Advances in Risk Management

YVES CRAMA, GEORGES HÜBNER AND JEAN-PHILIPPE PETERS 7

(see Fontnouvelle, Rosengren and Jordan, 2004, or Chapelle, Crama, Hübner
and Peters, 2005).
As a consequence, numerous authors propose to use alternative
approaches to improve the accuracy of the tail modeling. One of the
most common approaches relies on Extreme Value Theory (EVT), which
is presented in next section.

Modeling extreme losses

Extreme Value Theory (EVT) is a powerful theoretical tool to build statistical
models describing extreme events. It has been developed to answer the
crucial question: if things go wrong, how wrong can they go?
Two techniques are available: the Block Maxima method and the Peak
Over Threshold (POT) method. While the origins of the former date back
in the early twentieth century, it has been presented in a general context by
Gumbel (1958). It focuses on the modeling of maxima of different periods,
such as month or year (for example, thepobservations are the maximum
observed value of each of thepperiods considered). These extremes are then
modeled with the Generalized Extreme Value (GEV) distribution. While
useful in domains such as climatology, the Block Maxima approach is less
attractive for financial applications.
ThePOTapproachbuildsuponresultsofBalkemaanddeHaan(1974)and
Pickands (1975) which state that, for a broad class of distributions, the values
of the random variables above a sufficiently high threshold follow a Gener-
alized Pareto Distribution (GPD) with location parameterμ, scale parameter
βand shape parameterξ(also called the tail index). The GPD can thus be
thought of as the conditional distribution ofXgivenX>μ(see Embrechts
et al., 1997, for a comprehensive review). Its cdf can be expressed as:

F(x;ξ,β,μ)= 1 −

(

1 +ξ

(x−μ)

β

)−ξ^1

(1.7)

A major issue when applying POT is the determination of the threshold
μ. Astandard technique is based on the visual inspection of the Mean Excess
Function (MEF) plot (see Davidson and Smith, 1990, or Embrechts, Klüp-
perberg and Mikosch, 1997, for details). This graph plots the empirical mean
excess, defined as:

e(u)=

1

nu

∑nu

i= 1

(xi−u)

where thexi’s are thenuvalues ofXsuch thatxi>u. The MEF plot is a
plot ofe(u) againstu. The method is to detect a significant shift in slope
at some high point. When the empirical plot seems to follow a reasonably