Financial Risk Manager Handbook Second Editiondownload.e-bookshelf.de/download/0000/5840/82/L-G-0000584082... · Financial Risk Manager Handbook Second Edition Wiley John Wiley &

Financial Risk ManagerHandbook

Second Edition

Wiley

John Wiley & Sons, Inc.

Philippe Jorion

GARP

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Second Edition

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Second Edition

Wiley

John Wiley & Sons, Inc.

Philippe Jorion

GARP

Library of Congress Cataloging-in-Publication Data:

Copyright 2003 by Philippe Jorion, except for FRM sample questions, which arecopyright 19972001 by GARP. The FRM designation is a GARP trademark. All rightsreserved.

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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http://www.copyright.com

Philippe Jorion

Global Association of Risk Professionals

is Professor of Finance at the Graduate School of Management at the

University of California at Irvine. He has also taught at Columbia University, North-

western University, the University of Chicago, and the University of British Columbia.

He holds an M.B.A. and a Ph.D. from the University of Chicago and a degree in engi-

neering from the University of Brussels.

Dr. Jorion has authored more than seventy publications directed to academics

and practitioners on the topics of risk management and international finance. Dr.

Jorion has written a number of books, including

the first account of the largest municipal failure in U.S.

history, and which is

aimed at finance practitioners and has become an industry standard.

Philippe Jorion is a frequent speaker at academic and professional conferences.

He is on the editorial board of a number of finance journals and is editor in chief of

the .

The (GARP), established in 1996, is a not-

for-profit independent association of risk management practitioners and researchers.

Its members represent banks, investment management firms, governmental bodies,

academic institutions, corporations, and other financial organizations from all over

the world.

GARPs mission, as adopted by its Board of Trustees in a statement issued in Febru-

ary 2003, is to be the leading professional association for risk managers, managed by

and for its members dedicated to the advancement of the risk profession through

education, training and the promotion of best practices globally.

In just seven years the Associations membership has grown to over 27,000 indi-

viduals from around the world. In the just six years since its inception in 1997, the

FRM program has become the worlds most prestigious financial risk management

certification program. Professional risk managers having earned the FRM credential

are globally recognized as having achieved a minimum level of professional compe-

tency along with a demonstrated ability to dynamically measure and manage financial

risk in a real-world setting in accord with global standards. Further information about

GARP, the FRM Exam, and FRM readings are available at .

v

Big Bets Gone Bad: Derivatives and

Bankruptcy in Orange County,

Value at Risk: The New Benchmark for Managing Financial Risk,

Journal of Risk

www.garp.com

About the Author

About GARP

Contents

Preface xix

Introduction xxi

Ch. 1 Bond Fundamentals 3

Ch. 2 Fundamentals of Probability 31

Part I: Quantitative Analysis 1

vii

1.1 Discounting, Present, and Future Value . . . . . . . . . . . . 31.2 Price-Yield Relationship . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Valuation . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Taylor Expansion . . . . . . . . . . . . . . . . . . . . 71.2.3 Bond Price Derivatives . . . . . . . . . . . . . . . . . 91.2.4 Interpreting Duration and Convexity . . . . . . . . . . 161.2.5 Portfolio Duration and Convexity . . . . . . . . . . . . 23

1.3 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 26

2.1 Characterizing Random Variables . . . . . . . . . . . . . . . 312.1.1 Univariate Distribution Functions . . . . . . . . . . . 322.1.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Multivariate Distribution Functions . . . . . . . . . . . . . . 372.3 Functions of Random Variables . . . . . . . . . . . . . . . . 40

2.3.1 Linear Transformation of Random Variables . . . . . . 412.3.2 Sum of Random Variables . . . . . . . . . . . . . . . 422.3.3 Portfolios of Random Variables . . . . . . . . . . . . . 422.3.4 Product of Random Variables . . . . . . . . . . . . . . 432.3.5 Distributions of Transformations of Random Variables 44

2.4 Important Distribution Functions . . . . . . . . . . . . . . . 462.4.1 Uniform Distribution . . . . . . . . . . . . . . . . . . 462.4.2 Normal Distribution . . . . . . . . . . . . . . . . . . . 472.4.3 Lognormal Distribution . . . . . . . . . . . . . . . . . 512.4.4 Students Distribution . . . . . . . . . . . . . . . . . 542.4.5 Binomial Distribution . . . . . . . . . . . . . . . . . . 56


t

Ch. 3 Fundamentals of Statistics 63

Ch. 4 Monte Carlo Methods 83

Ch. 5 Introduction to Derivatives 105

Part II: Capital Markets 103

viii

3.1 Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1.1 Measuring Returns . . . . . . . . . . . . . . . . . . . 643.1.2 Time Aggregation . . . . . . . . . . . . . . . . . . . . 653.1.3 Portfolio Aggregation . . . . . . . . . . . . . . . . . . 66

3.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 693.3 Regression Analysis . . . . . . . . . . . . . . . . . . . . . . 71

3.3.1 Bivariate Regression . . . . . . . . . . . . . . . . . . 723.3.2 Autoregression . . . . . . . . . . . . . . . . . . . . . 743.3.3 Multivariate Regression . . . . . . . . . . . . . . . . . 743.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . 753.3.5 Pitfalls with Regressions . . . . . . . . . . . . . . . . 77


4.1 Simulations with One Random Variable . . . . . . . . . . . . 834.1.1 Simulating Markov Processes . . . . . . . . . . . . . . 844.1.2 The Geometric Brownian Motion . . . . . . . . . . . . 844.1.3 Simulating Yields . . . . . . . . . . . . . . . . . . . . 884.1.4 Binomial Trees . . . . . . . . . . . . . . . . . . . . . 89

4.2 Implementing Simulations . . . . . . . . . . . . . . . . . . . 934.2.1 Simulation for VAR . . . . . . . . . . . . . . . . . . . 934.2.2 Simulation for Derivatives . . . . . . . . . . . . . . . 934.2.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Multiple Sources of Risk . . . . . . . . . . . . . . . . . . . . 964.3.1 The Cholesky Factorization . . . . . . . . . . . . . . . 97


5.1 Overview of Derivatives Markets . . . . . . . . . . . . . . . . 1055.2 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . 107

5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.2 Valuing Forward Contracts . . . . . . . . . . . . . . . 1105.2.3 Valuing an Off-Market Forward Contract . . . . . . . . 1125.2.4 Valuing Forward Contracts with Income Payments . . . 113

5.3 Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.1 Definitions of Futures . . . . . . . . . . . . . . . . . . 1175.3.2 Valuing Futures Contracts . . . . . . . . . . . . . . . 119

5.4 Swap Contracts . . . . . . . . . . . . . . . . . . . . . . . . . 1195.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 120

CONTENTS

Financial Risk Manager Handbook, Second Edition

Ch. 6 Options 123

Ch. 7 Fixed-Income Securities 153

Ch. 8 Fixed-Income Derivatives 187

ix

6.1 Option Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . 1236.1.1 Basic Options . . . . . . . . . . . . . . . . . . . . . . 1236.1.2 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . 1266.1.3 Combination of Options . . . . . . . . . . . . . . . . 128

6.2 Valuing Options . . . . . . . . . . . . . . . . . . . . . . . . 1326.2.1 Option Premiums . . . . . . . . . . . . . . . . . . . . 1326.2.2 Early Exercise of Options . . . . . . . . . . . . . . . . 1346.2.3 Black-Scholes Valuation . . . . . . . . . . . . . . . . . 1366.2.4 Market vs. Model Prices . . . . . . . . . . . . . . . . . 142

6.3 Other Option Contracts . . . . . . . . . . . . . . . . . . . . . 1436.4 Valuing Options by Numerical Methods . . . . . . . . . . . . 1466.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 149

7.1 Overview of Debt Markets . . . . . . . . . . . . . . . . . . . 1537.2 Fixed-Income Securities . . . . . . . . . . . . . . . . . . . . . 156

7.2.1 Instrument Types . . . . . . . . . . . . . . . . . . . . 1567.2.2 Methods of Quotation . . . . . . . . . . . . . . . . . . 158

7.3 Analysis of Fixed-Income Securities . . . . . . . . . . . . . . 1607.3.1 The NPV Approach . . . . . . . . . . . . . . . . . . . 1607.3.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.4 Spot and Forward Rates . . . . . . . . . . . . . . . . . . . . 1657.5 Mortgage-Backed Securities . . . . . . . . . . . . . . . . . . . 170

7.5.1 Description . . . . . . . . . . . . . . . . . . . . . . . 1707.5.2 Prepayment Risk . . . . . . . . . . . . . . . . . . . . 1747.5.3 Financial Engineering and CMOs . . . . . . . . . . . . 177


8.1 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . 1878.2 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.2.1 Eurodollar Futures . . . . . . . . . . . . . . . . . . . 1908.2.2 T-bond Futures . . . . . . . . . . . . . . . . . . . . . 193

8.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 1958.3.2 Quotations . . . . . . . . . . . . . . . . . . . . . . . 1978.3.3 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.4.1 Caps and Floors . . . . . . . . . . . . . . . . . . . . . 2028.4.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . 2048.4.3 Exchange-Traded Options . . . . . . . . . . . . . . . . 206


CONTENTS


Ch. 9 Equity Markets 211

Ch. 10 Currencies and Commodities Markets 225

Ch. 11 Introduction to Market Risk Measurement 243

Part III: Market Risk Management 241

x

9.1 Equities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 2119.1.2 Valuation . . . . . . . . . . . . . . . . . . . . . . . . 2139.1.3 Equity Indices . . . . . . . . . . . . . . . . . . . . . . 214

9.2 Convertible Bonds and Warrants . . . . . . . . . . . . . . . . 2159.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 2159.2.2 Valuation . . . . . . . . . . . . . . . . . . . . . . . . 217

9.3 Equity Derivatives . . . . . . . . . . . . . . . . . . . . . . . 2199.3.1 Stock Index Futures . . . . . . . . . . . . . . . . . . . 2199.3.2 Single Stock Futures . . . . . . . . . . . . . . . . . . 2229.3.3 Equity Options . . . . . . . . . . . . . . . . . . . . . 2239.3.4 Equity Swaps . . . . . . . . . . . . . . . . . . . . . . 223


10.1 Currency Markets . . . . . . . . . . . . . . . . . . . . . . . . 22510.2 Currency Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 227

10.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 22710.2.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10.3 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.3.1 Products . . . . . . . . . . . . . . . . . . . . . . . . . 23110.3.2 Pricing of Futures . . . . . . . . . . . . . . . . . . . . 23210.3.3 Futures and Expected Spot Prices . . . . . . . . . . . . 235


11.1 Introduction to Financial Market Risks . . . . . . . . . . . . . 24311.2 VAR as Downside Risk . . . . . . . . . . . . . . . . . . . . . 246

11.2.1 VAR: Definition . . . . . . . . . . . . . . . . . . . . . 24611.2.2 VAR: Caveats . . . . . . . . . . . . . . . . . . . . . . 24911.2.3 Alternative Measures of Risk . . . . . . . . . . . . . . 249

11.3 VAR: Parameters . . . . . . . . . . . . . . . . . . . . . . . . 25211.3.1 Confidence Level . . . . . . . . . . . . . . . . . . . . 25211.3.2 Horizon . . . . . . . . . . . . . . . . . . . . . . . . . 25311.3.3 Application: The Basel Rules . . . . . . . . . . . . . . 255

11.4 Elements of VAR Systems . . . . . . . . . . . . . . . . . . . 25611.4.1 Portfolio Positions . . . . . . . . . . . . . . . . . . . 25711.4.2 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . 25711.4.3 VAR Methods . . . . . . . . . . . . . . . . . . . . . . 257

CONTENTS


Ch. 12 Identification of Risk Factors 265

Ch. 13 Sources of Risk 281

xi

11.5 Stress-Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 25811.6 Cash Flow at Risk . . . . . . . . . . . . . . . . . . . . . . . . 26011.7 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 261

12.1 Market Risks . . . . . . . . . . . . . . . . . . . . . . . . . . 26512.1.1 Absolute and Relative Risk . . . . . . . . . . . . . . . 26512.1.2 Directional and Nondirectional Risk . . . . . . . . . . 26712.1.3 Market vs. Credit Risk . . . . . . . . . . . . . . . . . . 26812.1.4 Risk Interaction . . . . . . . . . . . . . . . . . . . . . 268

12.2 Sources of Loss: A Decomposition . . . . . . . . . . . . . . . 26912.2.1 Exposure and Uncertainty . . . . . . . . . . . . . . . 26912.2.2 Specific Risk . . . . . . . . . . . . . . . . . . . . . . . 270

12.3 Discontinuity and Event Risk . . . . . . . . . . . . . . . . . . 27112.3.1 Continuous Processes . . . . . . . . . . . . . . . . . . 27112.3.2 Jump Process . . . . . . . . . . . . . . . . . . . . . . 27212.3.3 Event Risk . . . . . . . . . . . . . . . . . . . . . . . . 273

12.4 Liquidity Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 27512.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 278

13.1 Currency Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 28113.1.1 Currency Volatility . . . . . . . . . . . . . . . . . . . 28213.1.2 Correlations . . . . . . . . . . . . . . . . . . . . . . . 28313.1.3 Devaluation Risk . . . . . . . . . . . . . . . . . . . . 28313.1.4 Cross-Rate Volatility . . . . . . . . . . . . . . . . . . 284

13.2 Fixed-Income Risk . . . . . . . . . . . . . . . . . . . . . . . 28513.2.1 Factors Affecting Yields . . . . . . . . . . . . . . . . . 28513.2.2 Bond Price and Yield Volatility . . . . . . . . . . . . . 28713.2.3 Correlations . . . . . . . . . . . . . . . . . . . . . . . 29013.2.4 Global Interest Rate Risk . . . . . . . . . . . . . . . . 29213.2.5 Real Yield Risk . . . . . . . . . . . . . . . . . . . . . 29313.2.6 Credit Spread Risk . . . . . . . . . . . . . . . . . . . 29413.2.7 Prepayment Risk . . . . . . . . . . . . . . . . . . . . 294

13.3 Equity Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 29613.3.1 Stock Market Volatility . . . . . . . . . . . . . . . . . 29613.3.2 Forwards and Futures . . . . . . . . . . . . . . . . . . 298

13.4 Commodity Risk . . . . . . . . . . . . . . . . . . . . . . . . 29813.4.1 Commodity Volatility Risk . . . . . . . . . . . . . . . 29813.4.2 Forwards and Futures . . . . . . . . . . . . . . . . . . 30013.4.3 Delivery and Liquidity Risk . . . . . . . . . . . . . . . 301

CONTENTS


Ch. 14 Hedging Linear Risk 311

Ch. 15 Nonlinear Risk: Options 329

Ch. 16 Modeling Risk Factors 355

xii

13.5 Risk Simplification . . . . . . . . . . . . . . . . . . . . . . . 30213.5.1 Diagonal Model . . . . . . . . . . . . . . . . . . . . . 30213.5.2 Factor Models . . . . . . . . . . . . . . . . . . . . . . 30513.5.3 Fixed-Income Portfolio Risk . . . . . . . . . . . . . . . 306


14.1 Introduction to Futures Hedging . . . . . . . . . . . . . . . . 31214.1.1 Unitary Hedging . . . . . . . . . . . . . . . . . . . . . 31214.1.2 Basis Risk . . . . . . . . . . . . . . . . . . . . . . . . 313

14.2 Optimal Hedging . . . . . . . . . . . . . . . . . . . . . . . . 31514.2.1 The Optimal Hedge Ratio . . . . . . . . . . . . . . . . 31614.2.2 The Hedge Ratio as Regression Coefficient . . . . . . . 31714.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . 31914.2.4 Liquidity Issues . . . . . . . . . . . . . . . . . . . . . 321

14.3 Applications of Optimal Hedging . . . . . . . . . . . . . . . 32114.3.1 Duration Hedging . . . . . . . . . . . . . . . . . . . . 32214.3.2 Beta Hedging . . . . . . . . . . . . . . . . . . . . . . 324


15.1 Evaluating Options . . . . . . . . . . . . . . . . . . . . . . . 33015.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 33015.1.2 Taylor Expansion . . . . . . . . . . . . . . . . . . . . 33115.1.3 Option Pricing . . . . . . . . . . . . . . . . . . . . . . 332

15.2 Option Greeks . . . . . . . . . . . . . . . . . . . . . . . . 33315.2.1 Option Sensitivities: Delta and Gamma . . . . . . . . . 33315.2.2 Option Sensitivities: Vega . . . . . . . . . . . . . . . . 33715.2.3 Option Sensitivities: Rho . . . . . . . . . . . . . . . . 33915.2.4 Option Sensitivities: Theta . . . . . . . . . . . . . . . 33915.2.5 Option Pricing and the Greeks . . . . . . . . . . . . 34015.2.6 Option Sensitivities: Summary . . . . . . . . . . . . . 342

15.3 Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . 34615.3.1 Delta and Dynamic Hedging . . . . . . . . . . . . . . 34615.3.2 Implications . . . . . . . . . . . . . . . . . . . . . . . 34715.3.3 Distribution of Option Payoffs . . . . . . . . . . . . . 348


16.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . 35516.1.1 Why the Normal? . . . . . . . . . . . . . . . . . . . . 355

CONTENTS


Ch. 17 VAR Methods 371

Ch. 18 Introduction to Credit Risk 393

Part IV: Credit Risk Management 391

xiii

16.1.2 Computing Returns . . . . . . . . . . . . . . . . . . . 35616.1.3 Time Aggregation . . . . . . . . . . . . . . . . . . . . 358

16.2 Fat Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36116.3 Time-Variation in Risk . . . . . . . . . . . . . . . . . . . . . 363

16.3.1 GARCH . . . . . . . . . . . . . . . . . . . . . . . . . 36316.3.2 EWMA . . . . . . . . . . . . . . . . . . . . . . . . . . 36516.3.3 Option Data . . . . . . . . . . . . . . . . . . . . . . . 36716.3.4 Implied Distributions . . . . . . . . . . . . . . . . . . 368


17.1 VAR: Local vs. Full Valuation . . . . . . . . . . . . . . . . . . 37217.1.1 Local Valuation . . . . . . . . . . . . . . . . . . . . . 37217.1.2 Full Valuation . . . . . . . . . . . . . . . . . . . . . . 37317.1.3 Delta-Gamma Method . . . . . . . . . . . . . . . . . . 374

17.2 VAR Methods: Overview . . . . . . . . . . . . . . . . . . . . 37617.2.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 37617.2.2 Delta-Normal Method . . . . . . . . . . . . . . . . . . 37717.2.3 Historical Simulation Method . . . . . . . . . . . . . . 37717.2.4 Monte Carlo Simulation Method . . . . . . . . . . . . 37817.2.5 Comparison of Methods . . . . . . . . . . . . . . . . 379

17.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38117.3.1 Mark-to-Market . . . . . . . . . . . . . . . . . . . . . 38117.3.2 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . 38217.3.3 VAR: Historical Simulation . . . . . . . . . . . . . . . 38417.3.4 VAR: Delta-Normal Method . . . . . . . . . . . . . . . 386

17.4 Risk Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . 38817.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 389

18.1 Settlement Risk . . . . . . . . . . . . . . . . . . . . . . . . . 39418.1.1 Presettlement vs. Settlement Risk . . . . . . . . . . . 39418.1.2 Handling Settlement Risk . . . . . . . . . . . . . . . . 394

18.2 Overview of Credit Risk . . . . . . . . . . . . . . . . . . . . 39618.2.1 Drivers of Credit Risk . . . . . . . . . . . . . . . . . . 39618.2.2 Measurement of Credit Risk . . . . . . . . . . . . . . 39718.2.3 Credit Risk vs. Market Risk . . . . . . . . . . . . . . . 398

18.3 Measuring Credit Risk . . . . . . . . . . . . . . . . . . . . . 39918.3.1 Credit Losses . . . . . . . . . . . . . . . . . . . . . . 39918.3.2 Joint Events . . . . . . . . . . . . . . . . . . . . . . . 399

CONTENTS


Ch. 19 Measuring Actuarial Default Risk 411

Ch. 20 Measuring Default Risk from Market Prices 441

Ch. 21 Credit Exposure 459

xiv

18.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . 40118.4 Credit Risk Diversification . . . . . . . . . . . . . . . . . . . 40418.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 409

19.1 Credit Event . . . . . . . . . . . . . . . . . . . . . . . . . . . 41219.2 Default Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 414

19.2.1 Credit Ratings . . . . . . . . . . . . . . . . . . . . . . 41419.2.2 Historical Default Rates . . . . . . . . . . . . . . . . . 41719.2.3 Cumulative and Marginal Default Rates . . . . . . . . 41919.2.4 Transition Probabilities . . . . . . . . . . . . . . . . . 42419.2.5 Predicting Default Probabilities . . . . . . . . . . . . . 426

19.3 Recovery Rates . . . . . . . . . . . . . . . . . . . . . . . . . 42719.3.1 The Bankruptcy Process . . . . . . . . . . . . . . . . 42719.3.2 Estimates of Recovery Rates . . . . . . . . . . . . . . 428

19.4 Application to Portfolio Rating . . . . . . . . . . . . . . . . . 43019.5 Assessing Corporate and Sovereign Rating . . . . . . . . . . 433

19.5.1 Corporate Default . . . . . . . . . . . . . . . . . . . . 43319.5.2 Sovereign Default . . . . . . . . . . . . . . . . . . . . 433


20.1 Corporate Bond Prices . . . . . . . . . . . . . . . . . . . . . 44120.1.1 Spreads and Default Risk . . . . . . . . . . . . . . . . 44220.1.2 Risk Premium . . . . . . . . . . . . . . . . . . . . . . 44320.1.3 The Cross-Section of Yield Spreads . . . . . . . . . . . 44620.1.4 The Time-Series of Yield Spreads . . . . . . . . . . . . 448

20.2 Equity Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 44820.2.1 The Merton Model . . . . . . . . . . . . . . . . . . . . 44920.2.2 Pricing Equity and Debt . . . . . . . . . . . . . . . . . 45020.2.3 Applying the Merton Model . . . . . . . . . . . . . . . 45320.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . 455


21.1 Credit Exposure by Instrument . . . . . . . . . . . . . . . . . 46021.2 Distribution of Credit Exposure . . . . . . . . . . . . . . . . 462

21.2.1 Expected and Worst Exposure . . . . . . . . . . . . . 46321.2.2 Time Profile . . . . . . . . . . . . . . . . . . . . . . . 46321.2.3 Exposure Profile for Interest-Rate Swaps . . . . . . . . 46421.2.4 Exposure Profile for Currency Swaps . . . . . . . . . . 473

CONTENTS


Ch. 22 Credit Derivatives 491

Ch. 23 Managing Credit Risk 509

xv

21.2.5 Exposure Profile for Different Coupons . . . . . . . . 47421.3 Exposure Modifiers . . . . . . . . . . . . . . . . . . . . . . . 479

21.3.1 Marking to Market . . . . . . . . . . . . . . . . . . . 47921.3.2 Exposure Limits . . . . . . . . . . . . . . . . . . . . . 48121.3.3 Recouponing . . . . . . . . . . . . . . . . . . . . . . 48121.3.4 Netting Arrangements . . . . . . . . . . . . . . . . . 482

21.4 Credit Risk Modifiers . . . . . . . . . . . . . . . . . . . . . . 48621.4.1 Credit Triggers . . . . . . . . . . . . . . . . . . . . . 48621.4.2 Time Puts . . . . . . . . . . . . . . . . . . . . . . . . 487


22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 49122.2 Types of Credit Derivatives . . . . . . . . . . . . . . . . . . . 492

22.2.1 Credit Default Swaps . . . . . . . . . . . . . . . . . . 49322.2.2 Total Return Swaps . . . . . . . . . . . . . . . . . . . 49622.2.3 Credit Spread Forward and Options . . . . . . . . . . 49722.2.4 Credit-Linked Notes . . . . . . . . . . . . . . . . . . . 498

22.3 Pricing and Hedging Credit Derivatives . . . . . . . . . . . . 50122.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . 50222.3.2 Example: Credit Default Swap . . . . . . . . . . . . . 502

22.4 Pros and Cons of Credit Derivatives . . . . . . . . . . . . . . 50522.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 506

23.1 Measuring the Distribution of Credit Losses . . . . . . . . . . 51023.2 Measuring Expected Credit Loss . . . . . . . . . . . . . . . . 513

23.2.1 Expected Loss over a Target Horizon . . . . . . . . . . 51323.2.2 The Time Profile of Expected Loss . . . . . . . . . . . 514

23.3 Measuring Credit VAR . . . . . . . . . . . . . . . . . . . . . 51623.4 Portfolio Credit Risk Models . . . . . . . . . . . . . . . . . . 518

23.4.1 Approaches to Portfolio Credit Risk Models . . . . . . 51823.4.2 CreditMetrics . . . . . . . . . . . . . . . . . . . . . . 51923.4.3 CreditRisk+ . . . . . . . . . . . . . . . . . . . . . . . 52223.4.4 Moodys KMV . . . . . . . . . . . . . . . . . . . . . . 52323.4.5 Credit Portfolio View . . . . . . . . . . . . . . . . . . 52423.4.6 Comparison . . . . . . . . . . . . . . . . . . . . . . . 524


CONTENTS


Ch. 24 Operational Risk 533

Ch. 25 Risk Capital and RAROC 555

Ch. 26 Best Practices Reports 563

Ch. 27 Firmwide Risk Management 573

Part V: Operational and Integrated Risk Management 531

xvi

24.1 The Importance of Operational Risk . . . . . . . . . . . . . . 53424.1.1 Case Histories . . . . . . . . . . . . . . . . . . . . . . 53424.1.2 Business Lines . . . . . . . . . . . . . . . . . . . . . . 535

24.2 Identifying Operational Risk . . . . . . . . . . . . . . . . . . 53724.3 Assessing Operational Risk . . . . . . . . . . . . . . . . . . . 540

24.3.1 Comparison of Approaches . . . . . . . . . . . . . . . 54024.3.2 Acturial Models . . . . . . . . . . . . . . . . . . . . . 542

24.4 Managing Operational Risk . . . . . . . . . . . . . . . . . . . 54524.4.1 Capital Allocation and Insurance . . . . . . . . . . . . 54524.4.2 Mitigating Operational Risk . . . . . . . . . . . . . . . 547

24.5 Conceptual Issues . . . . . . . . . . . . . . . . . . . . . . . 54924.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 550

25.1 RAROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55625.1.1 Risk Capital . . . . . . . . . . . . . . . . . . . . . . . 55625.1.2 RAROC Methodology . . . . . . . . . . . . . . . . . . 55725.1.3 Application to Compensation . . . . . . . . . . . . . . 558

25.2 Performance Evaluation and Pricing . . . . . . . . . . . . . . 56025.3 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 562

26.1 The G-30 Report . . . . . . . . . . . . . . . . . . . . . . . . 56326.2 The Bank of England Report on Barings . . . . . . . . . . . . 56726.3 The CRMPG Report on LTCM . . . . . . . . . . . . . . . . . . 56926.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 571

27.1 Types of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 57427.2 Three-Pillar Framework . . . . . . . . . . . . . . . . . . . . . 575

27.2.1 Best-Practice Policies . . . . . . . . . . . . . . . . . . 57527.2.2 Best-Practice Methodologies . . . . . . . . . . . . . . 57627.2.3 Best-Practice Infrastructure . . . . . . . . . . . . . . . 576

27.3 Organizational Structure . . . . . . . . . . . . . . . . . . . . 57727.4 Controlling Traders . . . . . . . . . . . . . . . . . . . . . . . 581

27.4.1 Trader Compensation . . . . . . . . . . . . . . . . . . 58127.4.2 Trader Limits . . . . . . . . . . . . . . . . . . . . . . 582


CONTENTS


Ch. 28 Legal Issues 589

Ch. 29 Accounting and Tax Issues 605

Ch. 30 Regulation of Financial Institutions 629

Part VI: Legal, Accounting, and Tax Risk Management 587

Part VII: Regulation and Compliance 627

xvii

28.1 Legal Risks with Derivatives . . . . . . . . . . . . . . . . . . 59028.2 Netting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

28.2.1 G-30 Recommendations . . . . . . . . . . . . . . . . . 59328.2.2 Netting under the Basel Accord . . . . . . . . . . . . . 59428.2.3 Walk-Away Clauses . . . . . . . . . . . . . . . . . . . 59528.2.4 Netting and Exchange Margins . . . . . . . . . . . . . 596

28.3 ISDA Master Netting Agreement . . . . . . . . . . . . . . . . 59628.4 The 2002 Sarbanes-Oxley Act . . . . . . . . . . . . . . . . . 60028.5 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

28.5.1 General Legal Terms . . . . . . . . . . . . . . . . . . 60128.5.2 Bankruptcy Terms . . . . . . . . . . . . . . . . . . . 60228.5.3 Contract Terms . . . . . . . . . . . . . . . . . . . . . 602


29.1 Internal Reporting . . . . . . . . . . . . . . . . . . . . . . . 60629.1.1 Purpose of Internal Reporting . . . . . . . . . . . . . 60629.1.2 Comparison of Methods . . . . . . . . . . . . . . . . 60729.1.3 Historical Cost versus Marking-to-Market . . . . . . . 610

29.2 External Reporting: FASB . . . . . . . . . . . . . . . . . . . . 61229.2.1 FAS 133 . . . . . . . . . . . . . . . . . . . . . . . . . 61229.2.2 Definition of Derivative . . . . . . . . . . . . . . . . . 61329.2.3 Embedded Derivative . . . . . . . . . . . . . . . . . . 61429.2.4 Disclosure Rules . . . . . . . . . . . . . . . . . . . . 61529.2.5 Hedge Effectiveness . . . . . . . . . . . . . . . . . . . 61629.2.6 General Evaluation of FAS 133 . . . . . . . . . . . . . 61729.2.7 Accounting Treatment of SPEs . . . . . . . . . . . . . 617

29.3 External Reporting: IASB . . . . . . . . . . . . . . . . . . . . 62029.3.1 IAS 37 . . . . . . . . . . . . . . . . . . . . . . . . . . 62029.3.2 IAS 39 . . . . . . . . . . . . . . . . . . . . . . . . . . 621

29.4 Tax Considerations . . . . . . . . . . . . . . . . . . . . . . . 62229.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 623

30.1 Definition of Financial Institutions . . . . . . . . . . . . . . . 62930.2 Systemic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 63130.3 Regulation of Commercial Banks . . . . . . . . . . . . . . . . 632

CONTENTS


Ch. 31 The Basel Accord 641

Ch. 32 The Basel Market Risk Charges 669

Index 695

xviii

30.4 Regulation of Securities Houses . . . . . . . . . . . . . . . . 63530.5 Tools and Objectives of Regulation . . . . . . . . . . . . . . 63730.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 639

31.1 Steps in The Basel Accord . . . . . . . . . . . . . . . . . . . 64131.1.1 The 1988 Accord . . . . . . . . . . . . . . . . . . . . 64131.1.2 The 1996 Amendment . . . . . . . . . . . . . . . . . 64231.1.3 The New Basel Accord . . . . . . . . . . . . . . . . . 642

31.2 The 1988 Basel Accord . . . . . . . . . . . . . . . . . . . . . 64531.2.1 Risk Capital . . . . . . . . . . . . . . . . . . . . . . . 64531.2.2 On-Balance-Sheet Risk Charges . . . . . . . . . . . . . 64731.2.3 Off-Balance-Sheet Risk Charges . . . . . . . . . . . . . 64831.2.4 Total Risk Charge . . . . . . . . . . . . . . . . . . . . 652

31.3 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 65431.4 The New Basel Accord . . . . . . . . . . . . . . . . . . . . . 656

31.4.1 Issues with the 1988 Basel Accord . . . . . . . . . . . 65731.4.2 The New Basel Accord: Credit Risk Charges . . . . . . 65831.4.3 Securitization and Credit Risk Mitigation . . . . . . . . 66031.4.4 The Basel Operational Risk Charge . . . . . . . . . . . 661

31.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 66331.6 Further Information . . . . . . . . . . . . . . . . . . . . . . 665

32.1 The Standardized Method . . . . . . . . . . . . . . . . . . . 66932.2 The Internal Models Approach . . . . . . . . . . . . . . . . . 671

32.2.1 Qualitative Requirements . . . . . . . . . . . . . . . . 67132.2.2 The Market Risk Charge . . . . . . . . . . . . . . . . . 67232.2.3 Combination of Approaches . . . . . . . . . . . . . . 674

32.3 Stress-Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 67732.4 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

32.4.1 Measuring Exceptions . . . . . . . . . . . . . . . . . . 68032.4.2 Statistical Decision Rules . . . . . . . . . . . . . . . . 68032.4.3 The Penalty Zones . . . . . . . . . . . . . . . . . . . . 681


CONTENTS


PrefaceThe FRM Handbook provides the core body of knowledge for financial risk managers.

Risk management has rapidly evolved over the last decade and has become an indis-

pensable function in many institutions.

This Handbook was originally written to provide support for candidates taking the

FRM examination administered by GARP. As such, it reviews a wide variety of prac-

tical topics in a consistent and systematic fashion. It covers quantitative methods,

capital markets, as well as market, credit, operational, and integrated risk manage-

ment. It also discusses the latest regulatory, legal, and accounting issues essential to

risk professionals.

Modern risk management systems cut across the entire organization. This breadth

is reflected in the subjects covered in this Handbook. This Handbook was designed to

be self-contained, but only for readers who already have some exposure to financial

markets. To reap maximum benefit from this book, readers should have taken the

equivalent of an MBA-level class on investments.

Finally, I wanted to acknowledge the help received in the writing of this second ed-

ition. In particular, I would like to thank the numerous readers who shared comments

on the previous edition. Any comment and suggestion for improvement will be wel-

come. This feedback will help us to maintain the high quality of the FRM designation.

Philippe Jorion

April 2003

xix

IntroductionThe was first created in 2000 as a study support

manual for candidates preparing for GARPs annual FRM exam and as a general guide

to assessing and controlling financial risk in todays rapidly changing environment.

But the growth in the number of risk professionals, the now commonly held view

that risk management is an integral and indispensable part of any organizations man-

agement culture, and the ever increasing complexity of the field of risk management

have changed our goal for the Handbook.

This dramatically enhanced second edition of the Handbook reflects our belief

that a dynamically changing business environment requires a comprehensive text that

provides an in-depth overview of the various disciplines associated with financial risk

management. The Handbook has now evolved into the essential reference text for any

risk professional, whether they are seeking FRM Certification or whether they simply

have a desire to remain current on the subject of financial risk.

For those using the FRM Handbook as a guide for the FRM Exam, each chapter

includes questions from previous FRM exams. The questions are selected to provide

systematic coverage of advanced FRM topics. The answers to the questions are ex-

plained by comprehensive tutorials.

The FRM examination is designed to test risk professionals on a combination of

basic analytical skills, general knowledge, and intuitive capability acquired through

experience in capital markets. Its focus is on the core body of knowledge required

for independent risk management analysis and decision-making. The exam has been

administered every autumn since 1997 and has now expanded to 43 international

testing sites.

xxi

Financial Risk Manager Handbook

The FRM exam is recognized at the worlds most prestigious global certification

program for risk management professionals. As of 2002, 3,265 risk management pro-

fessionals have earned the FRM designation. They represent over 1,450 different com-

panies, financial institutions, regulatory bodies, brokerages, asset management firms,

banks, exchanges, universities, and other firms from all over the world.

GARP is very proud, through its alliance with John Wiley & Sons, to make this flag-

ship book available not only to FRM candidates, but to risk professionals, professors,

and their students everywhere. Philippe Jorion, preeminent in his field, has once again

prepared and updated the Handbook so that it remains an essential reference for risk

professionals.

Any queries, comments or suggestions about the Handbook may be directed to

frmhandbook garp.com. Corrections to this edition, if any, will be posted on GARPs

Web site.

Whether preparing for the FRM examination, furthering your knowledge of risk

management, or just wanting a comprehensive reference manual to refer to in a time

of need, any financial services professional will find the FRM Handbook an indispens-

able asset.

Global Association of Risk Professionals

April 2003

xxii INTRODUCTION



Second Edition

PART

QuantitativeAnalysis

one

Chapter 1

Bond Fundamentals

1.1 Discounting, Present, and Future Value

discounting factor interest rate

yield t

Risk management starts with the pricing of assets. The simplest assets to study are

fixed-coupon bonds, for which cash flows are predetermined. As a result, we can trans-

late the stream of cash flows into a present value by discounting at a fixed yield. Thus

the valuation of bonds involves understanding compounded interest, discounting, as

well as the relationship between present values and interest rates.

Risk management goes one step further than pricing, however. It examines poten-

tial changes in the value of assets as the interest rate changes. In this chapter, we

assume that there is a single interest rate that is used to discount to all bonds. This

will be our fundamental risk factor.

Even for as simple an instrument as a bond, the relationship between the price

and the risk factor can be complex. This is why the industry has developed a number

of tools that summarize the risk profile of fixed-income portfolios.

This chapter starts our coverage of quantitative analysis by discussing bond

fundamentals. Section 1.1 reviews the concepts of discounting, present values, and

future values. Section 1.2 then plunges into the price-yield relationship. It shows

how the Taylor expansion rule can be used to measure price movements. These

concepts are presented first because they are so central to the measurement of fi-

nancial risk. The section then discusses the economic interpretation of duration and

convexity.

An investor considers a zero-coupon bond that pays $100 in 10 years. Say that the

investment is guaranteed by the U.S. government and has no default risk. Because

the payment occurs at a future date, the investment is surely less valuable than an

up-front payment of $100.

To value the payment, we need a . This is also the ,

or more simply the . Define as the cash flow at time and the discounting

3

C t T

tenor present value

future value

internal rate of

return

effective annual rate (EAR)

TT

T

T

S

TS T

C

y TT

factor as . Here, is the number of periods until maturity, e.g. number of years, also

known as . The ( ) of the bond can be computed as

(1 1)(1 )

For instance, a payment of $100 in 10 years discounted at 6 percent is only

worth $55.84. This explains why the market value of zero-coupon bonds decreases

with longer maturities. Also, keeping fixed, the value of the bond decreases as the

yield increases.

Conversely, we can compute the of the bond as

(1 ) (1 2)

For instance, an investment now worth $100 growing at 6 percent will have a

future value of $179 08 in 10 years.

Here, the yield has a useful interpretation, which is that of an

on the bond, or annual growth rate. It is easier to deal with rates of returns

than with dollar values. Rates of return, when expressed in percentage terms and on an

annual basis, are directly comparable across assets. An annualized yield is sometimes

defined as the .

It is important to note that the interest rate should be stated along with the method

used for compounding. Equation (1.1) uses annual compounding, which is frequently

the norm. Other conventions exist, however. For instance, the U.S. Treasury market

uses semiannual compounding. If so, the interest rate is derived from

(1 3)(1 2)

where is the number of periods, or semesters in this case. Continuous compounding

is often used when modeling derivatives. If so, the interest rate is derived from

(1 4)

where , sometimes noted as exp( ), represents the exponential function. These are

merely definitions and are all consistent with the same initial and final values. One

has to be careful, however, about using each in the appropriate formula.

4

2

( )

PART I: QUANTITATIVE ANALYSIS


C

y T

PV

CPV .

y

C

T

FV PV y .

PV

FV .

y

CPV .

y

T

y

PV C e .

e

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