financial risk manager handbook second...
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Financial Risk ManagerHandbook
Second Edition
Wiley
John Wiley & Sons, Inc.
Philippe Jorion
GARP
Innodata0471474487.jpg
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Financial Risk ManagerHandbook
Second Edition
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Financial Risk ManagerHandbook
Second Edition
Wiley
John Wiley & Sons, Inc.
Philippe Jorion
GARP
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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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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
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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
2.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 57
t
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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
3.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 80
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
4.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 99
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
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Financial Risk Manager Handbook, Second Edition
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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
7.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 183
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
8.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 207
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Financial Risk Manager Handbook, Second Edition
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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
9.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 224
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
10.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 238
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
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Financial Risk Manager Handbook, Second Edition
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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
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Financial Risk Manager Handbook, Second Edition
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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
13.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 308
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
14.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 326
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
15.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 351
16.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . 35516.1.1 Why the Normal? . . . . . . . . . . . . . . . . . . . . 355
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Financial Risk Manager Handbook, Second Edition
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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
16.4 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 370
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
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Financial Risk Manager Handbook, Second Edition
-
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
19.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 437
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
20.3 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 457
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
Financial Risk Manager Handbook, Second Edition
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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
21.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 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
23.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 527
CONTENTS
Financial Risk Manager Handbook, Second Edition
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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
27.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 585
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Financial Risk Manager Handbook, Second Edition
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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
28.6 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 603
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
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Financial Risk Manager Handbook, Second Edition
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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
32.5 Answers to Chapter Examples . . . . . . . . . . . . . . . . . 684
CONTENTS
Financial Risk Manager Handbook, Second Edition
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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
Financial Risk Manager Handbook, Second Edition
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Financial Risk ManagerHandbook
Second Edition
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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
Financial Risk Manager Handbook, Second Edition
C
y T
PV
CPV .
y
C
T
FV PV y .
PV
FV .
y
CPV .
y
T
y
PV C e .
e