Automobile Insurance: Actuarial Models

246 Pages·1985·6.13 MB·English

by  Dr. Jean Lemaire (auth.)

Automobile Insurance
Huebner International Ser ies on
Risk, Insurance, and Economic
Security
J. David Cummins, Editor
The Wharton School
The S.S. Huebner Foundation for Insurance
Education
University of Pennsylvania
Philadelphia, Pennsylvania, U.S.A.
Series Advisors
Dr. Karl Borch, Norwegian School of Economics
and Business Administration
Dr. Phelim P. Boyle, University of Waterloo,
Canada
Dr. Jean Lemaire, Universite Libre de Bruxelles,
Belgium
Professor Akihiko Tsuboi, Kagawa University, Japan
Dr. Richard Zeckhauser, Harvard University,
U.S.A.
Previously published books in the series:
Cummins, J. David; Smith, Barry D.; Vance,
R. Neil; Van Derhei, Jack L.: Risk
Classification in Life Insurance
Mintel, Judith: Insurance Rate Litigation
Cummins, J. David: Strategic Planning and
Modeling in Property-Liability Insurance
The objective 01 the series is to publish original research and
advanced textbooks dealing with aII major aspects 01 risk bearlng
and economic securlty. The emphasis is on books that will be 01
interest to an international audience. Interdlsclpllnary topics as well
as those Irom traditional dlsciplines such as econom ies, rlsk and
Insurance, and actuarlal science are withln the scope 01 the serles.
The goal Is to provide an outlet lor imaginatlve approaches to
problems in both the theory and practice 01 risk and economic
security.
Automobile Insurance
Actuarial Models
Jean Lemaire
Springer-Science+Susiness Media, SV.
Ubrary of Congress Cataloging in Publicalion Data
Lemaire, Jean,
Automobile insurance.
(Huebner international series on risk, Insurance, and
economic security)
Translation 01: L'assurance automobile.
Bibliography: p.
Includes index.
1. Insurance, Automobile-Statistlcal methods.
2. Insurance, Automobile-Rates and tab Ies. 3. Insurance,
Automobile-Belgium. 1. Title. II. Series.
HG9970.15.L4513 1985 368.5'72 84-21329
ISBN 978-90481-5814-0 ISBN 978-94-015-7708-3 (eBook)
DOI 10.1007/978-94-015-7708-3
Copyright © 1985 by Springer Science+Business Media Oordrecht
Originally published by Kluwer-Nijhoff Publishing in 1985
Soltcover reprint olthe hardcover lst edition 1985
No pari 01 this book may be produced in any lorm by print,
photoprint, microlilm, or any other means without written permission
01 the publisher.
Contents
List of Figures ix
List of Tables xi
Preface xv
Part I
Third Party Liability Automobile Insurance in the World 1
1
Belgium 3
2
Europe 17
3
North America 39
Appendix I 57
Part II
A Priori Classification Criteria 63
4
Statistical Bases 65
5
Number or Amount of Claim? 69
6
Claim Frequency, Average Cost per Claim, and Pure Premium 71
7
Criticism of the Belgian Tariff 81
v
vi CONTENTS
8
Selection of the Significant Variables 87
9
Use of the Results of a Sample Survey 93
10
Criticism of Regression Analysis Selection Methods 101
11
Application: Improvement in Underwriting Procedures 107
Appendix II 111
Partlll
APosteriori Classification: Bonus-Malus Systems 115
12
Introduction: The Negative Binomial Model 117
13
Construction of an Optimal Bonus-Malus System 129
14
Other Loss Functions: Other Premium Calculation Principles 139
15
Penalization of Overcharges 151
16
Allowance for Severity of Claims 157
17
Efficiency Measures of a Bonus-Malus System 163
18
Analysis of the Hunger for Bonus 173
19
The Effect of Expense Loadings 185
20
Epilogue: Construction of The New Belgian Bonus-Malus System 193
Part IV
Some Statistical Methods of Evaluating Claims Provisions 203
21
The Main Statistical Methods 205
CONTENTS vii
22
An Example 231
References 241
Index 245
About the Author 251
List of Figures
Figure 1-1. Progression of the Average Premium 11
Figure 1-1. Index of Average Claim Cost from 1950 to 1981 59
Figure 1-2. Comparative Trend of Prices, Wages, and Average 61
Claim Cost from 1960 to 1981
Figure 7-1. Efficiency of Tariff According to Selected Variables 83
Figure 10-1. Claim Frequency According to Driver's Age 104
Figure 10-2. Effect of Linearity on the Variable "Driver's Age" 104
Figure 15-1. APosteriori Distributions of Claim Frequency 152
Figure 16-1. Estimates of the Proportions of Claims with Bodily 160
Injury
Figure 17-1. Efficiency as Defined by Loimaranta of the Belgian 168
Bonus-Malus System
Figure 17-2. Efficiency (Second Measure) of the Belgian Bonus- 171
Malus System
Figure 18-1. Algorithm to Obtain Optimal Retentions 176
Figure 18-2. Optimal Retention as a Function of Claim Frequency 180
Figure 18-3. Expected Total Payments for a New Policyholder as a 181
Function of Â..
Figure 18-4. Expected Cost of Nonreported Claims 182
Figure 18-5. Optimal Retention as a Function of the Interest Rate 183
Figure 18-6. Efficiency 184
ix
List of Tables
Table 1-1. The Belgian Bonus-Malus System 4
Table 1-2. Components of Claim Costs 7
Table 1-3. Development of the Average Discount Under the 9
Bonus-Malus System
Table 1-4. Distribution of Policyholders by Class over the Last Ten 10
Years (in %)
Table 1-5. Observed and Simulated Frequencies 11
Table 1-6. Number of People Benefitting from the Second 13
Restriction
Table 1-7. Average Premium in the Thirtieth Year 13
Table 2-1. Basic POints Rating 18
Table 2-2. Scale of Reductions Based on Policyholder's 19
Occupation
Table 2-3. Points Increases Based on Duration, Age and Sex 19
Table 2-4. Average Annual Mileage as a Function of Vehicle Age 20
Table 2-5. Example of a Britlsh No-Claims Discount Scale 25
Table 2-6. Example of a Polnts Rating System 27
Table 2-7. Netherlands Bonus-Malus System 29
Table 2-8. Starting Class by Age and Distance Travelled 30
Table 2-9. Classes for Annual Distance Travelled 31
Table 2-10. Swedish Bonus System 32
Table 2-11. Basic Premiums for New Entrants in 1984 34
Table 2-12. Swiss Bonus-Malus System 34
Table 2-13. German Bonus-Malus System 37
Table 2-14. German Insurance Companies' Distribution of Profit 38
Table 1-1. Results of Safety Belt Use 57
Table 1-2. Average Cost of a Victim, 1943 to 1978, Expressed in 60
Years of Wages
Table 5-1. Average Claim Cost According to Number of 70
Accidents
Table 6-1. Claim Frequency, Average Cost, and Pure Premium, 72
Using the Age Variable
Table 6-2. Claim Frequency, Average Cost, and Pure Premium, 73
Using the Bonus Class Variable
xi
xii LIST OF TAB LES
Table 6-3. Claim Frequency, Average Cost, and Pure Premium, 74
Using the Age of Vehicle Variable
Table 6-4. Claim Frequency, Average Cost, and Pure Premium, 74
Using the Power Variable
Table 6-5. Claim Frequency, Average Cost, and Pure Premium, 75
Using the Cubic Capacity Variable
Table 6-6. Claim Frequency, Average Cost, and Pure Premium, 75
Using the Original Value Variable
Table 6-7. Claim Frequency, Average Cost, and Pure Premium, 76
Using the Net Premium Variable
Table 6-8. Claim Frequency, Average Cost, and Pure Premium, 77
Using Class, Occupation, and Sex Variables
Table 6-9. Claim Frequency, Average Cost, and Pure Premium, 77
Using Geographical Area, Language, and Coverage
Category Variables
Table 7-1. Regression Analysis of Tariff According to Power. 82
Dependent Variable Xl
Table 7-2. Regression Analysis of Tariff According to Power. 83
Dependent Variable x
3
Table 7-3. Regression Analysis of Tariff According to Cubic 84
Capacity. Dependent Variable Xl
Table 7-4. Regression Analysis of Tariff According to Cubic 84
Capacity. Dependent Variable x3
Table 8-1. Selection of Significant Variables. Dependent Variable 88
Xl
Table 8-2. Regression Coefficients of the Optimal Equation 89
Table 8-3. Selection of Significant Variables. Dependent Variable 91
x
3
Table 9-1. Claim Frequency Using the Occupation Variable 94
Table 9-2. Claim Frequency Using the Nationality Variable 95
Table 9-3. Claim Frequency Using the Marital Status Variable 96
Table 9-4. Claim Frequency Using the Number of Children 96
Variable
Table 9-5. Claim Frequency Using the Annual Distance Travelled 97
Variable
Table 9-6. Selection of Significant Variables 98
Table 12-1. Observed Distribution of Number of Claims in 120
Portfolio
Table 12-2. Observed and Fitted Distribution of Number of Claims. 121
Poisson Model
Table 12-3. Observed and Fitted Distribution of Number of Claims. 123
Negative Binomial Model. Moments Method
Table 12-4. Observed and Fitted Distribution of Number of Claims. 124
Negative Binomial Model. Maximum Likelihood
Method
Table 12-5. Observed and Fitted Distribution of Number of Claims. 125
Generalized Geometric Distribution
Table 12-6. Observed and Fitted Distribution of Number of Claims. 126
Mixed Poisson Distribution