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Multi-event bonus-malus scales.

ABSTRACT

This article is devoted to the design of bonus-malus scales involving different types of claims. Typically, claims with or without bodily injuries, or claims with full or partial liability of the insured driver, are distinguished and entail different penalties. Under mild assumptions, claim severities can also be taken into account in this way. Numerical illustrations enhance the interest of the approach.

INTRODUCTION AND MOTIVATION

Many important factors cannot be taken into account a priori when pricing motor third party liability insurance products. For instance, swiftness of reflexes, aggressiveness behind the wheel, or knowledge of the highway code cannot be integrated into risk classification. Consequently, tariff cells are still quite heterogeneous despite the use of many classification variables. This residual heterogeneity typically causes overdispersion: data involving claim counts exhibit variability exceeding that explained by Poisson models. This can be modeled by a random effect in a statistical model.

It is reasonable to believe that the hidden characteristics are partly revealed by the number of claims reported by the policyholders. Hence the adjustment of the premium on the basis of the individual claims experience in order to restore fairness among policyholders. In that respect, the allowance of past claims in a rating model derives from an exogenous explanation of serial correlation for longitudinal data. In this case, correlation is only apparent and results from the revelation of hidden features in the risk characteristics.

Once estimated, the statistical model incorporating the portfolio heterogeneity can be used to perform prediction on longitudinal data and allows for experience rating in motor insurance. In an empirical Bayesian setting, the prediction is derived from the expectation of a random effect with respect to a posterior distribution taking into account the history of the individual. This is the topic of credibility theory. We refer the interested reader to Chapter 7 of Kaas et al. (2001) for more details, as well as to the papers by Dionne and Vanasse (1989, 1992) and Pinquet (2000).

In many countries, insurance companies integrate past claims histories in ratemaking with the help of bonus-malus systems. Such systems can be seen as commercial simplifications of credibility mechanisms. When a bonus-malus system is in force, the policyholders move inside a scale according to the number of claims they file (going up when claims are reported and down when they do not file any claim). The amount of premium is then obtained by multiplying the reference premium with a percentage attached to the level occupied by the policyholder in the scale. All policies in the same tariff class can then be partitioned according to the level they occupy in the bonus-malus scale. Therefore, bonus-malus scales refine the a priori risk classification.

As pointed out by Lemaire (1995), all bonus-malus systems in force throughout the world (with the exception of Korea) penalize the number of reported claims, without taking the cost of these claims into account. In Chapter 13 of his book, Lemaire (1995) applied a model proposed by Picard (1976) to Belgian data. This credibility model allows insurance companies to subdivide the claims into two categories, small and large losses. Instead of determining a limiting amount (such a criterion would lead to substantial practical problems, due to the time needed to evaluate the cost of the claim and endless arguments with policyholders who caused a claim slightly above the limit), Lemaire (1995) distinguished the accidents that caused property damage only from those that caused bodily injuries. Since the latter cost much more on average, this approach implicitly integrates the cost of the claim in a posteriori premium corrections. The credibility model proposed by Lemaire (1995) is based on a Poisson-Gamma mixture, and assumes that given the expected annual claim frequency of the policyholder, the frequency of claims with bodily injuries conforms to a Beta distribution. This approach can be extended to several categories of claims using a Dirichlet distribution (i.e., a suitable multivariate Beta distribution).

In this article, we work in the framework of bonus-malus scales (and not of credibility models) and we deal with several types of events (assuming a multinomial partitioning scheme, to avoid using multivariate Beta distributions). As mentioned above, all the classical bonus-malus systems are based on a single type of event: the occurrence of claims at fault, no matter their severity or whether the policyholders are only partially liable for them. This oversimplification can be regarded as problematic for commercial purposes: it seems desirable to integrate the severity of the claims and to recognize the partial liability of the policyholder. For example, the bonus-malus system in force in France entails a reduced penalty if the policyholder is only partially liable for the claim.

Prominent examples of a posteriori ratemaking mechanisms based on several types of events are provided by the experience rating systems in force in North America. These systems not only incorporate accidents at fault but also elements of the policyholders' driving record. Let us briefly present the Massachusetts safe driver insurance plan, one of the most sophisticated in force in North America. This program is mandated by state law and encourages safe driving by rewarding drivers who do not cause an accident, or incur a traffic law violation. Specifically, each policyholder is assigned a level between 9 and 35, based on his driving record during the previous six years. A new driver begins at level 15 (relativity of 100%). Occupying any level below 15 entails a premium discount, whereas above level 15 the driver pays a surcharge. For each incident-free year of driving, the policyholder goes down one level. The driver will move up a certain number of levels based on the type of incident: two levels for a minor traffic violation, three levels for a minor at-fault accident, four levels for a major at-fault accident, and five levels for a major traffic violation. The Massachusetts system forgets all incidents after six years.

This article addresses the actuarial modeling of such systems, with penalties depending on different types of events. The modeling uses the concept of Markov chains. We will see that under mild assumptions, the trajectory of each policyholder in the scale can be modeled with the aid of discrete-time Markov processes. The relativities associated to each level will then be computed using the maximum accuracy principle introduced by Norberg (1976). All the reasonings held in this article are based on the stationary distribution of the system. It is worth mentioning that extensions to transient distributions are nevertheless possible along the line of Borgan, Hoem, and Norberg (1981).

Let us now detail the contents of the article. The next section presents the actuarial modeling of claim frequencies. In the section "Markov Modeling," we model the policyholders' trajectories in the scale using Markov chains, and we recall some basic features of these stochastic processes. In the section "Determination of the Relativities," we compute the relativities for each level of the scale. The next section presents some numerical illustrations. The final section concludes.

THE MODEL

We consider a portfolio partitioned in [kappa] risk classes [C.sub.1], [C.sub.2], ..., [C.sub.[kappa]] on the basis of the a priori information. Each of these risk classes possesses its own expected annual claim frequency. We denote as [lambda]k the expected number of claims for policyholders in [C.sub.k], k = 1, 2, ..., [kappa]. Furthermore, [w.sub.k] is the relative weight of the kth risk class in the portfolio.

Let us pick a policyholder at random from the portfolio and denote as N the number of claims he reported during the year. Furthermore, let C be the (unknown) risk class to which this policyholder belongs. Clearly, Pr[C = [C.sub.k]] = [w.sub.k]. Denoting as [THETA] the (unknown) accident proneness of this policyholder, the conditional probability mass function of N is given by

Pr[N = j|[THETA] = [theta], C = [C.sub.k]] exp(-[[lambda].sub.k][theta])[(-[[lambda].sub.k][theta]).sup.j]j/j!, j [member of] N = {0, 1, 2, ...}.

The risk profile of the portfolio is described by the structure function u(*). More formally, u(*) is the probability density function of [THETA] and we assume that E[[THETA]] = 1. Since [THETA] represents the residual effect of unobserved characteristics, it seems reasonable to assume that [THETA] and C are mutually independent. Hence, the unconditional probability mass function of N is given by

Pr[N = j] = [summation over (k)] [w.sub.k] [[integral].sup.+[infinity].sub.0] Pr[N = j|[THETA] = [theta],C = [C.sub.k]]u([theta])d[theta], j [member of] N.

We distinguish among [tau] different types of claims caused by the policyholder. Each type of claims induces a specific penalty for the policyholder. For instance, one could think of

* claims with bodily injuries and claims with material damage only ([tau] = 2);

* claims with partial liability and claims with full liability ([tau] = 2); or

* introducing claim severities (e.g., claims with amount less than $1,000, between $1,000 and $10,000, and claims above $10,000, so that [tau] = 3). In this case, we have to assume that claim severities and claim frequencies are mutually independent.

We assume a multinomial scheme for the classification of the claims, and we denote as [q.sub.k1], [q.sub.k2], ..., [q.sub.k[tau]] the probability that the claim is of type 1, 2, ..., [tau], respectively, for a policyholder in risk class [C.sub.k]. The identity [q.sub.k1] + [q.sub.k2] + ... + [q.sub.k[tau]] = 1 obviously holds true. Now, let [N.sub.1], [N.sub.2], ..., [N.sub.[tau]] be the number of claims of type 1, 2, ..., [tau], respectively. Given [THETA] and C, the random variables [N.sub.1], [N.sub.2], ..., [N.sub.[tau]] are mutually independent, with respective conditional probability mass function

Pr[[N.sub.l] = j|[THETA] = [theta], C = [C.sub.k]] = exp(-[[lambda].sub.k][theta][q.sub.kl]) [(-[[lambda].sub.k][theta][q.sub.kl]).sup.j]/j!, j [member of] N,

for l = 1, ..., [tau].

MARKOV MODELING

Bonus-malus systems can be modeled using Markov chains. This route has been followed for a long time by numerous researchers, such as Norberg (1976), Gilde and Sundt (1989), Denuit and Dhaene (2001), Centeno and Silva (2001), and Pitrebois, Denuit, and Walhin (2003).

The scale is assumed to have s + 1 levels, numbered from 0 to s. A specified level is assigned to a new driver (often according to the use of the vehicle). Each claim-free year is rewarded by a bonus point (i.e., the driver goes down one level). Each type of claim entails a specific penalty, expressed as a fixed number of levels per claim.

We assume that the scale possesses the following Markovian property: The knowledge of the present level and of the number of claims of each type filed during the present year suffices to determine the level to which the policy is transferred. This ensures that the bonus-malus system may be represented by a Markov chain (at least conditionally on the observable characteristics and random effects).

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](v; q) be the probability of moving from level [l.sub.1] to level [l.sub.2] for a policyholder with annual mean claim frequency v and vector probability q = [([q.sub.1], ..., [q.sub.[tau]]).sup.t]; here [q.sub.j] is the probability that the claim be of type j. Furthermore, M(v; q) is the one-step transition matrix, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking the vth power of M(v; q) yields the v-step transition matrix whose element ([l.sub.1][l.sub.2]), denoted as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the probability of moving from level [l.sub.1] to level [l.sub.2] in 1; transitions.

The transition matrix M(v; q) associated with such a bonus-malus system is assumed to be regular, i.e., there exists some integer [xi].sub.0] [greater than or equal to] 1 such that all entries of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are strictly positive. Consequently, the Markov chain describing the trajectory of a policyholder with expected claim frequency v and vector probability q is ergodic and thus possesses a stationary distribution [pi](v; q) = ([[pi].sub.0](v; q), [[pi].sub.1](v; q), ...., [[pi].sub.s] [(v; q)).sup.t]; [pi].sub.l](v; q) is the stationary probability for a policyholder with mean frequency v to be in level l, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let e be a column vector of 1's and let E be the (s + 1) x (s + 1) matrix all of whose entries are 1, i.e., consisting of s + 1 column vectors e. Then, the stationary probabilities are directly obtained from the formula

[[pi].sup.t](v; q) = [e.sub.t][(I - M(v; q) + E).sup.-1]

that can be found, e.g., in Rolski et al. (1999).

Let [L.sub.v;q] {0, 1, ..., s} and conform to the distribution [[pi](v; q), i.e.,

Pr[[L.sub.v;q] = l] = [[pi].sub.l] (v'q), l = 0,1, ..., s.

The variable [L.sub.v;q] thus represents the level occupied by a policyholder with annual expected claim frequency v and probability vector q once the steady state has been reached.

Now, let L be the level occupied in the scale by a randomly selected policyholder once the steady state has been reached. The distribution of L can be written as

Pr[L = l] = [summation over (k)][w.sub.k] [[integral].sup.+[infinity].sub.0] [[pi].sub.l]([[lambda].sub.k][theta];[q.sub.k]u([theta])d[theta], l = 0, 1, ..., s. (1)

DETERMINATION OF THE RELATIVITIES

The relativity associated with level l is denoted as [r.sub.l]; the meaning is that an insured occupying that level pays an amount of premium equals to [r.sub.l] % of the reference premium determined on the basis of his observable characteristics.

Following Norberg (1976), our aim is to minimize the expected squared difference between the "true" relative premium [THETA] and the relative premium [r.sub.L] applicable to this policyholder (after the stationary state has been reached), i.e., the goal is to minimize

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

It is easily seen that E[[r.sub.L]] = 1, resulting in financial equilibrium once steady state is reached.

NUMERICAL ILLUSTRATIONS

In this section, we consider an application similar to the one developed in Chapter 13 of Lemaire (1995). As mentioned in the introduction, we do not use the Beta modeling of Picard (1976) and work with bonus-malus scales (and not with credibility formulas). Moreover, we allow explicitly for a priori risk classification.

A Priori Ratemaking

The data used to illustrate this article relate to a Belgian motor third party liability portfolio observed during the year 1997. The data set comprises 19,585 policies. The overall mean claim frequency is 19.5% (far above the European average). The annual frequency of claims with bodily injuries is 1.6%; the frequency of claims without bodily injuries (i.e., claims with material damage only) amounts to 17.9%.

In this portfolio, the two types of claims we consider are positively correlated. This can be seen from Table 1, where the conditional expectation of the number of claims of one type is computed given the number of claims of the other type. The more claims of one type reported, the higher this conditional expectation, resulting in positive dependence.

The following information is available on an individual basis: in addition to the number of claims filed by each policyholder (with the dichotomy bodily injuries/material damage only) and the exposure-to-risk from which these claims originate (i.e., the number of days the policy has been in force during 1997), we know the age of the policyholder in 1997 (18-22 years, 23-30, 31 and above), his/her gender (male-female), the kind of district where he/she lives (rural area or urban area) and the power of the vehicle in kilowatts (less than or above 66 KW).

In this section, the structure function is taken to be a gamma probability density function with unit mean, i.e.,

u([theta]) = 1/[GAMMA](a) [a.sub.a] [[theta].sub.a-1] exp(-a[theta]), [theta] > 0, for some a > 0. A segmented tariff has been built on the basis of a Negative Binomial regression model. More precisely, given [THETA] = [theta], the annual claim frequency N of the policyholder has probability function

Pr[N = k|[THETA] = [theta]] = exp (- d exp ([[beta].sub.0] + [[beta].sub.1][x.sub.1] + [[beta].sub.2][x.sub.2] + [[beta].sub.3][x.sub.31] + [[beta].sub.4][x.sub.4] + [[beta].sub.5][x.sub.5])) x [(d exp ([[beta].sub.0] + [[beta].sub.1][x.sub.1] + [[beta].sub.2][x.sub.2] + [[beta].sub.3][x.sub.31] + [[beta].sub.4][x.sub.4] + [[beta].sub.5][x.sub.5])).sup.k]/k!

for k = 0, 1, 2, ..., where d is the exposure-to-risk (in years), [x.sub.1] equals I if the policyholder's age is between 18 and 22, and 0 otherwise, [x.sub.2] equals 1 if the policyholder's age is between 23 and 30, and 0 otherwise, [x.sub.3] equals 1 if the policyholder is female, and 0 otherwise, [x.sub.4] equals 1 if the policyholder lives in a rural area, and 0 otherwise, and [x.sub.5] equals 1 if the vehicle is less than 66 kW, and 0 otherwise. This model has been fitted using the GENMOD procedure of SAS/STAT. All the regression coefficients were found significantly different from 0. This yields the twenty-four risk classes [C.sub.1], [C.sub.2], ..., [C.sub.24] described in Table 2. Table 3 gives, for each of these risk classes [C.sub.k], the specific annual expected claim frequency [[lambda].sub.k], the relative weight [w.sub.k], and the probabilities [q.sub.k1] and [q.sub.k2] that the claim will cause bodily injuries or not ([q.sub.k1] can be estimated as the ratio of the number of claims with bodily injuries and the total number of claims filed by the policyholders in [C.sub.k]).

Bonus-Malus Scale

Let us now consider the soft experience rating system defined in Taylor (1997) with the transition rules formerly used by the Japanese companies. Instead of considering one type of claim, we now penalize differently claims with bodily injuries and claims with material damage only. There are nine levels (numbered from 0 to 8). Level 6 is the starting level. A higher-level number indicates a higher premium. If no claims have been reported by the policyholder then he moves down one level. Claims with material damage only are penalized by two levels whereas claims with bodily injuries entail a penalty of four levels. If [n.sub.1] claims with bodily injuries and [n.sub.2] claims with material damage only are reported during the year then the policyholder moves 4[n.sub.1] + 2[n.sub.2] levels up. This system is abbreviated as -1/+2/+4, in obvious notations.

The transition matrix for a policyholder with annual mean claim frequency 0 and vector probability q = [([q.sub.1], [q.sub.2]).sup.t] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

and [SIGMA] represents the sum of all the elements in the same row.

Computation of the Relativities

Table 4 gives, for each of the nine levels of the bonus-malus scale, the proportion of the portfolio in that level (column 2) and the relativity attached to that level (column 3) for the system -1/+2/+4. About half of the portfolio is in level 0 and enjoys a discount of about 30%. The rest of the portfolio is spread out among levels 1-8. The [r.sub.l]s range from 69.38% to 209.82%.

In order to compare the results with those of traditional bonus-malus scales, we have also considered three other scales. For all of them, each claim-free year is rewarded by one level down in the scale. The first bonus-malus system penahzes each claim (with or without bodily injuries) by two levels up in the scale (this system is referred to as -1/+2), the second one by three levels up (this system is referred to as -1/+3) and the third one by four levels up (this system is referred to as -1/+4). The [r.sub.l]s for these scales are computed with the formulas derived in Pitrebois, Denuit, and Walhin (2003).

Clearly, the more the claims are penalized, the more the policyholders occupying the lowest levels are awarded discounts, and the less the policyholders in the upper part of the scale are penalized. The scale -1/+2/+4 is closer to scale -1/+2. This is because the majority of the claims only induce material damage. Nevertheless, [r.sub.8] is reduced from 218.03% to 209.82% when the claims with bodily injuries are more severely penalized.

Impact of the Average Claim Frequency

The numerical example discussed in the first three subsections of the section "Numerical Illustrations" uses the portfolio of a company that exhibits an average claim frequency much above the standard ones (that typically range in the interval 6-10%). This of course produces penalties that are not too high, as can be seen from Table 4.

The effect of the average claim frequency at the portfolio level on the relativities is explored in this section. More precisely, we have kept the twenty-four risk classes described in Table 2 but we have scaled the corresponding annual claim frequency to produce an average claim frequency of 6, 8, and 10%, respectively, at the portfolio level.

The results are displayed in Table 5. We can see there that when the average claim frequency increases from 6% to 10%, the proportion of policyholders in level 0 decreases from 85.93% to 76.01%. Moreover, the maluses increase as the average claim frequency of the portfolio decreases (the relativity r8 decreases from 264.38% to 247.16% as the average claim frequency increases from 6% to 10%). The penalties induced by the bonus-malus system remain nevertheless applicable in practice.

CONCLUSION

European directives have introduced complete rating freedom: Insurance companies operating in most EU countries are now free to set up their own rates, select their own classification variables and design their own bonus-malus system. In most European countries, companies have taken advantage of this freedom by introducing more rating variables. It can be expected that they will start to compete on the basis of bonus-malus system. In that respect, this article offers an alternative approach to traditional bonus-malus scales.

REFERENCES

Borgan, O, J. M. Hoem, and R. Norberg, 1981, A Nonasymptotic Criterion for the Evaluation of Automobile Bonus Systems, Scandinavian Actuarial Journal, 1981: 165-178.

Centeno, M., and J. M. A. Silva, 2001, Bonus Systems in an Open Portfolio, Insurance: Mathematics & Economics, 28: 341-350.

Denuit, M., and J. Dhaene, 2001, Bonus-Malus Scales Using Exponential Loss Functions, German Actuarial Bulletin, 25: 13-27.

Dionne, G., and C. Vanasse, 1989, A Generalization of Actuarial Automobile Insurance Rating Models: The Negative Binomial Distribution with a Regression Component, ASTIN Bulletin, 19: 199-212.

Dionne, G., and C. Vanasse, 1992, Automobile Insurance Ratemaking in the Presence of Asymmetrical Information, Journal of Applied Econometrics, 7: 149-165.

Gilde, V., and B. Sundt, 1989, On Bonus Systems with Credibility Scales, Scandinavian Actuarial Journal, 1989: 13-22.

Kaas, R., M. J. Goovaerts, J. Dhaene, and M. Denuit, 2001, Modern Actuarial Risk Theory (Dordrecht: Kluwer Academic Publishers).

Lemaire, J., 1995, Bonus-Malus Systems in Automobile Insurance (Boston: Kluwer Academic Publisher).

Norberg, R., 1976, A Credibility Theory for Automobile Bonus System, Scandinavian Actuarial Journal, 1976: 92-107.

Picard, P., 1976, Generalisation de l'etude sur La Survenance Des Sinistres En Assurance Automobile, Bulletin Trimestriel de l'Institut des Actuaires Francais, 385: 204-267.

Pinquet, J., 2000, Experience Rating through Heterogeneous Models. In G. Dionne, Handbook of Insurance (Dordrecht: Kluwer Academic Publishers).

Pitrebois, S., M. Denuit, and J.-F. Walhin, 2003, Setting a BMS in the Presence of Other Rating Factors: Taylor's Work Revisited, ASTIN Bulletin, 33: 419-436.

Rolski, T., H. Schmidli, V. Schmidt, and J. Teugels, 1999, Stochastic Processes for Insurance and Finance (New York: John Wiley & Sons).

Taylor, G., 1997, Setting a Bonus-Malus Scale in the Presence of Other Rating Factors, ASTIN Bulletin, 27: 319-327.

Sandra Pitrebois is at Secura, Avenue des Nerviens, 9-31 boite 6, B-1040 Bruxelles, Belgium. The author can be contacted via e-mail: Sandra.Pitrebois@secura-re.com Michel Denuit is at Institut de Statistique, Universite Catholique de Louvain, Voie du Roman Pays, 20, B-1348 Louvainla-Neuve, Belgium. Jean-Francois Walhin is at Institut des Sciences Actuarielles, Universite Catholique de Louvain, Grand-Rue, 54, B-1348 Louvain-la-Neuve, Belgium. The authors can be contacted via e-mail: denuit@stat.ucl.ac.be and walhin@actu.ucl.ac.be. The authors thank the referees for interesting comments which led to considerable improvements of this article. The authors gratefully acknowledge the financial support of the Belgian government under the "Projet d'Action de Recherches Concertees" 04/09-320.
TABLE 1
Expected Annual Frequency of Claims with Bodily Injuries (Resp.
with Material Damage Only) Given the Number of Claims with Material
Damage Only (Resp. with Bodily Injuries)

Conditional Expectation Conditional Expectation
of the Number of Claims of the Number of Claims
with Bodily Injuries with Material Damage Only

Given the number of claims Given the number of claims
with material damage only with bodily injuries

 =0 0.014 =0 0.177
 =1 0.019 =1 0.255
 =2 0.022 =2 0.701

TABLE 2
Description of the Twenty-four Risk Classes Composing the Portfolio

Risk Class Age Gender District Power

[C.sub.1] 18-22 Male Urban >66 kW
[C.sub.2] 18-22 Male Rural >66 kW
[C.sub.3] 18-22 Male Urban <66 kW
[C.sub.4] 18-22 Male Rural <66 kW
[C.sub.5] 23-30 Male Urban >66 kW
[C.sub.6] 23-30 Male Rural >66 kW
[C.sub.7] 23-30 Male Urban <66 kW
[C.sub.8] 23-30 Male Rural <66 kW
[C.sub.9] >30 Male Urban >66 kW
[C.sub.10] >30 Male Rural >66 kW
[C.sub.11] >30 Male Urban <66 kW
[C.sub.12] >30 Male Rural <66 kW
[C.sub.13] 18-22 Female Urban >66 kW
[C.sub.14] 18-22 Female Rural >66 kW
[C.sub.15] 18-22 Female Urban <66 kW
[C.sub.16] 18-22 Female Rural <66 kW
[C.sub.17] 23-30 Female Urban >66 kW
[C.sub.18] 23-30 Female Rural >66 kW
[C.sub.19] 23-30 Female Urban <66 kW
[C.sub.20] 23-30 Female Rural <66 kW
[C.sub.21] >30 Female Urban >66 kW
[C.sub.22] >30 Female Rural >66 kW
[C.sub.23] >30 Female Urban <66 kW
[C.sub.24] >30 Female Rural <66 kW

TABLE 3
Expected Annual Claim Frequencies for the Twenty-four Risk
Classes Described in Table 2, and Proportion of Claims with
and without Bodily Injuries

 Weight Frequency
 [[omega] [[lambda] Probability Probability
Risk Class .sub.k] .sub.k] [q.sub.k1] [q.sub.k2]

[C.sub.1] 0.0009 0.4138 0.0891 0.9109
[C.sub.2] 0.0030 0.3323 0.0835 0.9165
[C.sub.3] 0.0036 0.3909 0.0949 0.9051
[C.sub.4] 0.0115 0.3140 0.0890 0.9110
[C.sub.5] 0.0217 0.2811 0.0823 0.9177
[C.sub.6] 0.0526 0.2257 0.0771 0.9229
[C.sub.7] 0.0477 0.2655 0.0877 0.9123
[C.sub.8] 0.0874 0.2132 0.0822 0.9178
[C.sub.9] 0.0613 0.2082 0.0746 0.9254
[C.sub.10] 0.1308 0.1672 0.0699 0.9301
[C.sub.11] 0.0666 0.1967 0.0796 0.9204
[C.sub.12] 0.1510 0.1580 0.0746 0.9254
[C.sub.13] 0.0004 0.3874 0.0946 0.9054
[C.sub.14] 0.0006 0.3111 0.0887 0.9113
[C.sub.15] 0.0027 0.3660 0.1007 0.8993
[C.sub.16] 0.0075 0.2940 0.0945 0.9055
[C.sub.17] 0.0067 0.2632 0.0874 0.9126
[C.sub.18] 0.0133 0.2113 0.0819 0.9181
[C.sub.19] 0.0336 0.2486 0.0931 0.9069
[C.sub.20] 0.0797 0.1997 0.0873 0.9127
[C.sub.21] 0.0230 0.1949 0.0793 0.9207
[C.sub.22] 0.0404 0.1565 0.0743 0.9257
[C.sub.23] 0.0505 0.1842 0.0846 0.9154
[C.sub.24] 0.1035 0.1479 0.0792 0.9208

TABLE 4
Results for the Bonus-Malus Systems -1/+2/+4, -1/+2, -1/+3, and -1/+4

 -1/+2/+4 -1/+2

Level l [[pi].sub.l] % [r.sub.l] % [[pi].sub.l] % [r.sub.l] %

8 4.67 209.82 4.09 218.03
7 4.04 190.04 3.55 197.68
6 4.05 169.06 3.55 176.73
5 3.96 155.00 3.55 161.55
4 5.21 133.41 4.72 139.85
3 5.12 124.99 4.85 129.73
2 9.57 103.52 10.07 104.74
1 7.89 98.66 8.29 99.83
0 55.50 69.38 57.33 70.36

 -1/+3 -1/+4

Level l [[pi].sub.l] % [r.sub.l] % [[pi].sub.l] % [r.sub.l] %

8 7.44 187.55 10.37 169.50
7 6.16 170.10 8.49 152.90
6 6.14 148.81 7.16 139.75
5 5.68 136.07 6.16 129.11
4 5.24 126.22 8.77 104.65
3 8.88 101.98 7.21 98.96
2 7.34 96.95 6.00 93.89
1 6.13 92.40 5.05 89.35
0 46.99 64.38 40.79 61.34

TABLE 5
Results for the Bonus-Malus Systems -1+/+2/+4 with Annual Expected
Claim Frequency of 6, 8, and 10%, Respectively, at the Portfolio Level

 Freq = 0.06 Freq = 0.08

Level l [[pi].sub.l] % [r.sub.l] % [[pi].sub.l] % [r.sub.l] %

8 0.14 264.38 0.37 256.53
1 0.19 247.95 0.45 239.01
6 0.35 217.70 0.71 211.61
5 0.48 206.90 0.89 199.30
4 1.28 170.84 1.94 166.92
3 1.48 167.23 2.19 161.48
2 5.29 138.35 6.58 133.14
1 4.87 134.93 5.91 129.13
0 85.93 91.78 80.95 88.34

 Freq = 0.10

Level l [[pi].sub.l] % [r.sub.l] %

8 0.76 247.16
1 0.85 229.13
6 1.18 203.86
5 1.40 190.78
4 2.63 161.39
3 2.88 154.78
2 7.59 127.60
1 6.69 123.22
0 76.01 84.80
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Article Details
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Author:Pitrebois, Sandra; Denuit, Michel; Walhin, Jean-Francois
Publication:Journal of Risk and Insurance
Geographic Code:1USA
Date:Sep 1, 2006
Words:5187
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