**BONUS-MALUS SYSTEM APPLICABLE IN THE DEMOCRATIC REPUBLIC OF THE CONGO**

Corresponding author: \* bopatriciat.boluma@unikin.ac.cd

Received Date: \*date

Accepted Date: \*date

Published Date: \*date

**HIGHLIGHTS**

  - A posteriori pricing of the automobile portfolio will be based on the Bonus-Malus System.

  - *Model of a posteriori pricing, based on the Bonus-Malus system, applicable in the Democratic Republic of Congo*.

  - *Class model and the multiplicative model.*

ABSTRACT

*In the majority of the countries and in particular in the DRC, the Automobile Insurance is obligatory; it is also the most important branch: in the DRC for example, it covers more than 80% of the turnover of the National Insurance Company; If the automobile branch is poorly managed; this may even lead to the insolvency of the insurance company. A priori pricing does not improve the danger parameter (the variance) which measures the difference between the estimated model and the observed reality, since the tariff characteristics do not take into account the driver's experience, the portfolio remains heterogeneous. This is why post-season pricing is required that takes into account driver behavior. In this model, policyholders are divided into classes based on the frequency of claims, in order to preserve the solvency of the insurer. A posteriori pricing of the automobile portfolio will be based on the Bonus-Malus System (BMS): the driver who has declared less than one loss (no loss) receives a reduction in his premium in the year t<sub>n+1</sub> (Bonus) and the bad driver who has declared more than one claim will see his premium increased to the year t<sub>n+1</sub> (Malus). In this article, we will develop a model of a posteriori pricing, based on the Bonus-Malus system, applicable in the Democratic Republic of Congo. We will be inspired by two models: the one used in Belgium (called class model) and the one used in France (called the multiplicative model).*

*Keywords: Bonus-malus, Insurance policy, risk, frequency of claims, prior pricing, ex post facto pricing.*

# INTRODUCTION

According to J. Hémard, a French actuary (HÉMARD, J. (1925).), "insurance is an operation by which one party, the insured, is promised, in return for a remuneration, the premium, for himself and for a third party, in the event of the realization of a risk, a benefit by another party, the insurer, who, taking charge of a set of risks, composes them in accordance with the laws of probability and of statistics".

Insurance is a contract between the insurer and the insured where the latter, in return for a premium, protects himself against the occurrence of a risk defined in the contract. It is also a transfer of risk from the insured to the insurer (Charpentier, A. (2010).).

In motor insurance, the insured person is protected against all kinds of material damage to the insured vehicle (property insurance), and bodily injury to the driver of the vehicle. Depending on the type of policy taken out, motor insurance can also cover material damage or bodily injury caused by the insured vehicle to third parties, this is known as liability insurance (d'Haultfoeuille, X., Givord, P., & Boutin, X. (2014).).

In the DRC, as in most countries of the world, car insurance is compulsory for any vehicle that travels on a public road.

In 2016, the amount of premiums collected from the motor branch (portfolio) in Kinshasa covered more than 80% of the total volume of premiums collected in Kinshasa (motor, fire, life, marine, ARD ...). See compensation services and various Sonas branches in Kinshasa.

If the motor business is poorly managed, this may even lead to the insolvency of the insurance company.

The aim of car insurance companies is to make each insured pay a fair premium that is proportional to the risk to be insured. The problem that arises is to be able to determine certain criteria by which policyholders can be differentiated.

*In section 2, we present a* priori pricing, where the insurer tries to predict, from the moment a new policyholder joins, his future loss experience according to certain criteria adopted at the time of subscription. By making a statistical analysis of the claims reported in Kinshasa in 2016, we note a persistent heterogeneity of the motor portfolio and show the need to apply a posteriori pricing, which consists in charging a premium taking into account the insured's history (his or her number of reported claims) (Alberini, A., Bareit, M., Filippini, M., & Martinez-Cruz, A. L. (2018).). This personalization of the premium according to the number of reported claims is often called the "Bonus-malus system".

In some countries of the world, the bonus-malus system is imposed by the government, in which case all insurers must adopt the same system (number of classes, transition rules, etc.). In other countries, the market is completely free; each insurer builds its own system.

After having given, in section 3, some theoretical foundations on the construction of a Bonus Malus system, we construct, in the following section, a Bonus system based on the two types of Bonus Malus system practised throughout the world: Bonus Malus system with classes (Belgian type) and multiplicative Bonus Malus system (French type) (Habibi, S., Hugosson, M. B., Sundbergh, P., & Algers, S. (2019).).

We conclude this paper by showing in section 5 that the Bonus Malus Class System is more appropriate for the Democratic Republic of Congo, as it is fair and balanced.

**  
**

**THE MOTOR VEHICLE INDUSTRY'S PRICING SYSTEM**

**Pricing is the fair distribution of the total burden on the community.**

Its purpose is to estimate the risk of each insurance policy in order to distribute the total burden of the community fairly. That is, to calculate a risk premium for each policyholder according to a number of observed factors.

The actuary, on the basis of reliable statistical data, is able to determine the appropriate premium rates that will allow the constitution of reserves and technical provisions in order to safeguard the solvency of the company (Sumien, P. (1923).).

Since each policyholder must contribute according to the risk he or she poses to the community, the actuary must subdivide the insurance portfolio into homogeneous classes when it is heterogeneous (George, F., & Callewaert, V. (2013).). The problem is to determine some criteria for differentiating between policyholders. These criteria are called *tariff characteristics***.**

***A priori* pricing**

The insurer tries to predict, from the moment a new policyholder joins, his future loss experience according to certain criteria adopted at the time of subscription.

In motor insurance, *a priori* pricing is *based on a segmentation of the insurance portfolio into homogeneous classes: the aim is to classify policyholders according to their potential risk (*Wuthrich, M. V. (2020).)*. The aim is to classify policyholders according to their potential risk. The aim is therefore to select pricing criteria that are relevant and commercially usable. After segmentation, the insurance company has homogeneous classes where policyholders belonging to the same class pay the same premium.* This method consists of predicting the expected number of claims according to the characteristics observed a priori in the policyholders, such as the use of the vehicle, the age of the vehicle (seniority), the sex, the age of the policyholder, the power of the vehicle, etc. The observable characteristics of the policyholders are called classification variables or *a priori* variables (Wuthrich, M. V., & Buser, C. (2021).)*.*

A priori pricing therefore depends on the characteristics of the insured asset (vehicle) as well as the characteristics of the insured (driver profile). These observable characteristics are called *the classification variables or a priori variables or exogenous variables (*Callewaert, V. (2013).)*.*

It is difficult for statistical and practical reasons to take into account all the characteristics, so each company selects a few that it considers most significant (Callewaert, V. (2013).).

In the Democratic Republic of Congo, at the Société National d'Assurances (SONAS), only four criteria are taken into account for the a priori pricing:

  - Vehicle power: horsepower

  - Its use: commercial, taxi, rental or tourism

  - Its engine: diesel or petrol

  - The age of the vehicle.

Table 1: A priori pricing applied to SONAS/DRC

<table>
<thead>
<tr class="header">
<th>Class</th>
<th>Vehicle power</th>
<th>Annual premium in relation to the duration of the vehicle</th>
<th></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td></td>
<td></td>
<td><br /><span class="math display"> ≥ 6 ans</span><br /></td>
<td><br /><span class="math display"> &lt; 5 ans</span><br /></td>
</tr>
<tr class="even">
<td><p>1</p>
<p>2</p>
<p>3</p>
<p>4</p>
<p>5</p></td>
<td><p>1 to 5 V.C.</p>
<p>6 to 9 C.V</p>
<p>10 to 13 C.V</p>
<p>14 to 17 C.V.</p>
<p><span class="math inline">≥</span> 18</p></td>
<td><p>173 $</p>
<p>217 $</p>
<p>285 $</p>
<p>375 $</p>
<p>508 $</p></td>
<td><p>163 $</p>
<p>201 $</p>
<p>262 $</p>
<ol start="343" type="1">
<li><p>$</p></li>
</ol>
<p>466$</p></td>
</tr>
</tbody>
</table>

**Limit of use of a priori pricing; a posteriori pricing**

We collected data on claims in different Sonas branches in Kinshasa (Gombe, Limété, Ngaba, N'djili, Masina, Kasa- vubu, UPN), these data are shown in the claims distribution table below, on a sample of 6475 vehicles observed for the year 2016 drawn by the Simple Random Sampling method.

Table 2: Distribution of claims by frequency

| Number of claims | Workforce |
| ---------------- | --------- |
| 0                | 5797      |
| 1                | 564       |
| 2                | 96        |
| 3                | 12        |
| 4                | 5         |
| 5                | 1         |

To show the limit of using a priori pricing, we will experiment with two model cases and then choose the one that does not deviate from the real model: hence the model with a homogeneous portfolio (Poisson model) and model with a heterogeneous portfolio (Negative Binomial) (Besson, J. L. (1992).).

  - ***Poisson model (Homogeneous portfolio)***: in the first approximation, we assume that all policyholders are equal before the risk, i.e. the probability of having an accident is the same for all policyholders. In this case, the occurrence of claims is a random event and there is no need to penalize policyholders responsible for the claims. If, in addition, we make the following intuitive assumptions (Elmer, C. F., & Kemfert, C. (2021).):

<!-- end list -->

1)  The probability of having 1 claim in a time interval\] t , t+∆ t\[ is proportional to the length of this interval and does not depend on the number of claims at time t.

2)  The probability of having more than one accident during the time interval\] t , t+∆ t\[ is negligible

3)  The number of claims for two non-overlapping time intervals are independent.

Under assumptions (a), (b), (c), the distribution of the number of claims in the portfolio is a fish distribution with parameterλ .

By performing a statistical analysis of the homogeneous portfolio model :

  - we have, on the one hand, calculated the average number of claims and the variance which gave respectively \(\overline{x} = 0.126\) and s<sup>2</sup> =0.163; for a fish distribution the variance should be equal to the mean;

  - On the other hand, we can fit these observations by a fish distribution with parameterλ = \(\overline{x} =\) 0.126 whereλ is the mathematical expectation of the number of claims in the distribution (or claim frequency).

Table 3: Adjustment of the data by the fish distribution

| Number of claims | Numbers Observed n<sub>k</sub> | Theoretical numbers n.p<sub>k</sub> |
| ---------------- | ------------------------------ | ----------------------------------- |
| 0                | 5797                           | 5708.5                              |
| 1                | 564                            | 719.2                               |
| 2                | 96                             | 45.31                               |
| 3                | 12                             | 1.90                                |
| 4                | 5                              | 0.059                               |
| 5                | 1                              | 0                                   |

Where k is the number of accidents (claims)

n<sub>k</sub> represents the observed numbers

np<sub>k</sub> represents the theoretical number of employees

\(p_{\text{k\ }}\)=\(\ \text{\ e}^{- \lambda\ \ \ \ \ \ }\frac{\lambda^{k}}{k!}\)

By grouping neighboring classes so that the new classes have a theoretical size of more than 5, we have the following table:

Table 4: Aggregation of adjusted data

| Number of claims | Numbers Observed n<sub>k</sub> | Theoretical numbers n.p<sub>k</sub> |
| ---------------- | ------------------------------ | ----------------------------------- |
| 0                | 5797                           | 5708.5                              |
| 1                | 564                            | 719.2                               |
| 2                | 132                            | 39.26                               |

Using the Pearson goodness of fit test, we obtain

\({\chi^{2}}_{\text{calcul}ée} = 201,17\)

At the 0.01 threshold, using the tables of \(\chi^{2}\) tables at 2 ddl, we find 6.63; the fit is weak, the homogeneity hypothesis, according to which all policyholders are equal before the risk, is rejected. We are therefore led to reject model 1 (homogeneous portfolio) (Staudt, Y., & Wagner, J. (2019).).

***Model 2: Negative Binomial Model (Heterogeneous Portfolio)**:* We assume here that not all policyholders are equal in terms of risk, i.e. that each policyholder has its own claims distribution. The qualities of a driver are therefore entirely summarized by the value of his claims frequency λ.

Assuming once again assumptions a), b), c) of the previous model, the distribution of the number of claims of each insured is a fish distribution with parameterλ , parameter that varies from policy to policy and whose distribution function (structure function) is U(λ ).

If the random variableλ is distributed according to a gamma distribution (\(\Gamma\)) with frequency function \(\ \text{dU}\left( \lambda \right) = \frac{\tau^{\text{a\ }}e^{- \lambda\tau}\text{\ .\ \ }\lambda^{a - 1}\ }{\Gamma(a)}\) (a,\(\text{\ τ} > 0),\ \)then the distribution of the number of claims in the portfolio is a Negative Binomial, for the demonstration see.

Its probability distribution will be given by

\(p_{k} = \left( \frac{k + a - 1}{a - 1} \right)\left( \frac{\tau}{1 + \tau} \right)^{a}\left( \frac{1}{1 + \tau} \right)^{k}\ ,\ \) k=0,1,2,...

of mean \(m = \frac{a}{\tau}\) and variance \(\sigma^{2} = \frac{a}{\tau^{2}}(1 + \tau)\)

Let us also make a statistical analysis of this heterogeneous portfolio:

Let us estimate the parameters a and \(\tau\) of the model from the average number of claims \(\overline{x}\) and the variance \(\sigma^{2}\)

From \(\overline{x} = m = \frac{a}{\tau}\) and \(s^{2} = \sigma^{2} = \frac{a}{\tau^{2}}\left( 1 + \tau \right)\ \)we have :

a= \(\frac{{\overline{x}}^{\ \ 2}}{s^{\ \ 2} - \ \overline{x}}\) = 0.4290 and \(\tau\) = \(\frac{{\overline{x}}^{\text{\ \ }}}{s^{\ \ 2} - \ \overline{x}}\) = 3.4054

hence the following table, adjusting the data:

Table 5: Negative Binomial fitted data

<table>
<thead>
<tr class="header">
<th>Number of claims</th>
<th><p>Number of employees Observed</p>
<p>n<sub>k</sub></p></th>
<th>Theoretical numbers n.p<sub>k</sub></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>0</td>
<td>5797</td>
<td>5797.715</td>
</tr>
<tr class="even">
<td>1</td>
<td>564</td>
<td>564.60</td>
</tr>
<tr class="odd">
<td>2</td>
<td>96</td>
<td>91.57</td>
</tr>
<tr class="even">
<td>3</td>
<td>12</td>
<td>16.8</td>
</tr>
<tr class="odd">
<td>4</td>
<td>5</td>
<td>3.27</td>
</tr>
<tr class="even">
<td>5</td>
<td>1</td>
<td>0.6</td>
</tr>
</tbody>
</table>

Where k is the number of accidents; n<sub>k</sub> is the observed number; np<sub>k</sub> is the theoretical number

\(p_{k} = \left( \frac{k + a - 1}{a - 1} \right)\left( \frac{\tau}{1 + \tau} \right)^{a}\left( \frac{1}{1 + \tau} \right)^{k}\ ,\ \) k=0,1,2,...

By grouping neighbouring classes so that the new classes have a theoretical size of more than 5, we have the following table:

Table 6: Grouped adjusted data

| Number of claims | Numbers Observed n<sub>k</sub> | Theoretical numbers n.p<sub>k</sub> |
| ---------------- | ------------------------------ | ----------------------------------- |
| 0                | 5797                           | 5797.715                            |
| 1                | 564                            | 564.60                              |
| 2                | 96                             | 91.57                               |
| 3                | 18                             | 20.67                               |

Using the Pearson goodness of fit test, we obtain

\({\chi^{2}}_{calculée} = 0.56\)

At the 0.05 threshold, using the \(\chi^{2}\) tables at 2 ddl we find 3.84; the fit is good. The second model comes closest to reality, which demonstrates the heterogeneity of the portfolio and therefore ***the need to introduce a Bonus Malus system in the Democratic Republic of Congo.***

In this model, each policyholder will be characterized within the portfolio by their claims frequency and the number of accidents they cause per year.

Hence the following theorem:

*Theorem*: If the a priori distribution ofλ is a \(\Gamma\) of parameter a and \(\tau,\) the a posteriori distribution is also a \(\Gamma\)of parameter a'=a+k and \(\tau'\)=\(\ \tau + t\) ;Where  
\(k = \sum_{i = 1}^{t}k_{i}\) is the total number of claims with the policy observed for t years.

The proof of this theorem can be found in.

It follows from this theorem that the distribution ofλ improved by the knowledge of the driver's background is

\(\lambda_{t + 1}\left( k_{1},\ldots,k_{t} \right) = \frac{a^{'}}{t^{'}} = \frac{a + k}{\tau + t\ }\) (II.1)

If it is decided to construct a Bonus-Malus system based solely on the number of claims, formula (II.1) can be used as a basis for constructing this system, by comparing the expected frequencies \(\lambda_{t + 1}\ \)between the different groups.

This system has the property of being equitable (it makes each person pay at any time a premium proportional to his own estimated frequency \(\lambda_{t + 1}\) at time t+1) and balanced (the average of the estimated claim frequencies is constantly equal to \(\frac{a}{\tau}\) the overall average). In other words, the amount collected by the company is stationary, financial equilibrium is achieved every year (KELLE, M. (2000).).

### Construction of the Bonus-Malus System Model 

**Model *Assumptions ***

**An insurance company uses a bonus-malus system when :**

**The set of policies of a given group can be partitioned into a finite number of** \(s\) **classes** \(c_{\text{i\ }}\left( i = 1,\ \ldots,s \right)\ \)such that the annual premium depends only on the class.

1.  **The class at a given time is determined univocally by the class of the previous period and the number of claims reported during the period.**

2.  **There are two final classes, one where all policies after a sufficiently large number of years without claims and another where all policies with a sufficiently large number of accidents are found.**

**Such a system is determined by the following three factors:**

1.  **The number of classes (noted as** \(s\))

2.  The premium scale \(b_{i}\ \left( i = 1,\ldots,s \right)\) such that policyholders in the class \(i\) pay the premium \(b_{i}\) and \({\forall i,i = 1,\ldots,s\ on\ a\ :\ b}_{i} \leq b_{i + 1}\)

3.  The transition rules, i.e. the laws governing the transition from one class to another when the number of claims is known.

**These transition rules can be presented in the form of transformations** \(T_{k}\) **transformations such as** \(T_{k}\left( i \right) = j\) **which means that the font is transferred from class** \(c_{i}\) **class to the** \(c_{j}\ \)**if** \(\text{k\ \ }\)**claims have been reported.**

**These transformations can also be presented in matrix form** \(\left( t_{\text{ij}}^{\left( k \right)} \right).\)

The probability of an insured person moving from one class to another in the SBM depends on the transition rules predetermined in the system (Denuit, M., & Charpentier, A. (2005).).

Assuming that \(k\) accidents have been reported by the insured, the transition rules that allow the insured to move from one class to another are defined as follows:

\(t_{\text{ij}}^{\left( k \right)} = \left\{ \begin{matrix}
1\ \ \ \text{si}\ T_{k}\left( i \right) = j \\
\  \\
0\ \ \text{sinon}\text{\ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \)

Let \(\lambda\) the average annual frequency of claims in the portfolio and \(N_{t}\) the annual number of claims caused by an insured.

Consider an insurance company using a bonus-malus system. Each policyholder occupies a class in the bonus-malus scale which has \((s + 1)\) classes (numbered from 0 to s) (Francq, C., & Zakoian, J. M. (2019).).

Degree 0 gives the maximum bonus and the relative bonus increases with the level to reach its maximum in \(\text{s.}\)

Note:

\(L_{t}\) the class occupied by the insured between the times \(t\) and \(\ t + 1.\)

\(\{ L_{t},t\mathbb{\in N\}}\) the discrete-time stochastic process that represents the insured's trajectory.

The system is such that the degree of an insured for a given insurance period is determined by the degree of the previous period and the number of claims for that period (Staudt, Y., & Wagner, J. (2019).).

If the insured unconditionally moves down one class per year in the scale and each claim is penalized by a move up one \(\omega\) degree, the class \(L_{t + 1}\) in which the insured will be positioned at the time \(t + 1\) is given by :

\[L_{t + 1} = \max{\{\min{\{ L_{t}}} + \omega N_{t + 1} - 1,s\},\ 0\}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \]

In general, \(L_{t + 1} = \Psi(L_{t};\ N_{t + 1})\) where \(\Psi(\mathbf{.},\mathbf{.})\) is an increasing function in its two arguments.

Conditional on the quality of the risk

\({\mathbb{P\lbrack}L}_{t + 1} = l_{t + 1}\left| L_{t} = l_{t},\ldots,L_{0} = l_{0},\theta \right\rbrack = {\mathbb{P\lbrack}L}_{t + 1} = l_{t + 1}\left| L_{t} = l_{t},\theta \right\rbrack\) \(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 1 \right)\)

As long as the trajectory \(\text{\ l}_{0},\ldots,l_{t}\) is possible, i.e.

\[\mathbb{P}\left\lbrack L_{t} = l_{t},\ldots,L_{0} = l_{0} \right\rbrack > 0.\]

The relationship \(\left( III.1 \right)\ \)expresses the fact that the state currently occupied by the insured person in the scale summarises all the information useful to know his future evolution.

This means that the prediction of future developments is improved by knowing the levels occupied at times 1,2,...

It is this property that allows the evolution of an insured person to be modelled using Markov processes (RIMI, R., RIMI, O., & LATRECHE, A. (2015).).

This is because a Markov chain is a stochastic process in which future development depends only on the present state and not on the history of the process or how the present state was reached. It is a *memoryless* process such that the different states of the chain represent the different rungs of the bonus-malus system (Verbelen, R., Antonio, K., & Claeskens, G. (2018).).

Knowledge of the current level and the number of claims the insured has made during the year is sufficient to determine the level the insured will occupy in the following year. It is therefore not necessary to know how the insured reached the level he/she currently occupies (RIMI, R., RIMI, O., & LATRECHE, A. (2015).).

### **Probability of transition**

Let\(\text{\ \ ϑ}\) the annual frequency of claims and \(l_{1},l_{2}\) the classes of the bonus-malus scale. The probability of a policy being transferred from class \(l_{1}\) to class \(l_{2}\) in a period, for an insured with an annual claim frequency \(\vartheta\) is given by :

\[p_{l_{1}l_{2}}(\vartheta) = {\mathbb{P\lbrack}L}_{t + 1} = l_{2}\left| L_{t} = l_{1},\lambda\theta = \vartheta \right\rbrack\text{~\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 2 \right)\]

\(\forall\ \vartheta,\ p_{l_{1}l_{2}}(\vartheta) \geq 0\) and \(\sum_{l_{2}}^{s}{p_{l_{1}l_{2}}\left( \vartheta \right) = 1}\)

The transition matrix of the Markov chain discovering the insured's path through the ladder is represented by :

\(\mathbf{P}\left( \mathbf{\vartheta} \right)\mathbf{=}\begin{pmatrix}
p_{00}(\vartheta) & \cdots & p_{0s}(\vartheta) \\
 \vdots & \ddots & \vdots \\
p_{s0}(\vartheta) & \cdots & p_{\text{ss}}(\vartheta) \\
\end{pmatrix}\)

Since the annual loss frequency is independent of time, this Markov chain is homogeneous (Inoussa, R. A. (2013).).

The transition rules of a bonus-malus system are described using transformations \(T_{k}\) such as \(T_{k}\left( l_{1} \right) = l_{2}\) if the policy is transferred from class \(l_{1}\) to class \(l_{2}\) when a period of time has been declared \(k\) claims.

These transformations \(T_{k}\) can be described in matrix form:

\(T_{k} =\) \(\begin{pmatrix}
t_{00}^{(k)} & \cdots & t_{0s}^{(k)}) \\
 \vdots & \ddots & \vdots \\
t_{s0}^{(k)} & \cdots & t_{\text{ss}}^{(k)} \\
\end{pmatrix}\)

Where \(t_{l_{1}l_{2}}^{(k)} = \left\{ \begin{matrix}
1\ \text{si}\ T_{k}\left( l_{1} \right) = l_{2} \\
0\ \ \ \ \text{autrement} \\
\end{matrix} \right.\ \)

For the system to be consistent, for each class \(l_{1}\)there is one and only one \(l_{2}\) such that \(t_{l_{1}l_{2}}^{(k)} = 1.\ \)The probability \(p_{l_{1}l_{2}}(\vartheta)\) of a policy being transferred from class \(\ l_{1}\) à \(l_{2}\) for an insured characterised by a risk parameter \(\ \vartheta\) in one period is equal to :

\[p_{l_{1}l_{2}}\left( \vartheta \right) = \sum_{k = 0}^{+ \infty}{\mathbb{P}\left\lbrack N = k \middle| \lambda\theta = \vartheta \right\rbrack.}t_{l_{1}l_{2}}^{\left( k \right)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 3 \right)\]

Then the transition matrix can be written as :

\[\mathbf{P}\left( \mathbf{\vartheta} \right)\mathbf{=}\sum_{k = 0}^{+ \infty}{\mathbb{P}\left\lbrack N = k \middle| \lambda\theta = \vartheta \right\rbrack.T_{k}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 4 \right)\]

# Bonus-Malus system applicable in the DRC

# In the DRC, as in most countries of the world, 'third party liability' motor insurance to compensate victims of accidents caused by the insured vehicle is compulsory.

It is a fact that almost every year, Congolese motorists see their premiums increase and sometimes double (Chen, F., & Fan, H. Y. (2014).). This situation has always caused discontent among some drivers, who consider themselves 'rightly *or wrongly'* good drivers and who even refuse to pay these extra premiums.

It has been shown that the bonus-malus system can be used in the DRC.

Thus, by introducing this ex-post pricing system, this problem can be solved.

By comparing and criticizing the 2 types of Bonus Malus System applied in the world, we propose in this section a Bonus Malus System applicable in the Democratic Republic of Congo.

The (a priori) tariffs in force at the Société Nationale d'Assurance (SONAS) are broken down as follows

We base ourselves on the a priori tariffs practised by Sonas to construct a bonus-malus system.

This a priori pricing is done in a context of non-competition. Indeed, SONAS has a monopoly on insurance in the DRC. However, the insurance sector is in the process of being liberalized, which will allow private companies to create their own insurance companies, while Sonas will simply regulate the sector.

## *Case 1: Application to the class system*

The class system we propose has 23 classes:

Table 7: The class system / ROC

<table>
<thead>
<tr class="header">
<th>Class</th>
<th>Premium level</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><p>22</p>
<p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p>
<p>13</p>
<p>12</p>
<p>11</p>
<p>10</p>
<p>9</p>
<p>8</p>
<p>7</p>
<p>6</p>
<p>5</p>
<p>4</p>
<p>3</p>
<p>2</p>
<p>1</p>
<p>0</p></td>
<td><p>508</p>
<p>482</p>
<p>459</p>
<p>437</p>
<p>416</p>
<p>397</p>
<p>378</p>
<p>360</p>
<p>343</p>
<p>326</p>
<p>311</p>
<p>296</p>
<p>282</p>
<p>268</p>
<p>256</p>
<p>243</p>
<p>232</p>
<p>221</p>
<p>210</p>
<p>200</p>
<p>190</p>
<p>181</p>
<p>172</p></td>
</tr>
</tbody>
</table>

The premium level is expressed in dollars ($). For example, an insured in class 9 will pay a premium of $268; and an insured in class 14 will pay a premium of $343.

People whose vehicles have a power rating between 1 and 9 h.p. access the system in class 9 and the others in class 14.

The rules for transition from one class to another are as follows:

  - One degree drop per claim-free year

  - Per year with one or more claims :

<!-- end list -->

  - The rise of 4 classes for the first declared disaster

  - Increase of 5 classes for the following claims.

The restriction to this system is that :

  - Regardless of the number of accidents caused, the insured will not exceed classes 0 and 22.

The scale of bonuses and the transition rules are shown in the following table:

Table 8: The premium scale in relation to the number of claims reported by the insured

<table>
<thead>
<tr class="header">
<th>Class</th>
<th>Premium level</th>
<th><br /><span class="math display"><em>T</em><sub>0</sub></span><br /></th>
<th><br /><span class="math display"><em>T</em><sub>1</sub></span><br /></th>
<th><br /><span class="math display"><em>T</em><sub>2</sub></span><br /></th>
<th><br /><span class="math display"><em>T</em><sub>3</sub></span><br /></th>
<th><br /><span class="math display"><em>T</em><sub>4</sub></span><br /></th>
<th><br /><span class="math display"><em>T</em><sub>k </sub>(<em>k</em> ≥ 5)</span><br /></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><p>22</p>
<p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p>
<p>13</p>
<p>12</p>
<p>11</p>
<p>10</p>
<p>9</p>
<p>8</p>
<p>7</p>
<p>6</p>
<p>5</p>
<p>4</p>
<p>3</p>
<p>2</p>
<p>1</p>
<p>0</p></td>
<td><p>508</p>
<p>482</p>
<p>459</p>
<p>437</p>
<p>416</p>
<p>397</p>
<p>378</p>
<p>360</p>
<p>343</p>
<p>326</p>
<p>311</p>
<p>296</p>
<p>282</p>
<p>268</p>
<p>256</p>
<p>243</p>
<p>232</p>
<p>221</p>
<p>210</p>
<p>200</p>
<p>191</p>
<p>182</p>
<p>173</p></td>
<td><p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p>
<p>13</p>
<p>12</p>
<p>11</p>
<p>10</p>
<p>9</p>
<p>8</p>
<p>7</p>
<p>6</p>
<p>5</p>
<p>4</p>
<p>3</p>
<p>2</p>
<p>1</p>
<p>0</p>
<p>0</p></td>
<td><p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p>
<p>13</p>
<p>12</p>
<p>11</p>
<p>10</p>
<p>9</p>
<p>8</p>
<p>7</p>
<p>6</p>
<p>5</p>
<p>4</p></td>
<td><p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p>
<p>13</p>
<p>12</p>
<p>11</p>
<p>10</p>
<p>9</p></td>
<td><p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>21</p>
<p>20</p>
<p>19</p>
<p>18</p>
<p>17</p>
<p>16</p>
<p>15</p>
<p>14</p></td>
<td><p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>21</p>
<p>20</p></td>
<td><p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p>
<p>22</p></td>
</tr>
</tbody>
</table>

The premium level is in dollars ($). An insured person will pay $296 if he/she is in class 11.

\(T_{1}(\)11)=15 i.e. after reporting a claim, any insured in class 11 will be transferred to class 15.

Similarly, if an insured person is in class 7 and does not make a claim, he/she will be placed in class 6, otherwise, after a claim has been made, he/she will be placed in class 11, i.e. \(\ T_{1}\left( 7 \right) = 11\ ;\ \)if he declares 3 claims he will be transferred to the "malus" class 22 ; \(T_{3}\left( 7 \right) = 22\)

## *Case 2: Application to the Multiplicative System*

We know that in the multiplier system the premium to be paid by the insured is determined by multiplying the basic (reference) premium by a multiplier (reduction factor) (Kafková, S. (2015).).

This means that the multiplication coefficient is set at 1.25.

  - For each claim-free year, the premium is reduced by 5%, so the basic premium is multiplied by 0.95.

  - For each claim reported during the year, the premium is increased by 25%, i.e. the basic premium is multiplied by 1.25; the same applies to each additional claim.

The following restrictions apply to this system:

  - The coefficient cannot be less than 0.5, i.e. the insured can benefit from a bonus of up to 50% of his basic premium.

  - In no case may the proportionality coefficient be higher than 3.5, i.e. the insured cannot have a malus higher than 350% of his basic premium.

Applying this system, we have the following result:

Table 9: The multiplicative system / ROC

<table>
<thead>
<tr class="header">
<th><strong>Premium level</strong></th>
<th><strong>Number of claims</strong></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td></td>
<td><strong>0</strong></td>
<td><strong>1</strong></td>
<td><strong>2</strong></td>
<td><strong>3</strong></td>
<td><strong>4</strong></td>
<td><strong>5</strong></td>
</tr>
<tr class="even">
<td><p>508</p>
<p>488</p>
<p>473</p>
<p>458</p>
<p>443</p>
<p>428</p>
<p>413</p>
<p>398</p>
<p>383</p>
<p>368</p>
<p>353</p>
<p>338</p>
<p>323</p>
<p>308</p>
<p>293</p>
<p>278</p>
<p>263</p>
<p>248</p>
<p>233</p>
<p>218</p>
<p>203</p>
<p>188</p>
<p>173</p></td>
<td><p>482</p>
<p>463</p>
<p>449</p>
<p>435</p>
<p>420</p>
<p>406</p>
<p>392</p>
<p>378</p>
<p>363</p>
<p>349</p>
<p>335</p>
<p>321</p>
<p>306</p>
<p>292</p>
<p>278</p>
<p>264</p>
<p>249</p>
<p>235</p>
<p>221</p>
<p>207</p>
<p>192</p>
<p>178</p>
<p>164</p></td>
<td><p>635</p>
<p>610</p>
<p>591</p>
<p>572</p>
<p>553</p>
<p>535</p>
<p>516</p>
<p>497</p>
<p>478</p>
<p>460</p>
<p>441</p>
<p>422</p>
<p>403</p>
<p>385</p>
<p>366</p>
<p>347</p>
<p>328</p>
<p>310</p>
<p>291</p>
<p>272</p>
<p>253</p>
<p>235</p>
<p>216</p></td>
<td><p>793</p>
<p>762</p>
<p>739</p>
<p>715</p>
<p>692</p>
<p>668</p>
<p>645</p>
<p>621</p>
<p>598</p>
<p>575</p>
<p>551</p>
<p>528</p>
<p>504</p>
<p>481</p>
<p>457</p>
<p>434</p>
<p>410</p>
<p>387</p>
<p>364</p>
<p>340</p>
<p>317</p>
<p>293</p>
<p>270</p></td>
<td><p>991</p>
<p>952</p>
<p>923</p>
<p>893</p>
<p>865</p>
<p>835</p>
<p>806</p>
<p>776</p>
<p>747</p>
<p>718</p>
<p>688</p>
<p>660</p>
<p>630</p>
<p>601</p>
<p>571</p>
<p>542</p>
<p>512</p>
<p>483</p>
<p>455</p>
<p>425</p>
<p>396</p>
<p>366</p>
<p>337</p></td>
<td><p>1238</p>
<p>1190</p>
<p>1153</p>
<p>1116</p>
<p>1081</p>
<p>1043</p>
<p>1007</p>
<p>970</p>
<p>933</p>
<p>897</p>
<p>860</p>
<p>825</p>
<p>787</p>
<p>751</p>
<p>713</p>
<p>677</p>
<p>640</p>
<p>603</p>
<p>568</p>
<p>531</p>
<p>495</p>
<p>457</p>
<p>421</p></td>
<td><p>1547</p>
<p>1487</p>
<p>1441</p>
<p>1395</p>
<p>1351</p>
<p>1303</p>
<p>1258</p>
<p>1212</p>
<p>1166</p>
<p>1121</p>
<p>1075</p>
<p>1035</p>
<p>983</p>
<p>938</p>
<p>891</p>
<p>846</p>
<p>800</p>
<p>753</p>
<p>710</p>
<p>663</p>
<p>618</p>
<p>571</p>
<p>526</p></td>
</tr>
</tbody>
</table>

The premium is always expressed in $ (dollars). An insured whose basic premium is $308 will pay $292 the following year, i.e. a reduction of 5%, if he/she does not make a claim; but if he/she does make a claim, the premium will be $385, i.e. an increase of 25%.

## *Comparison of two systems and discussion*

After applying a priori pricing to these two systems, we find that :

For the class system, for example, an insured who pays $343 at the beginning of the system, if he does not declare a claim, will go to class 13 and pay $326 the following year, whereas if he declares a claim, he will go to class 18 and pay $416; for the multiplicative system, he will pay $343\*1.25=$438 in the case of an increase, and $343\*0.95=$326 in the case of a decrease.

Also, for the class system, if the insured declares 2 claims during the year, an insured with a base premium of $343 will pay $508 the following year; whereas in the multiplier system, he will pay $343\(\times 1.5625 = 536\ \$.\)

Let us summarize this in the following table:

Table 10: *Comparison of two bonus-malus models*

<table>
<thead>
<tr class="header">
<th></th>
<th><p>Disaster</p>
<p>Basic premium</p></th>
<th>0</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Class system</td>
<td>343 $</td>
<td>326 $</td>
<td>360 $</td>
<td>508 $</td>
</tr>
<tr class="even">
<td>Multiplicative system</td>
<td>343 $</td>
<td>326 $</td>
<td>438 $</td>
<td>536 $</td>
</tr>
</tbody>
</table>

We note that the malus varies less rapidly in the class system than in the multiplicative system; the multiplicative system is more severe in this case than the class system. In fact, in the multiplicative system, everything depends on the multiplication or increase coefficient set by the insurer, whereas in the class system, the number of classes being well defined, the decision rule is very precise for the insured's evolution in the system (Ágoston, K. C., & Gyetvai, M. (2020).).

In addition, the class system has the merit of being balanced and fair, qualities that the multiplicative system lacks.

So it is the class system that is best suited in the DRC.

**CONCLUSION**

In this paper, we have made a general presentation of the Bonus-Malus System, with the application of a Bonus-Malus System (BMS) in the Democratic Republic of Congo, based on the a priori pricing of the Société Nationale d'Assurance (SONAS).

The work of J. Lemaire and that of Manya and Malonda had shown that SBM can be introduced in the DRC.

For our part, the statistical analysis of accidents that occurred in the DRC in 2016 has enabled us to prove that the a priori pricing system practiced by SONAS does not improve the danger parameter (variance), which measures the difference between the estimated model and the observed reality, as the characteristics of the tariff do not take into account the driver's experience (number of claims over time), and the portfolio therefore remains heterogeneous. <span class="underline"> </span>

This is why there is a need for a posteriori pricing that takes into account the behavior of the driver.

In this work, we drew inspiration from two SBM models: SBM with classes (Belgian) and Multiplicative SBM (French) to propose an SBM applicable in the DRC.

Comparing the two models, we proposed the "class system" which is best used in the DRC because of the well-defined number of classes, the precise decision rule for the evolution of the insured in the system. In addition, the class system has the merit of being balanced and fair.

The development of a Bonus Malus system requires serious field studies: this is facilitated if the company's statistics on claims are well archived and updated; SONAS has great difficulty in archiving and consolidating data from the various agencies, it should create a body within the general services to centralize all the data into a database.

The SBM proposed in this paper should be given special attention by the SONAS authorities as it can be applied to optimize the management of motor vehicle claims.

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