Introduction Models for claim numbers and claim sizes

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1 Table of Preface page xiii 1 Introduction The aim of this book Notation and prerequisites Probability Statistics Simulation The statistical software package R 9 2 Models for claim numbers and claim sizes Distributions for claim numbers Poisson distribution Negative binomial distribution Geometric distribution Binomial distribution A summary note on R Distributions for claim sizes A further summary note on R Normal (Gaussian) distribution Exponential distribution Gamma distribution Fat-tailed distributions Lognormal distribution Pareto distribution Weibull distribution Burr distribution Loggamma distribution Mixture distributions 54 vii

2 Table of viii 2.4 Fitting models to claim-number and claim-size data Fitting models to claim numbers Fitting models to claim sizes 65 Exercises 83 3 Short term risk models The mean and variance of a compound distribution The distribution of a random sum Convolution series formula for a compound distribution Moment generating function of a compound distribution Finite mixture distributions Special compound distributions Compound Poisson distributions Compound mixed Poisson distributions Compound negative binomial distributions Compound binomial distributions Numerical methods for compound distributions Panjer recursion algorithm The fast Fourier transform algorithm Approximations for compound distributions Approximations based on a few moments Asymptotic approximations Statistics for compound distributions The individual risk model The mean and variance for the individual risk model The distribution function and moment generating function for the individual risk model Approximations for the individual risk model 139 Exercises Model based pricing setting premiums Premium calculation principles The expected value principle (EVP) The standard deviation principle (SDP) The variance principle (VP) The quantile principle (QP) The zero utility principle (ZUP) 150

3 Table of ix The exponential premium principle (EPP) Some desirable properties of premium calculation principles Other premium calculation principles Maximum and minimum premiums Introduction to credibility theory Bayesian estimation The posterior distribution The wider context of decision theory The binomial/beta model The Poisson/gamma model The normal/normal model Bayesian credibility theory Bayesian credibility estimates under the Poisson/gamma model Bayesian credibility premiums under the normal/normal model Empirical Bayesian credibility theory: Model 1 the Bühlmann model Empirical Bayesian credibility theory: Model 2 the Bühlmann Straub model 185 Exercises Risk sharing reinsurance and deductibles Excess of loss reinsurance Reinsurance claims Simulation results Aggregate claims model with excess of loss reinsurance Proportional reinsurance Deductibles (policy excesses) Retention levels and reinsurance costs Optimising the reinsurance contract Optimising reinsurance contracts based on maximising expected utility Excess of loss reinsurance Proportional reinsurance Optimising reinsurance contracts based on minimising the variance of aggregate claims Minimising Var[S I ] subject to fixed E[S I ] 235

4 Table of x Minimising Var[S R ] subject to fixed Var[S I ] Comparing stop loss and equivalent proportional reinsurance arrangements Minimising Var[S I ] + Var[S R ] Minimising the sum of variances when two independent risks are shared between two insurers 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 Optimal relative retentions in the case of excess of loss reinsurance Optimal relative retentions in the case of proportional reinsurance 251 Exercises Ruin theory for the classical risk model The classical risk model The relative safety loading Ruin probabilities Lundberg s inequality and the adjustment coefficient Properties of the adjustment coefficient Proof of Lundberg s inequality When does the adjustment coefficient exist? Equations for ψ(u) and ϕ(u): the ruin probability and the survival probability Compound geometric representations for ψ(u) and ϕ(u): the ruin probability and the survival probability Asymptotics for the probability of ruin Numerical methods for ruin quantities Numerical calculation of the adjustment coefficient Numerical calculation of the probability of ruin Statistics for ruin quantities 308 Exercises Case studies Case study 1: comparing premium setting principles Case 1 in the presence of an assumed model 316

5 Table of xi Case 2 without model assumptions, using bootstrap resampling Case study 2: shared liabilities who pays what? Case 1 exponential losses Case 2 Pareto losses Case 3 lognormal losses Case study 3: reinsurance and ruin Introduction Proportional reinsurance Proportional reinsurance with exponential claim sizes Excess of loss reinsurance in a layer 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