I PEARSON ALWAYS LEARNING i Financial Risk Manager (FRM®) Exam Part I Valuation and Risk Models Fifth Custom Edition for Global Association of Risk Professionals 2015 Excerpts taken from: Options, Futures, and Other Derivatives, Ninth Edition, by John C Hull Excerpts taken from: Options, Futures, and Other Derivatives, Ninth Edition by John C Hull Copyright © 2015, 2012, 2009, 2006, 2003, 2000 by Pearson Education, Inc Upper Saddle River, New Jersey 07458 Copyright © 2015, 2014, 2013, 2012, 2011 by Pearson Learning Solutions All rights reserved This copyright covers material written expressly for this volume by the editor/s as well as the compilation itself It does not cover the individual selections herein that first appeared elsewhere Permission to reprint these has been obtained by Pearson Learning Solutions for this edition only Further reproduction by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, must be arranged with the individual copyright holders noted Grateful acknowledgment is made to the following sources for permission to reprint material copyrighted or controlled by them: "Quantifying Volatility in VaR Models; Putting VaR to Work," by Linda Allen, Jacob Boudoukh and Anthony Saunders, reprinted from Understanding Market, Credit and Operational Risk: The Value at Risk Approach (2004), by permission of John Wiley & Sons, Inc "Measures of Financial Risk," by Kevin Dowd, reprinted from Measuring Market Risk, Second Edition (2005), by permission of John Wiley & Sons, Inc "Stress Testing," by Philippe Jorion, reprinted from Value at Risk: The New Benchmark for Financial Risk, Third Edition (2007), by permission of McGraw-Hili Companies "Principles for Sound Stress Testing Practices and Supervision" by Bank for International Settlements, by permission of the Basel Committee on Banking Supervision, (May 2009) Chapters 1-5 from Fixed Income Securities: Tools for Today's Markets, Third Edition (2011), by Bruce Tuckman, by permission of John Wiley & Sons, Inc "Assessing Country Risk; Country Risk Assessment in Practice," by Daniel Wagner, reprinted from Managing Country Risk: A Practitioner's Guide to Effective Cross-Border Risk Analysis (2012), by permission of Taylor & Francis (US) "External and Internal Ratings," by Arnaud De Servigny and Olivier Renault, reprinted from Measuring and Managing Credit Risk (2004), by permission of McGraw-Hili Companies "Capital Structure in Banks," by Gerhard Schroeck, reprinted from Risk Management and Value Creation in Financial Institutions (2002), by permission of John Wiley & Sons, Inc "Operational Risk," by John C Hull, reprinted from Risk Management and Financial Institutions (2010), by permission of John Wiley & Sons, Inc + Website, Third Edition Learning Objectives provided by the Global Association of Risk Professionals All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116 A Pearson Education Company www.pearsoned.com Printed in the United States of America 10 VO 11 19 18 17 16 15 000200010271921895 JH/KE PEARSON ISBN 10: 1-269-96316-3 ISBN 13: 978-1-269-96316-9 CHAPTER QUANTIFYING VOLATILITY IN V AR MODELS The Stochastic Behavior of Returns The Distribution of Interest Rate Changes Fat Tails Explaining Fat Tails Effects of Volatility Changes Can (Conditional) Normality Be Salvaged? , 4 Normality Cannot Be Salvaged VaR Estimation Approaches 10 Historical Standard Deviation Implementation Considerations Exponential SmoothingRiskMetrics™ Volatility Nonparametric Volatility Forecasting A Comparison of Methods The Hybrid Approach Return Aggregation and VaR 23 Long Horizon Volatility and VaR 26 Mean Reversion and Long Horizon Volatility 27 Correlation Measurement 28 Summary 29 Appendix 30 Backtesting Methodology and Results 30 10 Cyclical Volatility Implied Volatility as a Predictor of Future Volatility 10 11 11 13 16 19 20 22 CHAPTER PUTTING V AR TO WORK The VaR of DerivativesPreliminaries Linear Derivatives Nonlinear Derivatives Approximating the VaR of Derivatives Fixed Income Securities with Embedded Optionality "Delta-Normal" vs Full Revaluation 37 38 38 39 40 43 44 iii Structured Monte Carlo, Stress Testing, and Scenario Analysis Motivation Structured Monte Carlo Scenario Analysis 45 45 45 47 Worst-Case Scenario (WCS) 52 WCS vs VaR A Comparison of VaR to WCS Extensions 52 52 53 Summary 53 Appendix 54 Duration CHAPTER 54 MEASURES OF FINANCIAL RISK 59 The Mean-Variance Framework for Measuring Financial Risk 61 Value-at-Risk 65 Basics of VaR Determination of the VaR Parameters Limitations of VaR as a Risk Measure Coherent Risk Measures The Coherence Axioms and Their Implications The Expected Shortfall Spectral Risk Measures Scenarios as Coherent Risk Measures Summary CHAPTER STRESS TESTING 79 Why Stress Testing? 80 Principles of Scenario Analysis 82 Portfolio- versus Event-Driven 82 Generating Unidimensional Scenarios 82 Sensitivity Tests An Example: The SPAN System Multidimensional Scenario Analysis 82 83 84 Prospective Scenarios Historical Scenarios 84 86 Stress-Testing Models and Parameters 87 Policy Responses 88 Summary 88 65 CHAPTER 67 68 69 69 71 73 76 77 iv III Contents - PRINCIPLES FOR SOUND STRESS TESTING PRACTICES AND SUPERVISION 91 Introduction 92 Performance of Stress Testing During the Crisis 93 Use of Stress Testing and Integration in Risk Governance Stress Testing Methodologies Scenario Selection Stress Testing of Specific Risks and Products Changes in Stress Testing Practices Since the Outbreak of the Crisis 93 93 94 95 96 Principles for Banks Use of Stress Testing and Integration in Risk Governance Stress Testing Methodology and Scenario Selection Specific Areas of Focus Principles for Supervisors CHAPTER BINOMIAL TREES 96 96 99 102 103 107 Options on Currencies Options on Futures 118 118 Summary 119 Appendix 120 Derivation of the Black-ScholesMerton Option-Pricing Formula from a Binomial Tree CHAPTER THE BLACK-SCHOLESMERTON MODEL A One-Step Binomial Model and a No-Arbitrage Argument A Generalization Irrelevance of the Stock's Expected Return Risk-Neutral Valuation The One-Step Binomial Example Revisited Real World vs Risk-Neutral World Two-Step Binomial Trees A Generalization 108 109 110 110 111 111 112 113 A Put Example 113 American Options 114 Delta 114 Matching Volatility with u and d 115 Girsanov's Theorem 116 The Binomial Tree Formulas 116 Increasing the Number of Steps 116 Using DerivaGem 117 Options on Other Assets 117 Options on Stocks Paying a Continuous Dividend Yield Options on Stock Indices 120 123 Lognormal Property of Stock Prices 124 The Distribution of the Rate of Return 125 The Expected Return 126 Volatility 127 Estimating Volatility from Historical Data Trading Days vs Calendar Days 127 129 The Idea Underlying the BlackScholes-Merton Differential Equation 129 Assumptions 130 Derivation Of The Black-ScholesMerton Differential Equation 130 A Perpetual Derivative The Prices of Tradeable Derivatives 131 132 Risk-Neutral Valuation 132 Application to Forward Contracts on a Stock 133 117 118 Contents • v Black-Scholes-Merton Pricing Formulas 133 Understanding N(d,) and N(d) Properties of the Black-ScholesMerton Formulas 134 134 Cumulative Normal Distribution Function 135 Theta 153 Gamma 154 Making a Portfolio Gamma Neutral Calculation of Gamma 154 155 Relationship Between Delta, Theta, and Gamma 156 Vega 156 Warrants and Employee Stock Options 135 Rho 158 Implied Volatilities 136 The Realities of Hedging 158 Scenario Analysis 158 Extension of Formulas 159 The VIX Index 137 Dividends 137 European Options American Call Options Black's Approximation 138 139 139 Summary 140 Appendix 141 Proof Of The Black-Scholes-Merton Formula Using Risk-Neutral Valuation Key Result Proof of Key Resu It The Black-Scholes-Merton Result CHAPTER THE GREEK LETTERS 141 141 141 142 Delta of Forward Contracts Delta of a Futures Contract Portfolio Insurance 161 Use of Index Futures 162 Stock Market Volatility 162 Summary 163 Appendix 164 Taylor Series Expansions and Hedge Parameters 145 CHAPTER Illustration 146 Naked and Covered Positions 146 A Stop-Loss Strategy 147 Delta Hedging 148 Delta of European Stock Options Dynamic Aspects of Delta Hedging Where the Cost Comes From Delta of a Portfolio Transaction Costs vi • Contents 149 150 152 152 153 159 160 164 PRICES, DISCOUNT FACTORS, AND ARBITRAGE 167 The Cash Flows from Fixed-Rate Government Coupon Bonds 168 Discount Factors 169 The Law of One Price 169 Arbitrage and the Law of One Price 170 Application: STRIPS and the Idiosyncratic Pricing of U.S Treasury Notes and Bonds STRIPS The Idiosyncratic Pricing of U.S Treasury Notes and Bonds Accrued Interest Definition Pricing Implications Day-Count Conventions Appendix A Appendix B 173 174 175 175 176 176 177 The Equivalence of the Discounting and Arbitrage Pricing Approaches 10 172 176 Deriving Replicating Portfolios CHAPTER 172 177 Simple Interest and Compounding 179 180 181 Definitions of Spot, Forward, and Par Rates 182 Characteristics of Spot, Forward, and Par Rates Maturity and Price or Present Value Appendix A 189 189 Compounding Conventions Appendix B 190 Continuously Compounded Spot and Forward Rates 190 Appendix C 190 190 Flat Spot Rates Imply Flat Par Rates 190 Appendix D 190 A Useful Summation Formula Appendix E 191 191 Appendix F 191 The Relationship Between Spot and Par Rates and the Slope of the Term Structure 191 192 Appendix G Extracting Discount Factors from Interest Rate Swaps Spot Rates Forward Rates Par Rates Synopsis: Quoting Prices with Semiannual Spot, Forward, and Par Rates 186 The Relationship Between Spot and Forward Rates and the Slope of the Term Structure SPOT, FORWARD, AND PAR RATES Trading Case Study: Trading an Abnormally Downward-Sloping 10s-30s EUR Forward Rate Curve in Q2 2010 182 182 183 Maturity, Present Value, and Forward Rates CHAPTER 11 192 RETURNS, SPREADS, 195 AND YIELDS 196 Definitions 184 184 Realized Returns Spreads Yield-to-Maturity News Excerpt: Sale of Greek Government Bonds in March, 2010 196 197 198 201 185 Contents III vii Components of P&L and Return 201 A Sample P&L Decomposition 203 Carry-Roll-Down Scenarios 204 Realized Forwards Unchanged Term Structure Unchanged Yields Expectations of Short-Term Rates Are Realized Appendix A 205 205 206 206 207 Yield on Settlement Dates Other than Coupon Payment Dates Appendix B Yield-Based DV01 and Duration Yield-Based DV01 and Duration for Zero-Coupon Bonds, Par Bonds, and Perpetuities Duration, DV01, Maturity, and Coupon: A Graphical Analysis Duration, DV01, and Yield Yield-Based Convexity Application: The Barbell versus the Bullet CHAPTER 13 12 Key Rate '01s and Durations RISK METRICS 211 DV01 212 A Hedging Application, Part 1: Hedging a Futures Option 214 Duration 215 Convexity 216 A Hedging Application, Part II: A Short Convexity Position 218 Estimating Price Changes and Returns with DV01, Duration, and Convexity 219 Convexity in the Investment and Asset-Liability Management Contexts 221 Measuring the Price Sensitivity of Portfolios 221 viii • Key Rate Shifts Calculating Key Rate '01s and Durations Hedging with Key Rate Exposures 224 225 226 226 229 230 231 232 233 Partial '01s and PV01 235 Forward-Bucket '01s 236 Forward-Bucket Shifts and '01 236 Calculations Understanding Forward-Bucket '01s: A Payer Swaption 237 Hedging with Forward-Bucket '01s: A Payer Swaption 238 Multi-Factor Exposures and Measuring Portfolio Volatility 239 Appendix 239 Selected Determinants of ForwardBucket '01s Contents ._ -_ . ._ ._-_. . _ -_ _-_._-_ ._._._ _ _._._ -_ _ _. _ _. . ._ _ _ _ 223 RISK METRICS 207 ONE-FACTOR AND HEDGES 222 MULTI-FACTOR AND HEDGES CHAPTER 222 207 207 P&L Decomposition on Dates Other than Coupon Payment Dates Yield-Based Risk Metrics • • _ _._ _._ _.- - - - 239 CHAPTER 14 ASSESSING COUNTRY RISK Information Sources CHAPTER INTERNAL RATINGS 273 245 Ratings and External Agencies 274 246 Measuring Political Stability 248 CHAPTER 15 251 COUNTRY RISK ASSESSMENT IN PRACTICE 257 Creating a Risk Management Framework 258 Selecting Country Risk Management Tools 262 Economic Measures 263 Mapping Out a Country Risk Analysis Methodology 266 Alternative Measures of Country Risk 267 Corruption Perceptions Index Democracy Index Freedom in the World Gini Coefficient Global Peace Index Human Development Index Youth Unemployment Conclusion EXTERNAL AND 243 Are Rating Agencies' Ratings Worth Using? Comparing Indonesia and Vietnam 16 267 267 268 268 268 269 269 271 The Role of Rating Agenc~es in the Financial Markets 274 Comments and Criticisms about External Ratings 277 Ratings, Related Time Horizon, and Economic Cycles Industry and Geography Homogeneity Impact of Rating Changes on Corporate Security Prices 277 279 280 Approaching Credit Risk through Internal Ratings or Score-Based Ratings 282 Internal Ratings, Scores, and Time Horizons How to Build an Internal Rating System Granularity of Rating Scales Consequences 283 284 286 287 287 Summary CHAPTER 17 CREDIT RISK 289 Definition of Credit Risk Steps to Derive Economic Capital for Credit Risk Expected Losses (EL) Unexpected Losses (UL-Standalone) Unexpected Loss Contribution (ULC) Economic Capital for Credit Risk Problems with the Quantification of Credit Risk 290 290 291 293 294 296 298 Contents II ix CHAPTER 18 OPERATIONAL RISK What Is Operational Risk? Determination of Regulatory Capital Categorization of Operational Risks 302 303 304 Loss Severity and Loss Frequency 305 Implementation of AMA 305 Internal Data External Data Scenario Analysis 306 306 307 Business Environment and Internal Control Factors 308 Proactive Approaches Causal Relationships RCSA and KRls x 301 II Contents 308 308 309 Allocation of Operational Risk Capital 309 Use of Power Law 310 Insurance 310 Moral Hazard Adverse Selection 310 311 Sarbanes-Oxley 311 Summary 312 Sample Exam QuestionsValuation and Risk Models 315 Sample Exam Answers and Explanations-Valuation and Risk Models 319 Index 325 12 Answer: C Explanation: The 10 basis point shock to the 10-year yield is supposed to decline linearly to zero for the 20-xear yield _Thu?, the shock d~cr~ase_s by qne basi~ point peryear and will result in an increase of six basis points for the 14-year yield 13 Answer: A Explanation: When rates drop, the long position in the futures and the short position in the FRA both gain 14 Answer: A Explanation: DV01 may not be a reliable measure when changes in interest rates are not small Also, when applying DVOl we assume that the yield curve shifts are parallel 322 • Financial Risk Manager Exam Part I: Valuation and Risk Models accrued interest, 169, 174-176 actual stress loss (ASL), 85 actual/360 day-count convention, 176 actual/actual day-count convention, 175 adaptive expectations model, 14-15 adjusted duration, 222 advanced measurement approach (AM A), 303-304, 305-308 adverse selection, 311 agency arbitrage, 280 aleotoric risk, 244 Allen, Linda, 3-56 American call options, 139 American options, binomial trees and, 114 annuity factor, 183 annuity formula, 199 arbitrage, law of one price and, 170-172 arbitrage opportunity, 170 arbitrage pricing, 168, 177 ask price, 168-169 asset concentration, 50-51 asset returns, types of, asset-class-specific risks, 49-50 at-the-money (ATM) implied volatility, 24 at-the-money (ATM) put options, 51 at-the-point-in-time approach, 283-284, 286 average cycle, 277 AVG statistics, 31-34 back-testing, of rating system, 285-286 backtesting VaR, 30-34 Bank for International Settlements (BIS), 91-105, 245 banks, stress testing principles for, 96-103 barbell portfolio, 226-227 Basel Committee on Banking Supervision, 80, 95, 105, 303, 304, 307, 310 Basel II Accord, 92, 279-280, 286, 287, 302 basic indicator approach, 303 bias, 42 bid price, 168-169 binomial trees American options, 114 delta, 114-115 derivation of Black-Scholes-Merton option-pricing formula from, 120-121 formulas, 116 increasing the number of steps, 116-117 matching volatility with u and d, 115-116 one-step binomial model, no arbitrage argument and, 108-110 options on other assets, 117-119 overview, 108 put example, 113-114 risk-neutral valuation, 110-112 two-step, 112-113 using DerivaGem, 117 Black's approximation, 139-140 Black-Scholes implied volatility, 23-24 Black-Scholes option-pricing model, 10, 25, 38, 39, 41, 42 Black-Scholes-Merton model cumulative normal distribution function, 135 derivation from binomial tree, 120-121 differential equation, 129-133 distribution of rate of return, 125-126 dividends, 137-140 expected return, 126-127 implied volatilities, 136-137 lognormal property of stock prices, 124-125 overview, 124 pricing formulas, 133-135 proof using risk-neutral valuation, 141-142 volatility, 127-129 warrants and employee stock options, 135-136 Bolivia, 250 Bollereslev, Tim, 15 bond prices, effect of rating changes on, 280-281 325 Counterparty Risk Management Policy Group (CRMPG 111),96 boundary conditions, 131 Brady bonds, 47 country risk British Central Bank, 24 alternative measures of, 267-271 - -_ - -Gomparing-Indonesia-and VietRam, 251-254bulletinyestment,226-227 creating risk management framework, 258-262 business disruption, 302, 305 definition of risk, 244 business environment and internal control factors (BEICFs), 308 business practices, 302, 304 information sources, 245-246 management guidelines, 244-245 mapping out an analysis methodology, 266-267 calendar days, vs trading days, 129 measuring political stability, 248-251 calibration, of rating system, 285-286 capital allocation, 80