GCRI · Mathematics, statistics and data
Extreme value theory and tail risk
Extreme value theory and tail risk study the behaviour of rare, high-impact events. They provide methods to estimate the magnitude and frequency of extremes beyond the range of observed data. Reinsurance and Catastrophe stewards this page for the Nexus Consortium.
01Workspace
The workspace for extreme value theory and tail risk.
GCRI · GCRI · Tail fitIllustrative
Tail fit
The Global Centre for Risk and Innovation (GCRI) · Extreme value theory and tail risk · Run R-5442 · Example interface, not a record
Shape parameter uncertainGCRI · GCRI
Return level curve
Method card
- Method
- Extreme value theory and tail risk
- Approach
- peaks over threshold
- Threshold
- 90th percentile
- Label
- MODELLED
Parameters
generalised Pareto fit
| Shape ξ | 0.25 | −0.02 | 0.51 | heavy tail if above zero |
| Scale σ | 2.7 | 1.1 | 6.6 | units of the variable |
| Exceedances | 62 | declustered |
GCRI · Tail fit. Tail fit: example names and illustrative values, not a record.
02The method
What extreme value theory and tail risk covers.
- 01
Core methods
block maxima analysis; peaks over threshold modelling; return level estimation; multivariate extremes; non-stationary extreme value models; tail dependence analysis.
- 02
Data and instruments
long hydrometeorological records; insurance loss histories; financial return series; paleo and historical event records; climate model ensembles.
- 03
Validation
Fits are checked with diagnostic plots, goodness-of-fit tests and out-of-sample comparison. Sensitivity to threshold and record length is reported.
- 04
Limits
Extrapolation far beyond the data is highly uncertain. Changing climates and markets violate stationarity assumptions.
- 05
Use in risk decisions
Extreme value methods set design levels and capital buffers for rare events. Tail dependence analysis reveals when extremes in different systems tend to occur together.
03Reference practice
Extreme value analysis is standard in hydrology, climate and actuarial practice.
| Reference | What it sets |
|---|---|
| WMO guidance on extremes | Analysis of climate extremes |
| Flood frequency analysis practice | Design floods in national guidance |
| Actuarial and solvency standards | Tail risk measures |
04Process
How the process runs.
Process
Extreme value theory and tail risk: Catastrophe model pipeline
Catastrophe models usually have hazard, vulnerability, exposure and financial modules, and produce loss tables, average annual loss, exceedance curves, PML and TVaR.
01 Hazard
- 01Event catalogueDataset · Model developerThe hazard module holds a large catalogue of simulated events that represents a wide range of plausible scenarios.
- 03Hazard moduleModel · Model developerFor each event the hazard module gives the local hazard intensity that the vulnerability module reads.
02 Exposure and damage
- 02ExposureDataset · Competent authorityThe exposure module houses the portfolio data, such as location-specific information.
- 04Vulnerability moduleModel · Model developerThe vulnerability module quantifies the expected damage to the exposures from an event.
03 Financial
- 05Financial moduleModel · Competent authorityThe financial module calculates losses from all event scenarios. Ground-up, gross and net losses differ as shown below.
- 06Event loss tableDataset · Competent authorityAn event loss table lists each event with its annual occurrence probability and its loss for the portfolio.
- 07Year loss tableDataset · Competent authoritySome models also sample simulated years from event losses. This outline does not describe their format.
04 Metrics
- 08Exceedance probability curvesModel · Competent authorityThe occurrence curve gives the chance that the largest loss in a year exceeds an amount. The aggregate curve does so for the sum of losses.
- 09AAL, PML and TVaRModel · Competent authorityAverage annual loss sums event probability times loss. PML is one point on the occurrence curve.
05 Use of results
- 10Use of resultsDecision · Competent authorityThe tail metrics are used when considering a portfolio’s exposure and the level of catastrophe reinsurance an insurer requires.
Outline of an external standard or method, from its public sources. It describes what the framework says, not a record of any run. It does not show that Nexus or any partner performs, certifies or approves a step. A gate is a step of the method, never a Nexus approval.
05In systemic risk work
Tails that move together.
Extreme value theory estimates the probability of events beyond observed records. In systemic analysis two issues dominate: non-stationarity (the tail is moving) and tail dependence (extremes in different places or variables occurring together). Both widen uncertainty considerably.
- Fit non-stationary models where trends exist.
- Estimate tail dependence, not only correlation.
- Report return-level uncertainty.
- Applications on Water Nexus and Climate and Hazards.
07Stewardship
Who maintains this page.
Reinsurance and Catastrophe stewards extreme value theory and tail risk: it keeps this description current and reviews how the method is applied across the platforms.
08Sources
Where this page’s references lead.
The bodies and documents this page is read against. Status and editions change; each link goes to the issuing body.
09Questions
Questions and answers.
Why does one institution host every method page?
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