Reinsurance

Hansjörg Albrecher, PhD, is a professor in the Department of Actuarial Science at the University of Lausanne. Jan Beirlant, PhD, is a professor in the Department of Mathematics at the Katholieke Universiteit Leuven, Belgium and at the University of the Free State, South Africa. Jozef L. Teugels, PhD, is a professor in the Department of Mathematics at the Katholieke Universiteit Leuven, Belgium.

• Preface ix1 Introduction 11.1 What is Reinsurance? 11.2 Why Reinsurance? 21.3 Reinsurance Data 41.3.1 Case Study I: Motor Liability Data 51.3.2 Case Study II: Dutch Fire Insurance Data 101.3.3 Case Study III: Austrian Storm Claim Data 101.3.4 Case Study IV: European Flood Risk Data 111.3.5 Case Study V: Groningen Earthquakes 121.3.6 Case Study VI: Danish Fire Insurance Data 121.4 Notes and Bibliography 162 Reinsurance Forms and their Properties 19 2.1 Quota-share Reinsurance 192.1.1 Some Practical Considerations 202.2 Surplus Reinsurance 212.3 Excess-of-loss Reinsurance 242.3.1 Moment Calculations 252.3.2 Reinstatements 272.3.3 Further Practical Considerations 292.4 Stop-loss Reinsurance 302.5 Large Claim Reinsurance 312.6 Combinations of Reinsurance Forms and Global Protections 322.7 Facultative Contracts 332.8 Notes and Bibliography 333 Models for Claim Sizes 35 3.1 Tails of Distributions 353.2 Large Claims 363.3 Common Claim Size Distributions 403.3.1 Light-tailed Models 413.3.2 Heavy-tailed Models 443.4 Mean Excess Analysis 493.5 Full Models: Splicing 503.6 Multivariate Modelling of Large Claims 524 Statistics for Claim Sizes 594.1 Heavy or Light Tails: QQ- and Derivative Plots 604.2 Large Claims Modelling through Extreme Value AnalysisEVA for Pareto-type Tails 634.2.1 EVA for Pareto-type Tails 634.2.2 General Tail Modelling using EVA 824.2.3 EVA under Upper-truncation 914.3 Global Fits: Splicing, Upper-truncation and Interval Censoring 974.3.1 Tail-mixed Erlang Splicing 974.3.2 Tail-mixed Erlang Splicing under Censoring and Upper-truncation 994.4 Incorporating Covariate Information 1144.4.1 Pareto-type Modelling 1144.4.2 Generalized Pareto Modelling 1164.4.3 Regression Extremes with Censored Data 1194.5 Multivariate Analysis of Claim Distributions 1234.5.1  The Multivariate POT Approach 1244.5.2 Multivariate Mixtures of Erlangs 1254.6 Estimation of Other Tail Characteristics 1284.7 Further Case Studies 1324.8 Notes and Bibliography 1375 Models for Claim Counts 1395.1 General Treatment 1395.1.1 Main Properties of the Claim Number Process 1405.2 The Poisson Process and its Extensions 1415.2.1 The Homogeneous Poisson Process 1415.2.2 Inhomogeneous Poisson Processes 1435.2.3 Mixed Poisson Processes 1445.2.4 Doubly Stochastic Poisson Processes 1495.3 Other Claim Number Processes 1575.3.1 The Nearly Mixed Poisson Model 1575.3.2 Infinitely Divisible Processes 1585.3.3 The Renewal Model 1605.3.4 Markov Models 1615.4 Discrete Claim Counts 1615.5 Statistics of Claim Counts 1645.5.1 Modelling Yearly Claim Counts 1645.5.2 Modelling the Claim Arrival Process 1725.6  Claim Numbers under Reinsurance 1835.6.1  Number of Claims under Excess-loss Reinsurance 1835.7Notes and Bibliography 1876 Total Claim Amount 1896.1 General Formulas for Aggregating Independent Risks 1896.2 Classical Approximations for the Total Claim Size 1916.2.1 Approximations based on the First Few Moments 1916.2.2 Asymptotic Approximations for Light-tailed Claims 1936.2.3 Asymptotic Approximations for Heavy-tailed Claims 1986.3 Panjer Recursion 1996.4 Fast Fourier Transform 2006.5 Total Claim Amount under Reinsurance 2016.5.1 Proportional Reinsurance 2016.5.2 Excess-loss Reinsurance 2026.5.3 Stop-loss Reinsurance 2046.6 Numerical Illustrations 2066.7 Aggregation for Dependent Risks 2086.8 Notes and Bibliography 2127 Reinsurance Pricing 2177.1 Classical Principles of Premium Calculation 2197.2 Solvency Considerations 2197.2.1 The Ruin Probability 2237.2.2 One-year Time Horizon and Cost of Capital 2267.3 Pricing Proportional Reinsurance 2287.4 Pricing Non-proportional Reinsurance 2297.4.1 Exposure Rating 2297.4.2 Experience Rating 2327.4.3 Aggregate Pure Premium 2347.5 The Aggregate Risk Margin 2357.6 Leading and Following Reinsurers 2377.7 Notes and Bibliography 2388 Choice of Reinsurance 2418.1 Decision Criteria 2438.2 Classical Optimality Results 2458.2.1 Pareto-optimal Risk Sharing 2458.2.2 Stochastic Ordering 2478.2.3 Minimizing Retained Variance 2488.2.4 Maximizing Expected Utility 2518.2.5 Minimizing the Ruin Probability 2538.2.6 Combining Reinsurance Treaties over Subportfolios8.3 Solvency Constraints and Cost of Capital 2598.4 Minimizing Other Risk Measures 2618.5 Combining Reinsurance Treaties 2628.6 Reinsurance Chains 2638.7 Dynamic Reinsurance 2648.8 Beyond Piecewise Linear Contracts 2668.9 Notes and Bibliography 2689 Simulation 2739.1 The Monte Carlo Method 2739.2 Variance Reduction Techniques 2769.2.1 Conditional Monte Carlo 2779.2.2 Importance Sampling 2779.3 Quasi-Monte Carlo Techniques 2839.4 Notes and Bibliography 28810 Further Topics 291 10.1 More on Large Claim Reinsurance 29110.1.1 The Ordered Claims 29110.1.2 Large Claim Reinsurance 29610.1.3 ECOMOR 29810.2 Alternative Risk Transfer 30010.2.1 Notes and Bibliography 30410.3 Reinsurance and Finance 30510.4 Catastrophic Risk 306References 309Index 347