Chapter 12 Bonus-Malus

12.1 Introduction

12.1.1 Residual Heterogeneity

In a priori rating, premium reductions or surcharges are granted based on observations made by the insurance company in the recent past on insureds with identical profiles. The insured is, therefore, identified with a group of insureds who are indistinguishable based on the information held by the company and is subject to a premium based on the experience of this group. Obviously, the group of individuals to which the insured is identified is not entirely identical to the insured: there remains some heterogeneity, more or less strong, depending on the extent and relevance of the information available to the insurer. This can also be explained by the fact that many unobservable (or objectively unmeasurable) factors likely influence the claims experience; these factors include factors such as aggressive driving, consumption of alcoholic beverages or drugs, the accuracy of reflexes, and judgment. According to credibility theory, the premium amount is adjusted posteriori based on the observed claims experience, which reveals the unobservable factors mentioned earlier. This process of revelation can be seen as an indirect way of recovering non-accessible information a priori.

12.1.2 Nature of Dependency

The nature of dependency between the annual numbers of claims reported by an insured is complex:

In an endogenous approach, individuals’ history alters their risk profile (this is the statistical phenomenon of true contagion). A car accident generally changes the perception of the inherent danger of driving and encourages the insured to be more cautious, thereby reducing the probability of a claim. Moreover, the mechanisms of posterior premium customization incentivize prudence and induce negative contagion between annual numbers of claims.
In an exogenous approach, the dependency between the annual numbers of claims is only apparent and results exclusively from the revelation of the insured’s hidden characteristics. One can think of annual mileage or personality traits independent of history. This residual heterogeneity of individuals can be taken into account by a random effect in a statistical model.

Regardless of which explanation prevails, the number of claims reported in the past is often the most relevant variable for predicting future claims. Furthermore, the occurrence of a claim significantly increases the probability of reporting more claims in the future, which seems to support the aforementioned exogenous approach.

The theory of credibility is often reserved for legal entities due to its high technicality and contractual formalism. Most insurance companies that offer automobile insurance to individuals have adopted a posteriori pricing systems known as bonus-malus systems. In some countries, such as France, the government mandates such a system for insurers. The purpose of this system is to more equitably distribute costs between good and bad drivers. Various studies have shown that, more than age, gender, or vehicle use, it is the claims caused by an insured in the past that best predict their future claims.

12.1.3 Objectives of Bonus-Malus Systems

The implementation of a bonus-malus system essentially pursues three objectives. Firstly, it aims to make policyholders accountable and encourage them to be more cautious while driving. Under the bonus-malus system, policyholders who have caused a claim for which the insurance company had to intervene see their premium increase the following year. Policyholders, therefore, have an objective interest in taking as many precautions as possible while driving. In this regard, the bonus-malus system is a mechanism to combat moral hazard (i.e., the natural tendency of policyholders to take fewer precautions when they know they are covered by insurance).

Secondly, it aims to adjust the premium amount over time so that it reflects the insured’s actual risk. In this regard, the bonus-malus system’s mechanism falls within the framework of credibility theory. In a segmented environment, it aims to refine the insurer’s a priori assessment of the risk to be insured (as reflected by the tariff) based on observed claims.

Finally, it aims to respond to consumer pressures. The bonus-malus system increases the premium for policyholders who impose claims on the community and decreases the premium for good drivers.

12.1.4 Nature of Dependency

The nature of dependency between the annual numbers of claims reported by an insured is complex:

In an endogenous approach, individuals’ history alters their risk profile (this is the statistical phenomenon of true contagion). A car accident generally changes the perception of the inherent danger of driving and encourages the insured to be more cautious, thereby reducing the probability of a claim. Moreover, the mechanisms of posterior premium customization incentivize prudence and induce negative contagion between annual numbers of claims.
In an exogenous approach, the dependency between the annual numbers of claims is only apparent and results exclusively from the revelation of the insured’s hidden characteristics. One can think of annual mileage or personality traits independent of history. This residual heterogeneity of individuals can be taken into account by a random effect in a statistical model.

12.1.5 Objectives of Bonus-Malus Systems

Finally, it aims to respond to consumer pressures. The bonus-malus system increases the premium for policyholders who impose claims on the community and decreases the premium for good drivers.

12.1.6 Nature of Dependency

The nature of dependency between the annual numbers of claims reported by an insured is complex:

In an endogenous approach, individuals’ history alters their risk profile (this is the statistical phenomenon of true contagion). A car accident generally changes the perception of the inherent danger of driving and encourages the insured to be more cautious, thereby reducing the probability of a claim. Moreover, the mechanisms of posterior premium customization incentivize prudence and induce negative contagion between annual numbers of claims. 2.In an exogenous approach, the dependency between the annual numbers of claims is only apparent and results exclusively from the revelation of the insured’s hidden characteristics. One can think of annual mileage or personality traits independent of history. This residual heterogeneity of individuals can be taken into account by a random effect in a statistical model.

12.1.7 bjectives of Bonus-Malus Systems

Finally, it aims to respond to consumer pressures. The bonus-malus system increases the premium for policyholders who impose claims on the community and decreases the premium for good drivers.

12.1.8 Nature of Dependency

The nature of dependency between the annual numbers of claims reported by an insured is complex:

12.1.9 Objectives of Bonus-Malus Systems

Finally, it aims to respond to consumer pressures. The bonus-malus system increases the premium for policyholders who impose claims on the community and decreases the premium for good drivers.

12.1.10 Bonus Thirst

The penalties imposed by bonus-malus systems depend solely on the number of at-fault claims reported by the insured. Their cost is not taken into account. Several reasons can be advanced for this:

One consequence of implementing a bonus-malus system is the emergence of the “bonus hunger” phenomenon: since the system’s penalties are independent of the claim amounts, policyholders have every incentive to compensate for minor claims themselves. This has the advantage that low-value claims, for which the administrative costs would be disproportionate to the damages to be compensated, usually do not reach the insurance company.

12.1.11 Class-Based Systems and “French-Style” Systems

There are two types of bonus-malus mechanisms: class-based systems and “French-style” systems. In the latter case, the premium paid by the insured is reduced in the absence of a claim (by a fixed percentage), and a multiplicative penalty is applied for each reported claim.

In contrast to “French-style” systems, a class-based bonus-malus system places the insured within a scale comprising a certain number of levels. Formally, a class-based system is structured as follows:

Level 0 is the one associated with the highest discount, while level $s$ corresponds to the maximum penalty. Each level is associated with a percentage. If the insured occupies level $\ell$, the premium they have to pay is obtained by applying the percentage $r_\ell$ to the base premium (BP), which is freely set by the insurer (and can depend on the insured’s characteristics). A level is reserved for new insureds. After that, an adjustment of the insured’s position on this scale is made annually based on the insured’s claims experience and following the rules of the system. These rules are fixed and usually only take into account the insured’s level from the previous year and the number of claims incurred.

The application of this system will result in an increase in premium for the policyholder responsible for one or more accidents, while the insured who has not used the company’s coverage for a year will receive a premium reduction.

It should be noted that the implementation of bonus-malus systems in French-speaking countries was made possible thanks to a number of studies in the early 1960s, including those by (Fréchet 1959), (Delaporte 1959), (Thepaut 1959) – all presented at the ASTIN conference in La Baule in June 1959 (entitled “no claim bonus in motor insurance”) – (Franckx 1960) and (Bühlmann 1964).

12.1.12 A Brief History of the Bonus-Malus System in France

The origins of the French bonus-malus system date back to the late 1950s, although it wasn’t until 1976 that it became mandatory.

In France, the bonus-malus system is referred to as the “coefficient de réduction majoration” (CRM). A 50% bonus corresponds to a coefficient of 0.50. A 0% bonus corresponds to a coefficient of 1, and so on. Conversely, a coefficient of 1.25 represents a 25% malus. The coefficient is rounded down to two decimal places and changes based on reported claims. If the insured has not reported any claims during the year, the coefficient assigned to them from the previous year is multiplied by a coefficient of 0.95 (or 0.97 if the insured uses their vehicle for client visits, such as doctors, non-sedentary salespeople, etc.). If the insured reports a claim for which they are only partially responsible, the new coefficient is obtained by multiplying the old one by 1.125. In the case of a fully responsible accident, the multiplier coefficient increases to 1.25 (or 1.20 for client visits).

In any case, the coefficient can never be less than 0.50 or exceed 3.50. Furthermore, if the insured incurs a malus for 24 full months and does not cause any claims, their coefficient is automatically reset to 1, which means 0% bonus. After two consecutive years without any claims, the applicable coefficient cannot be higher than 1. No surcharge is applied for the first claim that occurs after a period of at least three years during which the coefficient de réduction majoration has been equal to 0.50.

Reported claims are centralized by insurance companies with the AGIRA database to prevent false declarations. The insurer provides the policyholder with an information statement at each annual contract renewal or, if not, upon request of the policyholder or when the contract is terminated by either party. This statement specifies the number, nature, date of occurrence, and responsible driver of claims that occurred during the previous five annual periods, as well as the level of responsibility and the coefficient de réduction majoration applied at the last annual renewal. A driver who wishes to be insured with a new insurer is required to provide this information statement issued by the insurer of the previous contract.

The French system applies to all policies in the market and has not been challenged by the European Commission.

12.1.13 A Brief History of the Bonus-Malus System in Belgium

The first form of experience rating in Belgium dates back to 1961 when a medium-sized insurance company introduced a policy with a clause known as NCD (No-Claim Discount) in the Belgian market. Under this contractual clause, the premiums paid by policyholders were reduced if they did not report any claims to the company. This innovation was so successful that the company doubled its market share in 5 years, attracting excellent drivers to its portfolio due to the well-known mechanism of anti-selection.

Due to the magnitude of the phenomenon, the Belgian government introduced an official bonus-malus mechanism as early as 1971. It consisted of a scale with 18 levels, numbered from 1 to 18. The insured at the minimum level paid 60% of the base premium, while those at the top of the scale paid 200%. The base level (corresponding to 100%) was level 10. Insured individuals who used their vehicles for leisure and commuting to work entered the system at level 6, granting them an implicit 15% reduction compared to those who used their vehicles for professional purposes, who were placed in class 10. Each year of coverage resulted in an unconditional descent of one level in the scale. Each reported claim was penalized by an increase of 3 levels. Moreover, an insured individual who did not report any claims for 4 consecutive years could not be at a level higher than 10. An essential detail was that an efficient system was implemented in the Belgian market to prevent evasion of the system: when changing insurers, the insured had to provide the new insurer with a certificate issued by the old insurer indicating the level occupied in the scale and any outstanding claims that could affect the insured’s position. Furthermore, policies at the time had a duration of 10 years, ensuring market stability.

However, it became evident that the system introduced in 1971 did not age well. One of the goals of this system was to differentiate between good and bad drivers. Intuitively, average drivers were expected to gravitate toward the center of the scale (classes 6-12), with bad drivers concentrated at the top of the scale and good drivers at the bottom. Since the system eliminated the effect of an at-fault claim after 3 years, average drivers would remain in the middle of the scale as long as their claim frequency was around 1/3. Those with a frequency exceeding 1/3 would concentrate at the top of the scale, and those with a frequency below this threshold would end up at the bottom of the scale. The entire system, therefore, required a claim frequency around 1/3 to function. However, the actual claim frequency in the market was already significantly below this value at the time. In 1971, the average annual claim frequency was only 22%. During the 1970s, following the first oil crisis, speed limits, anti-drunk driving measures, mandatory seatbelt use, and public awareness campaigns about road safety, the annual claim frequency dropped to 10%. As a result, Belgian policyholders inevitably concentrated in the lower ranks of the system, all enjoying significant premium reductions. This led to a substantial decrease in revenue for Belgian insurance companies, which had no choice but to increase the base premium. In 1987, 57% of Belgian policyholders were at level 1 (enjoying the highest discount), nearly 75% were at levels 1 to 3, and only 1% were in the malus zone, paying more than the base premium.

In 1992, the Belgian legislature decided to introduce a new bonus-malus scale with 23 levels described in Table ??. The 23 levels are numbered from 0 to 22. Level 0 is the most advantageous, with policyholders in this level paying only 54% of the base premium (equivalent to level 14). Level 22 is the least advantageous, with policyholders at this level having to bear 200% of the base premium. Depending on the vehicle’s use, a new driver is placed at level 14 (professional use) or level 11 (limited to leisure and commuting). This means that the penalty for vehicles used for professional purposes remains the same as in the previous system. Policyholders unconditionally move down one level in the scale each year. Each claimed event during a period is penalized by an increase of 5 levels. Under no circumstances can policyholders fall below level 0 or exceed level 22. Furthermore, the legislature stipulated that a policyholder without any claims for 4 consecutive years, who is at a level higher than 14, will be immediately reset to level 14. This is a concession to young drivers who had a difficult start but improved later. Thus, a policyholder at level 17 who reports no claims will be moved to level 16 or 14, depending on the number of claim-free years. This rule complicates the task of actuaries, as it requires keeping track of the claims history for the last 5 years for policyholders in the highest levels of the scale.

Technically, the new bonus-malus scale suffered from the same shortcomings as the old one. By the end of 1999, 50% of policyholders were at level 0, 75% were in levels 0 to 4, and only 1% were in the malus zone.

The Belgian system only allows insurers to set the amount of the base premium (corresponding to level 14 on the scale). The way this premium amount evolves based on observed policyholder claims is entirely determined by the official system. Therefore, the government believed that the Belgian bonus-malus system from 1992 constituted an impediment to the pricing freedom guaranteed by European insurance directives. To avoid the ire of the Commission, it allowed companies to define the post-assessment mechanism they intended to apply to their policyholders.

Of course, to avoid destabilizing the market, the removal of the official scale was done gradually. Since February 1, 2002, companies have been free to associate the percentage of their choice with the different levels of the official system. Only the confinement of policyholders in the Belgian 23-level scale and the transition rules are maintained until January 1, 2004. After that date, all references to the official scale will disappear permanently, and a claims history certificate covering the last 5 years must be provided by the policyholder to the new insurer in case of changing companies, to ensure some level of public information.

12.1.14 Chapter Outline

In this chapter, we will first show how to calibrate a bonus-malus system using Markov chain theory. It is essential to consider a priori pricing when establishing the bonus-malus scale. The percentages associated with the different levels must be determined based on the extent of differentiation performed a priori by the insurer. We will also see how to evaluate the performance of such a system. As the first criterion, we will use the average degree relative to the stable state. This is the degree that an average policyholder will occupy when the bonus-malus system reaches a steady state. The coefficient of variation of policyholder premiums will be a second criterion for evaluating a bonus-malus system. The more variable the premiums paid by policyholders, the less solidarity there is, and the more severe the bonus-malus system is. Next, we will examine two concepts of efficiency: Loimaranta’s efficiency and Lemaire’s efficiency. We will conclude by evaluating the optimal retention of policyholders subject to a bonus-malus mechanism.

12.2 Scales in Unsegmented Universes}

12.2.1 Introductory Example: the Good/Bad Driver Model

Let’s revisit the good/bad driver model from Example 3.7.3 of Tome I. An insured randomly selected from the portfolio generates a sequence of claim counts $N_1, N_2, N_3, \ldots$ during their years of driving. These random variables are assumed to follow a $\mathcal{Poiss}(\lambda, \Theta)$ distribution, where $\Theta$ takes two values: $\theta_1$ with probability $\varrho$ and $\theta_2$ with probability $1 - \varrho$, where $\theta_1 < \theta_2$. Thus, there are two types of drivers coexisting in the portfolio: good drivers with an annual claim frequency of $\lambda\theta_1$ and bad drivers with an annual frequency of $\lambda\theta_2$. The proportion of good drivers in the portfolio is $\varrho$. Conditional on $\Theta$, i.e., knowing the driver’s quality, the random variables $N_1, N_2, N_3, \ldots$ are assumed to be independent and identically distributed.

Suppose the insurer uses a three-level bonus-malus scale, numbered 0, 1, and 2. New insureds enter the system at level 1. Each claim-free year is rewarded with a downgrade of one level in the scale, while each claim results in an upgrade of one level.

If we denote $L_t$ as the level occupied by the insured during the period $(t, t+1)$, where $t = 0, 1, 2, \ldots$, we have $L_0 = 1$. The distribution of the level $L_1$ occupied during the second year for a good driver is given by \[\begin{eqnarray*} \Pr[L_1=0|\Theta=\theta_1] &=& \exp(-\lambda\theta_1)\\ \Pr[L_1=1|\Theta=\theta_1] &=& 0\\ \Pr[L_1=2|\Theta=\theta_1] &=& 1 - \exp(-\lambda\theta_1). \end{eqnarray*}\] Similarly, for a bad driver, \[\begin{eqnarray*} \Pr[L_1=0|\Theta=\theta_2] &=& \exp(-\lambda\theta_2)\\ \Pr[L_1=1|\Theta=\theta_2] &=& 0\\ \Pr[L_1=2|\Theta=\theta_2] &=& 1 - \exp(-\lambda\theta_2). \end{eqnarray*}\] Unconditionally, we have \[ \Pr[L_1=\ell] = \varrho\Pr[L_1=\ell|\Theta=\theta_1] + (1-\varrho)\Pr[L_1=\ell|\Theta=\theta_2], \hspace{2mm} \ell = 0, 1, 2. \]

The assumption of conditional independence of annual claim counts allows us to write \[ \Pr[L_{t+1}=\ell|L_t=0,\Theta=\theta_1] = \left\{\begin{array}{l} \exp(-\lambda\theta_1) \text{ for } \ell = 0\\ \lambda\theta_1\exp(-\lambda\theta_1) \text{ for } \ell = 1\\ 1 - \exp(-\lambda\theta_1)(1+\lambda\theta_1) \text{ for } \ell = 2 \end{array} \right. \] for good drivers, and \[ \Pr[L_{t+1}=\ell|L_t=0,\Theta=\theta_2] = \left\{\begin{array}{l} \exp(-\lambda\theta_2) \text{ for } \ell = 0\\ \lambda\theta_2\exp(-\lambda\theta_2) \text{ for } \ell = 1\\ 1 - \exp(-\lambda\theta_2)(1+\lambda\theta_2) \text{ for } \ell = 2 \end{array} \right. \] for bad drivers. Similarly, we have \[ \Pr[L_{t+1}=\ell|L_t=1,\Theta=\theta_1] = \left\{\begin{array}{l} \exp(-\lambda\theta_1) \text{ for } \ell = 0\\ 0 \text{ for } \ell = 1\\ 1 - \exp(-\lambda\theta_1) \text{ for } \ell = 2 \end{array} \right. \] and \[ \Pr[L_{t+1}=\ell|L_t=2,\Theta=\theta_1] = \left\{\begin{array}{l} 0 \text{ for } \ell = 0\\ \exp(-\lambda\theta_1) \text{ for } \ell = 1\\ 1 - \exp(-\lambda\theta_1) \text{ for } \ell = 2 \end{array} \right. \] with similar expressions for bad drivers.

We can arrange the probabilities $\Pr[L_{t+1}=\ell_2|L_t=\ell_1,\Theta=\theta_1]$ into a matrix \[ \boldsymbol{P}(\theta_1)=\left( \begin{array}{ccc} \exp(-\lambda\theta_1)&\lambda\theta_1\exp(-\lambda\theta_1)&1-\exp(-\lambda\theta_1)(1+\lambda\theta_1)\\ \exp(-\lambda\theta_1)&0&1-\exp(-\lambda\theta_1)\\ 0&\exp(-\lambda\theta_1)&1-\exp(-\lambda\theta_1) \end{array} \right) \] which has the property that the sum of each row is 1 (such a matrix is called stochastic). This matrix is called the transition matrix since it describes the transitions made by insured individuals between different levels of the bonus-malus system.

Similarly, for bad drivers, the probabilities of moving from level $\ell_1$ to level $\ell_2$ from one year to the next can be arranged into the matrix \[ \boldsymbol{P}(\theta_2)=\left( \begin{array}{ccc} \exp(-\lambda\theta_2)&\lambda\theta_2\exp(-\lambda\theta_2)&1-\exp(-\lambda\theta_2)(1+\lambda\theta_2)\\ \exp(-\lambda\theta_2)&0&1-\exp(-\lambda\theta_2)\\ 0&\exp(-\lambda\theta_2)&1-\exp(-\lambda\theta_2) \end{array} \right). \]

The evolution in the system is governed for a good driver by the relationship \[\begin{eqnarray*} \Pr[L_t=\ell_2|\Theta=\theta_1]&=&\sum_{\ell_1=0}^2\Pr[L_t=\ell_2|L_{t-1}=\ell_1,\Theta=\theta_1]\\ &&\hspace{35mm}\Pr[L_{t-1}=\ell_1|\Theta=\theta_1]. \end{eqnarray*}\] Using matrix formalism and denoting $\boldsymbol{p}^{(t)}(\theta_1)$ as the column vector with component $\ell$ being $\Pr[L_t=\ell|\Theta=\theta_1]$, we see that \[\begin{equation} \boldsymbol{p}^{(t)}(\theta_1)=\boldsymbol{P}^t(\theta_1)\boldsymbol{p}^{(t-1)}(\theta_1). \tag{12.1} \end{equation}\] By iterating (12.1), it follows that \[ \boldsymbol{p}^{(t)}(\theta_1)=\underbrace{\boldsymbol{P}^t(\theta_1)\boldsymbol{P}^t(\theta_1)\cdots\boldsymbol{P}^t(\theta_1)}_{\text{$t$ factors}} \left( \begin{array}{c} 0\\ 1\\ 0 \end{array} \right) \] since $\Pr[L_0=1]=1$, which implies $\boldsymbol{p}^{(0)}(\theta_1)=(0,1,0)^t$. It is sufficient to compute the various powers of the transpose $\boldsymbol{P}^t(\theta_1)$ of the transition matrix $\boldsymbol{P}(\theta_1)$ for good drivers to obtain the probability distribution of the levels $L_1,L_2,L_3,\ldots$ they will occupy in the scale over time.

One might wonder about the possible stabilization of the proportions of good drivers in the different levels of the scale. Mathematically, this amounts to studying the asymptotic behavior of $\boldsymbol{p}^{(t)}(\theta_1)$ as $t\to +\infty$, i.e., seeking the existence of a limit \[ \pi_\ell(\theta_1) = \lim_{t\to +\infty}\Pr[L_t=\ell|\Theta=\theta_1] = \lim_{t\to +\infty}\boldsymbol{p}^{(t)}(\theta_1). \] By denoting $\boldsymbol{\pi}(\theta_1)$ as the vector whose component $\ell$ is $\pi_\ell(\theta_1)$, passing to the limit as $t\to +\infty$ in both sides of (12.1), such a vector $\boldsymbol{\pi}(\theta_1)$, if it exists, must satisfy the relation \[ \boldsymbol{\pi}(\theta_1) = \boldsymbol{P}^t(\theta_1)\boldsymbol{\pi}(\theta_1), \] so that $\boldsymbol{\pi}(\theta_1)$ must be an eigenvector of $\boldsymbol{P}^t(\theta_1)$ associated with the eigenvalue 1, and such that $\sum_{\ell=0}^2\pi_\ell(\theta_1)=1$. Similarly, the vector $\boldsymbol{\pi}(\theta_2)$ that gives the distribution of bad drivers in the different levels of the scale after a sufficiently long time can be obtained by solving the system \[ \boldsymbol{\pi}(\theta_2) = \boldsymbol{P}^t(\theta_2)\boldsymbol{\pi}(\theta_2) \] with the constraint $\boldsymbol{e}^t\boldsymbol{\pi}(\theta_2)=1$. The vectors $\boldsymbol{\pi}(\theta_1)$ and $\boldsymbol{\pi}(\theta_2)$ obtained in this way describe the stationary distribution of the bonus-malus system. This term is used because once the proportions $\boldsymbol{\pi}(\theta_1)$ and $\boldsymbol{\pi}(\theta_2)$ are reached, the system indefinitely maintains them, thus reaching a stationary regime.

For example, let’s consider $\lambda=10\%$, $\theta_1=0.5$, $\theta_2=1.5$, and $\varrho=50\%$. This means that good drivers, representing half of the portfolio, have an annual claim frequency of $\lambda\theta_1=5\%$, while bad drivers have a frequency of $\lambda\theta_2=15\%$.

Table ?? describes, for these values, the distribution (as a percentage of the portfolio) of insured individuals among the different levels of the scale over time (assuming all insured individuals are at level 1 initially). These are the vectors $\boldsymbol{p}^{(t)}(\theta_1)$ and $\boldsymbol{p}^{(t)}(\theta_2)$ for $t=1,2,\ldots$. As can be observed, the distribution of insured individuals in the scale stabilizes quickly. We have $\boldsymbol{p}^{(t)}(\theta_1)\approx\boldsymbol{\pi}(\theta_1)$ and $\boldsymbol{p}^{(t)}(\theta_2)\approx\boldsymbol{\pi}(\theta_2)$ for $t\geq 5$. Since the stationary regime provides a good approximation of the probability distributions $\boldsymbol{p}^{(t)}(\theta_1)$ and $\boldsymbol{p}^{(t)}(\theta_2)$ in the example we are analyzing, we will base the subsequent calculations on the stationary probabilities $\boldsymbol{\pi}(\theta_1)$ and $\boldsymbol{\pi}(\theta_2)$.

Once the proportion of insured individuals in each level has stabilized, we denote $L$ as the level occupied by a driver in the scale. Therefore, we have \[ \Pr[L=\ell|\Theta=\theta_1]=\pi_\ell(\theta_1)\text{ and }\Pr[L=\ell|\Theta=\theta_2]=\pi_\ell(\theta_2) \] for $\ell=0,1,2$. The probability distribution of $L$ is described in Table ??.

Now, let’s consider the probability that an insured individual who occupies level $\ell$ once the system has stabilized is a good driver. We have \[2mm] $\Pr[\Theta=\theta_1|L=\ell]$ \[ =\frac{\Pr[L=\ell|\Theta=\theta_1]\Pr[\Theta=\theta_1]} {\Pr[L=\ell|\Theta=\theta_1]\Pr[\Theta=\theta_1]+\Pr[L=\ell|\Theta=\theta_2]\Pr[\Theta=\theta_2]}. \] Table ?? presents the probabilities $\Pr[\Theta=\theta_1|L=\ell]$ and $\Pr[\Theta=\theta_2|L=\ell]$ as a function of the levels occupied in the bonus-malus scale. It can be observed that the scale has good discriminatory power at higher levels: at level 2, there is a 9 times greater chance of being a bad driver than a good one. However, at the lowest level where nearly 90% of the insured individuals in the portfolio are concentrated, the scale poorly discriminates between the two categories of drivers.

If a priori (i.e., without information about the driver’s quality), there are 50 chances out of 100 that the insured individual is a good driver, a posteriori, the level occupied in the scale allows us to specify the driver’s quality (as it reflects their past claims history). Assuming average costs of individual claims, the a priori premium is \[ \Pr[\Theta=\theta_1]\lambda\theta_1+\Pr[\Theta=\theta_2]\lambda\theta_2. \] A posteriori, this premium changes to \[ \Pr[\Theta=\theta_1|L=\ell]\lambda\theta_1+\Pr[\Theta=\theta_2|L=\ell]\lambda\theta_2 \] for insured individuals occupying level $\ell$ of the scale.

We can express the a posteriori premium as \[ \lambda\left(\Pr[\Theta=\theta_1|L=\ell]\theta_1+\Pr[\Theta=\theta_2|L=\ell]\theta_2\right)= \lambda\mathbb{E}[\Theta|L=\ell], \] which helps us better understand how a bonus-malus system works (compared to what we studied in credibility theory in the previous chapter). The level $\ell$ occupied by the driver provides information about the distribution of $\Theta$, thus specifying the quality of the risk. The reevaluation of annual claim frequencies involves calculating the $\mathbb{E}[\Theta|L=\ell]$, for $\ell=0,1,2$. In the numerical example, they are as follows: \[\begin{eqnarray*} \mathbb{E}[\Theta|L=0]&=& 0.9679\\ \mathbb{E}[\Theta|L=1]&=&1.2352\\ \mathbb{E}[\Theta|L=2]&=&1.3956. \end{eqnarray*}\] These quantities, expressed as percentages, represent the relative premiums associated with different levels of the bonus-malus scale. The three-level bonus-malus system calibrated on the good/bad risk model with the given parameter values assigns a relative premium of $r_2=139.56\%$ to level 2, $r_1=123.52\%$ to level 1, and $r_0=96.79\%$ to level 0.

Based on this simple example, it is observed that the system provides very little discount to insured individuals occupying the lowest level (barely 3%), while the penalties incurred by insured individuals at levels 1 and 2 are substantial (23 and 40%, respectively). This can be explained by the lack of severity in the scale considered here.

12.2.2 Scales and Markov Chains

Let’s now generalize the approach used to handle the introductory example. This will be done using Markov chains.

12.2.2.1 Modeling the Insured Individual’s Progress in the Scale

Let $\lambda$ be the average annual claim frequency at the portfolio level. We consider that the annual number of claims caused by an insured individual selected at random from the portfolio follows a Poisson distribution with parameter $\lambda\Theta$, where $\Theta$ is a positive random variable with a mean of 1 (often assumed to have a $\Gamma(a,a)$ distribution).

Consider an insurance company that uses a bonus-malus system. Each insured individual occupies a degree in the bonus-malus scale, which is assumed to have $s+1$ levels (numbered from 0 to $s$). Level 0 entitles the insured individual to the maximum discount, with the relative premium increasing with the level and reaching its maximum at level $s$. From now on, we will denote $L_t$ as the degree occupied by the insured individual between times $t$ and $t+1$. The trajectory of the insured individual is thus represented by the discrete-time stochastic process $\{L_t, t \in \mathbb{N}\}$. We assume that the system is such that an insured individual’s degree for a given insurance period is determined by the degree for the previous period and the number of claims for that period. If the insured individual unconditionally moves down one level in the scale each year, and each claim results in an increase of $\omega$ degrees, then the level $L_{t+1}$ where the insured individual will be placed at time $t+1$ is given by \[ L_{t+1}=\max\{\min\{L_t+\omega N_{t+1}-1,s\},0\}. \] In general, we have $L_{t+1}=\Psi(L_t,N_{t+1})$ where $\Psi(\cdot,\cdot)$ is a non-decreasing function of its two arguments. Therefore, conditional on the quality of the risk, \[3mm] $\Pr[L_{t+1}=\ell_{t+1}|L_t=\ell_t,\ldots,L_0=\ell_0,\Theta]$ \[\begin{equation} =\Pr[L_{t+1}=\ell_{t+1}|L_t=\ell_t,\Theta] \tag{12.2} \end{equation}\] as long as the trajectory $\ell_0,\ldots,\ell_t$ is feasible, i.e. \[ \Pr[L_t=\ell_t,\ldots,L_0=\ell_0]>0. \]

The relationship (12.2) expresses the fact that the state currently occupied by the policyholder in the bonus-malus scale contains all the relevant information to predict their future evolution. In other words, knowledge of this state makes the prediction of future states independent of the knowledge of the states occupied at times 1, 2, and so forth. This property enables us to model the policyholder’s evolution using a Markov process.

Indeed, a Markov chain is a stochastic process in which future developments depend solely on the current state and not on the history of the process or how the current state was reached. It is a “memoryless” process where the different states in the chain represent various levels of the bonus-malus system. Knowing the level occupied at the current time and the number of claims made by the policyholder during the year is sufficient to determine the level the policyholder will occupy in the following year. Hence, it is unnecessary to know how the current level was attained.

Remark. The bonus-malus system in force in France can be represented using a Markov chain by associating a level with each percentage between 50 and 350 (see, for example, (Kelle 2000)). This allows us to treat the French system using the formalism of Markov chains introduced in this chapter, reducing it to a bonus-malus system with classes.

Remark. Many systems include so-called fast-return clauses. Typically, a policyholder who has had no claims for several consecutive years but still finds themselves at a higher level than the base level is automatically returned to the base level.

Such clauses often make the system non-Markovian because, in the malus zone, it is necessary to remember claims from several previous years to determine if the policyholder can benefit from these special clauses. In practice, it is usually sufficient to split certain states to recover the Markovian property.

Let’s illustrate this technique using the 1992 Belgian scale. Levels 16 to 21 are divided into sub-levels based on the number of consecutive years without accidents. Level $j.i$, $16\leq j\leq 21$, groups drivers at level $j$ with $i$ consecutive claim-free years. Table ?? describes the transition rules of the Belgian system, taking into account the fast-return clause. If we denote $n_j$ as the number of sub-levels needed for level $j$, these values are provided in Table ??.

12.2.2.2 Transition Probabilities

Let \[ p_{\ell_1\ell_2}(\vartheta)=\Pr[L_{t+1}=\ell_2|L_t=\ell_1,\lambda\Theta=\vartheta] \] be the probability that a policy is transferred from level $\ell_1$ to level $\ell_2$ during a period for an insured with an annual claim frequency of $\vartheta$. It is clear that $p_{\ell_1\ell_2}(\vartheta) \geq 0$ and $\sum_{\ell_2 =0}^s p_{\ell_1\ell_2}(\vartheta)=1$ for any $\vartheta$. The matrix \[ \boldsymbol{P}(\vartheta)=\left( \begin{array}{ccc} p_{00}(\vartheta) & \cdots & p_{0s}(\vartheta)\\ \vdots & \ddots & \vdots \\ p_{s0}(\vartheta) & \cdots & p_{ss}(\vartheta) \end{array} \right) \] represents the transition matrix of the Markov chain describing the insured’s progression in the scale. The fact that the annual claim frequency is independent of time makes this Markov chain homogeneous.

The transition rules of a bonus-malus system can be described using transformations $T_k$ such that $T_k(\ell_1)=\ell_2$ if the policy is transferred from level $\ell_1$ to level $\ell_2$ when $k$ claims have been reported during a period. These transformations $T_k$ can be described in matrix form as: \[ \boldsymbol{T}_k=\left( \begin{array}{ccc} t_{00}^{(k)} & \cdots & t_{0s}^{(k)}\\ \vdots & \ddots & \vdots \\ t_{s0}^{(k)} & \cdots & t_{ss}^{(k)} \end{array} \right) \] where $t_{\ell_1\ell_2}^{(k)}=1$ if $T_k(\ell_1)=\ell_2$ and $t_{\ell_1\ell_2}^{(k)}=0$ otherwise. For the system to be consistent, it is necessary that for each $\ell_1$, there exists only one $\ell_2$ such that $t_{\ell_1\ell_2}^{(k)}=1$.

The probability $p_{\ell_1\ell_2}(\vartheta)$ that a policy is transferred from level $\ell_1$ to level $\ell_2$ during a period for an insured characterized by a risk parameter $\vartheta$ is equal to \[ p_{\ell_1\ell_2}(\vartheta ) = \sum_{k=0}^{+\infty} \Pr[N=k|\lambda\Theta=\vartheta] t_{\ell_1\ell_2}^{(k)}. \] The transition matrix can then be written as \[ \boldsymbol{P}(\vartheta )= \sum_{k=0}^{+\infty} \Pr[N=k|\lambda\Theta=\vartheta] \boldsymbol{T}_k. \]

12.2.2.3 Transient Laws

Let $p_{\ell_1\ell_2}^{(\nu)}(\vartheta)$ be the probability that an insured with an annual claim frequency of $\vartheta$ is sent from level $\ell_1$ to level $\ell_2$ in $\nu$ years, i.e., \[ p_{\ell_1\ell_2}^{(\nu)}(\vartheta)=\Pr[L_{t+\nu}=\ell_2|L_t=\ell_1,\lambda\Theta=\vartheta]. \] Clearly, \[\begin{eqnarray*} p_{\ell_1\ell_2}^{(\nu)}(\vartheta)&=&\sum_{k=0}^s\Pr[L_{t+\nu}=\ell_2|L_{t+\nu-1}=k, L_t=\ell_1,\lambda\Theta=\vartheta]\\ &&\hspace{15mm}\Pr[L_{t+\nu-1}=k|L_t=\ell_1,\lambda\Theta=\vartheta]\\ &=&\sum_{k=0}^sp_{k\ell_2}(\vartheta)p_{\ell_1k}^{(\nu-1)}(\vartheta) \end{eqnarray*}\] where we recognize the formula for matrix multiplication. Thus, the $\nu$-th power of the transpose $\boldsymbol{P}^t(\vartheta)$ of the matrix $\boldsymbol{P}(\vartheta)$ provides the transition matrix at $\nu$ steps, whose element $(\ell_1\ell_2)$, denoted $p_{\ell_1\ell_2}^{(\nu)}(\vartheta)$, is the probability that an insured with an annual claim frequency of $\vartheta$ is sent from level ${\ell_1}$ to level ${\ell_2}$ in $\nu$ transitions.

If we denote \[ p_\ell^{(\nu)}(\vartheta)=\Pr[L_\nu=\ell|\lambda\Theta=\vartheta] \] and if we designate $\boldsymbol{p}^{(\nu)}(\vartheta)$ as the vector whose $\ell$-th component is $p_\ell^{(\nu)}(\vartheta)$, then we have \[\begin{equation} \boldsymbol{p}^{(\nu)}(\vartheta)=\boldsymbol{P}^t(\vartheta)\boldsymbol{p}^{(\nu-1)}(\vartheta). \tag{12.3} \end{equation}\] In general, $\boldsymbol{p}^{(0)}(\vartheta)$ is determined by the insurer (it is the entry level into the system), so a successive application of the last relation allows us to calculate $\boldsymbol{p}^{(\nu)}(\vartheta)$, which gives the percentages of insured individuals occupying the different levels of the scale after $\nu$ years.

12.2.2.4 Stationary Law

Practically all bonus-malus systems have a finite number of levels, among which there is a “super-bonus” level that an insured individual reaches after a sufficient number of claim-free years and remains in as long as they do not report a claim. The transition matrix $\boldsymbol{P}(\vartheta)$ describing the trajectory of an insured with an annual claim frequency of $\vartheta$ in such a scale is regular, i.e., there exists an integer $\xi_0\geq 1$ such that all elements of the matrix $\{\boldsymbol{P}(\vartheta)\}^{\xi_0}$ are strictly positive.

In this case, the Markov chain describing the insured’s journey in the scale is ergodic and therefore has a stationary law represented by the probability vector $\boldsymbol{\pi}(\vartheta)$. The $\ell$-th component $\pi_\ell(\vartheta)$ of this vector is the probability that an insured who has been in the portfolio for a sufficiently long time and has an annual claim frequency of $\vartheta$ occupies level $\ell$, i.e., \[ \boldsymbol{\pi}(\vartheta)=\lim_{\nu\to +\infty}\boldsymbol{p}^{(\nu)}(\vartheta). \] It is interesting to note that $\boldsymbol{\pi}(\vartheta)$ does not depend on the level at which new insured individuals are placed (this is a consequence of the Markovian nature of the system, which forgets its past).

Let’s recall how stationary probabilities $\pi_\ell(\vartheta)$ are obtained. Starting from equation (12.3), we see that if there is a limit, it must satisfy the equation \[ \lim_{\nu\to+\infty}\boldsymbol{p}^{(\nu)}(\vartheta)=\boldsymbol{P}^t(\vartheta) \lim_{\nu\to+\infty}\boldsymbol{p}^{(\nu-1)}(\vartheta) \] \[ \Leftrightarrow\boldsymbol{\pi}(\vartheta)=\boldsymbol{P}^t(\vartheta)\boldsymbol{\pi}(\vartheta). \] Therefore, the vector $\boldsymbol{\pi}(\vartheta)$ is a solution of the system \[\begin{equation} \left\{ \begin{array}{l} \boldsymbol{\pi}^t(\vartheta)=\boldsymbol{\pi}^t(\vartheta)\boldsymbol{P}(\vartheta),\\ \boldsymbol{\pi}^t(\vartheta)\boldsymbol{e}=1. \end{array} \right. \tag{12.4} \end{equation}\] The following property, which can be found in (Rolski et al. 2009), provides the explicit expression of $\boldsymbol{\pi}(\vartheta)$.

Proposition 12.1 Let $\boldsymbol{E}$ be the matrix of dimension $(s+1)\times(s+1)$ where all elements are equal to 1. We have \[ \boldsymbol{\pi}^t(\vartheta)=\boldsymbol{e}^t\big(\mathbf{I}-\boldsymbol{M}(\vartheta)+\boldsymbol{E}\big)^{-1}. \]

12.2.3 Norberg’s Method

12.2.3.1 Principle

Let’s consider an insured individual randomly selected from the portfolio. Their annual claim frequency is unknown. The number $N$ of claims they will report during the year follows a Poisson distribution $\mathcal{Poi}(\lambda,\Theta)$, where $\mathbb{E}[\Theta]=1$. (Norberg 1976) suggests determining the values of $r_\ell$ in a way that minimizes $\mathcal{Q}=\mathbb{E}[(\Theta-r_L)^2]$. The idea is to choose the $r_\ell$ that best approximate $\Theta$ in terms of least squares. By expanding the expression of $\mathcal{Q}$, we have \[\begin{eqnarray*} \mathcal{Q} &=&\sum_{\ell=0}^s\mathbb{E}\Big[(\Theta-r_\ell)^2\Big|L=\ell\Big]\Pr[L=\ell]\\ &=&\sum_{\ell=0}^s\int_{\theta>0}(\theta-r_\ell)^2u(\theta|\ell)d\theta\Pr[L=\ell] \end{eqnarray*}\] where $u(\cdot|\ell)$ represents the density of $\Theta$ given $L=\ell$. Clearly, \[ u(\theta|\ell)=\frac{\Pr[L=\ell|\Theta=\theta]u(\theta)}{\Pr[L=\ell]} \] from which we obtain \[ \mathcal{Q}= \int_{\theta >0}\sum_{\ell=0}^s(\theta-r_\ell)^2\pi_\ell(\lambda\theta)u(\theta)d\theta. \]

12.2.3.2 Relative Premium

Now, to obtain $\frac{\partial}{\partial r_\ell}\mathcal{Q}=0$, we get the equation \[ 0=\int_{\theta>0}(\theta-r_\ell)\pi_\ell(\lambda\theta) u(\theta)d\theta \] from which we ultimately obtain \[\begin{equation} r_\ell=\frac{\int_{\theta\geq 0}\theta \pi_\ell(\lambda\theta)u(\theta)d\theta} {\int_{\theta\geq 0}\pi_\ell(\lambda\theta)u(\theta)d\theta}. \tag{12.5} \end{equation}\]

It is worth noting that $r_\ell=\mathbb{E}[\Theta|L=\ell]$. In fact, \[\begin{eqnarray*} \mathbb{E}[\Theta|L=\ell]&=&\int_{\theta>0}\theta u(\theta|\ell)d\theta\\ &=&\frac{1}{\Pr[L=\ell]}\int_{\theta>0}\theta \pi_\ell(\theta\lambda)u(\theta)d\theta \end{eqnarray*}\] which coincides with the expression for $r_\ell$ given in (12.5). Therefore, the expectation of $\Theta$ conditioned on the insured occupying level $\ell$ is used to calculate the relative premiums associated with the different levels of the scale.

12.2.3.3 Financial Equilibrium

Initially, the insurance company does not use a {} pricing method and charges all insured individuals in the same rate class the same premium. The amount of this premium is set so that the average total collection is sufficient to compensate for the reported claims. Following the introduction of a bonus-malus system, the premiums charged to insured individuals will vary, but it is important that the total amount collected by the insurer remains constant, as the average total amount of claims is not changed. A bonus-malus system is said to have the property of financial equilibrium when the collection remains stable over time.

In our case, this translates to $\mathbb{E}[r_L]=1$. When the $r_\ell$ are calculated in accordance with (12.5), the system enjoys the property of financial equilibrium since \[\begin{eqnarray*} \mathbb{E}[r_L]&=&\sum_{\ell=0}^sr_\ell\Pr[L=\ell]\\ &=&\sum_{\ell=0}^s\mathbb{E}[\Theta|L=\ell]\Pr[L=\ell]=\mathbb{E}[\Theta]=1. \end{eqnarray*}\]

12.2.4 Gilde and Sundt’s Method

12.2.4.1 Principle

Equation (12.5) can produce $r_\ell$ values that are sometimes very irregular. For commercial reasons, it may be desirable to impose a structure on the percentages associated with different levels. An example is a linear scale of the form $r_\ell^{lin}=\alpha+\beta\ell$, $\ell=0,1,\ldots,s$. In this case, $r_0^{lin}=\alpha$ is the minimum premium level that the insured will have to pay, and moving up one level in the scale will result in a premium increase of $\beta$.

12.2.4.2 Relative Premium

(Gilde and Sundt 1989) proposed determining $\alpha$ and $\beta$ to minimize $\mathbb{E}[(\Theta-\alpha-\beta L)^2]$. Based on the theory of linear models studied in Chapter 9, the solution is given by \[\begin{equation} \beta=\frac{\lambda\mathbb{C}[L,\Theta]}{\mathbb{V}[L]} \mbox{ and }\alpha=\lambda-\frac{\lambda\mathbb{C}[L,\Theta]}{\mathbb{V}[L]} \mathbb{E}[L]. \tag{12.6} \end{equation}\] In the case of a linear scale, the percentage associated with level $\ell$ is therefore of the form \[ r_\ell^{lin}=1+\frac{\mathbb{C}[L,\Theta]}{\mathbb{V}[L]}(\ell-\mathbb{E}[L]) \] where \[\begin{eqnarray*} \mathbb{E}[L]&=&\sum_{\ell=0}^s\ell\int_{\theta\geq 0}\pi_\ell(\lambda\theta)u(\theta)d\theta\\ \mathbb{V}[L]&=&\sum_{\ell=0}^s\ell^2\int_{\theta\geq 0}\pi_\ell(\lambda\theta)u(\theta)d\theta-(\mathbb{E}[L])^2\\ \mathbb{C}[L,\Theta]&=&\sum_{\ell=0}^s\ell\int_{\theta\geq 0}\theta\pi_\ell(\lambda\theta)u(\theta)d\theta-\mathbb{E}[L]. \end{eqnarray*}\]

12.2.4.3 Financial Equilibrium

In this case as well, the system enjoys the property of financial equilibrium, since \[ \mathbb{E}[r_L^{lin}] = 1 + \frac{\mathbb{C}[L, \Theta]}{\mathbb{V}[L]}\mathbb{E}\Big[L-\mathbb{E}[L]\Big] = 1. \] Thus, the introduction of a bonus-malus scale will have no impact on the premium once the steady-state regime is reached.

12.3 Segmented Universe Scales

12.3.1 Introductory Example

Suppose the insurer has noticed that the accident frequency is higher in urban areas than in rural areas (due to traffic density) and decides to account for this difference in their pricing. Let $\lambda_A$ and $\lambda_R$ be the annual accident frequencies in urban and rural areas, respectively, and assume that the portfolio consists of a proportion of $w_A$ urban drivers and $w_R=1-w_A$ rural drivers.

In both urban and rural areas, there is still some heterogeneity, resulting in a 50% reduction in accident frequency for good drivers, and a 50% increase for bad drivers (both in urban and rural areas). Thus, the accident frequency of a randomly selected driver from the portfolio is $\Lambda\Theta$ (random as we do not know their driving quality or place of residence), with \[ \Lambda=\left\{ \begin{array}{l} \lambda_A\text{ with probability }w_A\\ \lambda_R\text{ with probability }w_R \end{array} \right. \] and \[ \Theta=\left\{ \begin{array}{l} 0.5\text{ with probability }\frac{1}{2}\\ 1.5\text{ with probability }\frac{1}{2} \end{array} \right. \] taking the values used in the numerical illustrations from Section 12.2.1. Note that $\Theta$ represents a residual effect independent of $\Lambda$: it is the residual effect of all other variables, except the place of residence. In this sense, $\Theta$ and $\Lambda$ can be considered independent. Moreover, $\mathbb{E}[\Theta]=1$.

In this case, the insurer is thus evaluating risk in two ways: initially, they recognize the higher risk of city dwellers by varying the premium based on the driving zone, and subsequently, they use the accident statistics to move the policyholder in the bonus-malus scale, and the premium varies accordingly. Of course, the scale should only correct the residual heterogeneity modeled by $\Theta$. The posterior corrections should therefore be based on $\mathbb{E}[\Theta|L=\ell]$. If this quantity is greater than 1, it means that the policyholder’s claims history indicates that they are riskier than the average drivers in the same zone (rural or urban). Conversely, a value less than 1 corresponds to a policyholder whose claims history, as reflected by their position in the scale, indicates a lower risk level than the average drivers in the same zone. In the first case, the policyholder will face a premium surcharge, while in the second case, the policyholder will receive a discount.

So, let’s calculate $\mathbb{E}[\Theta|L=\ell]$ for $\ell=0,1,2$. To do this, we write \[ \mathbb{E}[\Theta|L=\ell]=0.5\Pr[\Theta=0.5|L=\ell]+1.5\Pr[\Theta=1.5|L=\ell] \] and then \[\begin{eqnarray*} &&\Pr[\Theta=0.5|L=\ell]\\ &=&\frac{\Pr[L=\ell|\Theta=0.5]0.5}{\Pr[L=\ell]}\\ &=&\frac{1}{2\Pr[L=\ell]}\Big(\Pr[L=\ell|\Theta=0.5,\Lambda=\lambda_A]\Pr[\Lambda=\lambda_A|\Theta=0.5]\\ &&\hspace{25mm}+\Pr[L=\ell|\Theta=0.5,\Lambda=\lambda_B]\Pr[\Lambda=\lambda_B|\Theta=0.5]\Big)\\ &=&\frac{1}{2\pi_\ell}\Big(w_A\pi_\ell(0.5\lambda_A)+w_B\pi_\ell(0.5\lambda_B)\Big) \end{eqnarray*}\] where we have used the assumption of independence between $\Lambda$ and $\Theta$. Now, we need to calculate $\pi_\ell$. This probability is obtained through a double conditioning, with respect to $\Lambda$ and $\Theta$: \[\begin{eqnarray*} \pi_\ell=\Pr[L=\ell]&=&\Pr[L=\ell|\Theta=\theta_1,\Lambda=\lambda_A]\Pr[\Theta=\theta_1,\Lambda=\lambda_A]\\ &&+\Pr[L=\ell|\Theta=\theta_2,\Lambda=\lambda_A]\Pr[\Theta=\theta_2,\Lambda=\lambda_A]\\ &&+\Pr[L=\ell|\Theta=\theta_1,\Lambda=\lambda_B]\Pr[\Theta=\theta_1,\Lambda=\lambda_B]\\ &&+\Pr[L=\ell|\Theta=\theta_2,\Lambda=\lambda_B]\Pr[\Theta=\theta_2,\Lambda=\lambda_B]\\ &=&\pi_\ell(\lambda_A\theta_1)w_A\varrho+\pi_\ell(\lambda_A\theta_2)w_A(1-\varrho)\\ &&+\pi_\ell(\lambda_B\theta_1)w_B\varrho+\pi_\ell(\lambda_B\theta_2)w_B(1-\varrho). \end{eqnarray*}\] Similarly, \[ \Pr[\Theta=1.5|L=\ell]=\frac{1}{2\pi_\ell}\Big(w_A\pi_\ell(1.5\lambda_A)+w_B\pi_\ell(1.5\lambda_B)\Big). \] For instance, if we assume $\lambda_A=8\%$, $\lambda_B=12\%$, and $w_A=50\%$, we obtain $\mathbb{E}[\Lambda]=10\%$, which coincides with the value of $\lambda$ from the introductory example described in Section 12.2.1. Table ?? presents the results obtained with these numerical values.

The values of $\Pr[\Theta=0.5|L=\ell]$ and $\Pr[\Theta=1.5|L=\ell]$ are nearly identical to those obtained previously, as expected. The relative premiums $\mathbb{E}[\Theta|L=\ell]$ are slightly less dispersed than those obtained without the prior distinction between rural and urban areas.

To appreciate the extent to which the prior and posterior rating interact, we can calculate $\mathbb{E}[\Lambda|L=\ell]$. This expectation should increase with $\ell$ if the prior rating makes sense. Clearly, \[\begin{eqnarray*} \mathbb{E}[\Lambda|L=\ell]&=&\lambda_A\Pr[\Lambda=\lambda_A|L=\ell] +\lambda_B\Pr[\Lambda=\lambda_B|L=\ell]\\ &=&\lambda_A\frac{\Pr[L=\ell|\Lambda=\lambda_A]w_A}{\pi_\ell}+ \lambda_B\frac{\Pr[L=\ell|\Lambda=\lambda_B]w_B}{\pi_\ell}\\ &=&\frac{\lambda_Aw_A}{2\pi_\ell}\Big(\pi_\ell(0.5\lambda_A)+\pi_\ell(1.5\lambda_A)\Big)\\ &&+\frac{\lambda_Bw_B}{2\pi_\ell}\Big(\pi_\ell(0.5\lambda_B)+\pi_\ell(1.5\lambda_B)\Big). \end{eqnarray*}\]

The values of these conditional expectations are summarized in the last column of Table ??. Indeed, we observe a growth in the conditional mean of $\Lambda$ as a function of the level $\ell$ occupied.

12.3.2 Modeling Claims Frequency in Segmented Universe

Let us now attempt to generalize the approach followed to address the introductory example above. Suppose that the portfolio has been partitioned into risk classes based on the available a priori information. Within the $k$-th risk class, the annual number of claims per policy follows a Poisson distribution with parameters $\lambda_k$ and $\Theta$, where the random effect $\Theta$, whose distribution is independent of the risk class, models the residual heterogeneity of the portfolio.

If we randomly select an insured individual from the portfolio, their a priori annual claims frequency is denoted as $\Lambda$, which is a random variable since we do not know the risk class from which this insured individual originates. We denote $w_k$ as the relative importance of the $k$-th risk class, and we have $\Pr[\Lambda=\lambda_k]=w_k$. Conditional on $\Lambda=\lambda_k$, the annual number of claims follows a Poisson distribution with parameters $\lambda_k$ and $\Theta$.

12.3.3 Severity of Posterior Adjustments Depending on the Degree of A priori Differentiation

The posterior adjustment of the premium amount should depend on the extent to which the characteristics of insured individuals play a role in determining the premium amount at the contract’s inception. The bonus-malus system (as well as the entire credibility theory, for that matter) aims solely to differentiate “good” drivers from “bad” drivers based on observed claims experience. It is clear that an insurer who uses many characteristics of the insured individual to decide whether or not to provide coverage and to determine the premium amount can already predict the future claims experience of the insured individual quite accurately. In contrast, an insurer who does not use any characteristics of the insured individual to set the premium amount (and therefore has no knowledge of the insured’s profile) would have difficulty determining a priori whether an individual is a good or bad driver. Therefore, an insurer who extensively considers the insured individual’s characteristics to determine the premium at the beginning of the contract should only moderately adjust the premium afterward. On the other hand, an insurer who considers only a few factors a priori should resort to a more severe bonus-malus scale. Let us now justify these statements.

Since $\Theta$ represents a residual effect of variables not incorporated into the tariff, the random variables $\Lambda$ and $\Theta$ are assumed to be independent. To better understand how residual heterogeneity depends on the precision of the insurer’s a priori differentiation, let us examine the following formula, which indicates how the portfolio’s heterogeneity decomposes: \[ \mathbb{V}[N]=\lambda+\lambda^2\mathbb{V}[\Theta]+\mathbb{V}[\Lambda](1+\mathbb{V}[\Theta]). \] As $\lambda=\mathbb{E}[\Lambda]$ is constant regardless of the degree of a priori segmentation, we can observe that a more extensive tariff differentiation will increase $\mathbb{V}[\Lambda]$, resulting in a decrease in $\mathbb{V}[\Theta]$, since $\mathbb{V}[N]$ is constant. Thus, an insurer who segments more a priori will have lower residual variance and will need to consider claims statistics to a lesser extent when reevaluating the premiums paid by policyholders.

12.3.4 Norberg Method in a Segmented Universe

Let’s randomly select an insured individual from the portfolio. We denote by $L$ the level they occupy on the scale in a steady-state. The probability distribution of $L$ is given by: \[ \Pr[L=\ell]=\sum_kw_k\int_{\theta\geq 0}\pi_\ell(\lambda_k\theta)u(\theta)d\theta, \] where $w_k$ is the proportion of insured individuals belonging to the $k$-th risk class, with an a priori annual claims frequency of $\lambda_k$.

To determine the percentages associated with different levels of the scale, we again minimize the mean square deviation between $\Theta$ and $r_L$, which is: \[\begin{eqnarray*} \mathcal{Q}=\mathbb{E}\Big[(\Theta-r_L)^2\Big] &=&\sum_{\ell=0}^s\mathbb{E}\Big[(\Theta-r_\ell)^2\Big|L=\ell\Big]\Pr[L=\ell]\\ &=&\sum_{\ell=0}^s\int_{\theta>0}(\theta-r_\ell)^2\Pr[L=\ell|\Theta=\theta]u(\theta)d\theta\\ &=&\sum_kw_k\int_{\theta >0}\sum_{\ell=0}^s(\theta-r_\ell)^2\pi_\ell(\lambda_k\theta)u(\theta)d\theta. \end{eqnarray*}\] The solution is obtained by solving $\frac{\partial}{\partial r_\ell}\mathcal{Q}=0$ with: \[ \frac{\partial}{\partial r_\ell}\mathcal{Q}=-2\sum_k\int_{\theta>0}(\theta-r_\ell)\pi_\ell(\lambda_k\theta) u(\theta)d\theta, \] which yields: \[\begin{equation} r_\ell=\frac{\sum_kw_k\int_{\theta >0}\theta \pi_\ell(\lambda_k\theta)u(\theta)d\theta} {\sum_kw_k\int_{\theta >0}\pi_\ell(\lambda_k\theta)u(\theta)d\theta}. \tag{12.7} \end{equation}\]

It is interesting to note that $r_\ell=\mathbb{E}[\Theta|L=\ell]$ since: \[\begin{eqnarray} &&\mathbb{E}[\Theta|L=\ell]\nonumber\\ &=&\mathbb{E}\Big[\mathbb{E}[\Theta|L=\ell,\Lambda]\Big|L=\ell\Big]\nonumber\\ &=&\sum_k\mathbb{E}[\Theta|L=\ell,\Lambda=\lambda_k]\Pr[\Lambda=\lambda_k|L=\ell]\nonumber\\ &=&\sum_k\int_{\theta >0}\theta\frac{\Pr[L=\ell|\Theta=\theta,\Lambda=\lambda_k]w_k} {\Pr[L=\ell]}u(\theta)d\theta. \tag{12.8} \end{eqnarray}\]

It is easy to see that $\mathbb{E}[r_L]=1$, which ensures the financial equilibrium of the system once the steady-state is reached.

Finally, note that if the insurer does not differentiate premiums a priori, all $\lambda_k$ are equal to $\lambda$, and (12.7) yields (12.5). The method of (Gilde and Sundt 1989) also naturally extends to the segmented case.

12.3.5 Interaction between Posterior Adjustments Induced by the Scale and Prior Pricing

The degree of interaction between prior and posterior pricing is measured by the variations in: \[\begin{eqnarray} \mathbb{E}[\Lambda|L=\ell]&=&\sum_k\lambda_k\Pr[\Lambda=\lambda_k|L=\ell]\nonumber\\ &=&\sum_k\lambda_k\frac{\Pr[L=\ell|\Lambda=\lambda_k]w_k}{\pi_\ell}\nonumber\\ &=&\frac{\sum_k\lambda_kw_k\int_{\theta\geq 0}\pi_\ell(\lambda_k\theta)u(\theta)d\theta} {\sum_kw_k\int_{\theta\geq 0}\pi_\ell(\lambda_k\theta)u(\theta)d\theta}. \tag{12.9} \end{eqnarray}\] If prior pricing is relevant (in the sense that a high a priori frequency indeed reflects a high level of risk), $\mathbb{E}[\Lambda|L=\ell]$ should increase with the level $\ell$. This clearly shows that insured individuals with lower a priori claims frequencies tend to occupy the lower levels of the scale, while those with higher a priori claims frequencies are found in the higher levels of the scale. Therefore, insured individuals who received prior discounts (because their risk profile suggested they would cause fewer claims) will also be rewarded posteriorly (as they will gravitate to the lower levels of the scale). In contrast, insured individuals considered poor drivers a priori and thus penalized at the outset with higher premiums will again be penalized by the posterior customization system, as they tend to occupy the highest levels of the scale.

12.4 Numerical Illustrations

12.4.1 Prior Pricing

To illustrate the methods described above, we consider the automobile insurance portfolio observed during the year 1997, as analyzed in Chapter 9.

12.4.2 Scale “-1/top”

This is a scale with six levels, numbered from 0 to 5. A new insured is placed at level 5. Each year without a claim results in a one-level descent. The declaration of at least one claim has the effect of sending the insured back to level 5, regardless of the level they occupied at the beginning of the year. Note that the philosophy of such a system is fundamentally different from that of standard bonus-malus scales. Here, there is no intention to match the premium paid by the insured to their long-term risk: the goal is simply to penalize drivers responsible for a claim (thus combating moral hazard) and reward those who do not use the company’s guarantee.

The transition rules for this scale are described in the following table: {

}

The transition matrix $\boldsymbol{P}(\vartheta)$ for an insured with an annual claims frequency of $\vartheta$ is as follows: { \[ \boldsymbol{P}(\vartheta)=\left( \begin{array}{cccccc} \exp(-\vartheta)& 0& 0& 0& 0& 1-\exp(-\vartheta)\\ \exp(-\vartheta) & 0& 0& 0& 0& 1-\exp(-\vartheta)\\ 0& \exp(-\vartheta) & 0& 0& 0& 1-\exp(-\vartheta)\\ 0& 0& \exp(-\vartheta) & 0& 0& 1-\exp(-\vartheta)\\ 0& 0& 0& \exp(-\vartheta) & 0& 1-\exp(-\vartheta)\\ 0 & 0& 0& 0& \exp(-\vartheta) & 1-\exp(-\vartheta)\\ \end{array} \right). \] }

The calibration of the scale using the Norberg method (without considering and then considering a priori segmentation) yields the following results: {

}

The second column provides the proportions of insured individuals in the different levels in the steady-state. Figure ?? illustrates the transient laws associated with the -1/Top scale for an annual claims frequency of 10%. It can be observed that the steady-state is reached after 5 years.

The third column describes the percentages associated with the 6 levels if no premium differentiation is applied a priori. Insured individuals at level 0 receive a discount of around 30%, and those at level 5 are penalized by 66.6%. The fourth column presents these same percentages when the insurer segments a priori. This has the effect of limiting the magnitude of posterior adjustments. The discount at level 0 is reduced to 20%, and the penalty at level 5 is 43.2%. Finally, it can be seen that $\mathbb{E}[\Lambda|L=\ell]$ increases with $\ell$, reflecting the concentration of good drivers at the bottom of the scale and poor drivers at the top of the scale.

12.4.3 “-1/+2” Scale

This is the “soft” scale by (Taylor 1997), which can be considered a typical bonus-malus system. New insured individuals start at level 6. Each year without a claim results in a one-level descent in the scale, and each claim is penalized by moving up two levels.

The transition rules associated with this scale are described in the following table:

The transition matrix $\boldsymbol{P}(\vartheta)$ associated with this bonus-malus system is written as follows: { \[ \boldsymbol{P}(\vartheta)=\left( \begin{array}{ccccccccc} p_0& 0& p_1& 0& p_2&0& p_3&0&1-\Sigma\\ p_0& 0&0& p_1& 0& p_2&0& p_3&1-\Sigma\\ 0&p_0& 0&0& p_1& 0& p_2&0& 1-\Sigma\\ 0&0&p_0& 0&0& p_1& 0& p_2&1-\Sigma\\ 0&0&0&p_0&0& 0& p_1& 0& 1-\Sigma\\ 0&0&0&0&p_0&0& 0& p_1& 1-\Sigma\\ 0&0&0&0&0&p_0&0& 0& 1-p_0\\ 0&0&0&0&0&0&p_0& 0& 1-p_0\\ 0&0&0&0&0&0&0&p_0& 1-p_0\\ \end{array} \right). \] } where \[ p_k=\frac{\vartheta^k}{k!}\exp(-\vartheta) \] and $\Sigma$ represents the sum of the elements in columns 1 to 8 of the same row.

The calibration of the -1/+2 scale using the Norberg method (without considering and then considering prior segmentation) yields the following results: {

}

The conclusions are identical to those drawn for the -1/Top scale.

12.4.4 “-1/+4” Scale

Now let’s move on to the severe scale by (Taylor 1997). In this scale, each claim results in a penalty of moving up by 4 levels in the scale. The transition rules associated with this system are described in the following table: {

}

Now, let’s examine the values obtained for the -1/+4 scale using the Norberg method (without considering and then considering prior segmentation): {

}

12.5 Performance of Bonus-Malus Scales

12.5.1 The Relative Stationary Average Level (RSAL)

The Relative Stationary Average Level (RSAL) measures the average level an average driver occupies when the bonus-malus system reaches its stationary state. It assesses the concentration of policies in the lower levels of the bonus-malus system.

The strict stability of bonus-malus systems can only be achieved after an infinite number of years, once the stationary regime is reached. In practice, insurance companies are more interested in stabilizing the average premium level and its variance. Therefore, in the following, we will refer to stability as the condition where the mean and variance of the premium level have become more or less constant. Strict stability is then referred to as “total stability.”

12.5.2 The Relative Stationary Average Premium (RSAP)

The Relative Stationary Average Premium (RSAP) is defined as follows:

\[ \text{RSAP} = \frac{\text{Stationary average premium level} - \text{Minimum premium level}}{\text{Maximum premium level} - \text{Minimum premium level}} \]

Expressed as a percentage, this index determines the premium of an average insured relative to a minimum premium of 0 and a maximum premium of 100. A low RSAP value indicates that most policies concentrate in the lower classes, while a higher RSAP suggests a better distribution of insured individuals across all classes. Ideally, the RSAP index should be around 50%.

12.5.3 The Coefficient of Variation of Premiums

Insurance involves the transfer of risk from the insured to the insurance company. If we do not consider the claims history of insured individuals, this transfer is complete, meaning there is perfect solidarity among insured individuals, and premiums remain constant for all. However, when a bonus-malus system is introduced, premiums vary from year to year based on the claims an insured individual caused in the previous year.

The solidarity among insured individuals can be evaluated by measuring the variability of annual premiums, typically using the coefficient of variation, which is the ratio of the standard deviation to the mean premium.

Even for the most severe bonus-malus systems, these coefficients are still very close to 0, indicating that insured individuals only bear a small portion of the risk themselves.

12.5.4 Loimaranta’s Efficiency

There are numerous measures of the quality of a bonus-malus system, among which are the concepts of efficiency inherited from economics. The efficiency of a bonus-malus system measures the system’s reaction when the frequency of claims changes. Logically, premiums paid by policyholders should increase with the claim frequency. A relative increase in the claim frequency should result in the same relative increase in the premium. For example, consider two policyholders, one with a claim frequency of 0.10 and the other with 0.11; after a sufficiently long period, the second policyholder should pay 10% more in premiums than the first. A bonus-malus system having this property is considered perfectly elastic. In reality, the increase is often less than 10%. If it is, for example, 2% instead of 10%, the system’s elasticity is then 20%. In the following, we will examine an asymptotic concept of elasticity known as Loimaranta’s efficiency.

Let $b(\vartheta)$ be the average premium at the stationary state for a policyholder with an annual claim frequency of $\vartheta$. Ideally, an increase $\frac{d\vartheta}{\vartheta}$ in claim frequency should lead to an identical increase, $\frac{db(\vartheta)}{b(\vartheta)}$, in the premium. A bonus-malus system will be called perfectly elastic when:

\[ \frac{\frac{d\vartheta}{\vartheta}}{\frac{db(\vartheta)}{b(\vartheta)}} = 1 \]

The elasticity $\text{Eff}(\vartheta)$ of a bonus-malus system is defined as:

\[ \text{Eff}(\vartheta) = \frac{\frac{db(\vartheta)}{b(\vartheta)}}{\frac{d\vartheta}{\vartheta}} = \frac{d\ln b(\vartheta)}{d\ln\vartheta} \]

So, it is the elasticity of the asymptotic average premium $b(\vartheta)$ with respect to the claim frequency $\vartheta$.

Clearly,

\[ b(\vartheta) = \sum_{\ell=0}^s \pi_\ell(\vartheta)\cdot b_\ell \]

and

\[ \frac{db(\vartheta)}{d\vartheta} = \sum_{\ell=0}^s \frac{d\pi_\ell(\vartheta)}{d\vartheta} b_\ell \]

where $b_\ell$ is the premium associated with level $\ell$ of the scale. Equations to determine $\frac{d\pi_\ell(\vartheta)}{d\vartheta}$ can be obtained by differentiating the system (12.4), which defines the stationary distribution. Therefore, you need to solve the linear system:

\[ \frac{d\boldsymbol{\pi}^t(\vartheta)}{d\vartheta} = \frac{d\boldsymbol{\pi}^t(\vartheta)}{d\vartheta} \boldsymbol{P}(\vartheta) + \boldsymbol{\pi}^t (\vartheta) \frac{d\boldsymbol{P}(\vartheta)}{d\vartheta} \]

subject to the constraint:

\[ \sum_{\ell=0}^s \frac{d\pi_\ell(\vartheta)}{d\vartheta} = 0. \]

12.5.5 Lemaire’s Efficiency

Loimaranta’s efficiency has two major disadvantages:

Lemaire’s efficiency (which is, in fact, a particular case of De Pril’s efficiency) addresses these two drawbacks.

First, introduce a discount rate $\beta < 1$, and let $v_i^{(n)}(\vartheta)$ be the present value of all payments made over $n$ years by a policyholder with an annual claim frequency of $\vartheta$ at the beginning of the first insurance period at level $i$. The $\{v_i^{(n)}(\vartheta), n=1,2,\ldots\}$ satisfy the following recurrence scheme:

\[ v_i^{(n)}(\vartheta) = b_i + \beta \sum_{k=0}^{+\infty} \Pr[N=k|\lambda\Theta=\vartheta]v_{T_k(i)}^{(n-1)}(\vartheta), \quad i=0,1,\ldots,s. \]

There will be $n\cdot(s+1)$ different expected values to calculate, which can be laborious, especially for sophisticated bonus-malus systems and long contract durations. By transitioning from a finite horizon to an infinite horizon, we can simplify the calculations considerably, without significantly changing the values for contracts with sufficiently long durations. Let:

\[ v_i(\vartheta) = \lim_{n\rightarrow \infty}v_i^{(n)}(\vartheta), \]

be the infinite-horizon present value of all premiums paid by a policyholder with an annual claim frequency of $\vartheta$, initially at level $i$ of the scale. The components of the vector:

\[ \boldsymbol{v}(\vartheta) = (v_0(\vartheta), v_1(\vartheta),\ldots,v_s(\vartheta))^t, \]

then satisfy the following equations:

\[\begin{equation} v_i(\vartheta) = b_i + \beta \sum_{k=0}^{+\infty} \Pr[N=k|\lambda\Theta=\vartheta]v_{T_k(i)}(\vartheta), \quad i=0,1,\ldots,s. \tag{12.10} \end{equation}\]

(#prp:PropBMS11.5.1) The system of equations (12.10) has a unique solution.

Proof. Let $O$ be the transformation defined by $O\boldsymbol{v}= \wvec$, where:

\[ w_i(\vartheta) = b_i + \beta \sum_{k=0}^{\infty} \Pr[N=k|\lambda\Theta=\vartheta]v_{T_k(i)}(\vartheta), \quad i=0,1,\ldots,s. \]

Choose the norm $\| {\boldsymbol{v}} \| = \max _i |v_i|$. Then we have:

\[\begin{eqnarray*} \| O{\wvec}-O{\boldsymbol{v}}\| &=&\max_i \left| \beta \sum_{k=0}^{\infty } \Pr[N=k|\lambda\Theta=\vartheta] \left( w_{T_k(i)} (\vartheta )- v_{T_k(i)} (\vartheta )\right) \right| \\ &\leq &\beta \sum_{k=0}^{\infty } \Pr[N=k|\lambda\Theta=\vartheta] \max_i \left| w_{T_k(i)} (\vartheta )- v_{T_k(i)} (\vartheta )\right| \\ &\leq &\beta \max_j \left| w_j(\vartheta )- v_j(\vartheta ) \right| =\beta \| {\wvec}-{\boldsymbol{v}} \|, \end{eqnarray*}\]

by setting $j=T_k(i)$. Therefore, the transformation $O$ is a contraction, and there exists only one fixed point.

12.5.6 Lemaire’s Efficiency

Lemaire’s efficiency $\text{Eff}_i (\vartheta )$ can be defined in the same way as Loimaranta’s efficiency using $v_i(\vartheta )$ instead of $b(\vartheta )$. Thus:

\[ \text{Eff}_i (\vartheta )=\frac{\frac{dv_i(\vartheta )}{v_i(\vartheta ) }}{\frac{d\vartheta}{\vartheta } } =\frac{d\ln v_i(\vartheta )}{d\ln (\vartheta )}; \]

$\text{Eff}_i (\vartheta )$ is the elasticity of the present value of payments with respect to the claim frequency. This approach to efficiency depends on the starting class. Therefore, transient efficiency has an advantage over asymptotic efficiency: it can be used to define the initial class (by choosing the one that maximizes $\text{Eff}_i (\vartheta )$ for a mean value of $\vartheta$). Additionally, Lemaire’s efficiency gives more weight to what happens in the early years than what happens after many years (due to discounting).

To calculate $\text{Eff}_i (\vartheta )$, we need the derivatives $\frac{dv_i(\vartheta )}{d\vartheta }$, which we can obtain by solving the system:

\[ \frac{dv_i(\vartheta )}{d\vartheta } =\beta \sum_{k=0}^{\infty } \frac{\exp(-\vartheta)\vartheta ^k }{k !} \left( \left(\frac{k}{\vartheta } -1\right)v_{T_k(i)}(\vartheta ) + \frac{dv_{T_k(i)}(\vartheta )}{d\vartheta } \right). \]

Through a similar demonstration as that of Proposition @ref(prp:PropBMS11.5.1), we can show that the system has a unique solution.

12.5.7 Optimal Average Retention

12.5.7.1 Concept

The introduction of a bonus-malus system independent of the amount of claims in automobile insurance encourages policyholders to take on the costs of small claims themselves to avoid future premium increases. This phenomenon is called “bonus thirst.”

The significance of bonus thirst is related to the severity of a bonus-malus system: a severe bonus-malus system comes with a high number of undeclared claims. If the claim costs borne by the policyholder are reasonable, bonus thirst has the same effect as introducing a deductible and reduces administrative costs for the insurance company.

The following algorithm determines an optimal retention strategy for the policyholder; it is based on dynamic programming. For each class of the bonus-malus system, the algorithm determines the optimal retention level; below this amount, the policyholder has an interest in not reporting the claim to the insurance company and compensating the other party themselves.

The model is based on the following assumptions:

12.5.7.2 Determination of the Optimal Strategy

Let us now determine the optimal strategy. An insured party causing a claim of amount $x$ at time $t$, where $0 \leq t \leq 1$, has two strategies at their disposal:

They may choose not to report the claim, and the expected total cost discounted to the time of the claim is then given by \[ \beta^{-t} \overline{CT}_{x_i} + x + \beta^{1-t} \sum^{\infty}_{k=0} \overline{p}^i_k \big(\vartheta (1-t)\big) v_{T_{k+m}(i)}(\vartheta) \] where $m$ is the number of claims already reported during the period.
Alternatively, they may report the accident to the insurance company, and the expected total cost is \[ \beta^{-t} \overline{CT}_{x_i} + \beta^{1-t} \sum^{\infty}_{k=0} \overline{p}^i_k \big(\vartheta (1-t)\big)v_{T_{k+m+1}(i)}(\vartheta). \]

The retention limit $x_i$ is the claim amount $x$ for which the two strategies are equivalent. Therefore, \[\begin{equation} x_i = \beta^{1-t} \sum^{\infty}_{k=0} \overline{p}^i_k \big(\vartheta (1-t)\big) \Big(v_{T_{k+m+1}(i)}(\vartheta) - v_{T_{k+m}(i)}(\vartheta)\Big), \tag{12.11} \end{equation}\] for $i=0,1,\ldots,s$.

These equations form a system of $s$ equations with $s$ unknowns $x_1,x_2,\ldots,x_s$ (since the $x_i$ appear implicitly in the $\overline{p}^i_k (\vartheta (1-t))$). This system provides a new strategy $\mathbf{x}$ corresponding to a fixed cost vector $\mathbf{v} (\vartheta)$. The optimal policy \[ \mathbf{x}^* = (x_0^*,x_1^*, \ldots ,x_s^*) \] can then be determined by successive approximations using the following algorithm:

Part A:

Let’s choose an arbitrary strategy ${\boldsymbol{x}}^0$. The most interesting one, but not necessarily the one that leads to the optimal solution most quickly, is ${\boldsymbol{x}} ^0 = (0,0,\ldots ,0)$, the strategy of reporting all claims. This initial strategy will allow us to calculate the improvement in expected cost resulting from taking care of certain claims. The first system (??) simplifies to:

\[ v_i(\vartheta )=b_i+\beta \sum^{\infty }_{k=0} \Pr[N=k|\lambda\Theta=\vartheta] v_{T_k(i)}(\vartheta ) \]

and provides the cost vector ${\boldsymbol{v}} ^0 (\vartheta )$ corresponding to the initial strategy.

Part B:

12.5.7.3 Improved Strategy $\mathbf{x}^1$

An improved strategy $\mathbf{x}^1$ can then be obtained using the relationships (12.11), which reduce in this particular case to \[\begin{eqnarray*} x_i &=& \beta^{1-t} \sum^{\infty}_{k=0} \Pr[N=k|\lambda\Theta\\ &=& (1-t)\vartheta] \Big(v_{T_{k+m+1}(i)}(\vartheta) - v_{T_{k+m}(i)}(\vartheta)\Big), \end{eqnarray*}\] for $i=0,1,\ldots,s$.

Part A:

By inserting ${\mathbf{x}}^1$ into the system (??), we can determine the cost ${\mathbf{v}}^1 (\vartheta)$ for this strategy. This cost will be lower than that corresponding to the strategy ${\mathbf{x}}^0$.

Part B:

By inserting the new cost ${\mathbf{v}}^1 (\vartheta)$ into the system (12.11), we will find a better strategy ${\mathbf{x}}^2$. \end{description}

The successive insertion of ${\mathbf{x}}$ and ${\mathbf{v}} (\vartheta)$ into the two systems (??) and (12.11) allows us to find a sequence of strategies that continuously improve and reduce costs.

The cases considered in (Lemaire 1995) show that this sequence converges to the optimal solution ${\mathbf{x}}^*$ and ${\mathbf{v}}^*(\vartheta)$. The optimal retention strategy depends on:

the class $i$ occupied by the insured at the beginning of the period,
the discount rate $\beta$,
the claim frequency $\vartheta$,
the time $t$ of the claim, and
the number $m$ of claims already reported since the beginning of the period.

The optimal strategy is an increasing function of $t$: the retention limit increases as the policy expiration date (and the resulting premium reduction) approaches. However, the influence of $t$ on ${\mathbf{x}}$ is less significant than that of $C_i$, $\beta$, or $\vartheta$. Setting $t = 0$ (and consequently $m = 0$) greatly simplifies the calculations, but retentions are only slightly reduced.

12.6 Bibliographical Notes

The essential reference for bonus-malus systems is the book by (Lemaire 1995). This chapter is based on several recent articles by (Sandra Pitrebois, Denuit, and Walhin 2003c), (Sandra Pitrebois, Denuit, and Walhin 2003b), (Sandra Pitrebois, Denuit, and Walhin 2003a) and (SANDRA Pitrebois, Denuit, and Walhin 2004).

All the calculations in this chapter were based on the stationary distribution. In practice, it is sometimes interesting to also consider transient distributions, especially for systems that take many years to stabilize. In this case, the policy randomly selected from the portfolio is represented by the triplet $(\Lambda, \Theta, A)$, where $\Lambda$ and $\Theta$ are as previously described, and $A$ is the policy’s seniority in the portfolio. The probability distribution of $A$ describes the age of policies in the portfolio: thus, $\Pr[A=k]$ is the proportion of policies that have been in the portfolio for $k$ years. We then determine the percentages to be assigned to each level of the scale by minimizing $\mathbb{E}[(\Theta-r_{L_A})^2]$, where $L_A$ denotes the level occupied by the policy in the scale (in transient regime, that is). See, for example, (Borgan 1 and Hoem 1981).

It is essential to take into account policy cancellations, as they will primarily occur among insured parties occupying the highest levels of the scale and will disrupt the financial balance of the system (the discounts granted to drivers who do not report claims are exactly offset by the penalties imposed on those who do).

The acquisition of new policies and the cancellation of existing policies in the portfolio will be considered by adding a state to the Markov process, as suggested by (Lourdes Centeno and Silva 2001): transitions from the levels of the scale to this state will represent policy cancellations, while transitions from this state to one of the scale levels will indicate the conclusion of a new policy with the policyholder placed at the corresponding level, taking into account their claims history.

We should also mention the following works, which were not covered in this chapter. (Sandra Pitrebois, Walhin, and Denuit 2005) considered bonus-malus scales with different penalties based on the severity of claims. Additionally, bonus-malus systems are coupled with variable deductibles based on the position within the scale. This approach appears particularly attractive. Finally, (Jean François Walhin and Paris 2000) and (Jean-Francois Walhin and Paris 2001) examine certain technical aspects related to the use of bonus-malus scales, such as the censorship of claim counts that arises from it.

(Denuit, Walhin, and Pitrebois 2001) considered bonus-malus scales where the percentages were obtained using an exponential loss function (rather than quadratic as in this entire chapter). This provided corrections whose severity could be controlled with a parameter while maintaining the financial balance property.

(Brouhns et al. 2003) considered dynamic random effects and an evolving risk profile for insured drivers. In this more realistic scenario, only simulation methods can yield results.

12.7 Exercises

Exercise 12.1 Consider an automobile insurance portfolio. Let $N$ be the annual number of claims reported by an insured. The insurance company distinguishes between males and females. Based on its statistics, it has obtained the following probabilities: \[ \Pr[N=0|male]=0.8=1-\Pr[N=1|male] \] and \[ \Pr[N=0|female]=0.9=1-\Pr[N=1|female], \] using obvious notation. The portfolio consists of 60% males and 40% females.

Of course, not all males and females drive the same way. Among males, there are 50% good drivers with an annual claim frequency of 0.08 and 50% bad drivers with an annual frequency of 0.32. Similarly, among females, there are 50% good drivers with an annual claim frequency of 0.04 and 50% bad drivers with an annual frequency of 0.16.

Since the quality of male or female drivers is unobservable a priori, the company decides to apply a post-contractual customization mechanism based on the following scale:

The transition rules are as follows: a new insured is placed at level 3, every claim-free year results in a move down one level, while a claim is penalized by returning to level 3.

A new male insured presents himself to the company; what is the premium that will be charged to him if he perfectly matches the existing portfolio?
Assuming that the annual numbers of claims are independent and identically distributed conditional on gender and the driver’s quality who caused them, compare the distribution of men and women in the scale in a stationary regime? Comment.
Determine $r_1$, $r_2$, and $r_3$ for men (based on the stationary distribution).

Exercise 12.2 Consider an automobile insurance portfolio segmented by the driver’s gender. Let $N$ be the annual number of claims reported by an insured randomly selected from this portfolio. Conditional on $\Lambda\Theta$, $N$ follows a Poisson distribution with mean $\Lambda\Theta$, where \[ \Lambda=\left\{ \begin{array}{l} 0.1\text{ if the insured is female}\\ 0.2\text{ if the insured is male} \end{array} \right. \] and \[ \Theta=\left\{ \begin{array}{l} 0.5\text{ with probability }\frac{1}{2}\\ 1.5\text{ with probability }\frac{1}{2}. \end{array} \right. \] The random variables $\Lambda$ and $\Theta$ are assumed to be independent. The portfolio consists of 60% men and 40% women.

You are asked:

A woman insured for ten years has reported only one claim during this period. What is the reassessment of her annual claim frequency at the end of these 10 years?
The company decides to apply the same post-contractual customization mechanism based on the following scale to both men and women:

The transition rules are as follows: a new insured is placed at level 3, every claim-free year results in a move down one level, and any claim demotes the insured to level 3 (regardless of their previous level).

Determine $r_1$, $r_2$, and $r_3$ based on Norberg’s method (using the stationary distribution).
Determine $r_1$, $r_2$, and $r_3$ based on Gilde & Sundt’s method (using the stationary distribution).

Exercise 12.3 Let $N_t$ be the annual number of claims caused by a policy in the portfolio. Suppose that, given $\Theta$, the $N_t$ are independent and identically distributed random variables with the conditional distribution: \[ \Pr[N_t=1|\Theta=\theta]=1-\Pr[N_t=0|\Theta=\theta]=\theta \] where \[ \Theta=\left\{ \begin{array}{l} 0.1\text{ with probability }0.8\\ 0.2\text{ with probability }0.2. \end{array} \right. \]

You are asked:

Consider an insured who has not reported any claims during the first three years of coverage. How do you reassess the probability that they will cause 1 claim during year 4?
To correct the heterogeneity of the portfolio induced by $\Theta$, the actuary uses a three-level bonus-malus scale, numbered 0, 1, and 2. Entry is at level 1. Each claim-free year results in a move down one level in the scale. Each claim is penalized by moving up one level.

Provide the transition matrix given $\Theta=0.1$.
What is the distribution of insureds in the portfolio among the three levels in a stationary regime?
What relative premiums should be associated with the different levels?

Postface

Borgan 1, Ørnulf, and Jan M Hoem. 1981. “A Nonasymptotic Criterion for the Evaluation of Automobile Bonus Systems.” Scandinavian Actuarial Journal 1981 (3): 165–78.

Brouhns, Natacha, Montserrat Guillén, Michel Denuit, and Jean Pinquet. 2003. “Bonus-Malus Scales in Segmented Tariffs with Stochastic Migration Between Segments.” Journal of Risk and Insurance 70 (4): 577–99.

Bühlmann, Hans. 1964. “Optimale Prämienstufensysteme.” Mitteilungen Der VSVM 64 (2): 193–214.

Delaporte, Pierre. 1959. “Quelques Problèmes de Statistique Mathématique Posés Par l’assurance Automobile Et Le Bonus Pour Non Sinistre.” Bulletin Trimestriel de l’Institut Des Actuaires Français 227: 87–102.

Denuit, Michel, Jean-François Walhin, and Sandra Pitrebois. 2001. “Méthodes de Construction de Systemes Bonus-Malus En RC Auto.” Actu-L 1: 7.

Franckx, E. 1960. “Theorie Du Bonus, Consequence de l’etude de Mr Le Professeur Frechet.” ASTIN Bulletin: The Journal of the IAA 1: 113–22.

Fréchet, Maurice. 1959. “Essai d’une Etude Des Successions de Sinistres Considéres Comme Processus Stochastique.” Bulletin Trimestriel de l’Institut Des Actuaires Francais, Paris.

Gilde, Vibeke, and Bjørn Sundt. 1989. “On Bonus Systems with Credibility Scales.” Scandinavian Actuarial Journal 1989 (1): 13–22.

Kelle, Magali. 2000. “Modélisation Du Système de Bonus Malus Français.” Bulletin Français d’Actuariat 4: 45–64.

Lemaire, Jean. 1995. Bonus-Malus Systems in Automobile Insurance. Vol. 19. Springer.

Lourdes Centeno, Maria de, and Joao Manuel Andrade e Silva. 2001. “Bonus Systems in an Open Portfolio.” Insurance: Mathematics and Economics 28 (3): 341–50.

Norberg, Ragnar. 1976. “A Credibility Theory for Automobile Bonus Systems.” Scandinavian Actuarial Journal 1976 (2): 92–107.

Pitrebois, Sandra, Michel Denuit, and Jean-François Walhin. 2003a. “Setting a Bonus-Malus Scale in the Presence of Other Rating Factors: Taylor’s Work Revisited.” ASTIN Bulletin: The Journal of the IAA 33 (2): 419–36.

Pitrebois, Sandra, Michel Denuit, and JF Walhin. 2003b. “Marketing and Bonus-Malus Systems.” Berlin, Germany, Sn, 2–40.

Pitrebois, SANDRA, MICHEL Denuit, and JF Walhin. 2004. “Bonus-Malus Scales in Segmented Tariffs: Gilde & Sundt’s Work Revisited.” Australian Actuarial Journal 10 (1): 107–25.

Pitrebois, Sandra, M Denuit, and JF Walhin. 2003c. “Fitting the Belgian Bonus-Malus System.” Belgian Actuarial Bulletin 3 (1): 58–62.

Pitrebois, Sandra, Jean-François Walhin, and Michel Denuit. 2005. “Bonus-Malus Systems with Varying Deductibles.” ASTIN Bulletin: The Journal of the IAA 35 (1): 261–74.

Rolski, Tomasz, Hanspeter Schmidli, Volker Schmidt, and Jozef L Teugels. 2009. Stochastic Processes for Insurance and Finance. John Wiley & Sons.

Taylor, Gregory. 1997. “Setting a Bonus-Malus Scale in the Presence of Other Rating Factors.” ASTIN Bulletin: The Journal of the IAA 27 (2): 319–27.

Thepaut, André. 1959. “Aspect Politique Et Aspect Administratif Du Bonus Pour Non Sinistre.” Bulletin Trimestriel de l’Institut Des Actuaires Francais, Paris.

Walhin, Jean François, and Jose Paris. 2000. “The True Claim Amount and Frequency Distributions Within a Bonus-Malus System.” ASTIN Bulletin: The Journal of the IAA 30 (2): 391–403.

Walhin, Jean-Francois, and John Paris. 2001. “The Practical Replacement of a Bonus-Malus System.” ASTIN Bulletin: The Journal of the IAA 31 (2): 317–35.

Chapter 12 Bonus-Malus

12.1 Introduction

12.1.1 Residual Heterogeneity

12.1.2 Nature of Dependency

12.1.3 Objectives of Bonus-Malus Systems

12.1.4 Nature of Dependency

12.1.5 Objectives of Bonus-Malus Systems

12.1.6 Nature of Dependency

12.1.7 bjectives of Bonus-Malus Systems

12.1.8 Nature of Dependency

12.1.9 Objectives of Bonus-Malus Systems

12.1.10 Bonus Thirst

12.1.11 Class-Based Systems and “French-Style” Systems

12.1.12 A Brief History of the Bonus-Malus System in France

12.1.13 A Brief History of the Bonus-Malus System in Belgium

12.1.14 Chapter Outline

12.2 Scales in Unsegmented Universes}

12.2.1 Introductory Example: the Good/Bad Driver Model

12.2.2 Scales and Markov Chains

12.2.2.1 Modeling the Insured Individual’s Progress in the Scale

12.2.2.2 Transition Probabilities

12.2.2.3 Transient Laws

12.2.2.4 Stationary Law

12.2.3 Norberg’s Method

12.2.3.1 Principle

12.2.3.2 Relative Premium

12.2.3.3 Financial Equilibrium

12.2.4 Gilde and Sundt’s Method

12.2.4.1 Principle

12.2.4.2 Relative Premium

12.2.4.3 Financial Equilibrium

12.3 Segmented Universe Scales

12.3.1 Introductory Example

12.3.2 Modeling Claims Frequency in Segmented Universe

12.3.3 Severity of Posterior Adjustments Depending on the Degree of A priori Differentiation

12.3.4 Norberg Method in a Segmented Universe

12.3.5 Interaction between Posterior Adjustments Induced by the Scale and Prior Pricing

12.4 Numerical Illustrations

12.4.1 Prior Pricing

12.4.2 Scale “-1/top”

12.4.3 “-1/+2” Scale

12.4.4 “-1/+4” Scale

12.5 Performance of Bonus-Malus Scales

12.5.1 The Relative Stationary Average Level (RSAL)

12.5.2 The Relative Stationary Average Premium (RSAP)

12.5.3 The Coefficient of Variation of Premiums

12.5.4 Loimaranta’s Efficiency

12.5.5 Lemaire’s Efficiency

12.5.6 Lemaire’s Efficiency

12.5.7 Optimal Average Retention

12.5.7.1 Concept

12.5.7.2 Determination of the Optimal Strategy

12.5.7.3 Improved Strategy \(\mathbf{x}^1\)

12.6 Bibliographical Notes

12.7 Exercises

Postface