Nonlife Actuarial Models: Theory, Methods and Evaluation

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ISBN: 9780521764650 | 0521764653
Cover: Hardcover
Copyright: 10/12/2009

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Actuaries must pass exams, but more than that: they must put knowledge into practice. This coherent book gives complete syllabus coverage for Exam C of the Society of Actuaries (SOA) while emphasizing the concepts and practical application of nonlife actuarial models. Ideal for those approaching their professional exams, it is also a class-tested textbook for undergraduate university courses in actuarial science. All the topics that students need to prepare for Exam C are here, including modeling of losses, risk and ruin theory, credibility theory and applications, and empirical implementation of loss models. The book also covers more recent topics, such as risk measures and bootstrapping. Readers are assumed to have studied statistical inference and probability at the introductory undergraduate level. Numerous examples and exercises are provided, with many exercises adapted from past Exam C questions. Computational notes on the use of Excel are included. Teaching slides are available for download.

Preface	p. xiii
Notation and convention	p. xv
Loss models	p. 1
Claim-frequency distribution	p. 3
Claim frequency, claim severity, and aggregate claim	p. 4
Review of statistics	p. 4
Some discrete distributions for claim frequency	p. 6
Binomial distribution	p. 7
Geometric distribution	p. 8
Negative binomial distribution	p. 9
Poisson distribution	p. 11
The (a, b, 0) class of distributions	p. 15
Some methods for creating new distributions	p. 20
Compound distribution	p. 21
Mixture distribution	p. 31
Excel computation notes	p. 34
Summary and conclusions	p. 34
Exercises	p. 36
Claim-severity distribution	p. 41
Review of statistics	p. 42
Survival function and hazard function	p. 42
Mixed distribution	p. 44
Expected value of function of random variable	p. 45
Distribution of function of random variable	p. 46
Some continuous distributions for claim severity	p. 49
Exponential distribution	p. 49
Gamma distribution	p. 50
Weibull distribution	p. 51
Pareto distribution	p. 51
Some methods for creating new distributions	p. 52
Transformation of random variable	p. 53
Mixture distribution	p. 54
Splicing	p. 58
Tail properties of claim severity	p. 59
Effects of coverage modifications	p. 66
Deductible	p. 66
Policy limit	p. 72
Coinsurance	p. 73
Effects of inflation	p. 76
Effects of deductible on claim frequency	p. 77
Excel computation notes	p. 79
Summary and conclusions	p. 81
Exercises	p. 82
Aggregate-loss models	p. 86
Individual risk and collective risk models	p. 87
Individual risk model	p. 88
Exact distribution using convolution	p. 89
Exact distribution using the De Pril recursion	p. 92
Approximations of the individual risk model	p. 94
Collective risk model	p. 96
Properties of compound distributions	p. 96
Panjer recursion	p. 98
Approximations of the collective risk model	p. 100
Compound Poisson distribution and individual risk model	p. 102
Coverage modifications and stop-loss reinsurance	p. 103
Summary and conclusions	p. 108
Exercises	p. 108
Risk and ruin	p. 113
Risk measures	p. 115
Uses of risk measures	p. 116
Some premium-based risk measures	p. 117
Axioms of coherent risk measures	p. 118
Some capital-based risk measures	p. 120
Value-at-Risk (VaR)	p. 120
Conditional tail expectation and related measures	p. 123
More premium-based risk measures	p. 129
Proportional hazard transform and risk-adjusted premium	p. 129
Esscher transform and risk-adjusted premium	p. 132
Distortion-function approach	p. 133
Wang transform	p. 136
Summary and conclusions	p. 138
Exercises	p. 139
Ruin theory	p. 143
Discrete-time surplus and events of ruin	p. 144
Discrete-time ruin theory	p. 145
Ultimate ruin in discrete time	p. 146
Finite-time ruin in discrete time	p. 150
Lundberg's inequality in discrete time	p. 152
Continuous-time surplus function	p. 157
Continuous-time ruin theory	p. 159
Lundberg's inequality in continuous time	p. 159
Distribution of deficit	p. 163
Summary and conclusions	p. 165
Exercises	p. 165
Credibility	p. 169
Classical credibility	p. 171
Framework and notations	p. 172
Full credibility	p. 173
Full credibility for claim frequency	p. 173
Full credibility for claim severity	p. 177
Full credibility for aggregate loss	p. 179
Full credibility for pure premium	p. 181
Partial credibility	p. 182
Variation of assumptions	p. 184
Summary and discussions	p. 185
Exercises	p. 186
Buhlmann credibility	p. 190
Framework and notations	p. 191
Variance components	p. 192
Buhlmann credibility	p. 201
Buhlmann-Straub credibility	p. 208
Summary and discussions	p. 216
Exercises	p. 217
Bayesian approach	p. 223
Bayesian inference and estimation	p. 224
Posterior distribution of parameter	p. 225
Loss function and Bayesian estimation	p. 228
Some examples of Bayesian credibility	p. 230
Conjugate distributions	p. 234
The gamma-Poisson conjugate distribution	p. 235
The beta-geometric conjugate distribution	p. 235
The gamma-exponential conjugate distribution	p. 235
Bayesian versus Buhlmann credibility	p. 235
Linear exponential family and exact credibility	p. 242
Summary and discussions	p. 248
Exercises	p. 248
Empirical implementation of credibility	p. 253
Empirical Bayes method	p. 254
Nonparametric estimation	p. 255
Semiparametric estimation	p. 270
Parametric estimation	p. 271
Summary and discussions	p. 273
Exercises	p. 274
Model construction and evaluation	p. 279
Model estimation and types of data	p. 281
Estimation	p. 282
Parametric and nonparametric estimation	p. 282
Point and interval estimation	p. 282
Properties of estimators	p. 283
Types of data	p. 286
Duration data and loss data	p. 286
Complete individual data	p. 287
Incomplete individual data	p. 289
Grouped data	p. 294
Summary and discussions	p. 296
Exercises	p. 297
Nonparametric model estimation	p. 301
Estimation with complete individual data	p. 302
Empirical distribution	p. 302
Kernel estimation of probability density function	p. 306
Estimation with incomplete individual data	p. 311
Kaplan-Meier (product-limit) estimator	p. 311
Nelson-Aalen estimator	p. 319
Estimation with grouped data	p. 323
Excel computation notes	p. 326
Summary and discussions	p. 326
Exercises	p. 327
Parametric model estimation	p. 335
Methods of moments and percentile matching	p. 336
Method of moments	p. 336
Method of percentile matching	p. 341
Bayesian estimation method	p. 343
Maximum likelihood estimation method	p. 344
Complete individual data	p. 347
Grouped and incomplete data	p. 351
Models with covariates	p. 358
Proportional hazards model	p. 358
Generalized linear model	p. 364
Accelerated failure-time model	p. 365
Modeling joint distribution using copula	p. 366
Excel computation notes	p. 369
Summary and discussions	p. 371
Exercises	p. 372
Model evaluation and selection	p. 380
Graphical methods	p. 381
Misspecification tests and diagnostic checks	p. 385
Kolmogorov-Smirnov test	p. 386
Anderson-Darling test	p. 388
Chi-square goodness-of-fit test	p. 389
Likelihood ratio test	p. 391
Information criteria for model selection	p. 393
Summary and discussions	p. 394
Exercises	p. 395
Basic Monte Carlo methods	p. 400
Monte Carlo simulation	p. 401
Uniform random number generators	p. 402
General random number generators	p. 405
Inversion method	p. 406
Acceptance-rejection method	p. 408
Generation of correlated random variables	p. 411
Specific random number generators	p. 414
Some continuous distributions	p. 414
Some discrete distributions	p. 417
Accuracy and Monte Carlo sample size	p. 418
Variance reduction techniques	p. 421
Antithetic variable	p. 422
Control variable	p. 423
Importance sampling	p. 425
Excel computation notes	p. 426
Summary and discussions	p. 428
Exercises	p. 428
Applications of Monte Carlo methods	p. 435
Monte Carlo simulation for hypothesis test	p. 436
Kolmogorov-Smirnov test	p. 436
Chi-square goodness-of-fit test	p. 438
Bootstrap estimation of p-value	p. 439
Bootstrap estimation of bias and mean squared error	p. 442
A general framework of bootstrap	p. 445
Monte Carlo simulation of asset prices	p. 447
Wiener process and generalized Wiener process	p. 447
Diffusion process and lognormal distribution	p. 448
Jump-diffusion process	p. 453
Summary and discussions	p. 455
Exercises	p. 456
Appendix: Review of statistics	p. 458
Answers to exercises	p. 498
References	p. 518
Index	p. 521
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Nonlife Actuarial Models: Theory, Methods and Evaluation

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