedure. It emphasizes that extrapolated losses still require judgment about economic plausibility.
Key ideas
- A GPD fitted to losses above a threshold can imply a higher expected loss than the historical exceedance average.
- The model can extrapolate tail outcomes beyond the most extreme observation in the sample.
- There is no definitive threshold level that works for every dataset.
- Compare results across threshold choices because strong sensitivity may raise concerns about the model’s suitability.
- Plots can help inspect the fit, but the document gives no detailed diagnostic criteria.
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Full text
# Extreme value theory expected value of GPD # Extreme value theory expected value of GPD We're using extreme value theory to model tail risks on our portfolio. After we choose the threshold, we fit generalized Pareto distribution to our data over the threshold. The expected value of GPD is quite larger (10%) than the average value of our losses over the threshold. My question is, is this to be expected? Or does that mean that the GPD is a bad fit to the data and that we've chosen the wrong threshold? Also, is there a good way to check whether the GPD is a good fit? ## Answer by Helin (score 2) https://quant.stackexchange.com/a/28384 It is known that GPD can generate (significantly) more negative outcomes than realized. Whether that's a "bug" or "feature" is debated. I tend to think of GPD as an "extrapolation" method. Using empirical distribution, the worst case scenario your model can produce is the worst historical observation. Of course, we know we are frequently been surprised by "unprecedented" events in the market. GPD can generates much more extreme cases, effectively allowing you to extrapolate into "unknown territory". Whether the resulting value is economically sensible requires judgement. But in my mind, this is the whole point of using GPD. ## Answer by RiskyScientist (score 2) https://quant.stackexchange.com/a/29879 When using EVT and fitting the GPD the threshold level is a choice, and there is no definitive level at which it should be set, though people commonly initially try the most extreme 10% of the data. You should check that your result is not highly sensitive to the threshold, and note that there is no definitive level of sensitivity that is "correct." But this lack of sensitivity to threshold level is an indication of the appropriateness of using the GPD. It is not surprising that the expected value of your GPD is 10% higher than the average value of losses over the threshold, and that in itself does not mean the GPD is a bad fit. ## Answer by Nourhaine Nefzi (score 0) https://quant.stackexchange.com/a/28378 I can answer the second question : You can use the following command to see if the GPD fits well the exceedances ``` XFitGPD<-gpdFit(X,u= ...) plot(XFitGPD) ```
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