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.
Autorentext
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.
Klappentext
Presents a comprehensive treatment of the increasingly topical field of reinsurance
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.
Zusammenfassung
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.
Inhalt
Preface ix
1 Introduction 1
1.1 What is Reinsurance? 1
1.2 Why Reinsurance? 2
1.3 Reinsurance Data 4
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 10
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 16
2 Reinsurance Forms and their Properties 19
2.1 Quota-share Reinsurance 19
2.1.1 Some Practical Considerations 20
2.2 Surplus Reinsurance 21
2.3 Excess-of-loss Reinsurance 24
2.3.1 Moment Calculations 25
2.3.2 Reinstatements 27
2.3.3 Further Practical Considerations 29
2.4 Stop-loss Reinsurance 30
2.5 Large Claim Reinsurance 31
2.6 Combinations of Reinsurance Forms and Global Protections 32
2.7 Facultative Contracts 33
2.8 Notes and Bibliography 33
3 Models for Claim Sizes 35
3.1 Tails of Distributions 35
3.2 Large Claims 36
3.3 Common Claim Size Distributions 40
3.3.1 Light-tailed Models 41
3.3.2 Heavy-tailed Models 44
3.4 Mean Excess Analysis 49
3.5 Full Models: Splicing 50
3.6 Multivariate Modelling of Large Claims 52
4 Statistics for Claim Sizes 59
4.1 Heavy or Light Tails: QQ- and Derivative Plots 60
4.2 Large Claims Modelling through Extreme Value Analysis
EVA for Pareto-type Tails 63
4.2.1 EVA for Pareto-type Tails 63
4.2.2 General Tail Modelling using EVA 82
4.2.3 EVA under Upper-truncation 91
4.3 Global Fits: Splicing, Upper-truncation and Interval Censoring 97
4.3.1 Tail-mixed Erlang Splicing 97
4.3.2 Tail-mixed Erlang Splicing under Censoring and Upper-truncation 99
4.4 Incorporating Covariate Information 114
4.4.1 Pareto-type Modelling 114
4.4.2 Generalized Pareto Modelling 116
4.4.3 Regression Extremes with Censored Data 119
4.5 Multivariate Analysis of Claim Distributions 123
4.5.1 The Multivariate POT Approach 124
4.5.2 Multivariate Mixtures of Erlangs 125
4.6 Estimation of Other Tail Characteristics 128
4.7 Further Case Studies 132
4.8 Notes and Bibliography 137
5 Models for Claim Counts 139
5.1 General Treatment 139
5.1.1 Main Properties of the Claim Number Process 140
5.2 The Poisson Process and its Extensions 141
5.2.1 The Homogeneous Poisson Process 141
5.2.2 Inhomogeneous Poisson Processes 143
5.2.3 Mixed Poisson Processes 144
5.2.4 Doubly Stochastic Poisson Processes 149
5.3 Other Claim Number Processes 157
5.3.1 The Nearly Mixed Poisson Model 157
5.3.2 Innitely Divisible Processes 158
5.3.3 The Renewal Model 160
5.3.4 Markov Models 161
5.4 Discrete Claim Counts 161
5.5 Statistics of Claim Counts 164
5.5.1 Modelling Yearly Claim Counts 164
5.5.2 Modelling the Claim Arrival Process 172
5.6 Claim Numbers under Reinsurance 183
5.6.1 Number of Claims under Excess-loss Reinsurance 183
5.7Notes and Bibliography 187
6 Total Claim Amount 189
6.1 General Formulas for Aggregating Independent Risks 189
6.2 Classical Approximations for the Total Claim Size 191
6.2.1 Approximations based on the First Few Moments 191
6.2.2 Asymptotic Approximations for Light-tailed Claims 193
6.2.3 Asymptotic Approximations for Heavy-tailed Claims 198
6.3 Panjer Recursion 199
6.4 Fast Fourier Transform 200
6.5 Total Claim Amount under Reinsurance 201
6.5.1 Proportional Reinsurance 201
6.5.2 Excess-loss Reinsur...