Modern Actuarial Risk Theory (eBook)

Foreword to the First Edition 5
Preface to the Second Edition 6
Contents 14
Utility theory and insurance 18
1.1 Introduction 18
1.2 The expected utility model 19
1.3 Classes of utility functions 22
1.4 Stop-loss reinsurance 25
1.5 Exercises 30
The individual risk model 34
2.1 Introduction 34
2.2 Mixed distributions and risks 35
2.3 Convolution 42
2.4 Transforms 45
2.5 Approximations 47
2.5.1 Normal approximation 47
2.5.2 Translated gamma approximation 49
2.5.3 NP approximation 50
2.6 Application: optimal reinsurance 52
2.7 Exercises 53
Collective risk models 58
3.1 Introduction 58
3.2 Compound distributions 59
3.2.1 Convolution formula for a compound cdf 61
3.3 Distributions for the number of claims 62
3.4 Properties of compound Poisson distributions 64
3.5 Panjer’s recursion 66
3.6 Compound distributions and the Fast Fourier Transform 71
3.7 Approximations for compound distributions 74
3.8 Individual and collective risk model 76
3.9 Loss distributions: properties, estimation, sampling 78
3.9.1 Techniques to generate pseudo-random samples 79
3.9.2 Techniques to compute ML-estimates 80
3.9.3 Poisson claim number distribution 80
3.9.4 Negative binomial claim number distribution 81
3.9.5 Gamma claim severity distributions 83
3.9.6 Inverse Gaussian claim severity distributions 84
3.9.7 Mixtures/combinations of exponential distributions 86
3.9.8 Lognormal claim severities 88
3.9.9 Pareto claim severities 89
3.10 Stop-loss insurance and approximations 90
3.10.1 Comparing stop-loss premiums in case of unequal variances 93
3.11 Exercises 95
Ruin theory 104
4.1 Introduction 104
4.2 The classical ruin process 106
4.3 Some simple results on ruin probabilities 108
4.4 Ruin probability and capital at ruin 112
4.5 Discrete time model 115
4.6 Reinsurance and ruin probabilities 116
4.7 Beekman’s convolution formula 118
4.8 Explicit expressions for ruin probabilities 123
4.9 Approximation of ruin probabilities 125
4.10 Exercises 128
Premium principles and Risk measures 132
5.1 Introduction 132
5.2 Premium calculation from top-down 133
5.3 Various premium principles and their properties 136
5.3.1 Properties of premium principles 137
5.4 Characterizations of premium principles 139
5.5 Premium reduction by coinsurance 142
5.6 Value-at-Risk and related risk measures 143
5.7 Exercises 150
Bonus-malus systems 152
6.1 Introduction 152
6.2 A generic bonus-malus system 153
6.3 Markov analysis 155
6.3.1 Loimaranta ef.ciency 158
6.4 Finding steady state premiums and Loimaranta efficiency 159
6.5 Exercises 163
Ordering of risks 165
7.1 Introduction 165
7.2 Larger risks 168
7.3 More dangerous risks 170
7.3.1 Thicker-tailed risks 170
7.3.2 Stop-loss order 175
7.3.3 Exponential order 176
7.3.4 Properties of stop-loss order 176
7.4 Applications 180
7.4.1 Individual versus collective model 180
7.4.2 Ruin probabilities and adjustment coefficients 180
7.4.3 Order in two-parameter families of distributions 182
7.4.4 Optimal reinsurance 184
7.4.5 Premiums principles respecting order 185
7.4.6 Mixtures of Poisson distributions 185
7.4.7 Spreading of risks 186
7.4.8 Transforming several identical risks 186
7.5 Incomplete information 187
7.6 Comonotonic random variables 192
7.7 Stochastic bounds on sums of dependent risks 199
7.7.1 Sharper upper and lower bounds derived from a surrogate 199
7.7.2 Simulating stochastic bounds for sums of lognormal risks 202
7.8 More related joint distributions copulas
7.8.1 More related distributions association measures
7.8.2 Copulas 210
7.9 Exercises 212
Credibility theory 219
8.1 Introduction 219
8.2 The balanced Bühlmann model 220
8.3 More general credibility models 227
8.4 The Bühlmann-Straub model 230
8.4.1 Parameter estimation in the Bühlmann- Straub model 233
8.5 Negative binomial model for the number of car insurance claims 238
8.6 Exercises 243
Generalized linear models 246
9.1 Introduction 246
9.2 Generalized Linear Models 249
9.3 Some traditional estimation procedures and GLMs 252
9.4 Deviance and scaled deviance 260
9.5 Case study I: Analyzing a simple automobile portfolio 263
9.6 Case study II: Analyzing a bonus-malus system using GLM 267
9.6.1 GLM analysis for the total claims per policy 272
9.7 Exercises 277
IBNR techniques 280
10.1 Introduction 280
10.2 Two time-honored IBNR methods 283
10.2.1 Chain ladder 283
10.2.2 Bornhuetter-Ferguson 285
10.3 A GLM that encompasses various IBNR methods 286
10.3.1 Chain ladder method as a GLM 287
10.3.2 Arithmetic and geometric separation methods 288
10.3.3 De Vijlder’s least squares method 289
10.4 Illustration of some IBNR methods 291
10.4.1 Modeling the claim numbers in Table 10.1 292
10.4.2 Modeling claim sizes 294
10.5 Solving IBNR problems by R 296
10.6 Variability of the IBNR estimate 298
10.6.1 Bootstrapping 300
10.6.2 Analytical estimate of the prediction error 303
10.7 An IBNR-problem with known exposures 305
10.8 Exercises 307
More on GLMs 311
11.1 Introduction 311
11.2 Linear Models and Generalized Linear Models 311
11.3 The Exponential Dispersion Family 314
11.4 Fitting criteria 319
11.4.1 Residuals 319
11.4.2 Quasi-likelihood and quasi-deviance 320
11.4.3 Extended quasi-likelihood 322
11.5 The canonical link 324
11.6 The IRLS algorithm of Nelder and Wedderburn 326
11.6.1 Theoretical description 327
11.6.2 Step-by-step implementation 329
11.7 Tweedie’s Compound Poisson–gamma distributions 331
11.7.1 Application to an IBNR problem 332
11.8 Exercises 334
The ‘R’ in Modern ART 338
A.1 A short introduction to R 338
A.2 Analyzing a stock portfolio using R 345
A.3 Generating a pseudo-random insurance portfolio 351
Hints for the exercises 354
CHAPTER 1 354
CHAPTER 2 355
CHAPTER 3 357
CHAPTER 4 361
CHAPTER 5 363
CHAPTER 6 364
CHAPTER 7 364
CHAPTER 8 367
CHAPTER 9 368
CHAPTER 10 369
CHAPTER 11 369
Notes and references 370
CHAPTER 1 370
CHAPTER 2 370
CHAPTER 3 371
CHAPTER 4 371
CHAPTER 5 371
CHAPTER 6 372
CHAPTER 7 372
CHAPTER 8 372
CHAPTER 9 373
CHAPTER 10 373
CHAPTER 11 374
APPENDIX A 374
REFERENCES 374
Tables 380
Index 384