Table of Preface page xiii 1 Introduction 1 1.1 The aim of this book 1 1.2 Notation and prerequisites 2 1.2.1 Probability 2 1.2.2 Statistics 9 1.2.3 Simulation 9 1.2.4 The statistical software package R 9 2 Models for claim numbers and claim sizes 11 2.1 Distributions for claim numbers 12 2.1.1 Poisson distribution 13 2.1.2 Negative binomial distribution 16 2.1.3 Geometric distribution 18 2.1.4 Binomial distribution 20 2.1.5 A summary note on R 22 2.2 Distributions for claim sizes 23 2.2.1 A further summary note on R 24 2.2.2 Normal (Gaussian) distribution 24 2.2.3 Exponential distribution 25 2.2.4 Gamma distribution 28 2.2.5 Fat-tailed distributions 31 2.2.6 Lognormal distribution 35 2.2.7 Pareto distribution 40 2.2.8 Weibull distribution 45 2.2.9 Burr distribution 48 2.2.10 Loggamma distribution 51 2.3 Mixture distributions 54 vii
Table of viii 2.4 Fitting models to claim-number and claim-size data 58 2.4.1 Fitting models to claim numbers 60 2.4.2 Fitting models to claim sizes 65 Exercises 83 3 Short term risk models 90 3.1 The mean and variance of a compound distribution 91 3.2 The distribution of a random sum 93 3.2.1 Convolution series formula for a compound distribution 95 3.2.2 Moment generating function of a compound distribution 98 3.3 Finite mixture distributions 100 3.4 Special compound distributions 103 3.4.1 Compound Poisson distributions 103 3.4.2 Compound mixed Poisson distributions 108 3.4.3 Compound negative binomial distributions 110 3.4.4 Compound binomial distributions 114 3.5 Numerical methods for compound distributions 115 3.5.1 Panjer recursion algorithm 116 3.5.2 The fast Fourier transform algorithm 119 3.6 Approximations for compound distributions 124 3.6.1 Approximations based on a few moments 125 3.6.2 Asymptotic approximations 126 3.7 Statistics for compound distributions 128 3.8 The individual risk model 134 3.8.1 The mean and variance for the individual risk model 136 3.8.2 The distribution function and moment generating function for the individual risk model 137 3.8.3 Approximations for the individual risk model 139 Exercises 140 4 Model based pricing setting premiums 147 4.1 Premium calculation principles 148 4.1.1 The expected value principle (EVP) 148 4.1.2 The standard deviation principle (SDP) 149 4.1.3 The variance principle (VP) 149 4.1.4 The quantile principle (QP) 149 4.1.5 The zero utility principle (ZUP) 150
Table of ix 4.1.6 The exponential premium principle (EPP) 150 4.1.7 Some desirable properties of premium calculation principles 152 4.1.8 Other premium calculation principles 154 4.2 Maximum and minimum premiums 155 4.3 Introduction to credibility theory 156 4.4 Bayesian estimation 157 4.4.1 The posterior distribution 158 4.4.2 The wider context of decision theory 159 4.4.3 The binomial/beta model 161 4.4.4 The Poisson/gamma model 163 4.4.5 The normal/normal model 165 4.5 Bayesian credibility theory 169 4.5.1 Bayesian credibility estimates under the Poisson/gamma model 170 4.5.2 Bayesian credibility premiums under the normal/normal model 172 4.6 Empirical Bayesian credibility theory: Model 1 the Bühlmann model 176 4.7 Empirical Bayesian credibility theory: Model 2 the Bühlmann Straub model 185 Exercises 196 5 Risk sharing reinsurance and deductibles 205 5.1 Excess of loss reinsurance 206 5.1.1 Reinsurance claims 210 5.1.2 Simulation results 212 5.1.3 Aggregate claims model with excess of loss reinsurance 213 5.2 Proportional reinsurance 221 5.3 Deductibles (policy excesses) 223 5.4 Retention levels and reinsurance costs 226 5.5 Optimising the reinsurance contract 228 5.6 Optimising reinsurance contracts based on maximising expected utility 228 5.6.1 Excess of loss reinsurance 229 5.6.2 Proportional reinsurance 231 5.7 Optimising reinsurance contracts based on minimising the variance of aggregate claims 234 5.7.1 Minimising Var[S I ] subject to fixed E[S I ] 235
Table of x 5.7.2 Minimising Var[S R ] subject to fixed Var[S I ] 236 5.7.3 Comparing stop loss and equivalent proportional reinsurance arrangements 237 5.7.4 Minimising Var[S I ] + Var[S R ] 238 5.7.5 Minimising the sum of variances when two independent risks are shared between two insurers 239 5.8 Optimising reinsurance contracts for a group of independent risks based on minimising the variance of the direct insurer s net profit finding the optimal relative retentions 247 5.8.1 Optimal relative retentions in the case of excess of loss reinsurance 247 5.8.2 Optimal relative retentions in the case of proportional reinsurance 251 Exercises 253 6 Ruin theory for the classical risk model 267 6.1 The classical risk model 267 6.1.1 The relative safety loading 269 6.1.2 Ruin probabilities 270 6.2 Lundberg s inequality and the adjustment coefficient 272 6.2.1 Properties of the adjustment coefficient 272 6.2.2 Proof of Lundberg s inequality 276 6.2.3 When does the adjustment coefficient exist? 279 6.3 Equations for ψ(u) and ϕ(u): the ruin probability and the survival probability 282 6.4 Compound geometric representations for ψ(u) and ϕ(u): the ruin probability and the survival probability 291 6.5 Asymptotics for the probability of ruin 296 6.6 Numerical methods for ruin quantities 303 6.6.1 Numerical calculation of the adjustment coefficient 303 6.6.2 Numerical calculation of the probability of ruin 305 6.7 Statistics for ruin quantities 308 Exercises 310 7 Case studies 316 7.1 Case study 1: comparing premium setting principles 316 7.1.1 Case 1 in the presence of an assumed model 316
Table of xi 7.1.2 Case 2 without model assumptions, using bootstrap resampling 322 7.2 Case study 2: shared liabilities who pays what? 332 7.2.1 Case 1 exponential losses 333 7.2.2 Case 2 Pareto losses 338 7.2.3 Case 3 lognormal losses 344 7.3 Case study 3: reinsurance and ruin 348 7.3.1 Introduction 348 7.3.2 Proportional reinsurance 351 7.3.3 Proportional reinsurance with exponential claim sizes 353 7.3.4 Excess of loss reinsurance in a layer 356 7.3.5 Excess of loss reinsurance in a layer with exponential claim sizes 360 Appendix A Utility theory 368 Appendix B Answers to exercises 380 References 386 Index 389