Extreme Values
in Finance,
Telecommunications,
and the Environment
Edited by
Bärbel Finkenstädt
and
Holger Rootzén
CHAPMAN & HALL/CRC
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Contents
Contributors
Participants
Preface
1 StatisticsofExtremes,withApplicationsinEnvironment,
Insurance, and Finance
by Richard L. Smith
2 The Use and Misuse of Extreme Value Models in Practice
by Stuart Coles
3 Risk Management with Extreme Value Theory
by Claudia Kl
¨
uppelberg
4 Extremes in Economics and the Economics of Extremes
by Paul Embrechts
5 Modeling Dependence and Tails of Financial Time Series
by Thomas Mikosch
6 Modeling Data Networks
by Sidney Resnick
7 Multivariate Extremes
by Anne-Laure Foug
`
eres
©2004 CRC Press LLC
Contributors
Stuart Coles
Department of Mathematics
University of Bristol
Bristol, United Kingdom

Jenny Andersson, Gothenburg (Sweden),
Analysis of corrosion on aluminium and magnesium by statistics of extremes.
Bojan Basrak, Eindhoven (Netherlands),
On multivariate regular variation and some time-series models.
Nathana¨el Benjamin, Oxford (United Kingdom),
Bound on an approximation for the distribution of the extreme fluctuations of
exchange rates.
Paola Bortot, Bologna (Italy),
Extremes of volatile Markov chains.
Natalia Botchkina, Bristol (United Kingdom),
Wavelets and extreme value theory.
Leonardo Bottolo, Pavia (Italy),
Mixture models in Bayesian risk analysis.
Boris Buchmann, Munich (Germany),
Decompounding: an estimation problem for the compound Poisson distribution.
Adam Butler, Lancaster (United Kingdom),
The impact of climate change upon extreme sea levels.
Ana Cebrian, Louvain-La-Neuve (Belgium),
Analysis of bivariate extreme dependence using copulas with applications
to insurance.
Ana Ferreira, Eindhoven (Netherlands),
Confidence intervals for the tail index.
Christopher Ferro, Lancaster (United Kingdom),
Aspects of modelling extremal temporal dependence.
John Greenhough, Warwick (United Kingdom),
Characterizing anomalous transport in accretion disks from X-ray observations.
Viviane Grunert da Fonseca, Faro (Portugal),
Stochastic multiobjective optimization and the attainment function.
Janet Heffernan, Lancaster (United Kingdom),
A conditional approach for multivariate extreme values.

approach.
Paul Northrop, Oxford (United Kingdom),
An empirical Bayes approach to flood estimation.
Gr´egory Nuel, Evry (France),
Unusual word frequencies in Markov chains: the large deviations approach.
Fehmi
¨
Ozkan, Freiburg im Breisgau (Germany),
The defaultable L´evy term structure: ratings and restructuring.
Francesco Pauli, Trieste (Italy),
A multivariate model for extremes.
Olivier Perrin, Toulouse (France),
On a time deformation reducing stochastic processes to local stationarity.
©2004 CRC Press LLC
Martin Schlather, Bayreuth (Germany),
A dependence measure for extreme values.
Manuel Scotto, Algueir˜ao (Portugal),
Extremal behaviour of certain transformations of time series.
Scott Sisson, Bristol (United Kingdom),
An application involving uncertain asymptotic temporal dependence in the
extremes of time series.
Alwin Stegeman, Groningen (Netherlands),
Long-range dependence in computer network traffic: theory and practice.
Scherbakov Vadim, Glasgow (United Kingdom),
Voter model with mean-field interaction.
Yingcun Xia, Cambridge (United Kingdom),
A childhood epidemic model with birthrate-dependent transmission.
©2004 CRC Press LLC
Preface
The chapters in this volume are the invited papers presented at the fifth S´eminaire

answers a variety of questions of interest to an applied scientist in climatology, in-
surance, and finance. The chapter also reviews parts of univariate extreme value
theory and discusses estimation, diagnostics, multivariate extremes, and max-stable
processes.
©2004 CRC Press LLC
In the second chapter, Stuart Coles focuses on the particularly extreme event of
the 1999 rainfall in Venezuela that caused widespread distruction and loss of life. He
demonstrates that the probability for such an event would have been miscalculated
even by the standard extreme value models, and discusses the use of various options
available for extension in order to achieve a more satisfactory analysis.
The next three chapters consider applications of extreme value theory to risk man-
agement in finance and economics. First, in Chapter 3, Claudia Kl¨uppelberg reviews
aspects of Value-at-Risk (VaR) and its estimation based on extreme value theory.
She presents results of a comprehensive investigation of the extremal behavior of
some of the most important continuous and discrete time series models that are of
current interest in finance. Her discussions are followed by an historic overview
of financial risk management given by Paul Embrechts in Chapter 4. In Chapter
5, Thomas Mikosch introduces the stylized facts of financial time series, in par-
ticular the heavy tails exhibited by log-returns. He studies, in depth, their connec-
tion with standard econometric models such as the GARCH and stochastic volatil-
ity processes. The reader is also introduced to the mathematical concept of regular
variation.
Another important area where extreme value theory plays a significant role is data
network modelling. In Chapter 6, Sidney Resnick reviews issues to consider for data
network modelling, some of the basic models and statistical techniques for fitting
these models.
Finally, in Chapter 7, Anne-Laure Foug`eres gives an overview of multivariate
extreme value distributions and the problem of measuring extremal dependence.
The order in which the chapters are compiled approximately follows the order in
which they were presented at the conference. Naturally it is not possible to cover all

in Environment, Insurance, and Finance
Richard L. Smith
University of North Carolina
Contents
1.1 Motivating examples
1.1.1 Snowfall in North Carolina
1.1.2 Insurance risk of a large company
1.1.3 Value at risk in finance
1.2 Univariate extreme value theory
1.2.1 The extreme value distributions
1.2.2 Exceedances over thresholds
Poisson-GPD model for exceedances
1.2.3 Examples
The exponential distribution
Pareto-type tail
Finite upper endpoint
Normal extremes
1.2.4 The r largest order statistics model
1.2.5 Point process approach
1.3 Estimation
1.3.1 Maximum likelihood estimation
1.3.2 Profile likelihoods for quantiles
1.3.3 Bayesian approaches
1.3.4 Raleigh snowfall example
1.4 Diagnostics
1.4.1 Gumbel plots
1.4.2 QQ plots
1.4.3 The mean excess plot
1.4.4 Z- and W-statistic plots
1.5 Environmental extremes

ariver will be exceeded with probability 1/100 in a given year? — this quantity is
often called the 100-year return level). During the last 30 years, many new techniques
have been developed concerned with exceedances over high thresholds, the depen-
dence among extreme events in various types of stochastic processes, and multivariate
extremes.
These new techniques make it possible to answer much more complex questions
than simple distributions of extremes. Among those considered in the present review
are whether probabilities of extreme events are changing with time or corresponding
to other measured covariates (e.g., Section 1.5.1 through Section 1.5.3 and Section
1.6.4), the simultaneous fitting of extreme value distributions to several related time
series (Section 1.6.1 through Section 1.6.3), the spatial dependence of extreme value
distributions (Section 1.6.4) and the rather complex forms of extreme value calcu-
lations that arise in connection with financial time series (Section 1.8). Along the
way, we shall also review relevant parts of the mathematical theory for univariate
extremes (Section 1.2 through Section 1.4) and one recent approach (among several
©2004 CRC Press LLC
that are available) to the characterization of multivariate extreme value distributions
(Section 1.7).
For the rest of this section, we give some specific examples of data-oriented ques-
tions which will serve to motivate the rest of the chapter.
1.1.1 Snowfall in North Carolina
On January 25, 2000, a snowfall of 20.3 inches was recorded at Raleigh-Durham
airport in North Carolina. This is an exceptionally high snowfall for this part of the
U.S. and caused widespread disruption to travel, power supplies, and the local school
system. Various estimates that appeared in the press at the time indicated that such
an event could be expected to occur once every 100 to 200 years. The question we
consider here is how well one can estimate the probability of such an event based on
data available prior to the actual event. Associated with this is the whole question of
what is the uncertainty of such an assessment of an extreme value probability.
To simplify the question and to avoid having to consider time-of-year effects,

Year Day Amount Year Day Amount Year Day Amount
1948 24 1.0 1965 15 0.8 1977 7 0.3
1948 31 2.5 1965 16 3.7 1977 24 1.8
1954 11 1.2 1965 17 1.3 1979 31 0.4
1954 22 1.2 1965 30 3.8 1980 30 1.0
1954 23 4.1 1965 31 0.1 1980 31 1.2
1955 19 9.0 1966 16 0.1 1981 30 2.6
1955 23 3.0 1966 22 0.2 1982 13 1.0
1955 24 1.0 1966 25 2.0 1982 14 5.0
1955 27 1.4 1966 26 7.6 1985 20 1.7
1956 23 2.0 1966 27 0.1 1985 28 2.4
1958 7 3.0 1966 29 1.8 1987 25 0.1
1959 8 1.7 1966 30 0.5 1987 26 0.5
1959 16 1.2 1967 19 0.5 1988 7 7.1
1961 21 1.2 1968 10 0.5 1988 8 0.2
1961 26 1.1 1968 11 1.1 1995 23 0.7
1962 1 1.5 1968 25 1.4 1995 30 0.1
1962 10 5.0 1970 12 1.0 1996 6 2.7
1962 19 1.6 1970 23 1.0 1996 7 2.9
1962 28 2.0 1973 7 0.7 1997 11 0.4
1963 26 0.1 1973 8 5.7 1998 19 2.0
1964 13 0.4 1976 17 0.4
Table 1.2 The seven types of insurance claims, with the total number of claims and the mean
size of claim for each type.
Type Description Number Mean
1 Fire 175 11.1
2 Liability 17 12.2
3Offshore 40 9.4
4 Cargo 30 3.9
5 Hull 85 2.6

200
300
400
(c)
Years from start
Total size of claims
051015
0
500
1000
1500
2000
2500
3000
(d)
Threshold
Mean excess over threshold
0204060 80
0
50
100
150
200
250
300
Figure 1.1 Insurance data: (a) plot of raw data, (b) cumulative number of claims vs. time,
(c) cumulative claim amount vs. time, and (d) mean excess plot.
accounting for 26% of the total. In statistical terms, the data clearly represent a very
skewed, long-tailed distribution, though these features are entirely typical of insurance
data.

one year?
Published statistical analyses of insurance data often concentrate exclusively on
question 1, but it is arguable that the other three questions are all more important and
relevant than a simple characterisation of the probability distribution of claims, for a
company planning its future insurance policies.
1.1.3 Value at risk in finance
Much of the recent research in extreme value theory has been stimulated by the
possibility of large losses in the financial markets, which has resulted in a large amount
of literature on “value at risk” and other measures of financial vulnerability. As an
example of the types of data analysed and the kinds of questions asked, Figure 1.2
shows negative daily returns from closing prices of 1982 to 2001 stock prices in three
companies, Pfizer, General Electric, and Citibank. If X
t
is the closing price of a stock
or financial index on day t, then the daily return (in effect, the percentage loss or gain
on the day) is defined either by
Y
t
= 100

X
t
X
t−1
− 1

(1.1)
or, more conveniently for the present discussion, by
Y
t

0.2
Figure 1.2 Negative daily returns, defined by (1.3), for three stocks, 1982 to 2001, (a) Pfizer,
(b) General Electric, and (c) Citibank.
We are mainly interested in the possibility of large losses rather than large gains, so
we rewrite (1.2) in terms of negative returns,
Y
t
= 100 log
X
t−1
X
t
, (1.3)
which is the quantity actually plotted in Figure 1.2.
Typical problems here are:
1. Calculating the value at risk, i.e., the amount which might be lost in a portfolio of
assets over a specified time period with a specified small probability;
2. Describing dependence among the extremes of different series, and using this
description in the problem of managing a portfolio of investments; and
3. Modeling extremes in the presence of volatility — like all financial time series,
those in Figure 1.2 show periods when the variability or volatility in the series
is high, and others where it is much lower, but simple theories of extreme val-
ues in independent and identically distributed (i.i.d.) random variables or simple
stationary time series do not account for such behaviour.
In Section 1.8, we shall return to this example and suggest some possible approaches
to answering these questions.
1.2 Univariate extreme value theory
1.2.1 The extreme value distributions
In this section, we outline the basic theory that applies to univariate sequences of i.i.d.
random variables. This theory is by now very well established and is the starting point

> 0, b
n
such that
Pr

M
n
− b
n
a
n
≤ x

= F(a
n
x + b
n
)
n
→ H (x). (1.5)
The Three Types Theorem, originally stated without detailed mathematical proof
by Fisher and Tippett (1928), and later derived rigorously by Gnedenko (1943), asserts
that if a nondegenerate H exists (i.e., a distribution function which does not put all
its mass at a single point), it must be one of three types:
H(x) = exp(−e
−x
), all x, (1.6)
H(x) =

0, x < 0,

1 + ξ
x − µ
ψ

−1/ξ
+

, (1.9)
(y
+
= max(y, 0)) where µ is a location parameter, ψ>0isascale parameter, and ξ
is a shape parameter. The limit ξ → 0 corresponds to the Gumbel distribution, ξ>0
to the Fr´echet distribution with α = 1/ξ, and ξ<0tothe Weibull distribution with
α =−1/ξ.
In more informal language, the case ξ>0isthe “long-tailed” case for which
1 − H (x) ∝ x
−1/ξ
, ξ = 0isthe “medium-tailed” case for which 1 − H (x) decreases
exponentially for large x, and ξ<0isthe “short-tailed” case, in which the distribution
has a finite endpoint (the minimum value of x for which H (x) = 1) at x = µ −ψ/ξ.
©2004 CRC Press LLC
1.2.2 Exceedances over thresholds
Consider the distribution of X conditionally on exceeding some high threshold u (so
Y = X − u > 0):
F
u
(y) = Pr{Y ≤ y | Y > 0}=
F(u + y) − F(u)
1 − F(u)
.

for large x . This is reminiscent of the usual Pareto distribution,
G(x) = 1 − cx
−α
, with ξ = 1/α.Forξ = 0, we may take the limit as ξ → 0toget
G(y; σ, 0) = 1 − exp

−
y
σ

,
i.e., exponential distribution with mean σ .Forξ<0, the distribution has finite upper
endpoint at −σ/ξ. Some other elementary results about the GPD are
E(Y ) =
σ
1 − ξ
, (ξ<1),
Var(Y ) =
σ
2
(1 − ξ)
2
(1 − 2ξ)
,

ξ<
1
2

, (1.12)

The Poisson–GPD process is closely related to the GEV distribution for annual
maxima. Suppose x > u. The probability that the annual maximum of the Poisson–
GPD process is less than x is
Pr{ max
1≤i≤N
Y
i
≤ x}=Pr{N = 0}+
∞

n=1
Pr{N = n, Y
1
≤ x, Y
n
≤ x}
= e
−λ
+
∞

n=1
λ
n
e
−λ
n!
·

1 −

(1.13) reduces to the GEV form (1.9). Thus the GEV and GPD models are entirely
consistent with one another above the threshold u, and (1.14) gives an explicit rela-
tionship between the two sets of parameters.
The Poisson–GPD model is closely related to the peaks over threshold (POT)
model originally developed by hydrologists. In cases with high serial correlation,
the threshold exceedances do not occur singly but in clusters, and, in that case, the
method is most directly applied to the peak values within each cluster. For more
detailed discussion, see Davison and Smith (1990).
Another issue is seasonal dependence. For environmental processes in particular,
it is rarely the case that the probability of an extreme event is independent of the time
of year, so we need some extension of the model to account for seasonality. Possible
strategies include:
1. Remove seasonal trend before applying the threshold approach.
2. Apply the Poisson–GPD model separately to each season.
3. Expand the Poisson–GPD model to include covariates.
©2004 CRC Press LLC
All three approaches have been extensively applied in past discussions of threshold
methods. In the present chapter, we focus primarily on method 3. (e.g., Section 1.5 and
Section 1.6.4), though only after first rewriting the Poisson-GPD model in a different
form (Section 1.2.5).
1.2.3 Examples
In this section, we present four examples to illustrate how the extreme value and GPD
limiting distributions work in practice, given various assumptions on the distribution
function F from which the random variables are drawn. From a mathematical view-
point, these examples are all special cases of the domain of attraction problem, which
has been dealt with extensively in texts on extreme value theory, e.g., Leadbetter et al.
(1983) or Resnick (1987). Here we make no attempt to present the general theory, but
the examples serve to illustrate the concepts in some of the most typical cases.
The exponential distribution
Suppose F(x) = 1 − e

u
= 1. Then
F
u
(σ
u
z) =
F(u + z) − F(u)
1 − F(u)
=
e
−u
− e
−u−z
e
−u
= 1 − e
−z
so the exponential distribution of mean 1 is the exact distribution for exceedances in
this case.
Pareto-type tail
Suppose 1 − F(x) ∼ cx
−α
as x →∞, with c > 0 and α>0. Let b
n
= 0, a
n
=
(nc)
1/α

z) =
F(u + ubz) − F(u)
1 − F(u)
≈
cu
−α
− c(u + ubz)
−α
cu
−α
= 1 − (1 + bz)
−α
.
Set ξ =
1
α
and b = ξ to get the result in GPD form.
©2004 CRC Press LLC
Finite upper endpoint
Suppose ω
F
= ω<∞ and 1 − F(ω − y) ∼ cy
α
as y ↓ 0 for c > 0,α > 0. Set
b
n
= ω, a
n
= (nc)
−1/α

→ exp{−(−x)
α
},
which is of Weibull type.
For the threshold version, let u be very close to ω and consider σ
u
= b(ω −u) for
b > 0tobedetermined. Then for 0 < z <
1
b
,
F
u
(σ
u
z) =
F(u + σ
u
z) − F(u)
1 − F(u)
≈
c(ω − u)
α
− c(ω − u − σ
u
z)
α
c(ω − u)
α
= 1 − (1 − bz)

lim
u→∞
1 − (u + z/u)
1 − (u)
= lim
u→∞


1 +
z
u
2

−1
· exp

−
1
2

u +
z
u

2
+
1
2
u
2

x + b
n
)
}
=
1 − 
(
a
n
x + b
n
)
1 − 
(
b
n
)
→ e
−x
©2004 CRC Press LLC
and hence
lim
n→∞

n
(
a
n
x + b
n

The main result is as follows: if Y
n,1
≥ Y
n,2
≥···≥Y
n,r
are r largest order
statistics of i.i.d. sample of size n, and a
n
and b
n
are the normalising constants in
(1.5), then

Y
n,1
− b
n
a
n
, ,
Y
n,r
− b
n
a
n

converges in distribution to a limiting random vector (X
1


1 + ξ
x
j
− µ
ψ

. (1.15)
Some examples using this approach are the papers of Smith (1986) and Tawn (1988) on
hydrological extremes, and Robinson and Tawn (1995) and Smith (1997) for a novel
application to the analysis of athletic records. The latter application is discussed in
Section 1.3.3.
1.2.5 Point process approach
This was introduced as a statistical approach by Smith (1989), though the basic
probability theory from which it derives had been developed by a number of earlier
©2004 CRC Press LLC
authors. In particular, the books by Leadbetter et al. (1983) and Resnick (1987) contain
much information on point-process viewpoints of extreme value theory.
In this approach, instead of considering the times at which high-threshold ex-
ceedances occur and the excess values over the threshold as two separate processes,
they are combined into one process based on a two-dimensional plot of exceedance
times and exceedance values. The asymptotic theory of threshold exceedances shows
that under suitable normalisation, this process behaves like a nonhomogeneous Pois-
son process.
In general, a nonhomogeneous Poisson process on a domain D is defined by an
intensity λ(x), x ∈D, such that if A is a measurable subset of D and N(A) denotes
the number of points in A, then N(A) has a Poisson distribution with mean
(A) =

A

(A) = (t
2
− t
1
)

1 + ξ
y − µ
ψ

−1/ξ
provided y ≥ u, 1 + ξ(y − µ)/ψ > 0. (1.17)
•
•
•
•
•
•
•
A
y
u
t_1 t_2 T
0
Figure 1.3 Illustration of point process approach. Assume the process is observed over a time
interval [0, T ], and that all observations above a threshold level u are recorded. These points
are marked on a two-dimensional scatterplot as shown in the diagram. For a set A of the form
shown in the figure, the count N(A) of observations in the set A is assumed to be Poisson with
mean of the form given by (1.17).
©2004 CRC Press LLC