Value at Risk and Bank Capital Management This page intentionally left blank Value at Risk and Bank Capital Management Francesco Saita AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright © 2007, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data APPLICATION SUBMITTED British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 13: 978-0-12-369466-9 ISBN 10: 0-12-369466-3 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 07 08 09 10 9 8 7 6 5 4 3 2 1 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1: Value at Risk, Capital Management, and Capital Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 An Introduction to Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Capital Management and Capital Allocation: The Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: What Is “Capital” Management? . . . . . . . . . . . . . . . . . . . . . 7 2.1 Regulatory Capital and the Evolution of Basel II . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 The 1988 Basel I Accord and the 1996 Amendment . . . . . . . . . . . . . 8 2.1.2 The Concept of Regulatory Capital . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Overview of the Basel II Capital Accord . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Pillar 1: Minimum Capital Requirements — The Main Changes Introduced by Basel II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Box 2-1: Impact of the Basel II Accord on the Level of Minimum Regulatory Capital Requirements . . . . . . . . . . 12 2.2.2 Pillar 2: Supervisory Review Process . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Pillar 3: Market Discipline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.4 The Debate about Basel II Adoption and Implementation . . . . . . . . 16 2.3 Bank Estimates of Required Capital and the Different Notions of Bank Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Book Value of Capital and the Impact of IAS/IFRS . . . . . . . . . . . . . 17 2.3.2 Market Capitalization and the Double Perspective of Bank Managers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 The Impact of Alternative Notions of Capital on Capital Management and Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 vi CONTENTS 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3: Market Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 The Variance–Covariance Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 A Simplifi ed Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.2 The Choice of the Relevant Random Variables . . . . . . . . . . . . . . . . . 29 3.1.3 Mapping Exposures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Box 3-1: Mapping Equity Positions through Beta: An Example . . . 32 3.1.4 VaR for a Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Box 3-2: Calculating VaR for a Three-Stock Portfolio . . . . . . . . . 38 Box 3-3: Why Mapping Is Important . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.5 Estimating Volatility and Correlation: Simple Moving Averages . . . 39 3.1.6 Estimating Volatility and Correlation: Exponentially Weighted Moving Averages and GARCH Models . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.7 VaR Estimates and the Relevance of the Time Horizon . . . . . . . . . . 43 3.1.8 Implied Volatilities and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 44 Box 3-4: Deriving Implied Volatility from Option Prices . . . . . . . 45 3.2 Simulation Approaches: Historical Simulation and Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Historical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.4 Filtered Historical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Value at Risk for Option Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Problems in Option VaR Measurement . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Potential Solutions for Option VaR Measurement . . . . . . . . . . . . . . . 52 3.4 Extreme-Value Theory and Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.1 Extreme-Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.2 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Expected Shortfall and the Problem of VaR Nonsubadditivity . . . . . . . . . . 57 3.6 Back-Testing Market Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6.1 Which Series Should Be Considered? Actual versus Theoretical Portfolio Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6.2 Back-Testing VaR Forecasts: Unconditional Accuracy and Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.7 Internal VaR Models and Market Risk Capital Requirements . . . . . . . . . . . 62 3.8 Stress Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 4: Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Defi ning Credit Risk: Expected and Unexpected Losses . . . . . . . . . . . . . . . 67 4.2 Agency Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.1 External Rating Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Transition Matrixes and Cumulative and Marginal Default Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 CONTENTS vii 4.3 Quantitative Techniques for Stand-Alone Credit Risk Evaluation: Moody’s/KMV EDF and External Scoring Systems . . . . . . . . . . . . . . . . . . 74 4.3.1 Merton’s (1974) Model and Moody’s/KMV Expected Default Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Box 4-1: Deriving the Theoretical Credit Spread for Risky Bonds in the Merton (1974) Model . . . . . . . . . . . . . . . . . . 75 4.3.2 Credit-Scoring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Capital Requirements for Credit Risk under Basel II . . . . . . . . . . . . . . . . . . 80 4.4.1 Standardized Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4.2 Foundation and Advanced Internal Rating–Based Approaches . . . . 82 4.5 Internal Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.1 Internal Rating Assignment Process . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5.2 Rating Quantifi cation and the Defi nition of Default . . . . . . . . . . . . . 87 4.5.3 Point-in-Time versus Through-the-Cycle Internal Ratings . . . . . . . . 89 4.6 Estimating Loss Given Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.7 Estimating Exposure at Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.8 Interaction between Basel II and International Accounting Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.9 Alternative Approaches to Modeling Credit Portfolio Risk . . . . . . . . . . . . . 96 4.9.1 CreditMetricsTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.9.2 Moody’s/KMV PortfolioManagerTM . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.9.3 CreditPortfolio View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.9.4 CreditRisk+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.10 Comparison of Main Credit Portfolio Models . . . . . . . . . . . . . . . . . . . . . . . . 108 Box 4-2: Industry Practices Concerning Credit Portfolio Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Box 4-3: How Close Are Results Obtained from Credit Risk Portfolio Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.12 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter 5: Operational Risk and Business Risk . . . . . . . . . . . . . . . . . . 115 5.1 Capital Requirements for Operational Risk Measurement under Basel II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.1.1 Basic Indicator Approach (BIA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.1.2 Standardized Approach (SA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.1.3 Advanced Measurement Approach (AMA) . . . . . . . . . . . . . . . . . . . . 117 5.2 Objectives of Operational Risk Management . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Quantifying Operational Risk: Building the Data Sources . . . . . . . . . . . . . 119 5.3.1 Operational Risk Mapping and the Identifi cation of Key Risk Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3.2 Building an Internal Loss Database . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3.3 External Loss Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3.4 Scenario Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4 Quantifying Operational Risk: From Loss Frequency and Severity to Operational Risk Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4.1 Modeling Severity Based on Internal Loss Data . . . . . . . . . . . . . . . . 126 viii CONTENTS 5.4.2 Integrating Internal Severity Data with External Data and Scenario Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4.3 Estimating Operational Loss Frequency . . . . . . . . . . . . . . . . . . . . . . 128 5.4.4 Estimating Correlation or Dependence among Operational Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.4.5 Deriving Operational Risk Capital Estimates through Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4.6 Is Risk Measurement the Final Step? . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 Case Study: U.S. Bank Progress on Measuring Operational Risk, by Patrick de Fontnouvelle and Victoria Garrity (Supervision, Regulation, and Credit Department, Federal Reserve Bank of Boston) . . . 131 5.6 The Role of Measures of Business Risk and Earnings at Risk . . . . . . . . . . 134 5.7 Measuring Business Risk in Practice: Defi ning a Measure of Earnings at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.8 From Earnings at Risk to Capital at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 6: Risk Capital Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.1 The Need for Harmonization: Time Horizon, Confi dence Level, and the Notion of Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.2 Risk Aggregation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.2.1 Choosing the Components to Be Aggregated: Business Units versus Risk Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.2.2 Alternative Risk Aggregation Methodologies . . . . . . . . . . . . . . . . . . 149 6.3 Estimating Parameters for Risk Aggregation . . . . . . . . . . . . . . . . . . . . . . . . 153 Box 6-1: Some Examples of Linear Correlation Coeffi cient Estimates from Existing Studies and Their Implication on Aggregated Risk Capital . . . . . . . . . . . . . . . . . . . . . . . 158 6.4 Case Study: Capital Aggregation within Fortis (by Luc Henrard, Chief Risk Offi cer, Fortis, and Ruben Olieslagers, Director, Central Risk Management, Fortis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.5 A Synthetic Comparison of Alternative Risk Aggregation Techniques . . . . 164 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Chapter 7: Value at Risk and Risk Control for Market and Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.1 Defi ning VaR-Based Limits for Market Risk: Identifying Risk-Taking Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Box 7-1: C larifying VaR Measurement Limitations: Deutsche Bank’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.2 Managing VaR Limits for Market Risk: The Links between Daily VaR and Annual Potential Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 CONTENTS ix 7.2.1 Translating Actual Daily VaR Values into an Equivalent Ex Post Yearly VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Box 7-2: Daily VaR Fluctuations and Their Implications for Ex Post Yearly VaR: An Example Based on Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.2.2 Translating Yearly Ex Ante Acceptable Loss into an Equivalent Daily VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2.3 The Case of Variable VaR Limits and the Role of Cumulated Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.3 Managing VaR-Based Trading Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4 Identifying Risk Contributions and Internal Hedges: VaRDelta, Component VaR, and Incremental VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Box 7-3: Variant for the Calculation of Component VaR . . . . . . . . 188 7.5 Managing Risk and Pricing Limits for Credit Risk . . . . . . . . . . . . . . . . . . . 189 7.5.1 Setting Loan Autonomy Limits: From Notional Size to Expected Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.5.2 Setting Loan-Pricing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5.3 Case 1: Large Borrower Applying for a Loan from an Investment Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.5.4 Case 2: SME Applying for a Loan from a Smaller, Retail-Oriented Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Chapter 8: Risk-Adjusted Performance Measurement . . . . . . . . . . . . . 195 8.1 Business Areas, Business Units, and the Double Role of Risk-Adjusted Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.2 Checking the Measure of Profi t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.2.1 Transfer Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.2.2 Cost Attribution and Its Impact on RAP Measures . . . . . . . . . . . . . 198 8.3 Capital Investment versus Capital Allocation . . . . . . . . . . . . . . . . . . . . . . . . 199 8.4 Choosing the Measure of Capital at Risk: Allocated Capital versus Utilized Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.5 Choosing the Measure of Capital at Risk: Diversifi ed Capital versus Undiversifi ed Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.5.1 Comparison of Alternative Diversifi ed CaR Measures . . . . . . . . . . . 202 8.5.2 Criteria for Choosing between Diversifi ed and Undiversifi ed CaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.6 Choosing the Risk-Adjusted Performance Measure: EVA vs. RAROC . . . . 207 8.7 Variants and Potential Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.7.1 Differentiated Target Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.7.2 Alternative RAP Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.7.3 Expected Shortfall and Performance Measurement . . . . . . . . . . . . . 210 8.7.4 Operational Risk, Business Risk, and Performance Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.8 Risk-Adjusted Performances and Managers’ Performance Evaluation . . . . 212 8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216