lemaire 1988 a comparative

A Comparative Analysis of Most European and Japanese Bonus-malus SystemsAuthor(s): Jean LemaireSource: The Journal of Risk and Insurance, Vol. 55, No. 4 (Dec., 1988), pp. 660-681Published by: American Risk and Insurance AssociationStable URL: http://www.jstor.org/stable/253143 .

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INVITED ARTICLE

A Comparative Analysis of Most European And Japanese Bonus-malus Systems

Jean Lemaire

Abstract

The bonus-malus systems of 13 different countries are presented and compared, using the following criteria: (1) stationary distribution of the policyholders among the classes, (2) elasticity of the premiums with respect to the claim frequency (efficiency), and (3) magnitude of the hunger for bonus.

Introduction

The story of man and his motor car is one of the great love affairs of this century. The number of motor vehicles was estimated to exceed 371,100,000 in 1985, as compared to a few thousand at the turn of the century. An unfortunate consequence has been the parallel growth of accidents and casualties: 114,900 deaths in 1985. Automobile third-party liability insurance has consequently been made compulsory in most developed countries, and actuaries from all over the world face the problem of designing tariff structures that will fairly distribute the burden of claims among policyholders. While life insurance premiums are set with a fairly universal approach, such is not the case in automobile insurance. North American actuaries have chosen to develop a priori ratings, that use many classifying variables, such as age, sex, marital status, driving experience of the main driver, power and use of the car, geographical territory, annual mileage, and number of cars in the household. A posteriori rating is, however, comparatively restricted. European and Japanese actuaries have generally chosen the opposite approach: a few a priori classifying variables are selected, while much emphasis is placed on the a posteriori evaluation of drivers. Very sophisticated bonus-malus systems (BMS) induce premium modifications according to the driving history of the insured.

Jean Lemaire is Joseph Wharton Term Professor of Insurance at the University of Pennsylvania. This article is based on his book entitled Automobile Insurance: Actuarial Models which won the Clarence Arthur Kulp Memorial Award given by ARIA in 1987.

Thanks are extended to several persons who supplied the author with the tariff structures of their respective countries and to Rosella Piccaluga who did the programming for this research project.



Comparative Analysis of European And Japanese Bonus-malus Systems 661

A BMS (in some countries the terminology "merit-rating system" or "no-claim-discount system" is still in use) is a form of experience rating: the insureds responsible for one or more accidents are penalized by an additional premium or "malus"; claim-free policyholders are awarded a discount or "bonus". The main purpose of a BMS, besides encouraging policyholders to drive carefully, is to better assess individual risks so that everyone will pay, in the long run, a premium corresponding to his or her own claim frequency.

A BMS consists of a finite number of classes, each with its own premium level. New policyholders have access to a specified class, which may depend on the use of the car, the age of the policyholder, and other variables. After each policy period (usually a year), the policy moves up or down in the system, according to transition rules. In most countries, government regulations compel all insurers to use the same BMS. A policyholder then normally remains in the BMS throughout his or her driving lifetime; if a person decides to switch to another insurer, he or she must first obtain a certificate from the former insurer stating his or her attained bonus-malus level and whether pending claims could affect this level. The new insurer must then award the same discount or apply the same surcharge.

By definition, an insurer uses a BMS when (1) all policies of a given tariff group can be partitioned into a finite number of classes, so that the annual premium depends only on the class (the number of classes is denoted by "s"), and (2) the class for a given period of insurance is determined uniquely by the class for the preceding period and the number of claims reported during the preceding period.

Such a system is determined by three elements: (1) the premium scale b =

(bj,...,bs); bi is the premium or premium level attached to class i, (2) the access class io, and (3) the transition rules which determine the transfer from one class to another when the number of claims is known. These rules can be introduced in the form of tranformations Tk, such that Tk(i) = j if the policy is transferred from class i to class j when k claims have been reported.

Condition 2 of the definition assumes that a BMS follows a Markov (or memoryless) process: the knowledge of the present level and the number of claims of the present year suffice to determine the next class. For several countries (Belgium, France, Luxembourg, and Sweden), this condition is not met, and it is necessary to record the policyholder's past claims history for up to six years. In those cases it is possible to fictitiously render the process Markovian by subdividing some classes, adding an index that counts the number of claim-free years. An example of such a subdivision may be found in (Lemaire(1985), chapter 17).

This study analyses and compares the BMS of 13 different countries: Belgium, Finland, France, Germany, Italy, Japan, The Netherlands, Luxembourg, Norway, Spain, Sweden, Switzerland, and the United Kingdom. The tariff structures and the BMS of those countries are briefly presented, along with results and comparisons. First, the different tools used to compare the BMS are developed.



662 The Journal of Risk and Insurance

Tools for the Comparison of the Systems

The Relative Stationary Average Premium Level

An apparently inescapable consequence of the implementation of a BMS is a progressive decrease of the observed average premium level, due to a concentration of policyholders in the high-discount classes. With claim frequencies in Europe averaging 10 percent, it would be necessary to penalize each claim by nine classes to maintain a well-spread distribution of policyholders among the classes. Because such severe penalties seem commercially impossible to enforce, most policies tend to cluster in the lowest BMS classes. In Belgium, for instance, close to 60 percent of policyholders enjoy the maximum discount, over 73 percent find themselves in one of the three lower classes, while less than 1 percent of the policies are in the malus zone (premium level above 100). Consequently many insurers suffered great losses, especially since, in a regulated country like Belgium, it is most difficult (or it is time-consuming) to obtain an increase in the basic premium. For example, one large insurer, out of a premium income of Belgian Francs (BF) 2,365 million, allowed 793 million bonuses in 1985, while it recovered only 2.3 million in maluses. Its average insured thus benefited by a discount of 33.4 percent.

A crucial characteristic of a BMS is thus the distribution of policyholders among classes, once the system has reached the stationary state. Among other things this allows forecasting the average premium level. A simulation program was devised, based on the classical negative binomial model. The model assumes that the number of claims of a policyholder, characterized by his or her claim frequency X, conforms to a Poisson distribution

Pk(X) = e-XXk/k! k=0,1,...,

while X is distributed in the portfolio according to a gamma structure distribution:

u(X) = rae-TrXa-l/r(a)

It is well known that the resulting distribution of the number of claims in the portfolio conforms to a negative binomial distribution:

Pk = f pk(X)u(X)dX = (Vr ( 1:+ Tk

The simulation was used to compute the stationary distribution of the insureds among the classes, as well as the stationary average level. Given the wide variety of BMS in force, stationary levels are difficult to compare. Therefore, a "relative stationary average level", has been defined as:

stationary average level - minimum level maximum level - minimum level

Expressed as a percentage, this is an index which determines the relative position of the average policyholder, when the lowest premium level is set equal to zero and the highest one to 100.




The parameters of the gamma were chosen in such a way that its mean is equal to 0.10, while its variance equals 0.107. The annual percentage of new policies was set equal to 6 percent. These values roughly correspond to the averages observed in Western Europe in 1986. A specific model was developed for policy cancellations. Indeed in all countries cancellations are observed to be more important in the upper classes of the BMS. A linear fit of the observed cancellation rates in Belgium was used to estimate, for all classes of all BMS, a cancellation probability as a function of the expected claim frequency of the class.

The Efficiency of the Bonus-malus System

Let vi(X) denote the discounted expectation of all the premiums charged to a policyholder of claim frequency X starting in class i. The following system of s equations must be satisfied by vj(X):

vi(X) = bi + 3EPk(X)VTk(i)(X) i - 1,..., s, k = O

This assumes an infinite horizon and an annual discount factor 3. The main aim of the establishment of a BMS is to reduce the premium for

good drivers and to increase it for bad drivers. In order to make a BMS acceptable, the total premiums vi(X) must be an increasing function of X. Ideally, the dependence should be linear. An increment dX/X in the claim frequency should produce an equal change, dvj(X)/vj(X), in the premium. The system is called perfectly efficient if

dvi(X)/v(X)-1 dX/X

As a general rule, however, the change in premium is much less than the change in claim frequency. The efficiency (or the fairness) of a BMS is defined (see Lemaire(1985), chapter 17) as the elasticity of the discounted expectation of all payments with respect to the claim frequency:

- dvi(X)/vi(X) W(X) dX/X

The efficiency yit(X) is a function of X. In the comparison of table 1, only the values of the efficiency corresponding to the benchmark claim frequency of 10 percent are presented, for a new policyholder. The interest rate used for discounting is 7 percent.

The Average Optimal Retention

A well-publicized consequence of BMS is the so-called "hunger for bonus" effect: that is the tendency of policyholders to bear small claims themselves and not to report them to their insurer, in order to avoid future premium increases. The existence of such a phenomenon seems to be well accepted by all parties concerned. It has the same effect as a small deductible, and reduces




administrative costs. However, the bonus hunger should not be too great. If the main objective of a BMS is to achieve a better separation of the good and the bad risks (and, possibly, to persuade policyholders to drive more carefully), the objective is certainly not to transfer most claims from the insurer to the insureds. Any BMS that would force or induce a policyholder to bear a claim of, for example, $3000, might be considered to penalize claims excessively, while encouraging hit-and-run behaviors.

The optimal hunger for bonus associated with each BMS can be computed using an algorithm based on a dynamic programming approach (see Lemaire(1985), chapter 18). For each class the program computes the optimal retention level which is the amount under which it is in the policyholder's interest to not report a claim. As a major characteristic for comparative purposes, the average optimal retention was computed, weighted by the stationary class probabilities. Several assumptions were used in the algorithm. First, the observed industry-wide distribution of claims of Belgian taxis (1983 losses based on evaluations as of 1986, all figures indexed to 1986 prices) was used. Most observed claims distributions are already influenced by the hunger for bonus; hence they cannot be used in the algorithm. Fortunately, Belgian taxis are not linked to the BMS, because most of them are driven by several drivers. It should be noted that optimal retentions appear to be extremely insensitive to the claims distribution used. Secondly, the algorithm assumes the value of X to be 0.1364. The reason for this choice is that the observed claim frequency of 10 percent is already influenced by the hunger for bonus: the observed frequency is substantially smaller than the "real" one, due to the non-declaration of small claims. The computation of optimal retentions uses the estimated "true" frequency. Its value was chosen in such a way that the algorithm, applied to the Belgian BMS, forecasts an observed claim frequency of 0.10. Thirdly, the commercial premium at level 100 for the Belgian BMS was set equal to BF 24, 512, the observed value for 1986. In order to perform valid comparisons with systems in other countries, the premium charged at level 100 for the other BMS was computed so that the average premium, if all claims are reported, was the same for all countries. Indeed the class labelled "100" is situated at quite different positions, depending on the country: to have adopted the same premium would have drastically distorted the results. Lastly, the interest rate was set at 7 percent.

All figures related to bonus hunger are subsequently expressed in U.S. dollars, converted at the May 1988 exchange rate of $1 = BF 35.

Results

For each country, the regulatory environment is briefly described and the a priori classification variables are listed. The BMS is then defined and analyzed. To ensure a uniform presentation, it was necessary to modify the original labeling of the classes for some countries.

In all fairness to the different BMS, it must be noted that they have been compared under the same hypotheses, the most crucial being the assumption




of an average claim frequency of 10 percent. This figure is certainly acceptable as a European average; however, local discrepancies do occur. While the observed claim frequency in the Nordic countries has dropped to less than 5 percent, it is still close to 20 percent in some Mediterranean countries. Those significant differences should be kept in mind when comparing the systems. For instance, the Italian BMS has an extremely low efficiency, which makes it unsuitable for the average policyholder who is expected to be involved in an accident every tenth year. The efficiency would not have ranked so poorly had the analysis been performed with a higher claim frequency, close to the observed frequency in Italy.

Another important point is that very few of the countries explicitly use the age of the policyholder as an a priori classification variable. (None uses sex or marital status). There is, however, always an implicit penalty for new drivers, since the premium level corresponding to the access class of all BMS is in every case substantially higher than the average stationary premium level. Comparing the access level to the stationary level for the sub-population of the policyholders insured over a period of 20 years allows computation of this implicit surcharge.

Table 1 shows a comparison of the main BMS characteristics. Table 2 presents the stationary distribution of policyholders among the classes for all systems.

Belgium

In Belgium a tariff structure set by Ministerial Decree must be applied by all insurers. The classification variables include: (1) age of driver, that, however, is hardly used (drivers under age 23 have to pay a small deductible of $140

Table 1

Comparison of the Bonus-malus Systems: Main Characteristics.

Relative Surcharge Average Stationary Paid by Optimal Average Newcomers Efficiency Retention

Country Level (TO) (TO) (U.S. $)

Belgium 5.7 37.3 or 61.5 6.7 or 8.4 203 Finland 22.1 150.2 13.7 643 France 9.0 69.5 16.8 414 Germany 16.5 299.4 12.3 357 Italy 7.3 64.0 2.5 6 Japan 18.4 135.6 to 182.7 14.8 83 Luxembourg 10.1 86.0 10.1 214 Netherlands 27.0 85.4 to 164.9 20.1 to 23.4 637 Norway - 161.8 21.2 757 Spain 36.7 31.9 6.5 329 Sweden 30.1 204.5 17.7 900 Switzerland 11.9 82.2 22.2 420 United Kingdom 16.8 93.0 10.6 434




Table 2

Stationary Distributions (in percentages) of Policyholders Among Classes

Class BELG. FINL. GERM. ITALY JAPAN LUX. NETH. NORW. SPAIN SWED. SWIT. U.K.

22 0.07 0.00 0.30 21 0.07 0.00 0.31 20 0.09 0.00 0.27 19 0.09 0.02 0.33 18 0.02 0.13 0.14 0.02 0.41 17 0.03 0.17 0.19 0.02 0.46 16 0.02 0.89 0.58 0.26 0.02 0.65 15 0.03 6.40 0.82 0.64 0.01 0.83 14 0.07 0.54 1.12 1.13 0.64 3.46 1.10 0.93 13 0.13 1.08 8.75 0.02 1.63 1.50 3.38 0.10 1.88 12 0.25 6.13 7.36 0.07 2.11 1.91 9.29 0.36 2.01 11 0.38 2.31 7.82 0.29 8.62 8.51 7.81 0.45 2.21 10 1.60 8.57 6.90 0.90 7.04 7.45 7.00 1.89 8.16 9 1.57 8.67 5.67 7.50 6.68 6.65 6.56 2.23 7.18 8 1.81 8.32 5.11 7.25 5.71 5.50 5.88 8.81 6.58 7 2.06 8.59 4.55 6.61 5.39 5.25 5.43 7.22 9.58 6.08 1.77 6 7.05 7.12 3.99 5.62 4.65 4.88 6.00 7.61 8.52 5.46 8.84 5 6.04 6.00 4.73 5.26 4.81 4.50 5.31 6.96 15.36 7.69 4.92 8.75 4 5.09 5.17 4.17 4.99 4.40 4.40 4.53 8.10 12.15 7.41 5.91 11.15 3 7.90 4.45 3.40 5.10 5.55 5.50 3.74 6.80 9.73 8.97 5.38 9.64 2 7.02 3.74 3.24 7.28 4.60 4.96 3.25 5.80 8.07 16.10 4.42 7.64 1 58.90 29.26 25.57 49.08 36.24 36.74 28.32 42.93 54.67 41.71 35.27 52.19

FRANCE*

Classes 'Wo

200-350 0.48 190-199 0.14 180-189 0.18 170-179 0.24 160-169 0.14 150-159 0.37 140-149 1.06 130-139 0.32 120-129 1.20 110-119 3.33 100-109 11.11 90- 99 15.31 80- 89 11.37 70- 79 9.44 60- 69 11.36 50- 59 33.91

*Note: Because the French BMS consists of 301 classes, groupings were necessary.

when found guilty of an accident), (2) power of vehicle (the number of kilowatts DIN of the engine determines the basic premium), and (3) use of




vehicle (differentiated access to the BMS, according to use). Geographical area and so forth are not used. The BMS is presented in table 3.

With a penalty of two classes for the first claim, a claim frequency of 10 percent, and a very low starting class for non-business users (who constitute over 90 percent of the portfolio), it is no surprise that 58.90 percent of the policies will be found in class 1 (the highest among all systems) and that less than 1 percent of the policyholders are expected to pay a malus. The stationary average level is 67.92, which leads to a relative stationary average level of 5.7, the lowest among all systems. The average Belgian policyholder is, therefore, closer to the lowest class than are all of his or her European and Japanese colleagues.

The stationary average level for the sub-population of insureds who stay in the system for at least 20 years is 61.91, hardly more than the minimum of 60. Consequently, new non-business users, starting at level 85, pay an implicit surcharge of 37.3 percent. New business users have access to the BMS at level 100; their penalty is 61.5 percent. The efficiency of the system is consequently very low: 6.7 percent for non-business users, 8.4 percent for business users.

Table 3

Belgian Bonus-malus System*

Class after claims totaling Premium 6 or

Class Level 0 1 2 3 4 5 more

18 200 17 18 18 18 18 18 18 17 160 16 18 18 18 18 18 18 16 140 15 18 18 18 18 18 18 15 130 14 17 18 18 18 18 18 14 120 13 16 18 18 18 18 18 13 115 12 15 18 18 18 18 18 12 110 11 14 17 18 18 18 18 11 105 10 13 16 18 18 18 18 10 100 9 12 15 18 18 18 18 9 100 8 11 14 17 18 18 18 8 95 7 10 13 16 18 18 18 7 90 6 9 12 15 18 18 18 6 85 5 8 11 14 17 18 18 5 80 4 7 10 13 16 18 18 4 75 3 6 9 12 15 18 18 3 70 2 5 8 11 14 17 18 2 65 1 4 7 10 13 16 18 1 60 1 3 6 9 12 15 18

*Access: - Class 6 for non-business users - Class 10 for business users

Special rule: a policyholder who does not make a claim for four consecutive years, but who is nevertheless in a class higher than 10, is automatically brought down to class 10.




Average retention levels remains at an acceptable level, but wild differences between the classes are observed: the moderate average retention of $203 is strongly influenced by the low retention ($157) and the large population of class 1. The retentions are much higher in the upper classes: they exceed $575 for all classes but the first five; a class-15 policyholder, with three claim-free years, should bear the costs of a claim of $1740 in order to benefit from the special transition rule.

This system was introduced in 1971. At that time, the observed claim frequency was 21 percent for the non-business users, 26 percent for the business users, and the system was considered to be adequate. That obviously is not longer the case. Instead of separating the good and the bad drivers, the only achievement of the present system is to introduce a disguised penalty for young drivers.

Recognizing the face that their present BMS is totally obsolete, Belgian insurers have undertaken a major revision of their tariff structure. A more efficient BMS is likely to be introduced in the next few years.

Finland

The tariff structure in Finland is set by the Ministry of Social Affairs and Health. It has to be applied uniformly by all insurers. The a priori classification variables used are geographical area (four areas) and type and use of vehicle. Business users form a single group. Non-business users are partitioned into 14 groups, according to the vehicle model. Note that age of driver has not been selected. The BMS is presented in table 4.

Table 4

The Finnish Bonus-malus System*

Class after claims totaling

Premium 4 or Class Level 0 1 2 3 more

14 150 13 14 14 14 14 13 130 12 14 14 14 14 12 120 11 13 14 14 14 11 110 10 13 14 14 14 10 100 9 11 14 14 14 9 80 8 10 13 14 14 8 70 7 10 13 14 14 7 60 6 9 11 14 14 6 60 5 9 11 14 14 5 50 4 8 10 13 14 4 50 3 8 10 13 14 3 50 2 8 10 13 14 2 50 1 8 10 13 14 1 40 1 7 10 13 14

*Access: Class 12.




Due to a high access level and to very severe penalties for the first claim in the lower classes, all classes (except 13 and 14) are well populated. The forecast percentage of policies in class 1 is "only" 29.26 percent, a well-spread distribution of policyholders leading to an excellent relative stationary average level of 22.1. The average stationary premium level for the sub-population of insureds in the system for 20 years is 47.96. So newcomers, who have to enter the system in class 12, face an extremely high first-year surcharge of 150.2 percent.

The efficiency of the system amounts to 13.7 percent. It would have been much higher if the premium levels in the lowest classes had been more differentiated (four classes are at level 50, two at level 60).

The average optimal retention is very high: $643. In contrast to many other systems, it is not obtained as an average of low retentions in the populated lower classes and high retentions in the sparsely populated upper classes: optimal retentions are high in all classes, ranging from $569 (class 5) to $1022 (class 12). Retentions in classes 1 to 4 are larger than $575, due to the severe penalties.

France

The tariff structure in France gives relative freedom. Insurers are free to devise their own rates, provided they obey guidelines imposed by the Ministry of Economy, Finance and Budget. The guidelines specify that the computation of the basic premium shall use the following criteria: characteristics of the car, geographical area, use of the car, and annual mileage. Approval must be obtained for the use of other criteria. In practice, the insurers of the non-mutual sector (around two-thirds of the market) all apply a tariff structure that differs little from the one described below, which is recommended by the Groupement Technique Accidents.

The main a priori classification variables are type of vehicle (15 classes, based primarily on the power of the engine), occupation of the policyholder (nine groups), and geographical area (five areas). Other variables include the time lapse since passing the driving test, and the age of the vehicle, used as a proxy for annual mileage, which is impossible to enforce for practical reasons. Heavy penalties (from 50 percent to 200 percent) are enforced for exceptional risks (drunken driving, severe driving offenses, hit-and-run,. . .). A discount is awarded for restricted driving, when the policyholder limits driving his or her vehicle to himself and his or her spouse. Note that the use of the age of the driver is prohibited since 1984.

Bonus-malus System

The basic premium is reduced by 5 percent for each year without a claim, and is increased by 25 percent for each reported claim. In case of shared responsibility, the increase is reduced by half (12.5 percent). Those percentages are applied to the previous level. In other words, if the first claim causes the premium to pass from level 100 to 125, the second increases the premium to 156, the third to 195, and so on (all numbers are rounded down).




The highest level is 350. However, after two consecutive years without a claim, the insured goes back to the basic premium at level 100. The lowest level is 50, and is reached after 13 consecutive claim-free years.

This system was introduced in July 1984, so the recent decrease of the observed claim frequency was taken into account, and the penalties are rather severe. The positive effects of this strictness on the distribution of policyholders and on the efficiency are, however, partially erased by the special transition rule: no malus after two claim-free years. As a result, less than 3 percent of the policyholders have a premium level exceeding 130. Consequently, the relative average stationary level amounts only to 9 and the efficiency to 16.8. These are disappointing figures considering the strictness of the penalties. Had the special rule not been enforced, the efficiency would have been 25.1 percent, the best of all systems analyzed.

Another drawback of the special rule is that it induces a tremendous bonus hunger for drivers of the upper classes with one claim-free year. Retentions reach $3700 in several classes. The average optimal retention is $414.

The surcharge for newcomers in the system amounts to 69.5, a figure that would appear reasonable if it were not added to the 150 percent a priori surcharge that applies for three years to newly-licensed drivers, and the $350 deductible that applies for the first two driving years. Besides, it takes 10 claim-free years for a novice French driver to reach the same premium level as an average policyholder in the BMS for at least 20 years.

Federal Republic of Germany

No statutory tariff applies in Germany. In practice, however, all insurers apply the same tariff, that makes use of two a priori classification variables: occupation of the policyholder (civil servants and farmers are entitled to discounts) and power of the vehicle (11 categories, according to the power of the engine). The age of the driver is not used. However, novice drivers have a one-year surcharge of 50 percent, due to a differentiated access to the BMS. The BMS is presented in table 5.

The German BMS has the same number of classes as the Belgian BMS; yet it is superior in nearly every respect, due to the more severe penalties in the lower classes and the high access class. All classes are well populated; only the upper two have less than 1 percent of policyholders. Only 25.57 percent of policies (the lowest among all systems analyzed) are in class 1. Consequently, the relative stationary average level attains 16.8; the stationary average level is 66.5, which situates the average policyholder between classes 10 and 11 (whereas the average Belgian driver is between classes 2 and 3).

The implicit penalty for newcomers who cannot prove they have had a license for three years is the highest among all systems at 299.4 percent. After one claim-free year, however, the novice driver is transfered from class 15 to class 13, and his or her premium drops by 75 points. The surcharge decreases to 128.3 percent.

The efficiency is 12.3. This rather low figure is explained by the fact that the first claim is "forgiven" for occupants of class 1. They are transfered to class




Table 5

German Bonus-malus System*


Premium 4 or Class Level 0 1 2 3 more

18 200 13 18 18 18 18 17 200 13 18 18 18 18 16 175 13 17 18 18 18 15 175 13 16 17 18 18 14 125 13 16 17 18 18 13 100 12 14 16 17 18 12 85 11 13 14 16 18 11 70 10 13 14 16 18 10 65 9 12 13 14 18 9 60 8 11 13 14 18 8 55 7 11 13 14 18 7 50 6 11 13 14 18 6 45 5 11 13 14 18 5 40 4 10 12 13 18 4 40 3 8 11 13 18 3 40 2 7 11 13 18 2 40 1 6 11 13 18 1 40 1 5 10 12 18

*Access: Class 15, unless the new entrant can prove he or she held a valid driver's license for at least three years, in which case the driver is placed in class 14.

5, but the premium level remains at 40. A partial pardon is awarded to the policyholders of classes 2 to 4.

The strictness of the transition rules (large decreases for claim-free years in the upper classes coupled with severe penalties for the first claim in the middle classes) leads to high retentions, more than $850 for all classes between 5 and 18, with the exception of class 9. The optimal retentions in the lowest classes are small, due to the total or partial forgiveness of the first claim. Consequently the average retention remains at the acceptable level of $357.

Italy

The Italian tariff structure is set by ministerial decree. All insurers apply the same structure, which only uses power of vehicle (five categories) and geographical area (four areas) as a priori classification variables. Age of driver and use of vehicle are not used. The BMS is presented in table 6.

Italian insurers are very lenient to their policyholders; this can hardly be called a BMS. All claims are penalized by only one class, and those in class 1 must incur three claims in a single year in order to be "punished" by a premium increase of five points. Moreover, these nearly form a majority: 49 percent of the policies are in class 1, 61.5 percent pay the minimum premium level of 70. Classes 4 to 9 are fairly well populated due to the high access class.




Table 6

Italian Bonus-malus System*


Premium Class Level 0 1 2 3 4

13 200 1 1 13 13 13 13 12 175 10 13 13 13 13 1 1 152 9 12 13 13 13 10 132 8 1 1 12 13 13 9 115 8 10 1 1 12 13 8 100 7 9 10 1 1 12 7 92 6 8 9 10 11 6 85 5 7 8 9 10 5 80 4 6 7 8 9 4 75 3 5 6 7 8 3 70 2 4 5 6 7 2 70 1 3 4 5 6 1 70 1 2 3 4 5

*Notes

Access: Class 9.

A policyholder may elect not to be subject to the BMS, in which case he or she is subject to a fixed deductible for each claim.

After 20 years, the average policyholder pays a premium corresponding to a level of only 70.13, a trifle more than the minimum of 70. Newcomers thus have an implicit surcharge of 64 percent. The Italian BMS penalizes youth instead of claims.

As a consequence, all the evaluation characteristics provide abysmal figures: the efficiency is 2.5 percent, the relative stationary average level 7.3 (second-lowest, after the Belgian BMS only because of the high access class); the average optimal retention amounts to $5. It is in the interest of a class 1 policyholder to report any claim above $2. Indeed the probability of having three claims in a year is less than 0.0002.

Japan

The tariff structure in Japan is applied by all insurers and consists of only two a priori classification variables: age of policyholder, and type and use of the vehicle. Both of these variables influence the starting class of the BMS, which is presented in table 7.

One of the most often heard criticisms of BMS is that the transition rules penalize the number of reported claims and not their amounts: the penalties for bending a back bumper and for killing three persons are the same. Japan is the only country that has recently reacted against this criticism by adopting differentiated transition rules. The rules presented in this table are valid for claims involving only property damage. A claim with bodily injury is penalized as two claims with material damage. This has been taken into




Table 7

Japanese Bonus-malus System*

Class after claims totaling Premium

Class Level 0 1 2 3 4 5 6

16 150 15 16 16 16 16 16 16 15 140 14 16 16 16 16 16 16 14 130 13 16 16 16 16 16 16 13 120 12 15 16 16 16 16 16 12 110 11 14 16 16 16 16 16 1 1 100 10 13 15 16 16 16 16 10 90 9 12 14 16 16 16 16 9 80 8 11 13 15 16 16 16 8 70 7 10 12 14 16 16 16 7 60 6 9 11 13 15 16 16 6 50 5 8 10 12 14 16 16 5 45 4 7 9 11 13 15 16 4 42 3 6 8 10 12 14 16 3 40 2 5 7 9 11 13 15 2 40 1 4 6 8 10 12 14 1 40 1 3 5 7 9 11 13

*Access: Class 11, 12 or 13 depending on the age of the policyholder, and the type and use of the vehicle.

account in the simulation by assuming that 10 percent of the claims (the observed proportion in Belgium) involve bodily injury.

The penalties for material claims are moderate: only two classes. The access classes are high, however. They maintain some occupation in the middle classes and explain the relative stationary average level of 18.4. With lower access classes the repartition of the policyholders would have been disastrous, since the stationary average level after 20 years or more is 42.45 (it leads to a surcharge for new drivers of 135.6, 159.1 or 182.7 percent, depending on the starting class). This is easily explained by the mild penalties and the low premium differentials in the lowest classes: three classes at level 40, one at 42, one at 45. The efficiency amounts to 14.8, an average value.

Optimal retentions were only computed for claims involving property damage, since bodily injury claims necessarily must be reported to the police, and insurers learn about all claims reported to the police. Because the number of penalty classes for claims is uniform throughout the system, the retentions form a regular pattern, being very small in class 1 (the first claim has no financial consequence), and regularly increasing to a maximum of $2,389 in class 14. Since the upper classes are only sparsely populated, the average optimal retention is an extremely low $83.

Grand Duchy of Luxembourg

The tariff structure in Luxembourg is set by ministerial decree, and must be applied by all companies. The only a priori classification variable used is




power of the vehicle, with seven categories. Age of driver is not used, except for a small deductible for young drivers. The BMS is presented in table 8.

This sister country of Belgium (they share the same currency, and, for the most part, the same insurers) has adapted a similar BMS, even copying the special transition rule that eliminates all maluses after four claim-free years. Because the access class is relatively higher, the results are slightly better. The efficiency is 10.1 percent, the relative stationary average level 10.1. All classes up to the 11th are well populated, but only 5.6 percent of the policyholders are in the malus zone. Class 1 accounts for 36.74 percent of the policies, which compares favorably with the Belgian BMS, because it takes a longer time for the best newcomers to reach class 1. Note that, without the special rule, the efficiency would have been 10.8 percent, instead of 10.1 percent. The difference is slight because very few people are in a position to take advantage of this rule.

Given the similarity of the transition rules, the optimal retentions are similar to those observed in Belgium. Their average is $214, as compared to

Table 8

The Bonus-malus System of Luxembourg*


Premium Class Level 0 1 2 3 4 5 6

22 250 21 22 22 22 22 22 22 21 225 20 22 22 22 22 22 22 20 200 19 22 22 22 22 22 22 19 180 18 21 22 22 22 22 22 18 160 17 20 22 22 22 22 22 17 140 16 19 21 22 22 22 22 16 130 15 18 20 22 22 22 22 15 120 14 17 19 21 22 22 22 14 115 13 16 18 20 22 22 22 13 110 12 15 17 19 21 22 22 12 105 11 14 16 18 20 22 22 11 100 10 13 15 17 19 21 22 10 100 9 12 14 16 18 20 22 9 90 8 11 13 15 17 19 21 8 85 7 10 12 14 16 18 20 7 80 6 9 11 13 15 17 19 6 75 5 8 10 12 14 16 18 5 70 4 7 9 11 13 15 17 4 65 3 6 8 10 12 14 16 3 60 2 5 7 9 11 13 15 2 50 1 4 6 8 10 12 14 1 50 1 3 5 7 9 11 13

*Access: Class 11.

Special rule: A policyholder who does not make a claim for four consecutive years, but who is nevertheless in a class higher than 11, is automatically brought down to class 11.




$203. Very high retentions are observed in the upper classes, after three claim-free years (these classes have no equivalent in the Belgian BMS). The maximum is $3,826 in class 19, but very few people are in that class (0.14 percent), and the majority of those are such bad drivers that their chances of driving for three years without a claim are negligible.

The Netherlands

The tariff structure in the Netherlands allows complete freedom. A large majority of insurers, however, have adopted a structure that uses four a priori classification variables: weight of the vehicle (the basic premium is a linear function of the weight), geographical area (three areas), age of the driver and annual mileage (the latter two determine the starting class of the BMS). The BMS is presented in table 9.

This BMS was introduced at the end of 1981; hence it is geared to the present low claim frequencies and compares very favorably with other

Table 9

The Dutch Bonus-malus System*

Class after claims totaling Premium 3

Class Level 0 1 2 or more

14 120 13 14 14 14 13 100 12 14 14 14 12 90 11 14 14 14 11 80 10 14 14 14 10 70 9 13 14 14 9 60 8 12 14 14 8 55 7 11 14 14 7 50 6 10 14 14 6 45 5 9 13 14 5 40 4 8 12 14 4 37.5 3 8 12 14 3 35 2 7 11 14 2 32.5 1 7 11 14 1 30 1 6 10 14

*The access class depends on the age of the policyholder and on the annual mileage, according to the following rules:

Mileage Less than Between 7,500 Over

Age 7,500 and 12,500 12,500

less than 23 13 13 13 24 and 25 12 13 13 26and27 11 12 13 28 and more 10 11 13




systems. The occupation of every class exceeds 3 percent, less than 29 percent of policies are in the first class, and the relative stationary average level reaches 27. The efficiency depends on the starting class. It equals 23.4 percent (the highest among all BMS) for starters of class 10, and is above 20 percent for all others. The penalties for newcomers range from 85.4 percent to 164.9 percent.

The other side of the coin is of course the high optimal retentions: $637 on the average, $2,225 for the maximum in class 11. The lowest retention, in class 1, is as high as $505. The second-lowest is at the other end of the scale, in class 14: $666.

Norway

The following tariff structure is used by all insurers that adhere to the Association of Property and Liability insurers. Their market share is around 70 percent. The a priori classification variables are annual mileage (four classes), geographical area (three areas), and vehicle model. The BMS is presented in table 10.

This BMS has no upper class; hence, the relative stationary average level cannot be defined. In practice the occupation of all the upper classes is very small: less than 0.3 percent for the total population of the classes above 12, 5.2 percent for the total of all classes higher than the access class. The stationary average level is 56.54, i.e. the average policy is situated between classes 3 and 4, which is very low. The rather high access class induces a strong penalty for newcomers: 161.8 percent.

The excellent efficiency of 21.2 percent is explained by the strong penalties in the three lowest classes. An occupant of class 1 causing an accident is transfered to class 4, and his or her premium doubles. The Norwegian and the Swedish systems are the only ones for which the premium can double after a single claim. In both cases doubling occurs in class 1. An obvious consequence is that Norwegian (and Swedish) class 1 drivers have the strongest incentive to avoid reporting claims to their insurer. The retention in class 1 is $646, the average is $757.

After this research project was completed, a leading Norwegian insurer decided to launch a new BMS at the end of 1987. In this system, each claim-free year is rewarded by a discount of 13 percent of the previous premium, with a maximum total discount of 75 percent. The major innovation lies in the penalties: each claim induces a premium increase which is a fixed monetary amount. Drivers with at least 10 consecutive years in the lowest class are not penalized for their first claim. The penalty for drivers who have had between five and nine years in class 1 is reduced by half. A deductible is enforced for all claimants whose discount is less than 20 percent. New drivers who are at least 25 years old enjoy an initial discount of 20 percent. All other drivers start with no discount.




Table 10

Norwegian Bonus-malus System*



22 240 10 21 230 10 20 220 10 22 19 210 10 21 18 200 10 20 22 17 190 10 19 21 16 180 10 18 20 22 15 170 10 17 19 21 14 160 10 16 18 20 22 13 150 10 15 17 19 21 12 140 10 14 16 18 20 22 11 130 10 13 15 17 19 21 10 120 9 12 14 16 18 20 22 9 110 8 11 13 15 17 19 21 8 100 7 10 12 14 16 18 20 7 90 6 9 1 1 13 15 17 19 6 80 5 8 10 12 14 16 18 5 70 4 7 9 1 1 13 15 17 4 60 3 6 8 10 12 14 16 3 50 2 6 8 10 12 14 16 2 40 1 5 7 9 11 13 15 1 30 1 4 6 8 10 12 14

*Access: Class 8.

Special feature: There is no upper class or upper premium level. The number of classes is theoretically infinite.

Spain

All Spanish insurers use a tariff structure that has eight classes of vehicle models. No other a priori variables are used. The BMS is presented in table 11.

This system has several noteworthy characteristics: (1) it has only five classes, (2) the ratio between the premium levels of the two extreme classes is very low: 1.43, as compared to at least twice that figure for all the other systems, (3) it is a pure bonus system, in the sense that all newcomers enter in the upper class, and (4) all bonuses are erased after a single claim.

Since there are only five classes, the high value of the relative stationary average level (36.7 percent) is somewhat misleading; all classes are well populated because the starting class is the highest, but a heavy concentration of policies is observed in class 1 (54.67 percent). The penalty for new drivers is




Table 11

Spanish Bonus-malus System*


Premium 1 Class Level 0 or more

5 100 4 5 4 100 3 5 3 90 2 5 2 80 1 5 1 70 1 5

*Access: Class 5.

the lowest among all systems (31.9 percent) and the efficiency is very poor at 6.5.

Because any claim leads to class 5, the sequence of the optimal retentions is a decreasing function of the premium level. Retentions are $366 in classes 1 and 2, $317 in class 3, $226 in class 4, and only $103 in class 5.

Sweden

All Swedish insurers have to apply a tariff structure with the following a priori classification variables: vehicle model, annual mileage (five classes), and geographical area (seven classes). The BMS is presented in table 12.

The Swedish BMS is toughest for the best drivers: a single claim in class 1 doubles the premium, and six consecutive claim-free years are then required in order to bring the premium back to its former level. This leads to a high efficiency (17.7 percent) and to the largest retention in class 1: $906. All retentions are high, with a maximum of $1,443 and an average of $900. Table 13 shows the importance of class subdivision for non-Markovian BMS: the optimal retentions substantially increase with the number of claim-free years in class 2.

The BMS is a pure bonus system, with new entrants having access to the upper class. Hence, there is a very high surcharge for novice drivers (204.5 percent) and a well-balanced distribution of policies. Note, however, that the efficiency would have been higher with a lower access class: /4u5(X) = 0.2011, for instance. These factors concur to produce a very high relative stationary average level of 30.1.

Switzerland

All insurers must apply a tariff structure with only two a priori classification variables: cubic capacity of the engine (four classes), and age of the driver (a deductible of around $425 applies to drivers under 25, when found guilty of an accident). The BMS is presented in table 14.

In 1963, Switzerland was the first country to introduce a BMS. The present system differs little from the original one. It has some excellent features, and




Table 12

Swedish Bonus-malus System*


Premium 3 Class Level 0 1 2 or more

7 100 6 7 7 7 6 80 5 7 7 7 5 70 4 7 7 7 4 60 3 6 7 7 3 50 2 5 7 7 2 40 1 or 2 4 6 7 1 25 1 3 5 7

*Access: Class 7.

A special feature is that class 1 is only awarded after six consecutive claim-free years.

Table 13

Optimal Retentions in Class 2: Sweden

Number of claim-free years in class 2 Optimal retentions

1 $586 2 $700 3 $831 4 $980 5 $1149

a very high efficiency of 22.2 for the present claim frequency of 0.10. It must have seemed very severe to the policyholders back in the 1960's, with claim frequencies above 0.20. While the relative stationary average level is low (11.9), and optimal retentions are high (average: $420), they follow a regular pattern, with a peak of $3,977 in class 19. The average policyholder after at least 20 years in the BMS is very close to class 4, since his or her average premium level is 54.88. Consequently the penalty for newcomers, starting in class 10, is 82.2 percent.

United Kingdom

The United Kingdom offers complete freedom in the tariff structure. Most companies use age of driver, vehicle model, geographical area, use of vehicle and age of vehicle extensively. Discounts are usually allowed if the driving is restricted to the driver and his or her spouse. Since insurers enjoy complete freedom, many bonus-malus systems coexist. The system presented in table 15 is used by a leading insurer, and is typical.




Table 14

Swiss Bonus-malus System*



22 270 21 22 22 22 22 22 22 21 250 20 22 22 22 22 22 22 20 230 19 22 22 22 22 22 22 19 215 18 22 22 22 22 22 22 18 200 17 21 22 22 22 22 22 17 185 16 20 22 22 22 22 22 16 170 15 19 22 22 22 22 22 15 155 14 18 21 22 22 22 22 14 140 13 17 20 22 22 22 22 13 130 12 16 19 22 22 22 22 12 120 11 15 18 21 22 22 22 11 110 10 14 17 20 22 22 22 10 100 9 13 16 19 22 22 22 9 90 8 12 15 18 21 22 22 8 80 7 11 14 17 20 22 22 7 75 6 10 13 16 19 22 22 6 70 5 9 12 15 18 21 22 5 65 4 8 11 14 17 20 22 4 60 3 7 10 13 16 19 22 3 55 2 6 9 12 15 18 21 2 50 1 5 8 11 14 17 20 1 45 1 4 7 10 13 16 19

*Access: Class 10.

Since a posteriori rating plays a lesser role in the United Kingdom than on the continent, a BMS with few classes and a low efficiency of 10.6 is probably sufficient for British insurers, given all the other a priori criteria they use. However, beware that since newcomers have access to the system in class 6 their implicit penalty if 93 percent. Since, in addition, 17-year-olds pay an a priori surcharge of 113.3 percent, the total surcharge is 311.7 percent, the highest among all tariff structures analysed.

With the exception of the upper class, all other classes are well populated. The occupation of class 1, 54.67 percent, is excessive. The problem is further compounded by the recent introduction of "protected discount schemes," usually confined to policyholders who have earned entitlement to the highest rate of discount. Some insurers allow class 1-policyholders to make as many as two claims in three years without loss of bonus and without charging any additional premium. Other insurers charge an additional premium of 10 percent and allow an unlimited number of claims without loss of bonus.

The effects of those schemes have not been investigated in this analysis. It is however clear that they will further decrease the efficiency, increase the percentage of policyholders in class 1, and deeply modify optimal retentions:




Table 15

A British Bonus-malus System*


Premium 3 Class Level 0 1 2 or more

7 100 6 7 7 7 6 75 5 7 7 7 5 65 4 6 7 7 4 55 3 5 7 7 3 45 2 5 7 7 2 40 1 4 6 7 1 35 1 4 6 7

*Access: Class 6

bonus hunger will nearly disappear in class 1, and tremendously increase in class 2. Without protected schemes, the retentions are acceptable, averaging $434, and only exceeding $700 in class 6.

Conclusions

The preceding analysis led to the following rules of thumb for the construction of a good BMS:

1. Use a large number of classes. 2. Introduce penalties for the first claim as severe as commercially possible,

especially in the lower classes. 3. Never forgive the first claim, whatever the pressure of the marketing

department. 4. Do not introduce a special transition rule to erase maluses faster. 5. Do not introduce a priori surcharges for young drivers. Instead use a

high access class, especially in countries where malus evasion is not too difficult, but do not go as far as implementing a pure bonus system.

Reference

1. Lemaire,J(1985). Automobile Insurance: Actuarial Models. Boston, Kluwer.



lemaire 1988 a comparative

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