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1. Introduction 1; 1.1 What is Reinsurance? ; 1.2 Why Reinsurance? ; 1.3 Reinsurance Data; 1.3.1 Case Study I: Motor Liability Data ; 1.3.2 Case Study II: Dutch Fire Insurance Data ; 1.3.3 Case Study III: Austrian Storm Claim Data; 1.3.4 Case Study IV: European Flood Risk Data; 1.3.5 Case Study V: Groningen Earthquakes; 1.3.6 Case Study VI: Danish Fire Insurance Data; 1.4 Notes and Bibliography.

2 Reinsurance Forms and their Properties; 2.1 Quota-share Reinsurance; 2.1.1 Some Practical Considerations; 2.2 Surplus Reinsurance; 2.3 Excess-of-loss Reinsurance; 2.3.1 Moment Calculations; 2.3.2 Reinstatements; 2.3.3 Further Practical Considerations; 2.4 Stop-loss Reinsurance; 2.5 Large Claim Reinsurance; 2.6 Combinations of Reinsurance Forms and Global Protections; 2.7 Facultative Contracts; 2.8 Notes and Bibliography.

3 Models for Claim Sizes; 3.1 Tails of Distributions; 3.2 Large Claims; 3.3 Common Claim Size Distributions; 3.3.1 Light-tailed Models; 3.3.2 Heavy-tailed Models; 3.4 Mean Excess Analysis; 3.5 Full Models: Splicing; 3.6 Multivariate Modelling of Large Claims.

4 Statistics for Claim Sizes; 4.1 Heavy or Light Tails: QQ- and Derivative Plots; 4.2 Large Claims Modelling through Extreme Value Analysis; 4.2.1 EVA for Pareto-type Tails; 4.2.2 General Tail Modelling using EVA; 4.2.3 EVA under Upper-truncation; 4.3 Global Fits: Splicing, Upper-truncation and Interval Censoring; 4.3.1 Tail-mixed Erlang Splicing; 4.3.2 Tail-mixed Erlang Splicing under Censoring and Upper-truncation; 4.4 Incorporating Covariate Information; 4.4.1 Pareto-type Modelling; 4.4.2 Generalized Pareto Modelling; 4.4.3 Regression Extremes with Censored Data; 4.5 Multivariate Analysis of Claim Distributions; 4.5.1 The Multivariate POT Approach; 4.5.2 Multivariate Mixtures of Erlangs; 4.6 Estimation of Other Tail Characteristics; 4.7 Further Case Studies; 4.8 Notes and Bibliography.

5 Models for Claim Counts; 5.1 General Treatment; 5.1.1 Main Properties of the Claim Number Process; 5.2 The Poisson Process and its Extensions; 5.2.1 The Homogeneous Poisson Process; 5.2.2 Inhomogeneous Poisson Processes; 5.2.3 Mixed Poisson Processes; 5.2.4 Doubly Stochastic Poisson Processes; 5.3 Other Claim Number Processes; 5.3.1 The Nearly Mixed Poisson Model; 5.3.2 Infinitely Divisible Processes; 5.3.3 The Renewal Model; 5.3.4 Markov Models; 5.4 Discrete Claim Counts; 5.5 Statistics of Claim Counts; 5.5.1 Modelling Yearly Claim Counts; 5.5.2 Modelling the Claim Arrival Process; 5.6 Claim Numbers under Reinsurance; 5.6.1 Number of Claims under Excess-loss Reinsurance; 5.7 Notes and Bibliography.

6 Total Claim Amount; 6.1 General Formulas for Aggregating Independent Risks; 6.2 Classical Approximations for the Total Claim Size; 6.2.1 Approximations based on the First Few Moments; 6.2.2 Asymptotic Approximations for Light-tailed Claims; 6.2.3 Asymptotic Approximations for Heavy-tailed Claims; 6.3 Panjer Recursion; 6.4 Fast Fourier Transform; 6.5 Total Claim Amount under Reinsurance; 6.5.1 Proportional Reinsurance; 6.5.2 Excess-loss Reinsurance; 6.5.3 Stop-loss Reinsurance; 6.6 Numerical Illustrations; 6.7 Aggregation for Dependent Risks; 6.8 Notes and Bibliography.

7 Reinsurance Pricing; 7.1 Classical Principles of Premium Calculation; 7.2 Solvency Considerations; 7.2.1 The Ruin Probability; 7.2.2 One-year Time Horizon and Cost of Capital; 7.3 Pricing Proportional Reinsurance; 7.4 Pricing Non-proportional Reinsurance; 7.4.1 Exposure Rating; 7.4.2 Experience Rating; 7.4.3 Aggregate Pure Premium; 7.5 The Aggregate Risk Margin; 7.6 Leading and Following Reinsurers; 7.7 Notes and Bibliography.

8 Choice of Reinsurance; 8.1 Decision Criteria; 8.2 Classical Optimality Results; 8.2.1 Pareto-optimal Risk Sharing; 8.2.2 Stochastic Ordering; 8.2.3 Minimizing Retained Variance; 8.2.4 Maximizing Expected Utility; 8.2.5 Minimizing the Ruin Probability; 8.2.6 Combining Reinsurance Treaties over Subportfolios; 8.3 Solvency Constraints and Cost of Capital; 8.4 Minimizing Other Risk Measures; 8.5 Combining Reinsurance Treaties; 8.6 Reinsurance Chains; 8.7 Dynamic Reinsurance; 8.8 Beyond Piecewise Linear Contracts; 8.9 Notes and Bibliography.

9 Simulation; 9.1 The Monte Carlo Method; 9.2 Variance Reduction Techniques; 9.2.1 Conditional Monte Carlo; 9.2.2 Importance Sampling; 9.3 Quasi-Monte Carlo Techniques; 9.4 Notes and Bibliography.

10 Further Topics; 10.1 More on Large Claim Reinsurance; 10.1.1 The Ordered Claims; 10.1.2 Large Claim Reinsurance; 10.1.3 ECOMOR; 10.2 Alternative Risk Transfer; 10.2.1 Notes and Bibliography; 10.3 Reinsurance and Finance; 10.4 Catastrophic Risk.

Reinsurance: Actuarial and Statistical Aspects provides a survey of both the academic literature in the field as well as challenges appearing in reinsurance practice and puts the two in perspective. The book is written for researchers with an interest in reinsurance problems, for graduate students with a basic knowledge of probability and statistics as well as for reinsurance practitioners. The focus of the book is on modelling together with the statistical challenges that go along with it. The discussed statistical approaches are illustrated alongside six case studies of insurance loss data sets, ranging from MTPL over fire to storm and flood loss data. Some of the presented material also contains new results that have not yet been published in the research literature. An extensive bibliography provides readers with links for further study. https://www.wiley.com/en-us/Reinsurance%3A+Actuarial+and+Statistical+Aspects-p-9780470772683

ISBN: 978-0-470-77268-3

ALBRECHER, Hansjörg

Teugels, Jef L.

Beirlant, Jan

368:51-7

51-7:368

368:519.2

519.2:368