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.
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