# Reinsurance: Actuarial and Statistical Aspects

## Books

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.

1 INTRODUCTION 1

1.1 What is Reinsurance? 1

1.2 Why Reinsurance? 2

1.3 Reinsurance Data 5

1.3.1 Case Study I: Motor liability data 5

1.3.2 Case Study II: Dutch fire insurance data 10

1.3.3 Case Study III: Austrian storm claim data 11

1.3.4 Case Study IV: European flood risk data 11

1.3.5 Case Study V: Groningen earthquakes 12

1.3.6 Case Study VI: Danish fire insurance data 12

1.4 Notes and Bibliography 13

2 REINSURANCE FORMS AND THEIR PROPERTIES 17

2.1 Quota-share reinsurance 17

2.1.1 Some practical considerations 18

2.2 Surplus reinsurance . 19

2.3 Excess-of-loss reinsurance . 21

2.3.1 Moment calculations . 22

2.3.2 Reinstatements . 25

2.3.3 Further practical considerations . 27

2.4 Stop-loss reinsurance 28

2.5 Large claim reinsurance 29

2.6 Combinations of reinsurance forms and global protections . 30

2.7 Facultative Contracts 31

2.8 Notes and Bibliography 31

3 MODELS FOR CLAIM SIZES 33

3.1 Tails of distributions . 33

3.2 Large claims 34

3.3 Common claim size distributions 39

3.3.1 Models for Light Tails . 40

3.3.2 Models for Heavy Tails 42

3.4 Mean excess analysis 48

3.5 Full models: splicing 49

3.6 Multivariate modelling of large claims 51

4 STATISTICS FOR CLAIM SIZES 57

4.1 Heavy or light tails: QQ- and derivative plots 58

4.2 Large claims modelling through extreme value analysis 65

4.2.1 EVA for Pareto-type tails 65

4.2.2 General tail modelling using EVA 77

4.2.3 EVA under upper-truncation 84

4.3 Global fits: splicing, upper-truncation and interval censoring 90

4.3.1 Tail-Mixed Erlang splicing 90

4.3.2 Tail-Mixed Erlang splicing under censoring and upper-truncation 92

4.4 Incorporating covariate information . 100

4.4.1 Pareto-type modelling . 100

4.4.2 Generalized Pareto modelling 102

4.4.3 Regression extremes with censored data 105

4.5 Multivariate analysis of claim distributions . 107

4.5.1 The multivariate POT approach . 108

4.5.2 Multivariate mixtures of Erlangs 109

4.6 Estimation of other tail characteristics 111

4.7 Further case studies . 114

4.8 Notes and Bibliography 115

5 MODELS FOR CLAIM COUNTS 129

5.1 General treatment 129

5.1.1 Main Properties of the Claim Number Process 130

5.2 The Poisson process and its extensions 131

5.2.1 The homogeneous Poisson process 131

5.2.2 Inhomogeneous Poisson Processes 133

5.2.3 Mixed Poisson processes 134

5.2.4 Doubly Stochastic Poisson Processes 139

5.3 Other claim number processes 146

5.3.1 The Nearly Mixed Poisson Model 146

5.3.2 Infinitely Divisible Processes 147

5.3.3 The Renewal Model 149

5.3.4 Markov Models . 150

5.4 Discrete claim counts 151

5.5 Statistics of claim counts . 153

5.5.1 Modelling yearly claim counts . 153

5.5.2 Modelling the claim arrival process 159

5.6 Claim numbers under reinsurance 164

5.6.1 Number of Claims under Excess-Loss Reinsurance 165

5.7 Notes and Bibliography 168

6 TOTAL CLAIM AMOUNT 179

6.1 General formulas for aggregating independent risks 179

6.2 Classical approximations for the total claim size 181

6.2.1 Approximations based on the first few moments . 181

6.2.2 Asymptotic approximations for light tailed claims 184

6.2.3 Asymptotic approximations for heavy tailed claims 188

6.3 Panjer recursion 189

6.4 Fast Fourier Transform 191

6.5 Total claim amount under reinsurance 192

6.5.1 Proportional Reinsurance 192

6.5.2 Excess-Loss Reinsurance 192

6.5.3 Stop-Loss Reinsurance 194

6.6 Numerical Illustrations 197

6.7 Aggregation for Dependent Risks 199

6.8 Notes and Bibliography 202

7 REINSURANCE PRICING 205

7.1 Classical principles of premium calculation . 207

7.2 Solvency considerations 210

7.2.1 The ruin probability 211

7.2.2 One-year time horizon and cost of capital 214

7.3 Pricing proportional reinsurance 216

7.4 Pricing non-proportional reinsurance 217

7.4.1 Exposure Rating 218

7.4.2 Experience Rating 221

7.4.3 Aggregate Pure Premium 223

7.5 The aggregate risk margin . 224

7.6 Leading and following reinsurers 226

7.7 Notes and Bibliography 227

8 CHOICE OF REINSURANCE 231

8.1 Decision criteria 233

8.2 Classical optimality results 236

8.2.1 Pareto-optimal risk sharing 236

8.2.2 Stochastic ordering 238

8.2.3 Minimizing retained variance 239

8.2.4 Maximizing expected utility 242

8.2.5 Minimizing the ruin probability . 244

8.2.6 Combining reinsurance treaties over subportfolios 248

8.3 Solvency Constraints and Cost of Capital 251

8.4 Minimizing other risk measures 253

8.5 Combining reinsurance treaties 254

8.6 Reinsurance chains . 255

8.7 Dynamic Reinsurance 256

8.8 Beyond piecewise linear contracts 258

8.9 Notes and Bibliography 260

9 SIMULATION 265

9.1 The Monte Carlo Method . 265

9.2 Variance Reduction Techniques 268

9.2.1 Conditional Monte Carlo 269

9.2.2 Importance Sampling . 270

9.2.3 Control Variates . 273

9.3 Quasi-Monte Carlo Techniques 274

9.4 Notes and Bibliography 278

10 FURTHER TOPICS 281

10.1 More on Large Claim Reinsurance 281

10.1.1 The ordered claims 281

10.1.2 Large claim reinsurance 286

10.1.3 ECOMOR  288

10.2 Alternative risk transfer 290

10.3 Reinsurance and finance 295

10.4 Catastrophic risk 297

Index 333

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