Part I Introduction to operational risk modeling
1 Operational risk
1.1 Introduction
1.1.1 Basel 11 - General
1.1.2
1.2 Operational risk in insurance
1.3 The analysis of operational risk
1.4 The model-based approach
1.4.1 The modeling process
1.5 Organization of this book
Basel 11 - Operational risk
2 Basic probability concepts
2.1 Introduction
2.2
2.3 Moments
Distribution functions and related concepts
2.4 Quantiles of a distribution
2.5 Generating functions
2.6 Exercises
3 Measures of risk
3.1 Introduction
3.2 Risk measures
3.3 Tail- Value-at-Risk
Part 11 Probabilistic tools for operational risk modeling
4 Models for the size of losses: Continuous distributions
4.2 A n inventory of continuous distributions
4.2.1 One-parameter distributions
4.1 Introduction
4.2.2 Two-parameter distributions
4.2.3 Three-parameter distributions
4.2.4 Four-parameter distributions
4.2.5 Distributions with finite support
4.3 Selected distributions and their relationships
4.3. I Introduction
4.3.2 Two important parametric families
4.4 Limiting distributions
4.5 The role of parameters
4.5. I Parametric and scale distributions
4.5.2 Finite mixture distributions
4.5.3 Data-dependent distributions
4.6 Tails of distributions
4.6.1 Classification based on moments
4.6.2 Classification based on tail behavior
4.6.3 Classification based on hazard rate
function
4.7 Creating new distributions
4.7.1 Introduction
4.7.2 Multiplication by a constant
4.7.3 Transformation by raising to a power
4.7.4 Transformation by exponentiation
4.7.5 Continuous mixture of distributions
4.7.6 Frailty models
4.8 TVaR for continuous distributions
4.8.1 Continuous elliptical distributions
4.8.2 Continuous exponential dispersion
distributions
4.9 Exercises
4.7.7 Splicing pieces of distributions
5 Models for the number of losses: Counting distributions
5.1 Introduction
5.2 The Poisson distribution
5.3 The negative binomial distribution
5.4 The binomial distribution
5.5 The (a,b,O) class f
5.7 Compound frequency models
5.6 The (a,b , 1) class
5.8 Recursive calculation of compound probabilities
5.9 An inventory of discrete distributions
5.9.1 The (a,b,O) class
5.9.2 The (a,b , 1) class
5.9.3 The zero-truncated subclass
5.9.5 The compound class
5.10 A hierarchy of discrete distributions
5.11 Further properties of the compound Poisson class
5.12 Mixed frequency models
5.13 Poisson mixtures
5.14 Effect of exposure on loss counts
5.15 TVaR for discrete distributions
5.9.4 The zero-modified subclass
5.15.1 T VaR for discrete exponential dispersion
distributions
5.16 Exercises
6 Aggregate loss models
6.1 Introduction
6.2 Model choices
6.3 The compound model for aggregate losses
6.4 Some analytic results
6.5 Evaluation of the aggregate loss distribution
6.6 The recursive method
6.6.1 Compound frequency models
6.6.2 Underflow/overjlow problems
6.6.3 Numerical stability
6.6.4 Continuous severity
6.6.5 Constructing arithmetic distributions
6.7 Fast Fourier transform methods
6.8 Using approximating severity distributions
6.9 Comparison of methods
6.10 TVaR for aggregate losses
6.8.1 Arithmetic distributions
6.10.1 TVaR for discrete aggregate loss
distributions
6.10.2 TVaR for some frequency distributions
6.10.3 TVaR for some severity distributions
6.10.4 Summary
6.11 Exercises
7 Extreme value theory: The study of jumbo losses
7.1 Introduction
7.2 Extreme value distributions
7.3 Distribution of the maximum
7.3.1 From a fixed number of losses
7.3.2 From a random number of losses
Stability of the maximum of the extreme value
distribution
7.4
7.5 The Fisher- Tippett theorem
7.6 Maximum domain of attraction
7.7 Generalized Pareto distributions
7.8 The frequency of exceedences
7.8.1 From a fixed number of losses
7.8.2 From a random number of losses
7.9 Stability of excesses of the generalized Pareto
7.10 Mean excess function
7.11 Limiting distributions of excesses
7.12 TVaR for extreme value distributions
7.13 Further reading
7.14 Exercises
8 Multivariate models
8.1 Introduction
8.2 Sklar’s theorem and copulas
8.3 Measures of dependency
8.4 Tail dependence
8.5 Archimedean copulas
8.6 Elliptical copulas
8.7 Extreme value copulas
8.8 Archimax copulas
8.9 Exercises
Part . 1 III Statistical methods for calibrating models of operational
9 Review of mathematical statistics
9.1 Introduction
9.2 Point estimation
9.2.1 Introduction
9.2.2
9.3 Interval estimation
9.4 Tests of hypotheses
9.5 Exercises
Measures of quality of estimators
10 Parameter estimation
10.1 Introduction
10.2 Method of moments and percentile matching
10.3 Maximum likelihood estimation
10.3.1 Introduction
10.3.2 Complete, individual data
10.3.3 Complete, grouped data
10.3.4 Truncated or censored data
10.4 Variance and interval estimation
10.5 Bayesian estimation
10.5.1 Definitions and Bayes ’ theorem
10.5.2 Inference and prediction
10.5.3 Computational issues
10.6 Exercises
11 Estimation for discrete distributions
11.1 Introduction
11.2 Poisson distribution
11.3 Negative binomial distribution
1 1.4 Binomial distribution
11.5 The (a,b, 1) class
11.6 Compound models
11.7 EYgPect of exposure on maximum likelihood
11.8 Exercises
estimation
12 Model selection
12.1 Introduction
12.2 Representations of the data and model
12.3 Graphical comparison of the density and
distribution functions
12.4 Hypothesis tests
22.4.1 Kolmogorov-Smirnov test
12.4.2 Anderson-Darling test
12.4.3 Chi-square goodness-of-fit test
12.44 Likelihood ratio test
12.5.1 Introduction
12.5.2 Judgment-based approaches
12.5.3 Score- based approaches
12.5 Selecting a model
12.6 Exercises
13 Fitting extreme value models
13.1 Introduction
13.2 Parameter estimation
13.2.1 ML estimation from the extreme value
distribution
13.2.2 ML estimation from the generalized
Pareto distribution
13.2.3 Estimating the Pareto shape parameter
13.2.4 Estimating extreme probabilities
13.3.1 Mean excess plots
13.3 Model selection
14 Fitting copula models
14.1 Introduction
14.2 Maximum likelihood estimation
14.3 Semiparametric estimation of the copula
14.4 The role of thresholds
14.5 Goodness-of-fit testing
14.6 An example
Appendix A Gamma and related functions
Appendix B Discretization of the severity distribution
B,l The method of rounding
B. 2 Mean preserving
B.3 Undiscretization of a discretixed distribution
Appendix C Nelder-Mead simplex method
References
Index