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The Quantitative Risk ManagementExercise BookMarius HofertRüdiger FreyAlexander J. McNeil

The Quantitative Risk ManagementExercise BookMarius HofertRüdiger FreyAlexander J. McNeilPrinceton University PressPrinceton and Oxford

Copyright 2020 by Marius Hofert, Rüdiger Frey, Alexander J. McNeilRequests for permission to reproduce material from this workshould be sent to [email protected] by Princeton University Press41 William Street, Princeton, New Jersey 085406 Oxford Street, Woodstock, Oxfordshire OX20 1TRpress.princeton.eduAll Rights ReservedISBN 9780691206707The publisher would like to acknowledge the authors for providing this manual in PDF form

To SaisaiTo Catharina, Sebastian and MichaelaTo Janine, Alexander and Calliope

PrefaceOver the past fifteen years we have taught many courses on Quantitative Risk Management(QRM) to students, academics, financial practitioners and actuaries based on the textbookQuantitative Risk Management: Concepts, Techniques and Tools (McNeil, Frey and Embrechts;2005, 2015). Henceforth we refer to this book as the QRM textbook and use the abbreviatedcitations MFE (2005) and MFE (2015) for the two editions.We have used the QRM textbook to deliver lecture courses to students at our own academicinstitutions including ETH Zurich, Heriot-Watt University, the University of Leipzig, theVienna University of Economics and Business (WU), the University of Washington (UW), theNational University of Singapore (NUS) and the University of Waterloo. We are also awarethat many colleagues have used the QRM textbook to teach courses at universities around theworld.The QRM textbook has also formed the basis for numerous intensive training courses,workshops and summer schools which have been delivered to financial practitioners andactuaries. In particular, it has been used as part of the CERA (Chartered Enterprise RiskActuary) education programme of the European Actuarial Academy (EAA).Throughout this time, we have been conscious that the QRM textbook has lacked a formalexercise collection to assist instructors and to provide students with the opportunity to test,consolidate and extend their knowledge. The QRM Exercise Book aims to fill this gap. Itcontains a comprehensive selection of the exercises that we have amassed during our ownteaching experience, as well as a few exercises donated by colleagues.The structure of the exercise book follows Chapters 1–11 of MFE (2015). Within chapters,exercises are typically grouped into three categories: review, basic and advanced questions.Review questions are designed to help readers to test their understanding of the conceptspresented in the corresponding chapter of MFE (2015). As such, the solutions are usuallyfound in MFE (2015) or involve at most simple calculations based on material in MFE (2015).Basic questions are designed to test and enhance the understanding of the technical material.Typical examples might require students to calculate specific examples based on generaltechniques or might ask for simple proofs and minor extensions of results in MFE (2015). Takentogether, review and basic questions are intended to be at the level of examination questionsand should not require particularly long answers. Advanced exercises are either more lengthy,more difficult or take the reader somewhat beyond the textbook material.Occasionally, a programming exercise is included. We recommend that these are tackledusing the open source R language and environment for statistical computing. To facilitate thesolution of programming exercises we provide references to publicly available R datasets andpackages in the questions.In view of the more qualitative and discursive nature of Chapter 1 of the QRM textbook,5

Prefacethe exercises and solutions are structured in a slightly different way. Case study and discussionquestions take the place of basic and advanced questions respectively. Case study questionsinvite students to research a number of well-known company failures, market crashes andnatural catastrophes and consider their causes and/or their financial and economic implications.Discussion questions are more open-ended questions that address, for example, current trendsand controversies in risk management.The primary resource for solving these exercises is MFE (2015) although the exercise bookalso contains references to other books and papers that take the reader further in certain areas.An updated book of solutions to the exercises and a set of R scripts solving the programmingexercises are available on the QRM Tutorial website qrmtutorial.org. For instructors thereare some additional resources on the website including slides and extensive demonstrationscripts in R.We are grateful to Paul Embrechts who initiated the QRM project during his time as Professorof Insurance Mathematics at ETH Zurich. We have all benefited from his inspiration andmentoring over the years. We would also like to thank the Forschungsinstitut für Mathematik(FIM) at ETH Zurich for its financial support during multiple visits by the authors. Thanks arealso due to several doctoral and postdoctoral teaching assistants in Zurich who helped developthe stock of examples and exercises for teaching QRM, including Valeria Bignozzi, BikramjitDas, Catherine Donnelly, Edgars Jakobsons, Georg Mainik and Johanna G. Nešlehová, as wellas to colleagues, friends and students around the world who have contributed exercises andsolutions, in particular, Andrew Cairns and Jialing Han.6

ContentsPreface51 Risk in PerspectiveReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9910102 Basic ConceptsReview . . . . .Basic . . . . . .Advanced . . .13131418in Risk Management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Empirical PropertiesReview . . . . . . . .Basic . . . . . . . . .Advanced . . . . . .of Financial Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21212225Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272729325 Extreme Value TheoryReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353536406 Multivariate ModelsReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .434344497 Copulas and DependenceReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515152584 Financial TimeReview . . . . .Basic . . . . . .Advanced . . .7

Contents8 Aggregate RiskReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .616163649 Market RiskReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6969707310 Credit RiskReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7777788111 Portfolio Credit Risk ManagementReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85858691References938

1 Risk in PerspectiveReviewExercise 1.1 (Types of financial risk)Give definitions and examples for each of the following types of risk in the financial context:a) market risk,b) credit risk,c) operational risk,d) model risk, ande) systemic risk.Exercise 1.2 (Financial crisis of 2007–9)Give an outline of the financial crisis of 2007–9 describing the main events and reasons for thecrisis.Exercise 1.3 (Bank run)Explain what is meant by a bank run and give examples from history.Exercise 1.4 (Regulatory frameworks)a) Explain how the three-pillar concept works in the Basel and Solvency II regulatory frameworks.b) Compare and contrast the main features of the Basel and Solvency II frameworks.Exercise 1.5 (Procyclicality)Explain the meaning of procyclicality, particularly when used in relation to regulatory capitalrequirements.Exercise 1.6 (Basel III)Summarize the main changes to the Basel framework that are adopted in Basel III and explainhow these address shortcomings of the previous Basel II Accord.Exercise 1.7 (Capital adequacy, leverage and liquidity coverage ratios)Explain the essential differences between capital adequacy ratios, leverage ratios and liquiditycoverage ratios as applied in the Basel III accord.9

1 Risk in PerspectiveCase StudiesExercise 1.8 (Major failures)Select from the following list of well-known failures of financial risk management. In each case,summarize what happened and identify the risks that the company or organisation failed tomanage.a) Barings Bankb) Metallgesellschaft (MG)c) Sumitomod) Société Généralee) Orange Countyf) Long-Term Capital Management (LTCM)g) American International Group (AIG)h) Amaranth AdvisorsExercise 1.9 (Market crashes)Investigate what happened during the following well-known market crashes and describe anychanges that have resulted from these events, or any lessons that risk managers should draw.a) The Wall Street Crash of 1929.b) The ‘Black Monday’ event of 19 October 1987.c) The ‘Flash Crash’ of 6 May 2010.Exercise 1.10 (Natural catastrophes)Research the following natural catastrophes and describe their economic, financial and insuranceconsequences.a) The Kobe Earthquake (1995).b) Hurricane Katrina (2005).c) The Thailand Floods (2011).DiscussionExercise 1.11 (Risk and uncertainty)Attempt to formulate your own succinct definition of risk. What distinguishes risk fromuncertainty in your opinion?Exercise 1.12 (The Q in QRM)Discuss whether quantitative methodology should play a larger or a more reduced role in riskmanagement systems in the future.Exercise 1.13 (Trends in financial regulation)What are the current trends in financial regulation? In particular:10

Discussiona) Has risk regulation become too complex?b) Is there evidence of a movement away from internal models back to simpler standardizedapproaches?Exercise 1.14 (Regulation and credit provision)An often-heard criticism of Basel III by banking practitioners is that “increased regulationstrangles credit provision”. Discuss the arguments for and against this proposition.Exercise 1.15 (Shadow banking and insurance)a) What is meant by the shadow banking industry?b) Is there also a shadow insurance industry?11

2 Basic Concepts in Risk ManagementReviewExercise 2.1 (Notions of capital)In each of the following situations a certain notion of capital discussed in MFE (2015, Section 2.1.3) is most relevant for the decision maker. Explain which notion of capital thatis.a) A financial analyst who uses balance-sheet data to value a firm.b) A chief risk officer of an insurance company who has to decide on the appropriate level ofreinsurance.c) A regulator who has to decide on shutting down a bank with many bad loans on its book.Exercise 2.2 (Different notions of financial distress)a) Briefly explain the difference between illiquidity, insolvency, default and bankruptcy.b) Describe a scenario where a financial company is insolvent but has not defaulted.c) Describe a scenario where a financial company has defaulted but is not insolvent.Exercise 2.3 (Valuation of a real-estate investment)Suppose the manager of a real-estate fund has to value a particular flat in Zurich, Switzerland.Relate the following three methods for the valuation of the flat to the valuation methodsdiscussed in MFE (2015, Section 2.2.2).a) The manager takes the purchase price of the flat from several years ago and reduces it byan annual depreciation of 1% to allow for wear and tear.b) The manager finds transaction prices for similar flats in the neighborhood and uses them tocompute a price per square metre. She then multiplies the square-metre price by the size ofthe flat.c) The manager estimates prices per square meter from a broad Swiss property price indexand makes an ad hoc adjustment of 20% to account for the location of the flat in Zurich.Exercise 2.4 (Translation invariance of risk measures)Show that a translation-invariant risk measure can be interpreted as the amount of capitalthat needs to be added to a position so that it becomes acceptable to a regulator.Exercise 2.5 (Subadditivity of risk measures)Explain why subadditivity is often considered a desirable property of a risk measure.13

2 Basic Concepts in Risk ManagementExercise 2.6 (VaR and expected shortfall)a) Give mathematically precise definitions of value-at-risk VaRα (L) and expected shortfallESα (L) for a random loss L at confidence level α (0, 1).b) Explain the relative advantages of each risk measure over the other.Exercise 2.7 (Superadditivity scenarios for VaR)Describe some models for financial losses that can lead to situations where VaRα is superadditive.Exercise 2.8 (Additivity for two linearly dependent random variables)Consider an arbitrary random variable X and let Y aX b for constants a 0 and b R.Show that VaRα (X Y ) VaRα (X) VaRα (Y ) for α (0, 1).BasicExercise 2.9 (Risk-neutral valuation for interest-rate derivatives)Consider a two-period model. Denote by rt , t {0, 1}, the simple inte

The Quantitative Risk Management . Exercise Book . Marius Hofert . Rüdiger Frey . Alexander J. McNeil