Loss Models: From Data to Decisions, 2nd Edition

by Stuart A. Klugman (Drake University, Des Moines, IA ); Harry H. Panjer (Univ. of Waterloo, Ontario, Canada ); Gordon E. Willmot (Univ. of Waterloo, Ontario, Canada )

• ISBN13: 9780471215776
• ISBN10: 0471215775
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-08-01
• Publisher: Wiley-Interscience

Summary

Revised, updated, and even more useful to students, teachers, and practicing professionals The First Edition of Loss Models was deemed "worthy of classical status" by the Journal of the International Statistical Institute. While retaining its predecessor's thorough treatment of the concepts and methods of analyzing contingent events, this powerful Second Edition is updated and expanded to offer even more complete and flexible coverage of risk theory, loss distributions, and survival models. Beginning with a framework for model building and a description of frequency and severity loss data typically available, it shows readers how to combine frequency, severity, and loss models to build aggregate loss models and credibility-based pricing models, and how to analyze loss over multiple time periods. Important features of this new edition include: Thorough preparation for relevant parts of preliminary examinations of the Society of Actuaries (SOA) and Casualty Actuarial Society (CAS) Exercises based on past SOA and CAS exams Examples using actual insurance data Practical treatment of modern credibility theory Data files and more from an ftp site Loss Models, Second Edition is an important resource, providing a comprehensive, practically motivated toolkit and an excellent reference, for actuaries preparing for SOA and CAS preliminary examinations, students in actuarial science who need to understand loss and risk models, and practicing professionals involved in loss modeling.

Author Biography

STUART A. KLUGMAN, PhD, FSA, is Principal Financial Group Professor of Actuarial Science at Drake University, Des Moines, Iowa. <BR> HARRY H. PANJER, PhD, FSA, FCIA, HonFIA, is Professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Ontario, Canada. <BR> GORDON E. WILLMOT, PhD, FSA, FCIA, is Munich Re Professor in the Department of Statistics and Actuarial Science at the University of Waterloo.

Table of Contents

Preface	xvii
Acknowledgments	xxi
Part I Introduction
Modeling	3	(8)
The model-based approach	3	(3)
The modeling process	3	(2)
The modeling advantage	5	(1)
Organization of this book	6	(5)
Part II Actuarial models
Random variables	11	(14)
Introduction	11	(2)
Key functions and four models	13	(12)
Exercises	23	(2)
Basic distributional quantities	25	(14)
Moments	25	(9)
Exercises	33	(1)
Percentiles	34	(2)
Exercises	35	(1)
Generating functions and sums of random variables	36	(3)
Exercises	37	(2)
Classifying and creating distributions	39	(76)
Introduction	39	(1)
The role of parameters	40	(8)
Parametric and scale distributions	41	(1)
Parametric distribution families	42	(1)
Finite mixture distributions	43	(2)
Data-dependent distributions	45	(1)
Exercises	46	(2)
Tail weight	48	(8)
Existence of moments	48	(2)
Limiting ratios	50	(1)
Hazard rate and mean residual life patterns	50	(4)
Exercises	54	(2)
Creating new distributions	56	(13)
Introduction	56	(1)
Multiplication by a constant	56	(1)
Raising to a power	57	(1)
Exponentiation	58	(1)
Mixing	59	(3)
Frailty models	62	(2)
Splicing	64	(1)
Exercises	65	(4)
Selected distributions and their relationships	69	(3)
Introduction	69	(1)
Two parametric families	69	(1)
Limiting distributions	70	(2)
Exercises	72	(1)
Discrete distributions	72	(43)
Introduction	72	(1)
The Poisson distribution	73	(3)
The negative binomial distribution	76	(3)
The binomial distribution	79	(2)
The (a, b, 0) class	81	(2)
Truncation and modification at zero	83	(5)
Compound frequency models	88	(7)
Further properties of the compound Poisson class	95	(6)
Mixed frequency models	101	(2)
Poisson mixtures	103	(5)
Effect of exposure on frequency	108	(1)
An inventory of discrete distributions	109	(1)
Exercises	109	(6)
Frequency and severity with coverage modifications	115	(20)
Introduction	115	(1)
Deductibles	115	(6)
Exercises	120	(1)
The loss elimination ratio and the effect of inflation for ordinary deductibles	121	(3)
Exercises	123	(1)
Policy limits	124	(2)
Exercises	126	(1)
Coinsurance, deductibles, and limits	126	(3)
Exercises	128	(1)
The impact of deductibles on claim frequency	129	(6)
Exercises	133	(2)
Aggregate loss models	135	(74)
Introduction	135	(4)
Exercise	138	(1)
Model choices	139	(1)
Exercises	140	(1)
The compound model for aggregate claims	140	(13)
Exercises	148	(5)
Analytic results	153	(6)
Exercises	156	(3)
Computing the aggregate claims distribution	159	(2)
The recursive method	161	(13)
Applications to compound frequency models	162	(3)
Underflow/overflow problems	165	(1)
Numerical stability	166	(1)
Continuous severity	166	(1)
Constructing arithmetic distributions	167	(3)
Exercises	170	(4)
The impact of individual policy modifications on aggregate payments	174	(4)
Exercises	177	(1)
Calculations with approximate distributions	178	(6)
Arithmetic distributions	178	(3)
Empirical distributions	181	(1)
Piecewise linear cdf	182	(2)
Exercises	184	(1)
Inversion methods	184	(6)
Fast Fourier transform	185	(3)
Direct numerical inversion	188	(2)
Exercise	190	(1)
Comparison of methods	190	(2)
The individual risk model	192	(17)
Parametric approximation	192	(3)
Exact calculation of the aggregate distribution	195	(6)
Compound Poisson approximation	201	(4)
Exercises	205	(4)
Discrete-time ruin models	209	(14)
Introduction	209	(1)
Process models for insurance	210	(5)
Processes	210	(2)
An insurance model	212	(1)
Ruin	213	(2)
Discrete, finite-time ruin probabilities	215	(8)
The discrete-time process	215	(1)
Evaluating the probability of ruin	216	(6)
Exercises	222	(1)
Continuous-time ruin models	223	(42)
Introduction	223	(2)
The Poisson process	223	(2)
The continuous-time problem	225	(1)
The adjustment coefficient and Lundberg's inequality	225	(9)
The adjustment coefficient	226	(4)
Lundberg's inequality	230	(2)
Exercises	232	(2)
An integrodifferential equation	234	(5)
Exercises	239	(1)
The maximum aggregate loss	239	(5)
Exercises	242	(2)
Cramer's asymptotic ruin formula and Tijms' approximation	244	(8)
Exercises	250	(2)
The Brownian motion risk process	252	(4)
Brownian motion and the probability of ruin	256	(9)
Part III Construction of empirical models
Review of mathematical statistics	265	(18)
Introduction	265	(1)
Point estimation	266	(9)
Introduction	266	(1)
Measures of quality	267	(6)
Exercises	273	(2)
Interval estimation	275	(2)
Exercise	277	(1)
Tests of hypotheses	277	(6)
Exercise	281	(2)
Estimation for complete data	283	(14)
Introduction	283	(5)
The empirical distribution for complete, individual data	288	(4)
Exercises	291	(1)
Empirical distributions for grouped data	292	(5)
Exercises	295	(2)
Estimation for modified data	297	(34)
Point estimation	297	(8)
Exercises	304	(1)
Means, variances, and interval estimation	305	(11)
Exercises	315	(1)
Kernel density models	316	(6)
Exercises	321	(1)
Approximations for large data sets	322	(9)
Introduction	322	(1)
Kaplan-Meier type approximations	323	(1)
Multiple-decrement tables	324	(1)
Exercises	325	(6)
Part IV Parametric statistical methods
Parameter estimation	331	(88)
Method of moments and percentile matching	331	(6)
Exercises	335	(2)
Maximum likelihood estimation	337	(14)
Introduction	337	(3)
Complete, individual data	340	(1)
Complete, grouped data	341	(1)
Truncated or censored data	341	(4)
Exercises	345	(6)
Variance and interval estimation	351	(9)
Exercises	358	(2)
Bayesian estimation	360	(23)
Definitions and Bayes' theorem	360	(4)
Inference and prediction	364	(7)
Conjugate prior distributions and the linear exponential family	371	(3)
Computational issues	374	(3)
Exercises	377	(6)
Estimation for discrete distributions	383	(19)
Poisson	383	(3)
Negative binomial	386	(3)
Binomial	389	(3)
The (a, b, 1) class	392	(4)
Compound models	396	(2)
Effect of exposure on maximum likelihood estimation	398	(1)
Exercises	399	(3)
Bivariate models	402	(3)
Introduction	402	(1)
Copulas	403	(2)
Exercise	405	(1)
Models with covariates	405	(14)
Introduction	405	(2)
Proportional hazards models	407	(6)
The generalized linear and accelerated failure time models	413	(3)
Exercises	416	(3)
Model selection	419	(36)
Introduction	419	(1)
Representations of the data and model	420	(1)
Graphical comparison of the density and distribution functions	421	(6)
Exercises	426	(1)
Hypothesis tests	427	(13)
Kolmogorov--Smirnov test	428	(2)
Anderson--Darling test	430	(2)
Chi-square goodness-of-fit test	432	(4)
Likelihood ratio test	436	(2)
Exercises	438	(2)
Selecting a model	440	(15)
Introduction	440	(1)
Judgment-based approaches	441	(1)
Score-based approaches	442	(7)
Exercises	449	(6)
Five examples	455	(28)
Introduction	455	(1)
Time to death	455	(4)
The data	455	(2)
Some calculations	457	(2)
Exercise	459	(1)
Time from incidence to report	459	(2)
The problem and some data	460	(1)
Analysis	460	(1)
Payment amount	461	(5)
The data	462	(1)
The first model	463	(2)
The second model	465	(1)
An aggregate loss example	466	(5)
Another aggregate loss example	471	(5)
Distribution for a single policy	471	(1)
One hundred policies---excess of loss	471	(2)
One hundred policies---aggregate stop-loss	473	(1)
Numerical convolutions	474	(2)
Comprehensive exercises	476	(7)
Part V Adjusted estimates and simulation
Interpolation and smoothing	483	(32)
Introduction	483	(2)
Polynomial interpolation and smoothing	485	(4)
Exercises	488	(1)
Cubic spline interpolation	489	(11)
Construction of cubic splines	491	(8)
Exercises	499	(1)
Approximating functions with splines	500	(4)
Exercise	504	(1)
Extrapolating with splines	504	(1)
Exercise	505	(1)
Smoothing splines	505	(10)
Exercise	514	(1)
Credibility	515	(96)
Introduction	515	(2)
Statistical concepts	517	(13)
Conditional distributions	518	(2)
Conditional expectation	520	(3)
Nonparametric unbiased estimators	523	(6)
Exercises	529	(1)
Limited fluctuation credibility theory	530	(12)
Full credibility	532	(3)
Partial credibility	535	(4)
Problems with the approach	539	(1)
Notes and References	539	(1)
Exercises	539	(3)
Greatest accuracy credibility theory	542	(47)
Introduction	542	(3)
The Bayesian methodology	545	(8)
The credibility premium	553	(4)
The Buhlmann model	557	(3)
The Buhlmann--Straub model	560	(6)
Exact credibility	566	(3)
Linear versus Bayesian versus no credibility	569	(8)
Notes and References	577	(1)
Exercises	578	(11)
Empirical Bayes parameter estimation	589	(22)
Nonparametric estimation	592	(8)
Semiparametric estimation	600	(2)
Parametric estimation	602	(5)
Notes and References	607	(1)
Exercises	607	(4)
Simulation	611	(16)
Basics of simulation	611	(7)
The simulation approach	612	(5)
Exercises	617	(1)
Examples of simulation in actuarial modeling	618	(9)
Aggregate loss calculations	618	(1)
Examples of lack of independence or identical distributions	619	(1)
Simulation analysis of the two examples	620	(2)
Statistical analyses	622	(3)
Exercises	625	(2)
Appendix A An inventory of continuous distributions	627	(16)
A.1 Introduction	627	(4)
A.2 Transformed beta family	631	(4)
A.2.1 Four-parameter distribution	631	(1)
A.2.2 Three-parameter distributions	631	(2)
A.2.3 Two-parameter distributions	633	(2)
A.3 Transformed gamma family	635	(3)
A.3.1 Three-parameter distributions	635	(1)
A.3.2 Two-parameter distributions	636	(2)
A.3.3 One-parameter distributions	638	(1)
A.4 Other distributions	638	(2)
A.5 Distributions with finite support	640	(3)
Appendix B An inventory of discrete distributions	643	(8)
B.1 Introduction	643	(1)
B.2 The (a, b, 0) class	644	(1)
B.3 The (a, b, 1) class	645	(3)
B.3.1 The zero-truncated subclass	645	(2)
B.3.2 The zero-modified subclass	647	(1)
B.4 The compound class	648	(2)
B.4.1 Some compound distributions	648	(2)
B.5 A hierarchy of discrete distributions	650	(1)
Appendix C Frequency and severity relationships	651	(2)
Appendix D The recursive formula	653	(2)
Appendix E Discretization of the severity distribution	655	(4)
E.1 The method of rounding	655	(1)
E.2 Mean preserving	656	(1)
E.3 Undiscretization of a discretized distribution	657	(2)
Appendix F Numerical optimization and solution of systems of equations	659	(12)
F.1 Maximization using Solver	660	(4)
F.2 The simplex method	664	(1)
F.3 Using Excel® to solve equations	665	(6)
References	671	(10)
Index	681