Stochastic Mechanics Stochastic Modelling Random Media Signal Processing and Applied and Image Synthesis (Formerly: Mathematical and Finance Applications of ) Stochastic Optimization Stochastic Control Stochastic Models in Life Sciences 33 Edited by B. Rozovski˘ı P.W. Glynn

Advisory Board M. Hairer I. Karatzas F.P. Kelly A. Kyprianou Y. Le Jan B. Øksendal G. Papanicolaou E. Pardoux E. Perkins H.M. Soner For further volumes: http://www.springer.com/series/602 Paul Embrechts Claudia Klüppelberg Thomas Mikosch

Modelling Extremal Events for Insurance and Finance Paul Embrechts Thomas Mikosch Department of Mathematics Department of Mathematics ETH Zurich University of Copenhagen Zurich, Switzerland Copenhagen, Denmark Claudia Klüppelberg Zentrum Mathematik Technische Universität München Garching, Germany

Managing Editors Boris Rozovski˘ı Peter W. Glynn Division of Institute of Computational Brown University and Mathematical Providence, RI, USA Stanford University Stanford, CA, USA

ISSN 0172-4568 Stochastic Modelling and Applied Probability ISBN 978-3-642-08242-9 ISBN 978-3-642-33483-2 (eBook) DOI 10.1007/978-3-642-33483-2 Springer Heidelberg New York Dordrecht London

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Springer is part of Springer Science+Business Media (www.springer.com) Voor Gerda, Krispijn, Eline en Frederik. Na al dit werk blijft ´e´en vraag onbeantwoord: “Hoe kan ik jullie ooit danken voor de opoffering en steun?” Paul

Meinen Eltern Thomas 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

In a recent issue, The New Scientist ran a cover story under the title: “Mission improbable. How to predict the unpredictable”; see Matthews [448]. In it, 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– fitting and guesswork. My answer to the sceptics is that if people aren’t given 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 years we have been in contact with potential users of EVT, such as actuaries, VIII Preface risk managers, , .... Whatever theory can or cannot predict about extremal events, in practice the problems are there! As scientists, we cannot duck the question of the height of a sea dyke to be built in Holland, 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 calculated in such a way that in the event of a catastrophe, the company stays solvent. Finally, recent developments in the financial markets create products such as catastrophe–linked bonds where the repayment value is contingent on the occurrence of some well–defined catastrophe. These and many more examples benefit from a well–established body of theory which is now referred 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 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- sions and proofs. Plentiful examples and figures should make the introduction of new methodology more digestible. Those who have no time to read the book from cover to cover, and rather want a fairly streamlined introduction to EVT in practice, could immediately start with Chapter 6. Do however read the Guidelines first. From the applied 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 Comments should guide the reader to a wealth of available material. Though our list of references is long, as always it reflects our immediate interest. 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 doubtless multivariate extreme value theory. This is definitely a shortcoming! 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. Foreachoneproduced,numerouswereproposed,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 M¨uller, An- nette Sch¨arf 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, R¨udigerFrey, 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 sharing their data with us. We hope that the final outcome will also help them in their everyday handling of extremal events: Alois Gisler (Winterthur Ver- sicherungen), Ren´eHeld 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- tions and colleagues for their much appreciated support. One colleague means something special to all three of us: Hans B¨uhlmann.His stimulating enthu- XPreface 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. The subtitle of the book “For Insurance and Finance” hints at the poten- 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 f¨ur 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 academic environment facilitated the successful con- clusion of the book. CK worked on this project partly at ETH Z¨urich and partly at the Johannes Gutenberg University of Mainz. During most of the time she spent on the book in Z¨urich she was generously supported by the Schweizerische Lebensversicherungs– und Rentenanstalt, the Schweizerische R¨uckversicherungs–Gesellschaft (Swiss Re), Winterthur–Versicherungen, and the Union R¨uckversicherungs–Gesellschaft. Her sincere thanks go to these companies. TM remembers with nostalgia his time in New Zealand where he wrote his first parts of the book. The moral support of his colleagues at ISOR 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. We hope that their eagerness to learn and enthusiasm to communicate is felt throughout the pages of this book.

March, 1997 PE, Z¨urich CK, Mainz TM, Groningen Table of Contents

Reader Guidelines ...... 1

1RiskTheory...... 21 1.1 TheRuinProblem...... 22 1.2 The Cram´er–Lundberg Estimate ...... 28 1.3 RuinTheoryforHeavy–TailedDistributions...... 36 1.3.1 SomePreliminaryResults...... 37 1.3.2 Cram´er–Lundberg Theory for Subexponential Distributions ...... 39 1.3.3 The Total Claim Amount in the Subexponential Case . 44 1.4 Cram´er–Lundberg Theory for Large Claims: a Discussion .... 49 1.4.1 Some Related Classes of Heavy–Tailed Distributions . . 49 1.4.2 The Heavy–Tailed Cram´er–Lundberg Case Revisited . . 53

2 Fluctuations of Sums ...... 59 2.1 TheLawsofLargeNumbers...... 60 2.2 TheCentralLimitProblem...... 70 2.3 RefinementsoftheCLT...... 82 2.4 TheFunctionalCLT:BrownianMotionAppears...... 88 2.5 RandomSums...... 96 2.5.1 GeneralRandomlyIndexedSequences...... 96 2.5.2 RenewalCountingProcesses...... 103 2.5.3 Random Sums Driven by Renewal Counting Processes 106

3FluctuationsofMaxima ...... 113 3.1 Limit for Maxima ...... 114 3.2 Weak Convergence of Maxima Under Affine Transformations . 120 3.3 Maximum Domains of Attraction and Norming Constants . . . 128 3.3.1 The Maximum Domain of Attraction of the Fr´echet −α Distribution Φα(x)=exp{−x } ...... 130 XII Table of Contents

3.3.2 The Maximum Domain of Attraction of the Weibull α Distribution Ψα(x)=exp{−(−x) }...... 134 3.3.3 The Maximum Domain of Attraction of the Gumbel Distribution Λ(x)=exp{− exp{−x}} ...... 138 3.4 The Generalised Extreme Value Distribution and the GeneralisedParetoDistribution...... 152 3.5 AlmostSureBehaviourofMaxima...... 168

4 Fluctuations of Upper Order Statistics ...... 181 4.1 Order Statistics ...... 182 4.2 The Limit Distribution of Upper Order Statistics ...... 196 4.3 The Limit Distribution of Randomly Indexed Upper Order Statistics ...... 204 4.4 Some Extreme Value Theory for Stationary Sequences ...... 209

5 An Approach to Extremes via Point Processes ...... 219 5.1 Basic Facts About Point Processes ...... 220 5.1.1 DefinitionandExamples...... 220 5.1.2 DistributionandLaplaceFunctional...... 225 5.1.3 PoissonRandomMeasures...... 226 5.2 WeakConvergenceofPointProcesses...... 232 5.3 Point Processes of Exceedances ...... 237 5.3.1 TheIIDCase...... 238 5.3.2 The Stationary Case ...... 242 5.4 Applications of Point Process Methods to IID Sequences . . . . 247 5.4.1 Records and Record Times ...... 248 5.4.2 EmbeddingMaximainExtremalProcesses...... 250 5.4.3 The Frequency of Records and the Growth of Record Times ...... 254 5.4.4 InvariancePrincipleforMaxima...... 260 5.5 SomeExtremeValueTheoryforLinearProcesses...... 263 5.5.1 Noise in the Maximum Domain of Attraction of the Fr´echet Distribution Φα ...... 264 5.5.2 Subexponential Noise in the Maximum Domain of Attraction of the Gumbel Distribution Λ ...... 277

6 Statistical Methods for Extremal Events ...... 283 6.1 Introduction...... 283 6.2 Exploratory Data Analysis for Extremes ...... 290 6.2.1 Probability and Quantile Plots ...... 290 6.2.2 TheMeanExcessFunction...... 294 Table of Contents XIII

6.2.3 Gumbel’s Method of Exceedances ...... 303 6.2.4 TheReturnPeriod ...... 305 6.2.5 Records as an Exploratory Tool ...... 307 6.2.6 TheRatioofMaximumandSum...... 309 6.3 Parameter Estimation for the Generalised Extreme ValueDistribution...... 316 6.3.1 MaximumLikelihoodEstimation...... 317 6.3.2 Method of Probability–Weighted Moments...... 321 6.3.3 TailandQuantileEstimation,aFirstGo...... 323 6.4 Estimating Under Maximum Domain of Attraction Conditions ...... 325 6.4.1 Introduction...... 325 6.4.2 Estimating the Shape Parameter ξ ...... 327 6.4.3 EstimatingtheNormingConstants...... 345 6.4.4 TailandQuantileEstimation...... 348 6.5 FittingExcessesOveraThreshold...... 352 6.5.1 FittingtheGPD...... 352 6.5.2 An Application to Real Data ...... 358

7 Time Series Analysis for Heavy–Tailed Processes ...... 371 7.1 A Short Introduction to Classical Time Series Analysis ...... 372 7.2 Heavy–TailedTimeSeries...... 378 7.3 EstimationoftheAutocorrelationFunction...... 381 7.4 EstimationofthePowerTransferFunction ...... 386 7.5 Parameter Estimation for ARMA Processes ...... 393 7.6 Some Remarks About Non–Linear Heavy–Tailed Models. . . . . 403

8 Special Topics ...... 413 8.1 TheExtremalIndex...... 413 8.1.1 DefinitionandElementaryProperties...... 413 8.1.2 Interpretation and Estimation of the Extremal Index . 418 8.1.3 EstimatingtheExtremalIndexfromData...... 424 8.2 ALargeClaimIndex ...... 430 8.2.1 TheProblem ...... 430 8.2.2 TheIndex...... 431 8.2.3 SomeExamples...... 433 8.2.4 OnSumsandExtremes...... 436 8.3 WhenandHowRuinOccurs...... 439 8.3.1 Introduction...... 439 8.3.2 The Cram´er–Lundberg Case ...... 444 8.3.3 TheLargeClaimCase...... 449 XIV Table of Contents

8.4 PerpetuitiesandARCHProcesses...... 454 8.4.1 Stochastic Recurrence Equations and Perpetuities . . . . 455 8.4.2 BasicPropertiesofARCHProcesses...... 461 8.4.3 ExtremesofARCHProcesses...... 473 8.5 OntheLongestSuccess–Run...... 481 8.5.1 The Total Variation Distance to a PoissonDistribution...... 483 8.5.2 TheAlmostSureBehaviour ...... 486 8.5.3 TheDistributionalBehaviour...... 493 8.6 SomeResultsonLargeDeviations...... 498 8.7 Reinsurance Treaties ...... 503 8.7.1 Introduction...... 503 8.7.2 Probabilistic Analysis ...... 507 8.8 StableProcesses...... 521 8.8.1 Stable Random Vectors ...... 522 8.8.2 SymmetricStableProcesses ...... 526 8.8.3 Stable Integrals ...... 527 8.8.4 Examples ...... 532 8.9 Self–Similarity...... 541

Appendix ...... 551 A1 ModesofConvergence ...... 551 A1.1 ConvergenceinDistribution...... 551 A1.2 Convergence in Probability ...... 552 A1.3 AlmostSureConvergence ...... 553 A1.4 Lp–Convergence...... 553 A1.5 ConvergencetoTypes...... 554 A1.6 ConvergenceofGeneralisedInverseFunctions...... 554 A2 Weak Convergence in Metric Spaces ...... 555 A2.1 PreliminariesaboutStochasticProcesses...... 555 A2.2 The Spaces C [0, 1] and D [0, 1]...... 557 A2.3 The Skorokhod Space D (0, ∞) ...... 559 A2.4 WeakConvergence ...... 559 A2.5 TheContinuousMappingTheorem...... 561 A2.6 WeakConvergenceofPointProcesses...... 562 A3 RegularVariationandSubexponentiality...... 564 A3.1 BasicResultsonRegularVariation...... 564 A3.2 PropertiesofSubexponentialDistributions ...... 571 A3.3 The Tail Behaviour of Weighted Sums ofHeavy–TailedRandomVariables...... 583 A4 SomeRenewalTheory ...... 587 Table of Contents XV

References ...... 591

Index ...... 627

List of Abbreviations and Symbols ...... 643