Fitting a GPD to threshold exceedances is the vehicle; the destination is a VaR or ES number. This note closes the loop: it derives the closed-form VaR and ES formulas implied by a GPD fit, introduces the Smith tail estimator as the building block, and then presents the Hill estimator as an older alternative confined to Fréchet-class distributions. The comparison exposes a fundamental trade-off between variance and generality — knowledge that prevents misapplication at the modelling desk.
The second half covers conditional EVT, the McNeil-Frey procedure that glues GARCH-prewhitening to POT estimation. This is the industry-grade method for obtaining one-step-ahead VaR and ES estimates that respect both the volatility clustering of daily returns and the heavy-tailed structure of extreme losses — the combination that plain historical simulation and plain GARCH-normal both miss.
Learning objectives. By the end of this note you should be able to:
- derive VaR and ES from fitted GPD parameters using the Smith tail estimator;
- state the Hill estimator, explain why it is restricted to Fréchet MDA distributions, and read a Hill plot;
- describe the bias-variance trade-off in choosing the number of order statistics;
- implement the McNeil-Frey conditional EVT procedure step by step;
- choose between GPD and Hill estimation given the data characteristics.
In this lesson
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- The Smith tail estimator and EVT-based risk measures
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- The Hill estimator
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- Simulation study: GPD vs Hill
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- Conditional EVT: the McNeil-Frey procedure
Before you start
- The Generalised Pareto Distribution and Peaks-Over-Threshold
- Derivation — Domains of Attraction and the GEV Limit
- GARCH Extensions and Filtered Historical Simulation
The full lesson, with worked examples and exercise solutions, is for members.
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