Modelling Operational Risk Using Bayesian Inference (eBook)

Dr. Pavel V. Shevchanko is a Principal Research Scientist in the Division of Mathematics, Informatics and Statistics of CSIRO (The Commonwealth Scientific and Industrial Research Organisation of Australia). Dr Shevchenko joined CSIRO in 1999 to work in the area of financial risk. He leads research and commercial projects on the modelling of operational and credit risks; option pricing; insurance; modelling commodities and foreign exchange; and the development of relevant numerical methods and software. He received a MSc from Moscow Institute of Physics and Technology and Kapitza Institute for Physical Problems in 1994; and a PhD from The University of New South Wales in 1999 in theoretical physics. Dr Shevchenko has published extensively in academic journals, consults for major financial institutions and is a frequent presenter at industry and academic conferences.

Preface 5
Contents 8
List of Abbreviations and Symbols 13
1 Operational Risk and Basel II 14
1.1 Introduction to Operational Risk 14
1.2 Defining Operational Risk 17
1.3 Basel II Approaches to Quantify Operational Risk 17
1.4 Loss Data Collections 20
1.4.1 2001 LDCE 23
1.4.2 2002 LDCE 24
1.4.3 2004 LDCE 26
1.4.4 2007 LDCE 28
1.4.5 General Remarks 29
1.5 Operational Risk Models 30
2 Loss Distribution Approach 33
2.1 Loss Distribution Model 33
2.2 Operational Risk Data 34
2.3 A Note on Data Sufficiency 36
2.4 Insurance 37
2.5 Basic Statistical Concepts 38
2.5.1 Random Variables and Distribution Functions 38
2.5.2 Quantiles and Moments 41
2.6 Risk Measures 44
2.7 Capital Allocation 45
2.7.1 Euler Allocation 46
2.7.2 Allocation by Marginal Contributions 48
2.8 Model Fitting: Frequentist Approach 49
2.8.1 Maximum Likelihood Method 51
2.8.2 Bootstrap 54
2.9 Bayesian Inference Approach 55
2.9.1 Conjugate Prior Distributions 57
2.9.2 Gaussian Approximation for Posterior 58
2.9.3 Posterior Point Estimators 58
2.9.4 Restricted Parameters 59
2.9.5 Noninformative Prior 60
2.10 Mean Square Error of Prediction 61
2.11 Markov Chain Monte Carlo Methods 62
2.11.1 Metropolis-Hastings Algorithm 64
2.11.2 Gibbs Sampler 65
2.11.3 Random Walk Metropolis-Hastings Within Gibbs 66
2.11.4 ABC Methods 68
2.11.5 Slice Sampling 70
2.12 MCMC Implementation Issues 72
2.12.1 Tuning, Burn-in and Sampling Stages 72
2.12.2 Numerical Error 74
2.12.3 MCMC Extensions 77
2.13 Bayesian Model Selection 78
2.13.1 Reciprocal Importance Sampling Estimator 80
2.13.2 Deviance Information Criterion 80
Problems 81
3 Calculation of Compound Distribution 83
3.1 Introduction 83
3.1.1 Analytic Solution via Convolutions 84
3.1.2 Analytic Solution via Characteristic Functions 85
3.1.3 Compound Distribution Moments 88
3.1.4 Value-at-Risk and Expected Shortfall 90
3.2 Monte Carlo Method 91
3.2.1 Quantile Estimate 92
3.2.2 Expected Shortfall Estimate 94
3.3 Panjer Recursion 95
3.3.1 Discretisation 97
3.3.2 Computational Issues 99
3.3.3 Panjer Extensions 100
3.3.4 Panjer Recursion for Continuous Severity 101
3.4 Fast Fourier Transform 101
3.4.1 Compound Distribution via FFT 103
3.4.2 Aliasing Error and Tilting 104
3.5 Direct Numerical Integration 106
3.5.1 Forward and Inverse Integrations 106
3.5.2 Gaussian Quadrature for Subdivisions 110
3.5.3 Tail Integration 112
3.6 Comparison of Numerical Methods 115
3.7 Closed-Form Approximation 117
3.7.1 Normal and Translated Gamma Approximations 117
3.7.2 VaR Closed-Form Approximation 118
Problems 120
4 Bayesian Approach for LDA 122
4.1 Introduction 122
4.2 Combining Different Data Sources 125
4.2.1 Ad-hoc Combining 125
4.2.2 Example of Scenario Analysis 127
4.3 Bayesian Method to Combine Two Data Sources 128
4.3.1 Estimating Prior: Pure Bayesian Approach 130
4.3.2 Estimating Prior: Empirical Bayesian Approach 132
4.3.3 Poisson Frequency 132
4.3.4 The Lognormal LN(,) Severity with Unknown 137
4.3.5 The Lognormal LN(,) Severity withUnknown and 140
4.3.6 Pareto Severity 142
4.4 Estimation of the Prior Using Data 147
4.4.1 The Maximum Likelihood Estimator 147
4.4.2 Poisson Frequencies 148
4.5 Combining Expert Opinions with External and Internal Data 151
4.5.1 Conjugate Prior Extension 153
4.5.2 Modelling Frequency: Poisson Model 154
4.5.3 Modelling Frequency: Poisson with Stochastic Intensity 161
4.5.4 Lognormal Model for Severities 164
4.5.5 Pareto Model 167
4.6 Combining Data Sources Using Credibility Theory 170
4.6.1 Bühlmann-Straub Model 172
4.6.2 Modelling Frequency 174
4.6.3 Modelling Severity 177
4.6.4 Numerical Example 180
4.6.5 Remarks and Interpretation 181
4.7 Capital Charge Under Parameter Uncertainty 182
4.7.1 Predictive Distributions 182
4.7.2 Calculation of Predictive Distributions 184
4.8 General Remarks 186
Problems 188
5 Addressing the Data Truncation Problem 190
5.1 Introduction 190
5.2 Constant Threshold -- Poisson Process 192
5.2.1 Maximum Likelihood Estimation 193
5.2.2 Bayesian Estimation 197
5.3 Extension to Negative Binomial and Binomial Frequencies 199
5.4 Ignoring Data Truncation 203
5.5 Threshold Varying in Time 207
Problems 211
6 Modelling Large Losses 213
6.1 Introduction 213
6.2 EVT -- Block Maxima 214
6.3 EVT -- Threshold Exceedances 218
6.4 A Note on GPD Maximum Likelihood Estimation 222
6.5 EVT -- Random Number of Losses 224
6.6 EVT -- Bayesian Approach 226
6.7 Subexponential Severity 231
6.8 Flexible Severity Distributions 235
6.8.1 g-and-h Distribution 235
6.8.2 GB2 Distribution 237
6.8.3 Lognormal-Gamma Distribution 238
6.8.4 Generalised Champernowne Distribution 239
6.8.5 -Stable Distribution 240
Problems 242
7 Modelling Dependence 244
7.1 Introduction 244
7.2 Dominance of the Heaviest Tail Risks 247
7.3 A Note on Negative Diversification 249
7.4 Copula Models 250
7.4.1 Gaussian Copula 251
7.4.2 Archimedean Copulas 252
7.4.3 t-Copula 254
7.5 Dependence Measures 256
7.5.1 Linear Correlation 256
7.5.2 Spearman's Rank Correlation 257
7.5.3 Kendall's tau Rank Correlation 258
7.5.4 Tail Dependence 259
7.6 Dependence Between Frequencies via Copula 260
7.7 Common Shock Processes 261
7.8 Dependence Between Aggregated Losses via Copula 262
7.9 Dependence Between the k-th Event Times/Losses 262
7.10 Modelling Dependence via Lévy Copulas 262
7.11 Structural Model with Common Factors 263
7.12 Stochastic and Dependent Risk Profiles 265
7.13 Dependence and Combining Different Data Sources 269
7.13.1 Bayesian Inference Using MCMC 271
7.13.2 Numerical Example 273
7.14 Predictive Distribution 275
Problems 278
A List of Distributions 281
A.1 Discrete Distributions 281
A.1.1 Poisson Distribution, Poisson() 281
A.1.2 Binomial Distribution, Bin(n,p) 282
A.1.3 Negative Binomial Distribution, NegBin(r,p) 282
A.2 Continuous Distributions 283
A.2.1 Uniform Distribution, U(a,b) 283
A.2.2 Normal (Gaussian) Distribution, N(,) 283
A.2.3 Lognormal Distribution, LN(,) 283
A.2.4 t Distribution, T(, ,2) 284
A.2.5 Gamma Distribution, Gamma(,) 284
A.2.6 Weibull Distribution, Weibull(,) 284
A.2.7 Pareto Distribution (One-Parameter), Pareto(,x0) 285
A.2.8 Pareto Distribution (Two-Parameter), Pareto2(,) 285
A.2.9 Generalised Pareto Distribution, GPD(,) 286
A.2.10 Beta Distribution, Beta(,) 286
A.2.11 Generalised Inverse Gaussian Distribution, GIG(,,) 287
A.2.12 d-variate Normal Distribution, Nd(bold0mu mumu Raw,bold0mu mumu Raw) 288
A.2.13 d-variate t-Distribution, Td(,bold0mu mumu Raw,bold0mu mumu Raw) 288
B Selected Simulation Algorithms 289
B.1 Simulation from GIG Distribution 289
B.2 Simulation from -stable Distribution 290
Solutions for Selected Problems 291
Problems of Chapter 2 291
Problems of Chapter 3 293
Problems of Chapter 5 294
Problems of Chapter 6 295
Problems of Chapter 7 295
References 297
Index 307