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	206
	7.8.1 More related distributions	206
	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