ASTIN Bulletin BARAB Bulletin de l'association Royale des Actuaires Belges MVSV Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker

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1 References The following abbreviations are used throughout the references: AB ASTIN Bulletin BARAB Bulletin de l'association Royale des Actuaires Belges MVSV Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker SAJ Scandinavian Actuarial Journal Part I A.G.S.A.A. Recueil de statistiques sur l'assurance automobile en France, Astin Groep Nederland. New Motor Rating Structure in the Netherlands, Corlier, F., Lemaire, J., and D. Muhokolo. Simulation of an Automobile portfolio. Geneva Papers on Risk and Insurance, 1979, Institut National de Statistique. Accidents de la circulation sur la voie publique avec tues et blesses, Insurance Bureau of Canada. Classification and Rating Criteria for Automobile Insurance, Munich Reinsurance Company. Motor Insurance in the Federal Republic of Germany, Munich. Munich Reinsurance Company. TarijJfor Motor Insurance in the Federal Republic ofgermany, Munich. Sigma. Automobile Third Party Insurance: Trends ofdevelopment During the Last Decade. Swiss Reinsurance Company, Ziirich. Surintendant des Assurances du Quebec. La tarification en assurance automobile au Quebec, U.P.E.A. Rapport annuel, Part 1/ Hallin, M. Methodes statistiques de construction de tarif. MVSV, 1977, Hallin, M. Etude statistique des facteurs influen~ant un risque. BARAB, 1977, Hallin, M. and Ingenbleek, J.-F. Etude statistique de la probabilite de sinistre en assurance automobile. AB, 1981,

2 242 AUTOMOBILE INSURANCE Hallin, M. and Ingenbleek, J.-F. The Swedish automobile portfolio in 1977: A statistical study. SAJ, 1983, Lemaire, J. Critique du tarif automobile belge. BARAB, 1977, Lemaire, J. Selection procedures of regression analysis applied to automobile insurance. Part 1: MVSV, 1977, Part II: MVSV, 1979, Van Eeghen, J., Greup, E., and Nijssen, J. Rate Making. Nationale Nederlanden, Amsterdam, Part 11/ Bichsel, F., Erfahrung-Tarifierung in der Motorfahrzeughaftplichtversicherung. MVSV, 1964, Borgan, 0., Hoem, J., and Norberg, R. A non asymptotic criterion for the evaluation of automobile bonus systems. SAJ, 1981, Biihlmann, H. Optimale Prămienstufensysteme. MVSV, 1964, Biihlmann, H. Mathematical Methods in Risk Theory. Springer, Berlin. Depauw, J., De Rouck, M., Delbaen, F., Haezendonck, J., and Van Goethem, P. Etude matbematique de l'assurance automobile belge. BARAB, 1980, Derron, M. Mathematische Probleme der Automobilversicherung. MVSV, 1962, Ferreira, J. Identifying Equitable Insurance Premiums for Risk Classes. An Alternative to the Classical Approach. Paper presented at the 23rd International Meeting of the Institute of Management Sciences. Athens, Gerber, H. On additive premium calculation principles. AB, 1974, Gerber, H. On iterative premium calculation principles. MVSV, 1974, Gossiaux, A.M., and Lemaire, J. Methodes d'ajustement de distnbutions de sinistres. MVSV, 1981, Gregorius, F. Development of the Study. In New Motor Rating Structure in the Netherlands. Astin Groep Nederland, 1982, Jewell, W. Models in Insurance: Paradigms, Puzzles, Communications and Revolutions. Proceedings of the 2Ist International Congress of Actuaries. Ziirich-Lausanne, Lemaire, J. Si les assures connaissaient la programmation dynamique... BARAB, 1975, Lemaire, J. Driver versus company: Optimal behaviour of the Policyholder. SAJ, 1976, Lemaire. J. La soif du bonus. AB, 1977, Lemaire, J. Pour une reforme de I'assurance automobile. Prix F.N.R.S.-Royale BeIge-125e anniversaire, Bruxelles. Lemaire, J. How to defme a bonus-malus system with an exponential utility function. AB, 1979, Lemaire, J. L'assurance automobile: modeles matbematiques et statistiques. Fernand Nathan-Editions Labor, Bruxelles.

3 REFERENCES 243 Lemaire, J. The effect of expense loadings on the fairness of a tariff. AB, 1984, Lemaire, J., and Vandermeulen, E. Une propriete du principe de l'esperance mathematique. Bul!. Trimestriel Institut des Actuaires Franc;ais, 1983,5-14. Loimaranta, K. Some asymptotic properties of bonus systems. AB, 1972, Norberg, R A credibility theory for automobile bonus systems. SAJ, 1976, Picard, P. Generalisation de l'etude sur la survenance des sinistres en assurance automobile. Bull. Trimestriel Institut des Actuaires Franc;ais, 1976, Vepsălâinen, S. Applications to a theory of bonus systems. AB, 1972, PartlV De Vijlder, F. Estimation of IBNR claims by least squares. MVSV, 1978, Taylor, G. Separation of inflation and other effects from the distribution of non life insurance claim delays. AB, 1977, Taylor, G. Testing goodness of fit of an estimated run-off trlangie. AB, 1978, Van Eeghen, J. A survey of loss reserving methods, Nationale Nederlanden, Amsterdam, Verbeek, H. An approach to the analysis of claims experience in motor liability excess of loss reinsurance. AB, 1972,

4 Index A.G.S.A.A., 7, 59, 61 A verage cost per claim by age of driver, 72 by age of vehicle, 74 by bonus class, 73 by car value, 75 by coverage category, 77 by cubic capacity, 75 by engine power, 74 by geographical area, 77 by language, 77 by occupation, 77 by premium paid, 76 by sex, 77 by vehicle class, 77 Bar-le-Duc, 58 Bayes theorem, 131, 132 Belgium bonus-malus: Markovian presentation, 165 bonus-malus system, 4 construction of 'new tariff, 193 criticism of tariff, 81 description of tariff, 3 efficiency of bonus-malus system, 168, 171, 183 hunger for bonus, 176, 194 observed run-off triangle, 232 optimal a priori variables, 87, 98 optimal bonus-malus system, 136 real bonus-malus system, 191 Beta distribution, 158 Bichsel, F., 33 Bonus-malus (construction of new system), 193 Bonus-malus (corrected for loadings) levelloading, 187, 190 linear loading, 187, 191 Bonus-malus (defmition) Belgium, 4 comparison, 196 defmition, 163 France,21 Germany,37 245

5 246 Netherlands,29 Quebec, 53 Sweden,32 Switzerland, 34 United Kingdom, 25 Bonus-malus (efficiency) fust measure, 164 second measure, 169 Bonus-malus (optimal) absolute loss function, 139 allowance for severity, 161 construction of optimal system, 129 defmition, 118 expected value principle, 133 fourth degree loss function, 140 penalization of overcharges, 155 quadratic loss function, 131 variance principle, 142 zero-utility principle, 146 Borgan, 0., 171 Chain ladder method, 209 Claim frequency by age of driver, 72 by age of vehiele, 74 by annual mileage, 97 by bonus elass, 73 by car value, 75 by coverage category, 77 by cubic capacity, 75 by engine power, 74 by geographical area, 77 by language, 77 by marital status, 96 by nationality, 95 by number of children, 96 by occupation, 77, 94 by premium paid, 76 by sex, 77 by vehiele elass, 77 Credibility theory, 135, 159 Criticism of Belgian tariff, 81 of selection methods, 101 Dynamic programming, 173 INDEX Efficiency of bonus-malus systems fust measure (Loimaranta), 164 second measure, 169, 194 Elasticity, 170 Elimination selection method, 111 Expense loadings level, 187 linear, 187 Exponential utility function, 146, 153 Ferreira, J., 151 France bonus-malus system, 21 elaim statistics, 59 description of tariff, 17 Gamma distribution, 121, 158 Generalized geometric mode~ 124 Germany bonus-malus system, 37 description of tariff, 36 Greup, E., 103 Hallin, M., 103 Hoem, J., 171 Hunger for bonus, 173, 194 Independence (elaim number and amount),70 Ingenbleek, J.F., 103 Jewell, W., 185 Johnson, P., 22 Jung, J., 31 Loimaranta, K., 164 Loss function absolute, 139 defmition, 130 fourth degree, 140 quadratic, 131 Markov chain, 4, 164 Maximum likelihood method generalized geometric, 124 negative binomial, 123 Poisson, 120

6 INDEX separation method, 219 Mixed Poisson mode~ 125 Moments method generalized geometric, 124 mixed Poisson, 126 negative binomi~ 123 Poisson, 120 Multiplicative methods, 211 Negative binomial distribution, 122 Negative binomial model construction of bonus-malus, 132 defmition, 121 simulation, 8 Netherlands bonus-malus system, 29 description of tariff, 26 Non-declaration of small claims, 173 Norberg, R, 171 Nijssen, J., 103 O'Neil, M.-L., 39 Overcharges (penalization 00, 151 Penalization of overcharges, 151 Points rating system, 27 Poisson model, 119 Premium calcu1ation principles expected value, 133 variance, 142 zero-utility, 146 Professional driver-defmition, 5 ProfIt distribution Germany,38 Switzerland, 35 Progressive selection method, 112 Provisions (evaluation) chain ladder method, 208 multiplicative methods, 211 separation method, 217 Pure premium by age of driver, 72 by age of vehicle, 74 by bonus class, 73 by car value, 75 by coverage category, 77 by cubic capacity, 75 by engine power, 74 by geographical area, 77 bylanguage,77 by occupation, 77 by premium paid, 76 by sex, 77 by vehicle class, Quebec bonus-malus system, 53 description of tariff, 50 Regression analysis applications, 81, 88, 98 selection methods, 111 Retentions (optimal), 176, 196 Risk function, 130 Run-off triangle, 207 Safety belts effect, 57 Sample survey, 93 Sedentary drivers-defmition, 5 Selection methods (description) by elimioation, 111 progressive, 112 stepwise, 112 Selection of signiftcant variables, 87, 90,98 Separation method, 217 Severity of claims, 157 SÎmulatÎon of portfolio, 8, 194 Spurious correlations (examples), 78 Stationary probability distribution, 165,177 Statistical game, 129 Stepwise selection method, 112 Superimposed inflation, 7 Sweden bonus-malus system, 32 description of tariff, 31 Switzerland bonus-malus system, 34 description of tariff, 33 Tariff description Belgium, 3 France,17

7 248 Germany,36 Netherlands,26 Quebec, 50 Sweden,31 Switzerland, 33 United Kingdom, 22 United States, 39 Taylor, G., 217 United Kingdom bonus-malus system, 25 INDEX description of tariff, 22 United States, description of tariff, 39 U.P.E.A., 177, 193 Utility function, 146, 153 Van Eeghen, J., 103 Variables of statistical analysis, 66, 96 World's record (automobile accident), 58

8 About the Author Jean Lemaire holds master's degrees in applied mathematics and actuarial sciences and a Ph.D. in mathematics, ali obtained at the Free University of Brussels. He has published over 60 research papers, mostly in game theory and in actuarial science. He has won the tirst prize in the 1980 Boleslaw Monic Competition for an essay on net retentions, and the frrst prize in the 1979 competition of the Belgian National Fund of Scientitic Research "Royale Belge-125e anniversaire" for his work in automobile insurance. He is currently professor of actuarial science at the Free University of Brussels and Manager of the insurance company "La P.S." in Brussels, where he supervises the life and non-life insurance and reinsurance departments. He has been visiting professor at the Universities of Berkeley, California, Laval, Quebec, Kinshasa, Zaire and Brazzavllle, Congo. He is also president of AS TIN, president of A.A.Br. (Association des Actuaires issus de l'universite de Bruxelles), and editor ofthe newsletter of the International Actuarial Association.