CHAPTER 1
Operational Risk Is Not Just ‘‘Other’’ Risks 1
Effects of Globalization and Deregulation: Increased Risk
Exposures 2
Examples of High-Magnitude Operational Losses 4
Orange County, 1994, United States 5
Barings Bank, 1995, United Kingdom 5
Daiwa Bank, 1995, New York 7
Allied Irish Banks, 2002, Ireland 8
The Enron Scandal, 2001, United States 8
MasterCard International, 2005, United States 9
Terrorist Attack, September 11, 2001, New York and
Worldwide 10
Operational Losses in The Hedge Fund Industry 10
Summary of Key Concepts 12
References 12
CHAPTER 2
Operational Risk: Definition, Classification, and Its Place among
Other Risks 15
What Is Risk? 15
Definition of Operational Risk 16
Operational Risk Exposure Indicators 18
Classification of Operational Risk 19
Internal versus External Operational Losses 19
Direct versus Indirect Operational Losses 19
Expected versus Unexpected Operational Losses 22
Operational Risk Type, Event Type, and Loss Type 22
Operational Loss Severity and Frequency
Topology of Financial Risks 26
Capital Allocation for Operational, Market, and Credit Risks 29
Impact of Operational Risk on the Market Value of Bank
Equity 30
Effects of Macroeconomic Environment on Operational Risk 31
Summary of Key Concepts 31
References 32
CHAPTER 3
Basel II Capital Accord 35
The Basel Committee on Banking Supervision 35
The Basel Capital Accord 36
Pillar I: Minimum Capital Requirements for Operational
Risk 37
Decomposition of Capital 37
Capital for Expected and Unexpected Losses 39
Three Approaches to Assess the Operational Risk
Capital Charge 40
The Basic Indicator Approach 41
The Standardized Approach 42
The Advanced Measurement Approaches 44
Pillar II: Capital Adequacy and Regulatory Principles 47
Pillar III: Market Discipline and Public Disclosure 48
Overview of Loss Data Collection Exercises 49
The Role of Insurance 51
Which Operational Losses Should Be Transferred? 53
FIORI Insurance Policy by Swiss Re 54
Insurance Recoveries Data from the 2002 Loss Data
Collection Exercise 54
Limitations of Insurance 55
Policy Limit 55
High Costs of Insurance 55
Moral Hazard 55
Alternatives to Insurance 57
Catastrophe Options 58
Catastrophe Bonds 58
Compliance with Basel II in Practice 59
JPMorgan Chase 59
HBOS 61
Implementing Basel II: Some General Concerns 61
Summary of Key Concepts 63
References
CHAPTER 4
Key Challenges in Modeling Operational Risk 67
Operational Risk Models 67
Models Based on Top-Down Approaches 69
Multifactor Equity Pricing Models 69
Capital Asset Pricing Model 69
Income-Based Models 70
Expense-Based Models 70
Operating Leverage Models 70
Scenario Analysis and Stress Testing Models 70
Risk Indicator Models 71
Models Based on Bottom-Up Approaches 72
Process-Based Models 72
Actuarial Models 74
Proprietary Models 75
Specifics of Operational Loss Data 75
Scarcity of Available Historical Data 76
Data Arrival Process 77
Loss Severity Process 78
Dependence between Business Units 81
Summary of Key Concepts 81
References 82
CHAPTER 5
Frequency Distributions 85
Binomial Distribution 86
Geometric Distribution 87
Poisson Distribution 88
Negative Binomial Distribution 92
Nonhomogeneous Poisson Process (Cox Process) 92
Mixture Distributions 93
Nonhomogeneous Poisson Process with Stochastic
Intensity 93
Algorithm 1 93
Algorithm 2 94
Alternative Approach: Interarrival Times Distribution 94
Empirical Analysis with Operational Loss Data 95
Studies with Real Data 95
Cruz Study of Fraud Loss Data 95
Moscadelli Study of 2002 LDCE Operational
Loss Data
De Fontnouvelle, Rosengren, and Jordan Study of
2002 LDCE Operational Loss Data 98
Lewis and Lantsman Study of Unauthorized
Trading Data 99
Chernobai, Burnec¸ki, Rachev, Tr ¨ uck, and Weron
Study of U.S. Natural Catastrophe Insurance
Claims Data 99
Chernobai, Menn, Rachev, and Tr ¨ uck Study of
1980–2002 Public Operational Loss Data 99
Chernobai and Rachev Study of 1950–2002
Public Operational Loss Data 101
Studies with Simulated Data 103
Laycock Study of Mishandling Losses and
Processing Errors Data 103
Cruz Study with Internal Fraud Data 103
Summary of Key Concepts 105
Appendix: Basic Descriptive Techniques for Discrete
Random Variables 105
Sample 106
Population 106
References 109
CHAPTER 6
Loss Distributions 111
Nonparametric Approach: Empirical Distribution Function 113
Parametric Approach: Continuous Loss Distributions 114
Exponential Distribution 115
Lognormal Distribution 116
Weibull Distribution 117
Gamma Distribution 119
Beta Distribution 120
Pareto Distribution 122
Burr Distribution 123
Extension: Mixture Loss Distributions 125
A Note on the Tail Behavior 127
Empirical Evidence with Operational Loss Data 129
Studies with Real Data 129
M¨ uller Study of 1950–2002 Operational Loss
Data 129
Cruz Study of Legal Loss Data 130
Moscadelli Study of 2002 LDCE Operational
Loss Data
De Fontnouvelle, Rosengren, and Jordan Study of
2002 LDCE Operational Loss Data 134
Lewis Study of Legal Liability Loss Data 135
Studies with Simulated Data 135
Reynolds and Syer Study 135
Rosenberg and Schuermann Study 136
Summary of Key Concepts 136
Appendix: Basic Descriptive Techniques for Continuous
Random Variables 137
Sample 137
Population 140
Transformations of Random Variables 142
Parameter Estimation Methods 143
References 144
CHAPTER 7
Alpha-Stable Distributions 147
Definition of an Alpha-Stable Random Variable 148
Useful Properties of an Alpha-Stable Random Variable 150
Estimating Parameters of the Alpha-Stable Distribution 152
Sample Characteristic Function Approach 152
Numerical Approximation of the Density Function
Approach 153
Useful Transformations of Alpha-Stable Random Variables 153
Symmetric Alpha-Stable Random Variable 153
Log-Alpha-Stable Random Variable 154
Truncated Alpha-Stable Random Variable 154
Applications to Operational Loss Data 154
Chernobai, Menn, Rachev, and Tr ¨ uck Study of
1980–2002 Public Loss Data 155
Chernobai and Rachev Study of 1950–2002 Public
Loss Data 157
Summary of Key Concepts 158
Appendix: Characteristic Functions 158
Definition of Characteristic Functions 159
Some Properties of Characteristic Functions 160
Relation to Distribution Functions 161
CHAPTER 8
Extreme Value Theory 163
Block Maxima Model
Peak over Threshold Model 164
Generalized Pareto Distribution 164
Choosing the High Threshold 167
Value-at-Risk under Peak over Threshold Model 168
Estimating the Shape Parameter 169
Hill Estimator 169
Pickands Estimator 170
Advantages and Limitations of Extreme Value Theory 171
Empirical Studies with Operational Loss Data 171
Cruz Study of Fraud Loss Data 172
Moscadelli Study with 2002 LDCE Data 172
De Fontnouvelle, Rosengren, and Jordan Study with
2002 LDCE Data 175
Chavez-Demoulin and Embrechts Study of Operational
Loss Data 176
Dutta and Perry Study of 2004 LDCE Operational Loss
Data 177
Summary of Key Concepts 177
CHAPTER 9
Truncated Distributions 183
Reporting Bias Problem 183
Truncated Model for Operational Risk 184
Data Specification 185
Parameter Estimation 187
Constrained Maximum Likelihood Function
Approach 187
Expectation-Maximization Algorithm Approach 187
Comparison of Naive and Conditional
Approaches: Lognormal Example 188
Empirical Studies with Operational Loss Data 191
Lewis and Lantsman Study with Unauthorized Trading
Data 193
Baud, Frachot, and Roncalli Study with Cr´edit
Lyonnais Loss Data 193
CHAPTER 10
Testing for the Goodness of Fit 201
Visual Tests for the Goodness of Fit 201
Quantile-Quantile Plots 202
Mean Excess Plots 202
Common Formal Tests for the Goodness of Fit 204
Chi-Squared Tests 205
Pearson’s Chi-Squared Test 205
Likelihood Ratio Test 206
Empirical Distribution Function-Based Tests 206
Kolmogorov-Smirnov Test 207
Kuiper Test 209
Anderson-Darling Test 209
Cram´er-von Mises Test 210
Empirical Study with Operational Loss Data 211
Summary of Key Concepts 217
Appendix: Hypothesis Testing
CHAPTER 11
Value-at-Risk 221
Intuitively, What Is VaR? 221
Compound Operational Loss Models and Derivation of
Operational VaR 222
A Simple Actuarial Model 222
Computing the Aggregate Loss Distribution 224
Monte Carlo Approach 224
Direct Computation Approach 225
Panjer’s Recursive Method 225
Inversion Method 226
Operational VaR 226
VaR Sensitivity Analysis 228
Backtesting VaR 229
Kupiec’s Proportion of Failures Test 229
Lopez’s Magnitude Loss Function Test 230
Benefits and Limitations of VaR and Alternative Risk
Measures 231
Benefits of VaR 231
Pitfalls of VaR 234
Coherent Risk Measures 234
Empirical Studies with Operational Loss Data 237
De Fontnouvelle, Rosengren, and Jordan Study of 2002
LDCE Data 238
Chapelle, Crama, H¨ ubner, and Peters Study with
European Loss Data 239
Summary of Key Concepts
CHAPTER 12
Robust Modeling 245
Outliers in Operational Loss Data 246
Some Dangers of Using the Classical Approach 248
Overview of Robust Statistics Methodology 248
Formal Model for Robust Statistics 249
Traditional Methods of Outlier Detection 249
Examples of Nonrobust versus Robust Estimators 251
Outlier Detection Approach Based on Influence
Functions 251
Advantages of Robust Statistics 252
Outlier Rejection Approach and Stress Tests 252
Application of Robust Methods to Operational Loss Data 253
Summary of Key Concepts
CHAPTER 13
Modeling Dependence 259
Three Types of Dependence in Operational Risk 260
Linear Correlation 261

Correlation and Its Properties 262
Aggregate Loss Correlation When Only Frequencies
Are Dependent 264
Drawbacks of Linear Correlation 265
Alternative Dependence Measure: Rank Correlation 265

Definition of Copula 266
Examples of Copulas 267
Using Copulas to Aggregate Credit, Market, and Operational

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