Chapter
Chapter 2 Reinsurance Forms and their Properties
2.1 Quota-share Reinsurance
2.1.1 Some Practical Considerations
2.3 Excess-of-loss Reinsurance
2.3.1 Moment Calculations
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
Chapter 3 Models for Claim Sizes
3.1 Tails of Distributions
3.3 Common Claim Size Distributions
3.3.1 Light-tailed Models
3.3.2 Heavy-tailed Models
3.5 Full Models: Splicing
3.6 Multivariate Modelling of Large Claims
Chapter 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
4.8 Notes and Bibliography
Chapter 5 Models for Claim Counts
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
5.4 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
Chapter 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
6.4 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
Chapter 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
7.4.3 Aggregate Pure Premium
7.5 The Aggregate Risk Margin
7.6 Leading and Following Reinsurers
7.7 Notes and Bibliography
Chapter 8 Choice of Reinsurance
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
8.8 Beyond Piecewise Linear Contracts
8.9 Notes and Bibliography
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
Chapter 10 Further Topics
10.1 More on Large Claim Reinsurance
10.1.1 The Ordered Claims
10.1.2 Large Claim Reinsurance
10.2 Alternative Risk Transfer
10.2.1 Notes and Bibliography
10.3 Reinsurance and Finance
Reinsurance
：Actuarial and Statistical Aspects
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