Appendix: The theory of extreme value theory – an introduction © Con Keating

Appendix: The theory of extreme value theory –
an introduction © Con Keating
In 1900 Bachelier introduced the normal distribution to financial analysis (see also
Cootner, 1964); today most students of the subject would be able to offer a critique
of the shortcomings of this most basic (but useful) model. Most would point immediately
to the ‘fat tails’ evident in the distributions of many financial time series.
Benoit Mandelbrot (1997), now better known for his work on fractals, and his
doctoral student Eugene Fama published extensive studies of the empirical properties
of the distributions of a wide range of financial series in the 1960s and 1970s which
convincingly demonstrate this non-normality. Over the past twenty years, both
academia and the finance profession have developed a variety of new techniques,
such as the ARCH family, to simulate the observed oddities of actual series. The
majority fall short of delivering an entirely satisfactory result.
At first sight the presence of skewness or kurtosis in the distributions suggests
that of a central limit theorem failing but, of course, the central limit theorem should
only be expected to apply strongly to the central region, the kernel of the distribution.
Now this presents problems for the risk manager who naturally is concerned with the
more unusual (or extreme) behavior of markets, i.e. the probability and magnitudes of
the events forming the tails of the distributions.
There is also a common misunderstanding that the central limit theorem implies
that any mixture of distributions or samplings from a distribution will result in a
normal distribution, but a further condition exists, which often passes ignored, that
these samplings should be independent.
Extreme value theory (EVT) is precisely concerned with the analysis of tail behaviour.
It has its roots in the work of Fisher and Tippett first published in 1928 and a
long tradition of application in the fields of hydrology and insurance. EVT considers
the asymptotic (limiting) behavior of series and, subject to the assumptions listed
below, states which is read as: F is a realization in the maximum domain of attraction of H.
To illustrate the concept of a maximum domain of attraction, consider a fairground
game: tossing ping-pong balls into a collection of funnels. Once inside a funnel, the
ball would descend to its tip – a point attractor. The domain of attraction is the
region within a particular funnel and the maximum domain of attraction is any
trajectory for a ping-pong ball which results in its coming to rest in a particular
funnel. This concept of a stable, limiting, equilibrium organization to which dynamic
systems are attracted is actually widespread in economics and financial analysis.
The assumptions are that bn and an[0 (location and scaling parameters) exist
such that the financial time series, X, demonstrates regular limiting behavior and that
the distribution is not degenerate. These are mathematical technicalities necessary to
ensure that we do not descend inadvertently into paradox and logical nonsenses. m is
referred to as a shape parameter. There is (in the derivation of equation (A1)) an
inherent assumption that the realisations of X, x0, x1, . . . , xn are independently
and identically distributed. If this assumption were relaxed, the result, for serially
dependent data, would be slower convergence to the asymptotic limit.
The distribution Hm (x) is defined as the generalized extreme value distribution
(GEV) and has the functional form:

The distributions where the value of the tail index, m, is greater than zero, equal to
zero or less than zero are known, correspondingly, as Fre´chet, Gumbel and Weibull
distributions. The Fre´chet class includes Student’s T, Pareto and many other distributions
occasionally used in financial analysis; all these distributions have heavy
tails. The normal distribution is a particular instance of the Gumbel class where m
is zero.
This constitutes the theory underlying the application of EVT techniques but it
should be noted that this exposition was limited to univariate data.10 Extensions of
EVT to multivariate data are considerably more intricate involving measure theory,
the theory of regular variations and more advanced probability theory. Though much
of the multivariate theory does not yet exist, some methods based upon the use of
copulas (bivariate distributions whose marginal distributions are uniform on the
unit interval) seem promising.
Before addressing questions of practical implementation, a major question needs
to be considered. At what point (which quantile) should it be considered that the
asymptotic arguments or the maximum domain of attraction applies? Many studies
have used the 95th percentile as the point beyond which the tail is estimated. It is
far from clear that the arguments do apply in this still broad range and as yet there
are no simulation studies of the significance of the implicit approximation of this
choice.
The first decision when attempting an implementation is whether to use simply
the ordered extreme values of the entire sample set, or to use maxima or minima in
defined time periods (blocks) of, say, one month or one year. The decision trade-off
is the number of data points available for the estimation and fitting of the curve parameters versus the nearness to the i.i.d. assumption. Block maxima or minima
should be expected to approximate an i.i.d. series more closely than the peaks over
a threshold of the whole series but at the cost of losing many data-points and
enlarging parameter estimation uncertainty. This point is evident from examination
of the following block maxima and peaks over threshold diagrams.
Implementation based upon the whole data series, usually known as peaks over
threshold (POT), uses the value of the realization (returns, in most financial applications)
beyond some (arbitrarily chosen) level or threshold. Anyone involved in the
insurance industry will recognize this as the liability profile of an unlimited excess
of loss policy. Figures 8A.1 and 8A.2 illustrate these two approaches:

Figure 8A.1 shows the minimum values in each 25-day period and may be
compared with the whole series data-set below. It should be noted that both series
are highly autocorrelated and therefore convergence to the asymptotic limit should
be expected to be slow.
Figure 8A.2 shows the entire data-set and the peaks under an arbitrary value (1.5).
In this instance, this value has clearly been chosen too close to the mean of the
distribution – approximately one standard deviation. In a recent paper, Danielsson
and De Vries (1997) develop a bootstrap method for the automatic choice of this cutoff
point but as yet, there is inadequate knowledge of the performance of small
sample estimators. Descriptive statistics of the two (EVT) series are given in Table
8A.1.

Notice that the data-set for estimation in the case of block minima has declined to
just 108 observations and further that neither series possesses ‘fat tails’ (positive
kurtosis). Figure 8A.3 presents these series.

The process of implementing EVT is first to decide which approach, then the level
of the tail boundary and only then to fit a parametric model of the GEV class to the
processed data. This parametric model is used to generate values for particular VaR
quantiles. It is standard practice to fit generalized Pareto distributions (GPD) to POT
data:

omitting the u and x subscripts. The numerical MLE solution should not prove
problematic provided m[ñ1
2 which should prove the case for most financial data. A
quotation from R. L. Smith is appropriate: ‘The big advantage of maximum likelihood
procedures is that they can be generalized, with very little change in the basic
methodology, to much more complicated models in which trends or other effects may
be present.’ Estimation of the parameters may also be achieved by either linear (see,
for example, Kearns and Pagan, 1997) or non-linear regression after suitable
algebraic manipulation of the distribution function.
It should be immediately obvious that there is one potential significant danger for
the risk manager in using EVT; that the estimates of the parameters introduce
error non-linearly into the estimate of the VaR quantile. However, by using profile
likelihood, it should be possible to produce confidence intervals for these estimates,
even if the confidence interval is often unbounded.
Perhaps the final point to make is that it becomes trivial to estimate the mean
expected loss beyond VaR in this framework; that is, we can estimate the expected
loss given a violation of the VaR limit – an event which can cause changes in
management behavior and cost jobs.
This brief appendix has attempted to give a broad overview of the subject. Of
necessity, it has omitted some of the classical approaches such as Pickand’s and
Hill’s estimators. An interested reader would find the introductory texts10 listed far
more comprehensive.
There has been much hyperbole surrounding extreme value theory and its application
to financial time series. The reality is that more structure (ARCH, for example)
needs to be introduced into the data-generating processes before it can be said that
the method offers significant advantages over conventional methods. Applications,
however, do seem most likely in the context of stress tests of portfolios.