Modelling Extremal Events: for Insurance and Finance

659 Pages·1997·10.947 MB·English

by  Paul Embrechts, Claudia Klüppelberg, Thomas Mikosch (auth.)

Stochastic Modelling
StochasticMechanics
RandomMedia
and Applied Probability
SignalProcessing
andImageSynthesis (Formerly:
MathematicalEconomicsandFinance ApplicationsofMathematics)
StochasticOptimization
StochasticControl
33
StochasticModelsinLifeSciences
Editedby B. Rozovski˘ı
P.W. Glynn
AdvisoryBoard M. Hairer
I. Karatzas
F.P. Kelly
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H.M. Soner
Forfurthervolumes:
http://www.springer.com/series/602
Paul Embrechts
Claudia Klüppelberg
Thomas Mikosch
Modelling
Extremal Events
for Insurance and Finance
PaulEmbrechts ThomasMikosch
DepartmentofMathematics DepartmentofMathematics
ETHZurich UniversityofCopenhagen
Zurich,Switzerland Copenhagen,Denmark
ClaudiaKlüppelberg
ZentrumMathematik
TechnischeUniversitätMünchen
Garching,Germany
ManagingEditors
BorisRozovski˘ı PeterW.Glynn
DivisionofAppliedMathematics InstituteofComputational
BrownUniversity andMathematicalEngineering
Providence,RI,USA StanfordUniversity
Stanford,CA,USA
ISSN0172-4568 StochasticModellingandAppliedProbability
ISBN978-3-540-60931-5 ISBN978-3-642-33483-2(eBook)
DOI10.1007/978-3-642-33483-2
SpringerHeidelbergNewYorkDordrechtLondon
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Preface
Qui scribunt libros caveant a judice multo
Cum multus judex talibus immineat
Abelard (1121)
Writers of books should beware of the verdict of the crowd
For the verdict of the crowd is prejudiced against them
Translation by Christopher Platt
Inarecentissue,TheNewScientistranacoverstoryunderthetitle:“Mission
improbable.Howtopredicttheunpredictable”;seeMatthews[448].Init,the
author describes a group of mathematicians who claim that extreme value
theory (EVT) is capable of doing just that: predicting the occurrence of rare
events, outside the range of available data. All members of this group, the
three of us included, would immediately react with: “Yes, but, ...”, or, “Be
aware ...”. Rather than at this point trying to explain what EVT can and
cannot do, we would like to quote two members of the group referred to in
[448]. Richard Smith said, “There is always going to be an element of doubt,
as one is extrapolating into areas one doesn’t know about. But what EVT
is doing is making the best use of whatever data you have about extreme
phenomena.” Quoting from Jonathan Tawn, “The key message is that EVT
cannot do magic – but it can do a whole lot better than empirical curve–
fittingandguesswork.Myanswertothescepticsisthatifpeoplearen’tgiven
well–founded methods like EVT, they’ll just use dubious ones instead.”
These two quotes set the scene for the book you are holding. Over many
yearswehavebeenincontactwithpotentialusersofEVT,suchasactuaries,
VIII Preface
risk managers, engineers, .... Whatever theory can or cannot predict about
extremal events, in practice the problems are there! As scientists, we cannot
duckthequestionoftheheightofaseadyketobebuiltinHolland,claiming
that this is an inadmissible problem because, to solve it, we would have to
extrapolate beyond the available data. Likewise, reinsurers have for a long
time known a great deal about extremal events; in their case, premiums
have to be set which both cover the insured in case of a claim, and also are
calculatedinsuchawaythatintheeventofacatastrophe,thecompanystays
solvent.Finally,recentdevelopmentsinthefinancialmarketscreateproducts
such as catastrophe–linked bonds where the repayment value is contingent
on the occurrence of some well–defined catastrophe. These and many more
examplesbenefitfromawell–establishedbodyoftheorywhichisnowreferred
to as EVT. Our book gives you an introduction to the mathematical and
statistical theory underlying EVT. It is written with a broad audience of
potential users in mind. From the subtitle however, it is clear that the main
target group is in the financial industry. A reason for this emphasis is that
the latter have been less exposed to EVT methodology. This is in contrast
to hydrologists and reliability engineers, for instance, where for a long time
EVT has belonged to the standard toolkit.
While our readership is expected to be broad, we do require a certain
mathematical level. Through the availability of standardised software, EVT
can be at the fingertips of many. However, a clear understanding of its capa-
bilities and limitations demands a fair amount of mathematical knowledge.
Basic courses in linear algebra, calculus, probability and statistics are essen-
tial. We have tried hard to keep the technical level minimal, stressing the
understanding of new concepts and results rather than their detailed discus-
sionsandproofs.Plentifulexamplesandfiguresshouldmaketheintroduction
of new methodology more digestible.
Thosewhohavenotimetoreadthebookfromcovertocover,andrather
wantafairlystreamlinedintroductiontoEVTinpractice,couldimmediately
startwithChapter6.DohoweverreadtheGuidelinesfirst.Fromtheapplied
techniques presented in Chapter 6, you will eventually discover relevant ma-
terial from other chapters.
A long list of references, together with numerous sections of Notes and
Commentsshouldguidethereadertoawealthofavailablematerial.Though
ourlistofreferencesislong,asalwaysitreflectsourimmediateinterest.Many
important papers which do not fit our presentation have been omitted. Even
in more than 600 pages, one cannot achieve completeness; the biggest gap is
doubtlessmultivariateextremevaluetheory.Thisisdefinitelyashortcoming!
We feel that mathematical theory has to go hand in hand with statistical
Preface IX
theory and computer software before it can safely be presented to the end–
user, but for the multivariate case, despite important recent progress, we
do not feel that the theory has reached a stage as well–established as the
one–dimensional one.
As with any major project, we owe thanks to lots of people. First of all,
there are those colleagues and friends who have helped us in ways which go
far beyond what normally can be hoped for. Charles Goldie was a constant
source of inspiration and help, both on mathematical issues, as well as on
stylistic ones. He realised early on that three authors who are not native
English speakers, when left alone, will produce a Flemish–Dutch–German–
Swiss version of the English language which is bound to bemuse many. In
his typical diplomatic manner, Charles constructed confidence bands around
proper English which he hoped we would not overstep too often. The fine
tuning and final decisions were of course always in our hand, hence also the
full responsibility for the final outcome.
Gabriele Baltes, Jutta Gonska and Sigrid Hoffmann made an art out of
producing numerous versions in LATEX of half readable manuscripts at var-
ious stages. They went far beyond the support expected from a secretary.
The many computer graphs in the book show only the tip of the iceberg.
For each one produced, numerous were proposed, discussed, altered, .... We
owe many thanks, also for various other support throughout the project,
to Franco Bassi, Klemens Binswanger, Milan Borkovec, Hansj¨org Furrer,
Natascha Jung, Anne Kampovsky, Alexander McNeil, Patricia Mu¨ller, An-
nette Scha¨rf and Alfred Sch¨ottl. For the software used we thank Alexander
McNeil, John Nolan and Richard Smith.
Many colleagues helped in proofreading parts of the book at various
stages: Gerd Christoph, Daryl Daley, Ru¨diger Frey, Jan Grandell, Maria
Kafetzakis, Marcin Kotulski, Frank Oertel, Sid Resnick, Chris Rogers, Gen-
nady Samorodnitsky, Hanspeter Schmidli and Josef Steinebach. Their crit-
ical remarks kept us on our toes! Obviously there has been an extensive
exchange with the finance industry as potential end–user, in the form of
informal discussions, seminars or lectures. Moreover, many were generous in
sharingtheirdatawithus.Wehopethatthefinaloutcomewillalsohelpthem
in their everyday handling of extremal events: Alois Gisler (Winterthur Ver-
sicherungen), Ren´e Held and Hans Fredo List (Swiss Reinsurance), Richard
Olsen (Olsen and Associates), Mette Rytgaard (Copenhagen Reinsurance)
and Wolfgang Schmidt (Deutsche Bank).
All three of us take pleasure in thanking our respective home institu-
tionsandcolleaguesfortheirmuchappreciatedsupport.Onecolleaguemeans
something special to all three of us: Hans Bu¨hlmann. His stimulating enthu-
X Preface
siasm for the beauty and importance of actuarial mathematics provided the
ideal environment for our project to grow. We have benefitted constantly
from his scholarly advice and warm friendship.
Thesubtitleofthebook“ForInsuranceandFinance”hintsatthepoten-
tial financial applications. The “real thing”, be it either Swiss Francs, Ger-
man Marks or Dutch Guilders, was provided to us through various forms of
support. Both the Forschungsinstitut fu¨r Mathematik (ETH) and the Math-
ematisches Forschungsinstitut Oberwolfach provided opportunities for face–
to–face meetings at critical stages.
PE recalls fondly the most stimulating visit he had, as part of his sab-
batical in the autumn of 1996, at the School of ORIE at Cornell University.
The splendid social and academicenvironment facilitatedthe successful con-
clusion of the book. CK worked on this project partly at ETH Zu¨rich and
partly at the Johannes Gutenberg University of Mainz. During most of the
time she spent on the book in Zu¨rich she was generously supported by the
Schweizerische Lebensversicherungs– und Rentenanstalt, the Schweizerische
Ru¨ckversicherungs–Gesellschaft(SwissRe),Winterthur–Versicherungen,and
the Union Ru¨ckversicherungs–Gesellschaft. Her sincere thanks go to these
companies. TM remembers with nostalgia his time in New Zealand where he
wrotehisfirstpartsofthebook.ThemoralsupportofhiscolleaguesatISOR
of the Victoria University of Wellington allowed him to concentrate fully on
writing. He gratefully acknowledges the financial support of a New Zealand
FRST Grant.
Last but not least, we thank our students! One of the great joys of being
an academic is being able to transfer scientific knowledge to young people.
Their questions, projects and interest made us feel we were on the right
track.Wehopethattheireagernesstolearnandenthusiasmtocommunicate
is felt throughout the pages of this book.
March, 1997 PE, Zu¨rich
CK, Mainz
TM, Groningen