Almost by definition of risk, rare events play a crucial role. We tackle this problem by presenting some basic tools from extreme value theory (EVT). From a statistical point of view, the workhorses are the block maxima method (BMM) and the peaks over threshold method (POTM). Besides giving the mathematical formulation, we exemplify both approaches via simulated examples. Once these tools are in place, we can provide estimators of the relevant risk measures such as high-exceedance probabilities, quantiles and return periods. In a crucial part of the book, we then estimate these quantities for sea-level data at Hoek van Holland near Rotterdam. We obtain estimates, including confidence intervals, for a necessary dike height withstanding a required 1 in 10 000 years storm event. Further applications concern financial data and data from the L’Aquila earthquake. For the latter, we present dynamic models for earthquake aftershocks. After an excursion to the world of records in athletics, we present the signature application of EVT through the story of the sinking of the MV Derbyshire. We show how an application of EVT techniques has saved many lives at sea.

“In this chapter, we present some key results from extreme value theory (EVT) and illustrate how EVT can be used to supplement traditional statistical analysis. We use EVT when we are concerned about the impact of very rare, very large losses. Because they are rare, we are unlikely to have much data, but using EVT we can infer the extreme tail behaviour of most distributions.

There are two different, but related types of models for extreme value analysis. The first considers the distribution of the maximum value in a random sample of losses. These are called the block maxima models. The second comes from analysing the rare, very large losses, defined as the losses exceeding some high threshold. These are the points over threshold models.

We present the key results for both of these, and show how they are connected. We derive formulas for the VaR and Expected Shortfall risk measures using EVT that are useful when the loss distribution is fat-tailed, and the parameter a is close to 1.0. We use examples throughout to highlight the potential uses in practical applications.”