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### Extreme Value Theory 1. Introduction

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Extreme Value Theory (EVT) attempts to provide a complete characterisation of the tail behaviour of all types of probability distributions, arguing that this behaviour can in practice in the limit only take a small number of possible forms. When applied to single return series or loss series, it appears to offer a conceptually appealing approach to analysing extreme events and to calculating risk measures such as Value-at-Risk involving high severity low frequency confidence levels. It suggests that we can identify likelihoods of very extreme events ‘merely’ by using the following prescription:

(1)    Identify the apparent type of tail behaviour being exhibited by the variable in question.

(2)    Estimate the (small number of) parameters that then characterise the tail behaviour.

(3)    Estimate the likelihood of occurrence however far into the distributional tail, by inserting the desired quantile or confidence level into the tail distribution estimated in step 2

EVT provides a set of limiting results that potentially enable one to analyse unusual events. It involves two broad sets of results, one applying to ‘block maxima’ (or ‘block minima’) and one applying to ‘threshold exceedances’.

When EVT applies, the more traditional ‘block maxima’ results provide information on the distribution of the maximum value of the series in given blocks, e.g. daily losses over a 25 business day period (here the 25 business day period is the discrete ‘block’ of data). The distribution of the maxima converges to one of three different limiting forms that in aggregate can be represented by different parameterisations of the generalised extreme value (GEV) family of probability distribution.

The newer ‘threshold exceedances’ results provide an indication of the likelihood of outcomes exceeding a given threshold level. If the threshold is pushed out into the tail of the distribution then if EVT applies these likelihoods converge asymptotically to random variables from a simple family, the generalised Pareto distribution (GPD). The distributions resulting from the ‘threshold exceedances’ results are also termed ‘peaks-over-thresholds’ distributions. Our focus in most of these pages will be on the ‘threshold exceedances’ results as they are less wasteful of the data than the ‘block maxima’ results.