Operational Risk Modeling Analytics

by Panjer, Harry H.

• ISBN13: 9780471760894
• ISBN10: 0471760897
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-07-28
• Publisher: Wiley-Interscience

Summary

This book provides a highly detailed development of the many probabilistic and statistical methods that are used in operational risk. These methods have traditionally been developed in the insurance/actuarial field, but they are now applicable to operational risk measurement and management. The book is premised on the availability of data for analysis. In the past few years, there have been several initiatives to gather such data in individual organizations; it is intended that this volume present such data for more detailed analysis than what is currently being done in the literature. The book is written by one of the foremost authorities in the world on risk modelling and its effects in the business environment.

Author Biography

HARRY H. PANJER, PHD, FSA, FCIA, HonFIA, is Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Ontario, Canada. He is past president of both the Canadian Institute of Actuaries and the Society of Actuaries, and he has published numerous articles and books on the subject of risk modeling over the years in the fields of finance and actuarial science.

Table of Contents

Preface	xiii
Acknowledgments	xv
Part I Introduction to operational risk modeling
Operational risk	3	(16)
Introduction	3	(8)
Basel II - General	6	(2)
Basel II - Operational risk	8	(3)
Operational risk in insurance	11	(1)
The analysis of operational risk	12	(2)
The model-based approach	14	(2)
The modeling process	15	(1)
Organization of this book	16	(3)
Basic probability concepts	19	(26)
Introduction	19	(1)
Distribution functions and related concepts	20	(10)
Moments	30	(8)
Quantiles of a distribution	38	(1)
Generating functions	38	(3)
Exercises	41	(4)
Measures of risk	45	(12)
Introduction	45	(1)
Risk measures	46	(4)
Tail- Value-at-Risk	50	(7)
Part II Probabilistic tools for operational risk modeling
Models for the size of losses: Continuous distributions	57	(50)
Introduction	57	(1)
An inventory of continuous distributions	58	(10)
One-parameter distributions	58	(1)
Two-parameter distributions	59	(5)
Three-parameter distributions	64	(3)
Four-parameter distributions	67	(1)
Distributions with finite support	68	(1)
Selected distributions and their relationships	68	(2)
Introduction	68	(1)
Two important parametric families	69	(1)
Limiting distributions	70	(3)
The role of parameters	73	(7)
Parametric and scale distributions	74	(1)
Finite mixture distributions	75	(3)
Data-dependent distributions	78	(2)
Tails of distributions	80	(4)
Classification based on moments	80	(1)
Classification based on tail behavior	81	(1)
Classification based on hazard rate function	82	(2)
Creating new distributions	84	(9)
Introduction	84	(1)
Multiplication by a constant	84	(1)
Transformation by raising to a power	85	(2)
Transformation by exponentiation	87	(1)
Continuous mixture of distributions	88	(2)
Frailty models	90	(2)
Splicing pieces of distributions	92	(1)
TVaR for continuous distributions	93	(9)
Continuous elliptical distributions	94	(3)
Continuous exponential dispersion distributions	97	(5)
Exercises	102	(5)
Models for the number of losses: Counting distributions	107	(54)
Introduction	107	(1)
The Poisson distribution	108	(2)
The negative binomial distribution	110	(4)
The binomial distribution	114	(1)
The (a, b, 0) class	114	(4)
The (a, b, 1) class	118	(4)
Compound frequency models	122	(4)
Recursive calculation of compound probabilities	126	(4)
An inventory of discrete distributions	130	(6)
The (a, b, 0) class	130	(2)
The (a, b, 1) class	132	(1)
The zero-truncated subclass	132	(2)
The zero-modified subclass	134	(1)
The compound class	135	(1)
A hierarchy of discrete distributions	136	(1)
Further properties of the compound Poisson class	137	(5)
Mixed frequency models	142	(2)
Poisson mixtures	144	(5)
Effect of exposure on loss counts	149	(1)
TVaR for discrete distributions	150	(6)
TVaR for discrete exponential dispersion distributions	151	(5)
Exercises	156	(5)
Aggregate loss models	161	(44)
Introduction	161	(1)
Model choices	162	(1)
The compound model for aggregate losses	163	(5)
Some analytic results	168	(3)
Evaluation of the aggregate loss distribution	171	(3)
The recursive method	174	(9)
Compound frequency models	175	(3)
Underflow/overflow problems	178	(1)
Numerical stability	179	(1)
Continuous severity	179	(1)
Constructing arithmetic distributions	180	(3)
Fast Fourier transform methods	183	(4)
Using approximating severity distributions	187	(3)
Arithmetic distributions	187	(3)
Comparison of methods	190	(1)
TVaR for aggregate losses	191	(7)
TVaR for discrete aggregate loss distributions	191	(1)
TVaR for some frequency distributions	192	(2)
TVaR for some severity distributions	194	(4)
Summary	198	(1)
Exercises	198	(7)
Extreme value theory: The study of jumbo losses	205	(28)
Introduction	205	(2)
Extreme value distributions	207	(1)
Distribution of the maximum	208	(5)
From a fixed number of losses	208	(2)
From a random number of losses	210	(3)
Stability of the maximum of the extreme value distribution	213	(1)
The Fisher- Tippett theorem	214	(3)
Maximum domain of attraction	217	(2)
Generalized Pareto distributions	219	(2)
The frequency of exceedences	221	(5)
From a fixed number of losses	221	(1)
From a random number of losses	222	(4)
Stability of excesses of the generalized Pareto	226	(1)
Mean excess function	227	(1)
Limiting distributions of excesses	228	(1)
TVaR for extreme value distributions	229	(1)
Further reading	230	(1)
Exercises	230	(3)
Multivariate models	233	(34)
Introduction	233	(1)
Sklar's theorem and copulas	234	(3)
Measures of dependency	237	(2)
Tail dependence	239	(1)
Archimedean copulas	240	(13)
Elliptical copulas	253	(4)
Extreme value copulas	257	(5)
Archimax copulas	262	(1)
Exercises	263	(4)
Part III Statistical methods for calibrating models of operational risk
Review of mathematical statistics	267	(16)
Introduction	267	(1)
Point estimation	268	(7)
Introduction	268	(1)
Measures of quality of estimators	269	(6)
Interval estimation	275	(2)
Tests of hypotheses	277	(3)
Exercises	280	(3)
Parameter estimation	283	(46)
Introduction	283	(3)
Method of moments and percentile matching	286	(3)
Maximum likelihood estimation	289	(8)
Introduction	289	(2)
Complete, individual data	291	(2)
Complete, grouped data	293	(1)
Truncated or censored data	293	(4)
Variance and interval estimation	297	(7)
Bayesian estimation	304	(12)
Definitions and Bayes' theorem	304	(3)
Inference and prediction	307	(8)
Computational issues	315	(1)
Exercises	316	(13)
Estimation for discrete distributions	329	(20)
Introduction	329	(1)
Poisson distribution	329	(4)
Negative binomial distribution	333	(3)
Binomial distribution	336	(2)
The (a, b, 1) class	338	(5)
Compound models	343	(1)
Effect of exposure on maximum likelihood estimation	344	(1)
Exercises	345	(4)
Model selection	349	(34)
Introduction	349	(1)
Representations of the data and model	350	(1)
Graphical comparison of the density and distribution functions	351	(5)
Hypothesis tests	356	(11)
Kolmogorov-Smirnov test	357	(3)
Anderson-Darling test	360	(1)
Chi-square goodness-of-fit test	360	(5)
Likelihood ratio test	365	(2)
Selecting a model	367	(8)
Introduction	367	(1)
Judgment-based approaches	368	(1)
Score-based approaches	368	(7)
Exercises	375	(8)
Fitting extreme value models	383	(12)
Introduction	383	(1)
Parameter estimation	384	(8)
ML estimation from the extreme value distribution	384	(3)
ML estimation from the generalized Pareto distribution	387	(2)
Estimating the Pareto shape parameter	389	(2)
Estimating extreme probabilities	391	(1)
Model selection	392	(3)
Mean excess plots	392	(3)
Fitting copula models	395	(12)
Introduction	395	(1)
Maximum likelihood estimation	396	(2)
Semiparametric estimation of the copula	398	(1)
The role of thresholds	399	(2)
Goodness-of-fit testing	401	(1)
An example	402	(5)
Appendix A Gamma and related functions	407	(4)
Appendix B Discretization of the severity distribution	411	(4)
The method of rounding	411	(1)
Mean preserving	412	(1)
Undiscretization of a discretized distribution	413	(2)
Appendix C Nelder-Mead simplex method	415	(2)
References	417	(9)
Index	426