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A Course in Credibility Theory and Its Applications

Publisher: 
Springer Verlag
Number of Pages: 
331
Price: 
59.95
ISBN: 
3-540-25753-5
Date Received: 
Tuesday, October 11, 2005
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Hans Bühlmann and Alois Gisler
Series: 
Universitext
Publication Date: 
2005
Format: 
Paperback
Category: 
Monograph

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Rating of Risks and Claims Experience in Insurance . . . . . . . . . 1

1.2 Mathematical Formulation of the Problem of Rating Risks . . . . 7

1.2.1 Individual Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 The Correct Individual Premium . . . . . . . . . . . . . . . . . . . . 8

1.2.3 The Risk Rating Problem in the Collective . . . . . . . . . . . 9

1.2.4 Formulating the Rating Problem in the Language of

Bayesian Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 The Bayes Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Basic Elements of Statistical Decision Theory . . . . . . . . . . . . . . . 15

2.2 Bayes Risk and Bayes Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Bayesian Statistics and the Premium Rating Problem . . . . . . . . 18

2.4 The Bayes Premium in Three Special Cases . . . . . . . . . . . . . . . . 21

2.4.1 The Poisson—Gamma Case. . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2 The Binomial—Beta Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.3 The Normal—Normal Case . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Common Features of the Three Special Cases Considered 36

2.5 Conjugate Classes of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 The Exponential Class and Their Associated

Conjugate Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.2 Construction of Conjugate Classes . . . . . . . . . . . . . . . . . . . 46

2.6 Another Type of Example: the ParetoGamma Case . . . . . . . . 47

2.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Credibility Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 Credibility Estimators in a Simple Context . . . . . . . . . . . . . . . . . 56

3.1.1 The Credibility Premium in a Simple Credibility Model 56

3.1.2 A General Intuitive Principle . . . . . . . . . . . . . . . . . . . . . . . 58

XVI Contents

3.1.3 The Quadratic Loss of the Credibility Premium . . . . . . . 59

3.1.4 The Simple Bühlmann Model and the Homogeneous

Credibility Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Credibility Estimators in a General Set-Up . . . . . . . . . . . . . . . . . 64

3.2.1 Credibility Estimators as Orthogonal Projections in

the L2 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Orthogonality Conditions and Normal Equations . . . . . . . . . . . . 71

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 The Bühlmann—Straub Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 The Credibility Premium in the Bühlmann—Straub Model . . . . 81

4.4 Discussion and Interpretation of the Credibility Estimator . . . 84

4.5 Quadratic Loss of the Credibility Estimator . . . . . . . . . . . . . . . . . 86

4.6 The Homogeneous Credibility Estimator in the Bühlmann—

Straub Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Quadratic Loss of the Homogeneous Credibility Estimator . . . . 91

4.8 Estimation of the Structural Parameters σ2 and τ 2 . . . . . . . . . . 93

4.9 Empirical Credibility Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.10 Credibility for Claim Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.11 Credibility for Claim Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.12 Credibility for Risk Groups of Known Individual Contracts . . . 110

4.13 Modification for the Case of Known a Priori Differences . . . . . . 111

4.14 Example of Another Kind of Application . . . . . . . . . . . . . . . . . . 113

4.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Treatment of Large Claims in Credibility . . . . . . . . . . . . . . . . . . 125

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Semi-Linear Credibility with Truncation in the Simple

Bühlmann Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Semi-Linear Credibility with Truncation in a Model with

Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4 Further Methods for Treating Large Claims . . . . . . . . . . . . . . . . . 135

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 Hierarchical Credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 The Hierarchical Credibility Model . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3 Relevant Quantities and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4 Credibility Estimator in the Hierarchical Model . . . . . . . . . . . . . 148

6.5 Quadratic Loss in the Hierarchical Model . . . . . . . . . . . . . . . . . . . 159

6.6 Estimation of the Structural Parameters in the Hierarchical

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Contents XVII

7 Multidimensional Credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2 The Abstract Multidimensional Credibility Model . . . . . . . . . . . 169

7.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.2.2 The (Inhomogeneous) Multidimensional Credibility

Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.2.3 The Homogeneous Credibility Estimator . . . . . . . . . . . . . 173

7.2.4 The Quadratic Loss of the Multidimensional

Credibility Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.3 The Multidimensional Bühlmann—Straub Model . . . . . . . . . . . . 177

7.3.1 Motivation and Interpretation. . . . . . . . . . . . . . . . . . . . . . . 177

7.3.2 Definition of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.3.3 Credibility Formulae in the Multidimensional

Bühlmann—Straub Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.3.4 Quadratic Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3.5 Estimation of Structural Parameters . . . . . . . . . . . . . . . . . 185

7.4 General Remarks About Data Compression and Its

Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.4.2 General Multidimensional Data Structure . . . . . . . . . . . . 187

7.4.3 Optimal Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8 Credibility in the Regression Case . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.2 The Classical Statistics Point of View . . . . . . . . . . . . . . . . . . . . . . 199

8.3 The Regression Credibility Model . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.3.1 The Standard Regression Model . . . . . . . . . . . . . . . . . . . . . 202

8.3.2 The General Regression Case (Hachemeister) . . . . . . . . . 205

8.3.3 Homogeneous Credibility Estimator and Quadratic Loss 208

8.4 The Simple Linear Regression Case (Linear Trend Model) . . . . 208

8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

9 Evolutionary Credibility Models and Recursive Calculation 219

9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

9.2 Recursive Credibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9.3 Evolutionary Credibility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.4 Evolutionary Models and Recursive Credibility . . . . . . . . . . . . . . 226

9.5 Recursive Calculation Method (Kalman Filter) . . . . . . . . . . . . . . 230

9.6 The Evolutionary Credibility Regression Model. . . . . . . . . . . . . . 238

9.7 Recursive Calculation in the Evolutionary Regression Model . . 239

XVIII Contents

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

10.2 On the Embedding of the Individual Risk in a Collective . . . . . 252

10.3 Multidimensional Evolutionary Models . . . . . . . . . . . . . . . . . . . . . 253

10.4 Modelling a Collective with Both Joint and Individual

Movements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10.5 Multidimensional Evolutionary Regression Models . . . . . . . . . . . 262

10.6 Decomposition into an Individual and a Common Component . 264

A Appendix A:

Basic Elements from Probability Theory . . . . . . . . . . . . . . . . . . . 275

A.1 Random Variables, Distribution Functions and Moments . . . . . 275

A.2 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

A.3 Multidimensional Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A.4 Conditional Probability and Conditional Expectation . . . . . . . . 278

A.5 Two Useful Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

B Appendix B:

The Hilbert Space L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

C Appendix C:

Solutions to the Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

C.1 Exercises to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

C.2 Exercises to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

C.3 Exercises to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

C.4 Exercises to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

C.5 Exercises to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

C.6 Exercises to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

C.7 Exercises to Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

10 Multidimensional Evolutionary Models and Recursive

Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Publish Book: 
Modify Date: 
Saturday, August 29, 2009

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