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
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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
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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
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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: Clarifying VaR Measurement Limitations: Deutsche
Bank’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172