Bonus-Malus Premiums Based on Claim Frequency and the Size of Claims

Abstract

The bonus-malus system (BMS) is one of the most widely used tools in merit-rating automobile insurance, with the primary goal of ensuring that fair premiums are paid by all policyholders. The traditional BMS is dependent only on the claim frequency. Thus, an insured person who makes a claim with a small severity is penalized unfairly compared to an individual who makes a large severity claim. This study proposes a model for estimating the bonus-malus premium by employing a limit value (monetary unit) which distinguishes claim size into small and large based on claim frequency and claim severity distributions. This assists in determining the penalties for policyholders with claim sizes falling above and below the limit value. The number of claims is assumed to follow a Poisson distribution, and the total number of claims with a size greater than the limit value is considered a binomial distribution. The underlying risk of each policyholder is assumed to follow a beta Lindley distribution and is referred to as the prior distribution. Each policyholder’s claim size is also assumed to follow a gamma distribution, with the Lindley distribution considered as the prior distribution. Bonus-malus premiums are calculated following the Bayesian method. Practical examples using an actual data set are provided, and the results generated are compared to those produced using the traditional Poisson binomial-exponential beta model. This methodology provides a more equitable mechanism for penalizing policyholders in the portfolio.

1. Introduction

One crucial actuarial task is designing a charge structure that fairly distributes the responsibility of claims among policyholders. This is computed using the best model available to determine insurance premiums. The bonus-malus system (BMS) adjusts the premiums according to individual claim history and is popularly used, especially in automobile insurance, to determine that fair premiums are paid by all policyholders. As a reward for accident-free driving, a bonus is offered as a discount, whereas a malus results in an increase in the premium. The introduction of a BMS that links the premium to the number of reported claims will cause a tendency for policyholders to shoulder small claims on their own and not report them to their company to avoid a premium increase. “Hunger for bonus” is the term used to describe this phenomenon (Lemaire 1985). Lemaire (1976, 1977) studied the hunger for bonus and proposed a dynamic programming algorithm to determine the optimal claiming behavior. The BMS is used by insurance companies for two main reasons. The first is to encourage insureds to drive more cautiously throughout the year to avoid claims, and the latter is to ensure that insureds pay premiums that are proportional to their risks based on their claim experiences (Bühlmann 1967). The basic principle of this system is that greater claim frequencies result in higher premiums. The traditional BMS was solely dependent on the random variable of number of claims (Déniz 2016). However, not all events result in the same claim amount for an insured person. Because of different claim sizes, it makes sense to design a BMS based on both claim frequency and claim severity.

Several methodologies under the BMS framework have been used to determine fair premiums paid by policyholders. One of the most popular ways to determine premiums is the Bayesian method. Each policyholder has a constant as an unequal underlying risk. This unknown constant is referred to as a risk parameter and is treated as a random variable, with a specific probability distribution known as a prior distribution or structure-function distribution. This method is useful for calculating premiums in the BMS based on specific transition rules that distinguish policyholders as a bonus or malus. By dividing a posterior mean of the parameter by a prior mean, the basic Bayesian tool is easily applied to Bayes’ theorem. This tool employs the net premium principle and provides an estimate of risk parameters to distinguish between low- and high-risk policyholders.

The Poisson distribution is normally used to explain an independent random event of claims in vehicle insurance and expresses the behavior probability of individual policyholders (Déniz 2016). However, it cannot adequately describe the number of claims in an insurance portfolio. The negative binomial distribution, a type of mixing distribution with a Poisson distribution and a gamma distribution, was studied by Greenwood and Yule (1920). This marked the start of the process of constructing mixed Poisson distributions. A mixed Poisson was proposed for claim frequency distribution by Tremblay (1992). He designed an optimal BMS by mixing the Poisson distribution with the inverse Gaussian distribution, which has a wide variety of applications in medicine, finance, business, survival analysis and even in the maximum durability problem of investment in the gold market (Moumeesri and Klongdee 2019). In the automobile insurance sector, the random number of claims was also assumed to be Poisson distributed, while the randomly expected inherent risks of each insured person followed an inverse Gaussian distribution. Here, the Bayesian method was applied to estimate the posterior portfolio distribution function for a scenario covering the past t years. The expected premium value was calculated based on the BMS principle. Several papers have discussed mixing other distributions to obtain an optimal bonus-malus premium for the claim frequency (Dionne and Vanasse 1992; Lemaire 1995; Walhin and Paris 1999; Tzougas and Frangos 2014; Tzougas et al. 2019; Tzougas 2020). However, premium payments based on the BMS show no difference between a claim made by a policyholder for a small loss and another with a big loss. This phenomenon is called “hunger for bonus” (Lemaire 1985). For instance, a claim of USD 50 by a policyholder should not be penalized by the same increase in premium as a claim of USD 500. Déniz (2016) and Hernawati et al. (2017) proposed a model to determine premiums based on a BMS that distinguished two types of claims: those less than the limit value and those that exceeded this value. The result was a fairer method of penalizing all policyholders in the portfolio. Déniz and Calderín-Ojeda (2018) further developed the proposal of Déniz (2016). They studied a trivariate model where claims were distinguished into three types, while Pongsart et al. (2022) proposed a model for computing Bayesian bonus-malus premiums by distinguishing different multiple types of claims.

Our analysis identified situations where this concept should not be regarded as the fairest approach for penalizing policyholders. These included cases where only a few claims were made but the claim amount was large, or where a policyholder made a higher number of claims but the total claim amount was small. Therefore, the premium calculation based on the BMS should also assess the total claim amount. An optimal method to determine the premium charged to an insured must take both frequency and severity components into account (Frangos and Vrontos 2001). Many studies have attempted to determine premiums based on the BMS by including more factors than using only claim frequency in a model (Frangos and Vrontos 2001; Mert and Saykan 2005; Ibiwoye et al. 2011; Ni et al. 2014; Tzougas et al. 2015, 2017, 2018; 2020; Santi et al. 2016; Oh et al. 2019; Moumeesri et al. 2020; Jacob and Wu 2020). They proposed a model in which claim frequency and claim severity were included jointly to compute the bonus-malus premiums. This model was constructed by multiplying the bonus-malus premium based only on the number of claims with the bonus-malus premium based on the individual claim size. In addition, there was the work done to integrate the BMS into the farm insurance product. Boucher (2022) suggested developing unique BMSs using recursive partitioning techniques to each type of insured.

Here, we propose a model in which claim frequency and claim severity are included jointly to determine bonus-malus premiums by distinguishing two types of claims: those below a limit value classified as small, and the rest classified as large. The number of claims was assumed to be Poisson distributed, while the total number of claims with claim size larger than the limit value was assumed to follow a binomial distribution. The underlying risk of each policyholder was taken to be Lindley and beta distributed for the prior distributions. For claim severity distribution, we assume that the claim size of each policyholder follows a gamma distribution. The prior distribution was introduced as a Lindley distribution. To calculate the posterior structure functions for claim frequency and claim severity distributions, we followed the Bayesian approach. The mean of these functions was used to calculate the premiums paid by a policyholder.

The remainder of this paper is structured as follows. Section 2 discusses the methodology, divided into two parts as claim frequency distribution and claim severity distribution. Mixing distributions, the Bayesian method and the premium calculation are also explained. Section 3 illustrates the numerical applications and discusses the results using real claim data separated as claim frequency components and claim severity components. The results obtained are compared with those derived from the traditional Poisson-Lindley model that does not distinguish the type of claims (see Moumeesri et al. 2020 for details), and those achieved under the Poisson binomial-exponential beta model (see Hernawati et al. 2017 for details). The conclusions drawn are presented in Section 4.

2. Methodology

We assumed that the claim frequency and severity of each policyholder were independent. Claim frequency and claim severity distributions were divided into subsections as described below.

2.1. Claim Frequency Distribution

2.1.1. Mixing Distribution

In vehicle insurance, the Poisson distribution is used to describe the random occurrence of claims and represents the probability of individual policyholder behavior. Assume that the number of claims $K$ of each policyholder, given the parameter $\theta >0$, follows a Poisson distribution. Then, its probability mass function (pmf) can be represented as:

$$f\left(k|\theta \right)=\frac{e^{-\theta }{\theta }^k}{k!}, k=0,1,2,\dots .$$

(1)

The expected value of the Poisson random variable is $E\left[K|\theta \right]=\theta .$

When the $j$th policyholder causes a total claim amount $w_j$, this has a certain size, $x_i$, that can be considered as a small or large claim. To distinguish the two types of claims, the limit value denoted by $\mathsf{\Phi }$ separates the two sub-events as follows.

Let $Z_i$ be the random variable corresponding to the claim size over the limit values.

$$Z_i=\{\begin{array}{c}1, x_i>\mathsf{\Phi }\\ 0, x_i<\mathsf{\Phi }\end{array}$$

where $\mathsf{\Phi }$ is a positive real number and $x_i>0.$

$Z_i$ is modeled as a mutually independent and identically distributed random variable (i.i.d.) with a parameter of $0<p<1$ and Bernoulli distribution. Then, its pmf can be given as follows:

$$f\left(z_i|p\right)=p^{z_i}{\left(1-p\right)}^{1-z_i}, z_i=0,1.$$

We also assume that $Z={\sum }_{i=1}^k{Z}_i$ is the total number of claims with claim size larger than $\mathsf{\Phi }$. Then, the remainder, $K-Z$, is the total number of claims with claim size below $\mathsf{\Phi }$. Assume that $Z$ has a binomial distribution. The conditional pmf of each $Z_i$ can be provided in the following form:

$$f\left(z|k,p\right)=\left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z}, z=0,1,2,\dots ,k.$$

(2)

The expected value of the binomial random variable is $E\left[Z|k,p\right]=kp.$

Thus, the joint distribution of the number of claims $\left(K\right)$ and the number of claims corresponding to the limit value $\left(Z\right)$ can be represented as:

$$f\left(k,z|\theta ,p\right)=f\left(k|\theta \right)f\left(z|k,p\right)=\frac{e^{-\theta }{\theta }^k}{k!}\cdot \left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z},$$

(3)

where $k=0,1,2,\dots$ and $z=0,1,2,\dots ,k.$ This is the joint distribution of Poisson ($\theta$) and binomial ($k,p$).

All policyholders have a constant representing their expected inherent risk. This is the mean of the number of claims of each insured, denoted by $\theta$. The parameter $\theta$ in Equation (1) denotes the different underlying risks of each insured having an accident and varies from one individual to another. This parameter is assumed to be a random variable $\mathsf{\Theta }$ according to the Lindley distribution, with parameter $\delta$. Then, the probability density function (pdf) of $\theta$ can be represented as:

$${\pi }_1\left(\theta \right)=\frac{{\delta }^2}{\delta +1}\left(\theta +1\right)e^{-\delta \theta }, \theta >0, \delta >0.$$

The expected value of the Lindley random variable is $E\left[\mathsf{\Theta }\right]=\frac{\delta +2}{\delta \left(\delta +1\right)}.$

The parameter $p$ in Equation (2) varies from one individual to another and is assumed to be a random variable $P$ that follows a beta distribution, with parameters $\alpha >0$ and $\beta >0.$ Thus, its pdf can be represented as:

$${\pi }_2\left(p\right)=\frac{1}{\mathrm{B}\left(\alpha ,\beta \right)}{p}^{\alpha -1}{\left(1-p\right)}^{\beta -1}, 0<p<1,$$

where $\mathrm{B}\left(\alpha ,\beta \right)$ is the beta function defined by $\mathrm{B}\left(\alpha ,\beta \right)=\frac{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\alpha +\beta \right)}$, and $\mathsf{\Gamma }(\cdot )$ is the gamma function defined as $\mathsf{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt$. The expected value of the beta random variable is $E\left[P\right]=\frac{\alpha }{\alpha +\beta }.$

When the random variables $\mathsf{\Theta }$ and $P$ are assumed to be independent, then the joint prior distribution for two random variables can be given by:

$$\pi \left(\theta ,p\right)={\pi }_1\left(\theta \right){\pi }_2\left(p\right)=\frac{{\delta }^2}{\delta +1}\left(\theta +1\right)e^{-\delta \theta }\cdot \frac{1}{\mathrm{B}\left(\alpha ,\beta \right)}{p}^{\alpha -1}{\left(1-p\right)}^{\beta -1}.$$

(4)

The mixed Poisson binomial distribution in Equation (3) with Lindley beta distribution in Equation (4) leads to in the following equations:

$$\begin{array}{cc}\hfill f\left(k,z\right)& =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}f\left(k,z|\theta ,p\right)\pi \left(\theta ,p\right) dpd\theta \hfill \\ & =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}\frac{e^{-\theta }{\theta }^k}{k!}\cdot \left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z}\cdot \frac{{\delta }^2}{\delta +1}\left(\theta +1\right)e^{-\delta \theta }\cdot \frac{1}{\mathrm{B}\left(\alpha ,\beta \right)}{p}^{\alpha -1}{\left(1-p\right)}^{\beta -1} dpd\theta \hfill \\ & =\frac{{\delta }^2}{\delta +1}\cdot \frac{1}{k!}\left(\begin{array}{c}k\\ z\end{array}\right)\frac{\mathrm{B}\left(\alpha +z,\beta +k-z\right)}{\mathrm{B}\left(\alpha ,\beta \right)}\cdot \left[\frac{\mathsf{\Gamma }\left(k+2\right)}{{\left(1+\delta \right)}^{k+2}}+\frac{\mathsf{\Gamma }\left(k+1\right)}{{\left(1+\delta \right)}^{k+1}}\right]\hfill \\ & =\frac{\mathrm{B}\left(\alpha +z,\beta +k-z\right)}{\mathrm{B}\left(\alpha ,\beta \right)\left(k-z\right)!z!}\cdot \frac{{\delta }^2\mathsf{\Gamma }\left(k+1\right)\left(k+\delta +2\right)}{{\left(1+\delta \right)}^{k+3}}\hfill \\ & =\left(\begin{array}{c}k\\ z\end{array}\right)\frac{\mathrm{B}\left(\alpha +z,\beta +k-z\right)}{\mathrm{B}\left(\alpha ,\beta \right)}\cdot \frac{{\delta }^2\left(k+\delta +2\right)}{{\left(1+\delta \right)}^{k+3}},\hfill \end{array}$$

(5)

where $\alpha >0,\beta >0,\delta >0,k=0,1,2,\dots$ and $z=0,1,2,\dots ,k.$

2.1.2. Bayesian Method

The bonus-malus premium calculation has been rigorously studied, with the Bayesian method being one of the most popular computing options (Déniz et al. 2014). The main purpose of this approach is to obtain the posterior distribution function. Bayesian methodology can be applied when data for all policyholders are available, either based on claims history or insured profiles.

Let $\left(k,z\right)=\left\{\left(k_1,z_1\right),\left(k_2,z_2\right),\dots ,\left(k_t,z_t\right)\right\}$ be a sample, where $t$ is the sample size. Then, $N={\sum }_{i=1}^t{k}_i$ is the total number of claims that a policyholder made in $t$ years, $k_i$ is the number of claims that the policyholder made in the year $i$, $i=1,2,\dots ,t,$ and $M={\sum }_{i=1}^t{z}_i$ represents the total number of claims where the claim size was larger than $\mathsf{\Phi }$ in $t$ years.

The posterior distribution function of random variables $\mathsf{\Theta }$ and $P$ can be considered according to Bayes’ theorem as proportional to the product of the prior distribution and the likelihood function.

The likelihood function considered by Equation (3) is proportional to

$$\begin{array}{cc}\hfill L\left(\theta ,p;\left(k,z\right)\right)& =f(\left(k,z\right)|\theta ,p)\hfill \\ & ={\displaystyle \prod _{i=1}^t}\frac{e^{-\theta }{\theta }^{k_i}}{k_i!}\cdot \left(\begin{array}{c}{k}_i\\ z_i\end{array}\right)p^{z_i}{\left(1-p\right)}^{k_i-z_i}\hfill \\ & =p^{{{\displaystyle \sum }}_{i=1}^t{z}_i}{\left(1-p\right)}^{{{\displaystyle \sum }}_{i=1}^t{k}_i-{{\displaystyle \sum }}_{i=1}^t{z}_i}{e}^{-t\theta }{\theta }^{{{\displaystyle \sum }}_{i=1}^t{k}_i}{\displaystyle \prod _{i=1}^t}\frac{1}{k_i!}\left(\begin{array}{c}{k}_i\\ z_i\end{array}\right)\hfill \\ & \hspace{1em}\propto p^M{\left(1-p\right)}^{N-M}{e}^{-t\theta }{\theta }^N.\hfill \end{array}$$

(6)

The prior distribution considered by Equation (4) is proportional to

$$\pi \left(\theta ,p\right)\propto p^{\alpha -1}{\left(1-p\right)}^{\beta -1}\left(\theta +1\right)e^{-\delta \theta }.$$

(7)

We then obtain the posterior distribution function of $\mathsf{\Theta }$ and $P$ for a policyholder or a group of policyholders with claim history $\left(k_1,z_1\right),\left(k_2,z_2\right),\dots ,\left(k_t,z_t\right)$. Then, the posterior distribution function which is proportional to the product of the likelihood function in Equation (6) and the prior distribution in Equation (7) can be represented as

$${\pi }^{*}\left(\theta ,p|\left(k,z\right)\right)\propto p^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N.$$

Consider

$$\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\pi }^{*}\left(\theta ,p|\left(k,z\right)\right) dpd\theta \propto \underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{p}^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N dpd\theta$$

then

$$\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}{\pi }^{*}\left(\theta ,p|\left(k,z\right)\right) dpd\theta =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}A p^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N dpd\theta =1,$$

where $A$ is constant.

Let $u=\left(t+\delta \right)\theta ,\theta =\frac{u}{t+\delta },du=\left(t+\delta \right)d\theta ,d\theta =\frac{du}{t+\delta },$ thus

$$\begin{array}{r}A\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N\underset{0}{\overset{1}{{\displaystyle \int }}}{p}^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1} dpd\theta =1\\ A \mathrm{B}\left(M+\alpha ,N-M+\beta \right)\left[\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-u}\frac{u^{N+1}}{{\left(t+\delta \right)}^{N+1}}\cdot \frac{du}{t+\delta }+\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-u}\frac{u^N}{{\left(t+\delta \right)}^N}\cdot \frac{du}{t+\delta }\right]=1\\ A \mathrm{B}\left(M+\alpha ,N-M+\beta \right)\left[\frac{\mathsf{\Gamma }\left(N+2\right)}{{\left(t+\delta \right)}^{N+2}}+\frac{\mathsf{\Gamma }\left(N+1\right)}{{\left(t+\delta \right)}^{N+1}}\right]=1\\ A \mathrm{B}\left(M+\alpha ,N-M+\beta \right)\frac{\mathsf{\Gamma }\left(N+1\right)}{{\left(t+\delta \right)}^{N+1}}\left[\frac{N+1+t+\delta }{t+\delta }\right]=1.\end{array}$$

then

$$A=\frac{{\left(t+\delta \right)}^{N+2}}{\mathsf{\Gamma }\left(N+1\right)\mathrm{B}\left(M+\alpha ,N-M+\beta \right)\left(N+1+t+\delta \right)}.$$

Therefore, the posterior distribution function can be represented as the following equation:

$${\pi }^{*}\left(\theta ,p|\left(k,z\right)\right)=\frac{{\left(t+\delta \right)}^{N+2}{p}^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N}{\mathsf{\Gamma }\left(N+1\right)\mathrm{B}\left(M+\alpha ,N-M+\beta \right)\left(N+1+t+\delta \right)}.$$

(8)

2.1.3. Premium Calculation

Many principles are involved in pricing insurance premiums. In this study, the net premium is denoted by $H\left[X\right]$. This is the basic principle in the sense that premiums should cover the expected value of losses. The net premium is calculated by expected values or means of claims or the number of claims to be paid by the insurers.

A premium calculation principle is a rule for assigning a premium to an insurance risk. This principle also assigns any risk $X$ with probability density function $f_X(x|\theta )$, where $x$ takes values in the sample space $X$ and $\theta$ is considered a realization of a parameter space $\mathsf{\Theta }$. A premium is a real number. Heilmann (1989), Landsman and Makov (1998) and Young (2000) introduced the net premium principle, which can be represented as

$$P\left(\theta \right)=E_f\left[X|\theta \right]={\int }_{X}x{f}_X\left(x|\theta \right)dx, \theta \in \mathsf{\Theta }.$$

For the loss function $L: \mathsf{\Theta }\times \mathbb{P}\to ℝ$, the risk function $R: \mathsf{\Theta }\times \mathbb{P}\to ℝ$ is given by

$$R(\theta ,\mathcal{P})=E_f\left[L(\theta ,\mathcal{P})\right]={\int }_{X}L(\theta ,\mathcal{P})f_X\left(x|\theta \right)dx.$$

The number of claims made by a policyholder in a given period is assigned by two random variables $\left(K,Z\right),$ with the pdf $f(k,z|\theta ,p)$ depending on an unknown risk parameter $\left(\theta ,p\right)$ that has a prior distribution $\pi \left(\theta ,p\right)$. In the actuarial field, an unknown premium $P\left(\theta ,p\right)$, called the risk premium, is calculated by minimizing the expected loss E_f[L(g(k,z),$\mathcal{P}$)], where $g\left(k,z\right)$ is a suitable function of the number of claims with claim size relating to the limit values and $\mathcal{P}$ is the action space. If this experience is not available, then the collective premium is computed by minimizing the risk function E_π(θ,p)[L(P(θ,p),$\mathcal{P}$)], where $\pi \left(\theta ,p\right)$ is the prior distribution of the unknown parameter $\left(\theta ,p\right)$. Conversely, if the experience is available, then the sample $\left(k,z\right)$ will be used by the actuary to estimate the unknown risk premium $P^{*}$, called the Bayesian premium. Déniz et al. (2000) and Heilmann (1989) further developed the premium calculation principle as follows:

• The risk premium:

  $$P\left(\theta ,p\right)=E_f\left[g\left(k,z\right)\right]={\displaystyle \sum }_K^{}{\displaystyle \sum }_Z^{}g\left(k,z\right)f\left(k,z|\theta ,p\right).$$
• The collective premium:

  $$P=E_{\pi }\left[P\left(\theta ,p\right)\right]=\underset{\theta }{\overset{}{{\displaystyle \int }}}\underset{p}{\overset{}{{\displaystyle \int }}}P\left(\theta ,p\right)\pi \left(\theta ,p\right)dp d\theta .$$
• The Bayesian premium:

  $$P^{*}=\underset{\theta }{\overset{}{{\displaystyle \int }}}\underset{p}{\overset{}{{\displaystyle \int }}}P\left(\theta ,p\right){\pi }^{*}\left(\theta ,p|\left(k,z\right)\right)dp d\theta .$$

  $g\left(k,z\right)$ can be represented as

  $$g\left(k,z\right)=p_{l}z+p_s\left(k-z\right),$$

  where $p_l$ and $p_s$ are suitable weights related to the number of claims, with sizes above and below the limit value, respectively. We also assume that $0\le p_s\le p_l\le 1$.

The risk premium can be obtained using the net premium principle as shown below:

$$\begin{array}{cc}\hfill P\left(\theta ,p\right)& =E\left[g\left(k,z\right)\right]\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \sum _{z=0}^k}g\left(k,z\right)f\left(k,z|\theta ,p\right)\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \sum _{z=0}^k}\left(p_{l}z+p_s\left(k-z\right)\right)\frac{e^{-\theta }{\theta }^k}{k!}\left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z}\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}\frac{e^{-\theta }{\theta }^k}{k!}\left[p_l{\displaystyle \sum _{z=0}^k}z\left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z}+p_s{\displaystyle \sum _{z=0}^k}\left(k-z\right)\left(\begin{array}{c}k\\ z\end{array}\right)p^z{\left(1-p\right)}^{k-z}\right]\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}\frac{e^{-\theta }{\theta }^k}{k!}\left(p_{l}kp+p_{s}k-p_{s}kp\right)\hfill \\ & =\left[\left(p_l-p_s\right)p+p_s\right]\theta .\hfill \end{array}$$

(9)

The risk premium can be reduced to $P\left(\theta ,p\right)=\theta$ if $p_l=p_s=1$, and considered under the net premium principle or the traditional model obtained by the mean of the Poisson distribution.

If the value of $\left(\theta ,p\right)$ is known, then the premium is a fair amount to penalize a policyholder. However, these values cannot be observed in practice since the risk premium is a theoretical amount that cannot be determined exactly but must be estimated from the data. The average premium for all possible risk premiums is called a prior or collective premium. This can be represented by the equations below.

$$\begin{array}{cc}\hfill P& =E\left[P\left(\theta ,p\right)\right]\hfill \\ & =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}P\left(\theta ,p\right)\pi \left(\theta ,p\right)dp d\theta \hfill \\ & =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}\left[\left(p_l-p_s\right)p+p_s\right]\theta \frac{{\delta }^2}{\delta +1}\left(\theta +1\right)e^{-\delta \theta }\frac{1}{\mathrm{B}\left(\alpha ,\beta \right)}{p}^{\alpha -1}{\left(1-p\right)}^{\beta -1} dp d\theta \hfill \\ & =\frac{p_l\alpha +p_s\beta }{\alpha +\beta }\cdot \frac{\delta +2}{\delta \left(\delta +1\right)}.\hfill \end{array}$$

(10)

If all weights are set to be $p_l=p_s=1$, then the collective premium calculated under the traditional distribution or net premium principle is $P=\frac{\delta +2}{\delta \left(\delta +1\right)}$, where $P$ is the mean of the Lindley distribution.

The posterior premium or Bayesian premium can be represented as the following equations:

$$\begin{array}{cc}\hfill P_t^{*}\left(k,z\right)& =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{1}{{\displaystyle \int }}}P\left(\theta ,p\right){\pi }^{*}\left(\theta ,p|\left(k,z\right)\right)dp d\theta \hfill \\ & ={{\displaystyle \int }}_0^{\infty }{{\displaystyle \int }}_0^1\left[\left(p_l-p_s\right)p+p_s\right]\theta \frac{{\left(t+\delta \right)}^{N+2}{p}^{M+\alpha -1}{\left(1-p\right)}^{N-M+\beta -1}{e}^{-\left(t+\delta \right)\theta }\left(\theta +1\right){\theta }^N}{\mathsf{\Gamma }\left(N+1\right)\mathrm{B}\left(M+\alpha ,N-M+\beta \right)\left(N+1+t+\delta \right)}dp d\theta \hfill \\ & =\frac{p_l\left(M+\alpha \right)+p_s\left(N-M+\beta \right)}{N+\alpha +\beta }\cdot \frac{N+1}{t+\delta }\cdot \frac{N+2+t+\delta }{N+1+t+\delta },\hfill \end{array}$$

(11)

where $N={\sum }_{i=1}^t{k}_i$ represents the total number of claims and $M={\sum }_{i=1}^t{z}_i$ represents claims with size larger than $\mathsf{\Phi }$. Note that $P_t^{*}\left(k,z\right)=P_0^{*}\left(0,0\right)=P$. Thus, the Bayesian premium in Equation (11) is similar to the prior premium in Equation (10) when no information is available.

If we assume that the initial premium or base premium at time $t=0$ is 100, then, at time $t+1$, the Bayesian bonus-malus premium can be determined from the number of claims and expressed as:

$${\mathrm{Premium}}_{t+1}=\frac{P_t^{*}\left(k,z\right)}{P_0^{*}\left(0,0\right)}100=\frac{P_t^{*}\left(k,z\right)}{P}100,$$

i.e.,

$${\mathrm{Premium}}_{t+1}=\frac{\alpha +\beta }{p_l\alpha +p_s\beta }\cdot \frac{p_l\left(M+\alpha \right)+p_s\left(N-M+\beta \right)}{N+\alpha +\beta }\cdot \frac{N+1}{t+\delta }\cdot \frac{N+2+t+\delta }{N+1+t+\delta } \cdot \frac{\delta \left(\delta +1\right)}{\delta +2}100.$$

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2.2. Claim Severity Distribution

2.2.1. Mixing Distribution

Suppose that the amount $x$ is the claim size of each policyholder. We assume that the conditional distribution of the claim size, given by the parameter ${\beta }_1,$ follows a gamma distribution. Its pdf can then be given by

$$f\left(x|{\beta }_1\right)=\frac{{\beta }_1^{\tau }}{\mathsf{\Gamma }\left(\tau \right)}{x}^{\tau -1}{e}^{-{\beta }_{1}x}, x>0, \tau >0, {\beta }_1>0.$$

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The expected value of the gamma random variable is $E\left[X|{\beta }_1\right]=\frac{\tau }{{\beta }_1}$.

The parameter ${\beta }_1$ is not the same for all insureds. Therefore, our prior belief for $\lambda$ in Equation (13) can be expressed in the form of a distribution. If the parameter ${\beta }_1$ is distributed according to the Lindley distribution with the parameter ${\delta }_1$, then its pdf is provided by

$$\pi \left({\beta }_1\right)=\frac{{\delta }_1^2}{{\delta }_1+1}\left({\beta }_1+1\right)e^{-{\delta }_1{\beta }_1}, {\beta }_1>0, {\delta }_1>0.$$

The expected value of ${\beta }_1$ will be $E\left[{\beta }_1\right]=\frac{{\delta }_1+2}{{\delta }_1\left({\delta }_1+1\right)}.$

Then, the unconditional distribution of claim size $x$ can be obtained as follows

$$\begin{array}{cc}\hfill f\left(x\right)& =\underset{0}{\overset{\infty }{{\displaystyle \int }}}f\left(x|{\beta }_1\right)\pi \left({\beta }_1\right)d\beta \hfill \\ & =\underset{0}{\overset{\infty }{{\displaystyle \int }}}\frac{{\beta }_1^{\tau }}{\mathsf{\Gamma }\left(\tau \right)}{x}^{\tau -1}{e}^{-{\beta }_{1}x}\cdot \frac{{\delta }_1^2}{{\delta }_1+1}\left({\beta }_1+1\right)e^{-{\delta }_1{\beta }_1}d{\beta }_1\hfill \\ & =\frac{{\delta }_1^2}{{\delta }_1+1}\cdot \frac{x^{\tau -1}}{\mathsf{\Gamma }\left(\tau \right)}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\left(e^{-\left(x+{\delta }_1\right){\beta }_1}{\beta }_1^{\tau +1}+e^{-\left(x+{\delta }_1\right){\beta }_1}{\beta }_1^{\tau }\right)d{\beta }_1\hfill \\ & =\frac{\tau {\delta }_1^2{x}^{\tau -1}}{{\delta }_1+1}\cdot \frac{x+\tau +{\delta }_1+1}{{\left(x+{\delta }_1\right)}^{\tau +2}},\hfill \end{array}$$

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where $x>0,\tau >0$ and ${\delta }_1>0$. Figure 1 illustrates the pdf plots of gamma-Lindley distribution for the unconditional distribution of claim size $x$.

2.2.2. Bayesian Method

Consider that an insured is in the portfolio for $t$ years and that the number of claims made in the year $i$ is denoted by $k_i$. Let $N={\sum }_{i=1}^t{k}_i$ be the total number of claims that a policyholder made in $t$ years. Then, claim size history can be represented as vector $x=\left(x_1,x_2,\dots ,x_N\right)$. The total claim size over $t$ years made by a policyholder in the portfolio can be denoted by $S={\sum }_{k=1}^N{x}_k$.

Bayes’ theorem can be applied to consider the likelihood function and the prior distribution to obtain the posterior distribution function of the parameter ${\beta }_1$ given the claim size history of the insured $x_1,x_2,\dots ,x_N$ as follows:

The likelihood function is

$$\begin{array}{cc}\hfill L\left({\beta }_1;x_1,x_2,\dots ,x_N\right)& =f(x_1,x_2,\dots ,x_N|{\beta }_1)\hfill \\ & ={\displaystyle \prod _{k=1}^N}\frac{{\beta }_1^{\tau }}{\mathsf{\Gamma }\left(\tau \right)}{x}_k^{\tau -1}{e}^{-{\beta }_1{x}_k}\hfill \\ & \propto {\beta }_1^{\tau N}{e}^{-{\beta }_{1}S}.\hfill \end{array}$$

(15)

The prior distribution is

$$\pi \left({\beta }_1\right)=\frac{{\delta }_1^2}{{\delta }_1+1}\left({\beta }_1+1\right)e^{-{\delta }_1{\beta }_1}\propto \left({\beta }_1+1\right)e^{-{\delta }_1{\beta }_1}.$$

(16)

The posterior distribution for a policyholder or a group of policyholders with claim history $x_1,x_2,\dots ,x_N$ is proportional to the product of the likelihood function in Equation (15) and the prior distribution in Equation (16) and can be represented as:

$$\begin{array}{cc}\hfill {\pi }^{*}\left({\beta }_1|x_1,x_2,\dots ,x_N\right)& \propto f(x_1,x_2,\dots ,x_N|{\beta }_1)\pi \left({\beta }_1\right)\hfill \\ & ={\beta }_1^{\tau N}{e}^{-{\beta }_{1}S}\left({\beta }_1+1\right)e^{-{\delta }_1{\beta }_1}\hfill \\ & =e^{-\left(S+{\delta }_1\right){\beta }_1}\left({\beta }_1+1\right){\beta }_1^{\tau N}\hfill \end{array}$$

Consider

$$\underset{0}{\overset{\infty }{{\displaystyle \int }}}{\pi }^{*}\left({\beta }_1|x_1,x_2,\dots ,x_N\right)d{\beta }_1\propto \underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\left(S+{\delta }_1\right){\beta }_1}\left({\beta }_1+1\right){\beta }_1^{\tau N}d{\beta }_1,$$

then

$$\underset{0}{\overset{\infty }{{\displaystyle \int }}}{\pi }^{*}\left({\beta }_1|x_1,x_2,\dots ,x_N\right)d{\beta }_1=\underset{0}{\overset{\infty }{{\displaystyle \int }}}B e^{-\left(S+{\delta }_1\right){\beta }_1}\left({\beta }_1+1\right){\beta }_1^{\tau N}d{\beta }_1=1.$$

where $B$ is a constant.

Let $u=\left(S+{\delta }_1\right){\beta }_1,{\beta }_1=\frac{u}{S+{\delta }_1},du=\left(S+{\delta }_1\right)d{\beta }_1,d{\beta }_1=\frac{du}{S+{\delta }_1},$ then

$$\begin{array}{r}B\left[\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\left(S+{\delta }_1\right){\beta }_1}{\beta }_1^{\tau N+1}d{\beta }_1+\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\left(S+{\delta }_1\right){\beta }_1}{\beta }_1^{\tau N}d{\beta }_1\right]=1\\ B\left[\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-u}\frac{u^{\tau N+1}}{{\left(S+{\delta }_1\right)}^{\tau N+1}}\cdot \frac{du}{S+{\delta }_1}+\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-u}\frac{u^{\tau N}}{{\left(S+{\delta }_1\right)}^{\tau N}}\cdot \frac{du}{S+{\delta }_1}\right]=1\\ B\left[\frac{\mathsf{\Gamma }\left(\tau N+2\right)}{{\left(S+{\delta }_1\right)}^{\tau N+2}}+\frac{\mathsf{\Gamma }\left(\tau N+1\right)}{{\left(S+{\delta }_1\right)}^{\tau N+1}}\right]=1\\ B\cdot \frac{\mathsf{\Gamma }\left(\tau N+1\right)}{{\left(S+{\delta }_1\right)}^{\tau N+1}}\left[\frac{\tau N+1+S+{\delta }_1}{S+{\delta }_1}\right]=1.\end{array}$$

Hence,

$$B=\frac{{\left(S+{\delta }_1\right)}^{\tau N+2}}{\mathsf{\Gamma }\left(\tau N+1\right)\left(\tau N+1+S+{\delta }_1\right)}.$$

Therefore, the posterior distribution function results in the following Equation (17).

$${\pi }^{*}\left({\beta }_1|x_1,x_2,\dots ,x_N\right)=\frac{{\left(S+{\delta }_1\right)}^{\tau N+2}}{\mathsf{\Gamma }\left(\tau N+1\right)\left(\tau N+1+S+{\delta }_1\right)}{e}^{-\left(S+{\delta }_1\right){\beta }_1}\left({\beta }_1+1\right){\beta }_1^{\tau N}.$$

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2.2.3. Premium Calculation

Many principles are involved in pricing insurance premiums. In this article, we determined the net premium principle for both claim severity distribution and claim frequency distribution. The basic principle is that premiums should be the expected value of losses. The expected value of the posterior distribution function in Equation (17) for the gamma-Lindley distribution was

$$\begin{array}{cc}\hfill {\widehat{\beta }}_1& =E\left[{\beta }_1|x_1,x_2,\dots ,x_N\right]\hfill \\ & =\underset{0}{\overset{\infty }{{\displaystyle \int }}}{\beta }_1\frac{{\left(S+{\delta }_1\right)}^{\tau N+2}}{\mathsf{\Gamma }\left(\tau N+1\right)\left(\tau N+1+S+{\delta }_1\right)}{e}^{-\left(S+{\delta }_1\right){\beta }_1}\left({\beta }_1+1\right){\beta }_1{}^{\tau N}d{\beta }_1\hfill \\ & =\frac{\left(\tau N+1\right)\left(\tau N+2+S+{\delta }_1\right)}{\left(S+{\delta }_1\right)\left(\tau N+1+S+{\delta }_1\right)}.\hfill \end{array}$$

From $E\left[{\beta }_1|x_1,x_2,\dots ,x_N\right]={\widehat{\beta }}_1$ then $E\left[x_1,x_2,\dots ,x_N|{\beta }_1\right]=\frac{\tau }{{\widehat{\beta }}_1}.$

Therefore,

$$E\left[x_1,x_2,\dots ,x_N|{\beta }_1\right]=\frac{\tau \left(S+{\delta }_1\right)\left(\tau N+1+S+{\delta }_1\right)}{\left(\tau N+1\right)\left(\tau N+2+S+{\delta }_1\right)}.$$

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The Bayesian bonus-malus premium that must be paid by a specific group of insureds will be equal to the product of the Bayesian premium based on the frequency component in Equation (11) and the severity component in Equation (18). This can be expressed by

$$\mathrm{Premium}=\left[\frac{p_l\left(M+\alpha \right)+p_s\left(N-M+\beta \right)}{N+\alpha +\beta }\cdot \frac{N+1}{t+\delta }\cdot \frac{N+2+t+\delta }{N+1+t+\delta }\right]\times \left[\frac{\tau \left(S+{\delta }_1\right)\left(\tau N+1+S+{\delta }_1\right)}{\left(\tau N+1\right)\left(\tau N+2+S+{\delta }_1\right)}\right].$$

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The model was constructed considering only the frequency component to set the premium for each policyholder and considering the premium in the severity component as a constant. Therefore, this model was not fair when penalizing policyholders. For instance, a policyholder making a claim of AUD 100 should not be penalized by the same amount as other policyholders making claims of AUD 1000. The total amount of claims should be considered as the main factor for setting fair premiums for all policyholders. The model in Equation (19) offers fairer penalties to all policyholders than a model considering only the frequency component.

To determine the bonus-malus premiums that must be paid according to the proposed model, we need to know the number of claims, age of the policy and total claim amounts. All these values are generally shown in the portfolio.

The Bayesian bonus-malus premium at time $t=0$ is called the base premium. This is the initial premium paid by a new policyholder who joins the insurance scheme. The base premium is given by

$${\mathrm{Premium}}_0=\left[\frac{p_l\alpha +p_s\beta }{\alpha +\beta }\cdot \frac{\delta +2}{\delta \left(1+\delta \right)}\right]\times \left[\frac{\tau {\delta }_1\left(1+{\delta }_1\right)}{2+{\delta }_1}\right].$$

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3. Numerical Applications

3.1. Data

The data set used to calculate model premiums introduced in this paper was based on 1-year automobile insurance policies taken out in 2004 or 2005. This data set can be found on the website of the Faculty of Business and Economics, Macquarie University (Sydney, Australia), see also De Jong and Heller (2008). Out of 67,856 policies in the total portfolio, 4624 claims were made. Histograms of the data set and histograms in log scale are shown in Figure 2 and Figure 3, respectively. There were 4333 policyholders who made claims once, 271 twice, 18 three times and 2 four times. The limit value was taken as $\mathsf{\Phi }=$ 500 monetary units, and data were derived from Déniz (2016) to distinguish between the two types of claims. The number of policyholders that made claims once with claim size above the limit value was 2615, 123 twice, 7 three times and 1 four times, as shown in

Table 1. Observed claim frequency for the number of claims ($k$) and the total number of claims with claim size larger than a limit value ($z$) with $\mathsf{\Phi }$ = 500.

$\mathit{k}$	$\mathit{z}$					Total
	0	1	2	3	4
0	63,232					63,232
1	1840	2493				4333
2	37	117	117			271
3	1	5	5	7		18
4	0	0	1	0	1	2
Total	65,110	2615	123	7	1	67,856

.

3.2. Parameter Estimation and Data Fitting

3.2.1. Claim Frequency Distribution

Model parameters can be estimated after the prior distributions. The parameters were estimated from the data using the unconditional distribution of $\left(K,Z\right)$ provided in Equation (5). Let ${\omega }_f=\left(\alpha ,\beta ,\delta \right)$ be the vector of the parameters in Equation (5).

Let $\left(k,z\right)=\left\{\left(k_1,z_1\right),\left(k_2,z_2\right),\dots ,\left(k_n,z_n\right)\right\}$ be a sample with $n$ observations taken from the pdf in Equation (5). A parameter estimation method for the frequency component, called the minimum chi-square estimator (MCSE), was used to minimize the statistical value of the chi-square goodness of fit test (${\chi }^2$) for the frequency distribution. The advantage of this method is that it gives a better fit to the data set. The ${\chi }^2$ formula is defined by

$${\chi }^2={\displaystyle \sum }_{i=1}^n\frac{{\left(O_i-E_i\right)}^2}{E_i},$$

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where ${\chi }^2$ is the value of the chi-square goodness of fit test, $O_i$ is the observed frequency count for the $i^{th}$ level of the categorical variable and $E_i$ is the expected frequency count for the $i^{th}$ level of the categorical variable.

A randomized neighborhood search (RNS) technique was used to minimize the statistical value of ${\chi }^2$. Let ${\omega }_i$ be the vector of the parameters for some iteration $i$ = 0, 1, 2, …. Let ${\vartheta }_i$ be the representative of the statistical value (${\chi }^2$) corresponding to the parameter ${\omega }_i$. The RNS process (see Pongsart et al. (2022) for detail) is explained by the following steps.

Step 1: Set the initial parameter $\omega$ and compute the statistical value $\vartheta$ with the parameter $\omega$.

Step 2: Randomly change the parameter ${\omega }_i$ to ${\omega }_i^{*}$:

$${\omega }_i^{*}={\omega }_i+r_i{\omega }_i,$$

where $r_i\in \left[-0.5,0.5\right].$

Step 3: Compute the statistical value ${\vartheta }^{*}$ with the parameter ${\omega }_i^{*}$.

Step 4: Compare ${\vartheta }^{*}$ and $\vartheta$. If ${\vartheta }^{*}<\vartheta$, then we set ${\omega }_i={\omega }_i^{*}$ and go to Step 5. Else go to Step 2.

Step 5: If $\left|{\vartheta }^{*}-\vartheta \right|\le {10}^{-5}$ then the process is completed. Else go to Step 2.

MATLAB programming was used for parameter estimation.

Claim frequency distribution was applied using the Poisson binomial-Lindley beta distribution introduced in this article. The estimated parameter values of the Poisson binomial-Lindley beta distribution were $\widehat{\alpha }=$ 4.1061, $\widehat{\beta }=$ 2.9352 and $\widehat{\delta }=$ 14.5654 with chi-square goodness of fit test ${\chi }^2$ = 17.4035 and AIC = 42,582.3175. Values of observed claim frequency and expected frequency were compared in

Table 2. Observed frequency and expected frequency for estimated parameter values for the number of claims ($k$) and the total number of claims with claim size larger than a limit value ($z$) with $\mathsf{\Phi }$ = 500.

$\mathit{k}$	$\mathit{z}$	Observed Frequency	Expected Frequency
			Poisson Binomial-Exponential Beta	Poisson Binomial-Lindley Beta
0	0	63,232	63,262.4760	63,234.5099
1	0	1840	1768.2668	1795.7044
1	1	2493	2514.2978	2512.0406
2	0	37	59.1654	59.6707
2	1	117	121.0755	124.5243
2	2	117	109.6681	108.3118
3	0	1	2.2262	2.2053
3	1	5	5.3370	5.5043
3	2	5	6.9573	7.1421
3	3	7	5.1049	4.9526
4	0	0	0.0910	0.0880
4	1	0	0.2387	0.2436
4	2	1	0.3645	0.3780
4	3	0	0.3849	0.3910
4	4	1	0.2493	0.2367
Total		67,856	67,856	67,856
Estimated parameter values			$\widehat{\alpha }=$ 3.6490 $\widehat{\beta }=$ 2.5663 $\widehat{\lambda }=$ 13.7721	$\widehat{\alpha }=$ 4.1061 $\widehat{\beta }=$ 2.9352 $\widehat{\delta }=$ 14.5654
${\chi }^2$			18.0696	17.4035
AIC			42,586.2414	42,582.3175

between the traditional Poisson binomial-exponential beta distribution (Hernawati et al. 2017) and Poisson binomial-Lindley beta distribution. Results showed that our proposed Poisson binomial-Lindley beta distribution gave a better fit to the data than the traditional Poisson binomial-exponential beta distribution.

3.2.2. Claim Severity Distribution

The maximum likelihood estimation (MLE) is widely used for estimating the model parameters for claim severity distribution. The unconditional distribution of $X$ is provided in Equation (14).

Let $X={\left(x_1,x_2,\dots ,x_n\right)}^T$ be an identically independent observation for the gamma-Lindley distribution with pdf in Equation (14). To find the most likely value of the parameters ${\delta }_1$ and $\tau$, we require maximizing the likelihood function $L$ where

$$L\left({\delta }_1,\tau ;x_i\right)={\displaystyle \prod }_{i=1}^{n}f\left(x_i;{\delta }_1,\tau \right)={\displaystyle \prod }_{i=1}^n\left[\frac{\tau {\delta }_1^2{x}_i^{\tau -1}}{{\delta }_1+1}\cdot \frac{x_i+\tau +{\delta }_1+1}{{\left(x_i+{\delta }_1\right)}^{\tau +2}}\right].$$

Then, the log-likelihood function is

$$\begin{array}{cc}\hfill \ln L\left({\delta }_1,\tau ;x_i\right)=& n\ln \tau +2n\ln {\delta }_1+\left(\tau -1\right){\displaystyle \sum _{i=1}^n}\ln x_i+{\displaystyle \sum _{i=1}^n}\ln \left(x_i+\tau +{\delta }_1+1\right)-n\ln \left({\delta }_1+1\right)\hfill \\ & -\left(\tau +2\right){\displaystyle \sum _{i=1}^n}\ln \left(x_i+{\delta }_1\right).\hfill \end{array}$$

The estimators ${\widehat{\delta }}_1$ and $\widehat{\tau }$ of the parameters ${\delta }_1$ and $\tau$ respectively can be obtained by solving the equation:

$$\frac{d}{d{\delta }_1}\ln L\left({\delta }_1,\tau ;x_i\right)=0\mathrm{and}\frac{d}{d\tau }\ln L\left({\delta }_1,\tau ;x_i\right)=0.$$

where

$$\frac{2n}{{\delta }_1}+{\displaystyle \sum }_{i=1}^n\left(\frac{1}{x_i+\tau +{\delta }_1+1}\right)-\frac{n}{{\delta }_1+1}-\left(\tau +2\right){\displaystyle \sum }_{i=1}^n\frac{1}{x_i+{\delta }_1}=0$$

and

$$\frac{n}{\tau }+{\displaystyle \sum }_{i=1}^n\ln x_i+{\displaystyle \sum }_{i=1}^n\left(\frac{1}{x_i+\tau +{\delta }_1+1}\right)-{\displaystyle \sum }_{i=1}^n\ln \left(x_i+{\delta }_1\right)=0.$$

The parameters ${\delta }_1$ and $\tau$ cannot be estimated in closed form and the numerical iteration technique is used to solve the equations above.

To evaluate whether a given distribution is suited to a data set, the Kolmogorov-Smirnov test (K-S test) represents the goodness of fit test for the claim severity distribution. The K-S test statistic is defined by

$$D=\max \left|F^n\left(x\right)-F\left(x\right)\right|,$$

where $F\left(x\right)$ is the theoretical cumulative distribution of claim severity distributions and $F^n\left(x\right)$ is denoted by

$$F^n\left(x\right)=\frac{1}{n}[\mathrm{number}\mathrm{of}\mathrm{observations} \le x],$$

where $n$ is the sample size.

The results gave the maximum likelihood estimators ${\widehat{\delta }}_1=$ 1501.5620 and $\widehat{\tau }=$ 0.8012, with D-value = 0.0339 and AIC = 82,083.9448. Figure 4 shows the P-P plot of the Gamma-Lindley distribution.

3.3. Bonus-Malus Premiums

3.3.1. Claim Frequency Component

The Bayesian bonus-malus premiums based only on the frequency component were determined and calculated from Equation (12), with results shown in

Table 3. Bonus-malus premiums based on the Poisson binomial-Lindley beta distribution for the frequency component where $N$ represents the total number of claims and $M$ represents those that the claim size is larger than $\mathsf{\Phi }=$ 500.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	100.00	93.23	87.32	82.11	77.49	73.35	69.64	66.28
$\mathit{N}=$ 1	$M=$ 0		182.92	171.38	161.20	152.16	144.07	136.80	130.22
	$M=$ 1		187.97	176.11	165.65	156.35	148.04	140.57	133.81
$\mathit{N}=$ 2	$M=$ 0		270.16	253.19	238.21	224.90	212.98	202.26	192.56
	$M=$ 1		276.87	259.48	244.13	230.48	218.27	207.28	197.35
	$M=$ 2		283.58	265.77	250.04	236.07	223.56	212.31	202.13
$\mathit{N}=$ 3	$M=$ 0		355.63	333.37	313.72	296.24	280.59	266.50	253.76
	$M=$ 1		363.66	340.90	320.80	302.93	286.93	272.52	259.49
	$M=$ 2		371.70	348.43	327.89	309.62	293.26	278.54	265.22
	$M=$ 3		379.73	355.96	334.97	316.31	299.60	284.56	270.95
$\mathit{N}=$ 4	$M=$ 0		439.76	412.32	388.09	366.52	347.21	329.82	314.08
	$M=$ 1		448.87	420.87	396.13	374.12	354.41	336.66	320.59
	$M=$ 2		457.98	429.41	404.17	381.71	361.60	343.49	327.10
	$M=$ 3		467.09	437.95	412.21	389.30	368.79	350.32	333.60
	$M=$ 4		476.20	446.49	420.25	396.89	375.98	357.16	340.11

. For comparison, a limit value $\mathsf{\Phi }$ was assumed to be 500 monetary units, with weights $p_l=$ 0.8 and $p_s=$ 1.

From the results in Table 3, a bonus with no claims in the first year represented 6.77% of the base premium. Policyholders who made one claim with a claim size below and above the limit value during the first year must pay a malus at 82.92% and 87.97%, respectively of the base premium. Premiums decrease if policyholders have a claim-free year and increase if claims occur. The Bayesian bonus-malus premiums achieved under the traditional Poisson-Lindley model were compared (see Moumeesri et al. 2020 for details). The bonus-malus premiums were the same as the bonus ($N=$ 0) and different for the remainder of the malus ($N\ge 1$). For these malus classes, policyholders with a claim size below the limit value are rewarded compared to those with premiums determined under the traditional Poisson-Lindley model. The remaining premiums may be larger or smaller than those obtained in this model. A comparison of premiums between the traditional Poisson-Lindley model and the proposed Poisson binomial-Lindley beta model in the first year for various numbers of claims is shown in Figure 5.

For comparative purposes, the Bayesian bonus-malus premiums achieved under the Poisson binomial-exponential beta model were computed (see Hernawati et al. 2017 for details), with the results shown in

Table 4. Bonus-malus premiums based on the Poisson binomial-exponential beta distribution for the frequency component where $N$ represents the total number of claims and $M$ represents those that the claim size is larger than $\mathsf{\Phi }=$ 500.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	100.00	93.23	87.32	82.11	77.49	73.36	69.65	66.30
$\mathit{N}=$ 1	$M=$ 0		183.15	171.54	161.31	152.24	144.13	136.84	130.25
	$M=$ 1		188.79	176.82	166.28	156.92	148.56	141.05	134.26
$\mathit{N}=$ 2	$M=$ 0		270.98	253.80	238.66	225.23	213.24	202.45	192.71
	$M=$ 1		278.40	260.75	245.20	231.40	219.08	208.00	197.98
	$M=$ 2		285.82	267.70	251.74	237.57	224.92	213.54	203.26
$\mathit{N}=$ 3	$M=$ 0		357.38	334.72	314.77	297.06	281.23	267.01	254.15
	$M=$ 1		366.21	342.99	322.54	304.39	288.17	273.60	260.43
	$M=$ 2		375.03	351.25	330.31	311.72	295.12	280.19	266.70
	$M=$ 3		383.85	359.51	338.08	319.05	302.06	286.78	272.98
$\mathit{N}=$ 4	$M=$ 0		442.79	414.72	389.99	368.05	348.44	330.82	314.89
	$M=$ 1		452.74	424.03	398.75	376.31	356.27	338.25	321.97
	$M=$ 2		462.69	433.35	407.51	384.58	364.10	345.68	329.04
	$M=$ 3		472.63	442.67	416.27	392.85	371.92	353.11	336.11
	$M=$ 4		482.58	451.99	425.04	401.12	379.75	360.55	343.19

.

In Table 4, a policyholder with no claims from the first year to the seventh year is awarded by bonus classes that equal the premiums determined under the Poisson binomial-Lindley beta model. An insured who made one claim with a claim size below and above the limit value of during the first year must pay a malus at 83.15% and 88.79%, respectively of the base premium. The results in Table 4 show that malus values based on the Poisson binomial-Lindley beta model both below and above the limit value are less than those based on the Poisson binomial-exponential beta model.

The Bayesian bonus-malus premiums computed from the traditional Poisson binomial-exponential beta model were stricter for high-risk policyholders than the proposed Poisson binomial-Lindley beta model in the case of claim sizes both below and above the limit value. Premiums obtained from the traditional model were similar to the proposed model for low-risk policyholders.

3.3.2. Claim Frequency and Claim Severity Components

Bonus-malus premiums using the proposed model are shown in Equation (19). To find the bonus-malus premiums that must be paid, we need to know the number of claims made in these years, the number of claims corresponding to the limit value, the age of the policy and the total claim amount from a portfolio.

In some cases, the total claim amount corresponding to the limit value was equal to 400, 1500 and 2500. To better understand how the BMS operates,

Table 5. Bonus-malus premiums based on Poisson binomial-Lindley beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 400.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.52	75.07	70.31	66.12	62.39	59.06	56.07	53.37
$\mathit{N}=$ 1	$M=$ 0		103.57	97.04	91.27	86.15	81.57	77.46	73.73
	$M=$ 1		106.43	99.71	93.79	88.53	83.82	79.59	75.76
$\mathit{N}=$ 2	$M=$ 0		105.87	99.22	93.35	88.13	83.47	79.26	75.46
	$M=$ 1		108.50	101.69	95.67	90.32	85.54	81.23	77.34
	$M=$ 2		111.13	104.15	97.99	92.51	87.61	83.20	79.21
$\mathit{N}=$ 3	$M=$ 0		106.56	99.89	94.00	88.76	84.08	79.85	76.04
	$M=$ 1		108.97	102.15	96.12	90.77	85.97	81.66	77.75
	$M=$ 2		111.37	104.40	98.25	92.77	87.87	83.46	79.47
	$M=$ 3		113.78	106.66	100.37	94.78	89.77	85.27	81.19
$\mathit{N}=$ 4	$M=$ 0		106.66	100.01	94.13	88.90	84.21	80.00	76.18
	$M=$ 1		108.87	102.08	96.08	90.74	85.96	81.65	77.76
	$M=$ 2		111.08	104.15	98.03	92.58	87.70	83.31	79.34
	$M=$ 3		113.29	106.22	99.98	94.42	89.45	84.97	80.91
	$M=$ 4		115.50	108.29	101.93	96.26	91.19	86.63	82.49

shows the premiums that must be paid by the policyholder for various numbers of claims when the age of the policy is up to 7 years. The base premium, an initial premium paid by all new policyholders, is 80.52, and this decreases with a claim-free year. Conversely, after an accident with a claim size of 400 in the first year, the policyholder will pay 103.57.

If an accident occurs during the second year with a claim size of 1100, then a surcharge will be enforced and the insured will have to pay 160.54 as the premium for two accidents, with a claim amount that exceeds the limit value one time ($M$ = 1) for a total claim size of 1500 in 2 years, as shown in

Table 6. Bonus-malus premiums based on Poisson binomial-Lindley beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 1500.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.52	75.07	70.31	66.12	62.39	59.06	56.07	53.37
$\mathit{N}=$ 1	$M=$ 0		163.52	153.20	144.10	136.02	128.79	122.29	116.40
	$M=$ 1		168.03	157.42	148.07	139.77	132.34	125.66	119.61
$\mathit{N}=$ 2	$M=$ 0		167.15	156.65	147.38	139.14	131.77	125.14	119.14
	$M=$ 1		171.30	160.54	151.04	142.60	135.05	128.25	122.10
	$M=$ 2		175.45	164.43	154.70	146.06	138.32	131.36	125.06
$\mathit{N}=$ 3	$M=$ 0		168.23	157.71	148.41	140.14	132.74	126.07	120.04
	$M=$ 1		172.03	161.27	151.76	143.30	135.73	128.92	122.75
	$M=$ 2		175.83	164.83	155.11	146.47	138.73	131.77	125.47
	$M=$ 3		179.64	168.39	158.46	149.63	141.73	134.62	128.18
$\mathit{N}=$ 4	$M=$ 0		168.39	157.89	148.61	140.35	132.96	126.30	120.27
	$M=$ 1		171.88	161.16	151.69	143.26	135.71	128.91	122.76
	$M=$ 2		175.37	164.43	154.76	146.16	138.46	131.53	125.25
	$M=$ 3		178.86	167.70	157.84	149.07	141.22	134.15	127.74
	$M=$ 4		182.35	170.97	160.92	151.98	143.97	136.76	130.24

.

If no accident occurs in the third year, then the premium will reduce with a claim-free year and the insured must pay 151.04, which is the premium for two accidents of total claim size 1500 in 3 years (see Table 6).

If two accidents occur during the fourth year with claim sizes 900 and 100, then a surcharge will be enforced of 194.88, which is the premium for four accidents with a claim amount that exceeds the limit value two times ($M$ = 2) for total claim size 2500 in 4 years, as shown in

Table 7. Bonus-malus premiums based on Poisson binomial-Lindley beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 2500.

Number of Claims		$t$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.52	75.07	70.31	66.12	62.39	59.06	56.07	53.37
$\mathit{N}=$ 1	$M=$ 0		218.01	204.26	192.13	181.35	171.71	163.04	155.20
	$M=$ 1		224.02	209.89	197.42	186.35	176.44	167.53	159.48
$\mathit{N}=$ 2	$M=$ 0		222.86	208.86	196.50	185.52	175.69	166.85	158.85
	$M=$ 1		228.39	214.04	201.38	190.12	180.05	170.99	162.79
	$M=$ 2		233.93	219.23	206.26	194.73	184.42	175.13	166.73
$\mathit{N}=$ 3	$M=$ 0		224.30	210.26	197.87	186.84	176.97	168.09	160.05
	$M=$ 1		229.37	215.01	202.34	191.06	180.97	171.89	163.66
	$M=$ 2		234.44	219.76	206.81	195.28	184.97	175.68	167.28
	$M=$ 3		239.50	224.51	211.28	199.50	188.97	179.48	170.89
$\mathit{N}=$ 4	$M=$ 0		224.52	210.51	198.13	187.12	177.27	168.39	160.35
	$M=$ 1		229.17	214.87	202.24	191.00	180.94	171.88	163.67
	$M=$ 2		233.82	219.23	206.34	194.88	184.61	175.37	167.00
	$M=$ 3		238.47	223.59	210.45	198.75	188.28	178.85	170.32
	$M=$ 4		243.12	227.95	214.55	202.63	191.96	182.34	173.64

.

The Bayesian bonus-malus premiums were compared under the traditional Poisson binomial-exponential beta distribution for the frequency component and the gamma-Lindley distribution for the severity component. The results are shown in

Table 8. Bonus-malus premiums based on Poisson binomial-exponential beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 400.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.09	74.67	69.93	65.76	62.06	58.76	55.78	53.10
$\mathit{N}=$ 1	$M=$ 0		103.14	96.60	90.84	85.73	81.17	77.06	73.35
	$M=$ 1		106.32	99.58	93.64	88.37	83.66	79.43	75.61
$\mathit{N}=$ 2	$M=$ 0		105.62	98.92	93.03	87.79	83.11	78.91	75.11
	$M=$ 1		108.51	101.63	95.57	90.20	85.39	81.07	77.17
	$M=$ 2		111.41	104.34	98.12	92.60	87.67	83.23	79.23
$\mathit{N}=$ 3	$M=$ 0		106.51	99.76	93.81	88.53	83.81	79.58	75.74
	$M=$ 1		109.14	102.22	96.12	90.72	85.88	81.54	77.61
	$M=$ 2		111.77	104.68	98.44	92.90	87.95	83.50	79.48
	$M=$ 3		114.40	107.14	100.76	95.09	90.02	85.47	81.35
$\mathit{N}=$ 4	$M=$ 0		106.82	100.05	94.08	88.79	84.06	79.81	75.96
	$M=$ 1		109.22	102.29	96.19	90.78	85.95	81.60	77.67
	$M=$ 2		111.62	104.54	98.31	92.78	87.83	83.39	79.38
	$M=$ 3		114.02	106.79	100.42	94.77	89.72	85.18	81.08
	$M=$ 4		116.42	109.04	102.54	96.77	91.61	86.98	82.79

,

Table 9. Bonus-malus premiums based on Poisson binomial-exponential beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 1500.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.09	74.67	69.93	65.76	62.06	58.76	55.78	53.10
$\mathit{N}=$ 1	$M=$ 0		162.84	152.52	143.42	135.35	128.14	121.66	115.80
	$M=$ 1		167.85	157.21	147.83	139.52	132.08	125.40	119.37
$\mathit{N}=$ 2	$M=$ 0		166.75	156.18	146.87	138.60	131.22	124.58	118.59
	$M=$ 1		171.32	160.46	150.89	142.40	134.81	128.00	121.83
	$M=$ 2		175.89	164.73	154.91	146.20	138.41	131.41	125.08
$\mathit{N}=$ 3	$M=$ 0		168.15	157.49	148.10	139.77	132.32	125.63	119.58
	$M=$ 1		172.31	161.38	151.76	143.22	135.59	128.73	122.54
	$M=$ 2		176.46	165.27	155.41	146.67	138.86	131.83	125.49
	$M=$ 3		180.61	169.16	159.07	150.12	142.12	134.93	128.44
$\mathit{N}=$ 4	$M=$ 0		168.64	157.95	148.53	140.17	132.71	126.00	119.93
	$M=$ 1		172.43	161.50	151.87	143.32	135.69	128.83	122.62
	$M=$ 2		176.22	165.05	155.21	146.47	138.67	131.66	125.32
	$M=$ 3		180.01	168.59	158.54	149.62	141.65	134.49	128.01
	$M=$ 4		183.80	172.14	161.88	152.77	144.63	137.32	130.71

and

Table 10. Bonus-malus premiums based on Poisson binomial-exponential beta distribution for the frequency component and gamma-Lindley distribution for the severity component with a total claim size of $S=$ 2500.

Number of Claims		$\mathit{t}$
		0	1	2	3	4	5	6	7
$\mathit{N}=$ 0	$M=$ 0	80.09	74.67	69.93	65.76	62.06	58.76	55.78	53.10
$\mathit{N}=$ 1	$M=$ 0		217.11	203.35	191.22	180.46	170.85	162.21	154.40
	$M=$ 1		223.79	209.60	197.10	186.01	176.10	167.20	159.15
$\mathit{N}=$ 2	$M=$ 0		222.32	208.23	195.81	184.80	174.95	166.10	158.11
	$M=$ 1		228.41	213.93	201.18	189.86	179.74	170.65	162.44
	$M=$ 2		234.50	219.64	206.54	194.92	184.53	175.20	166.77
$\mathit{N}=$ 3	$M=$ 0		224.20	209.98	197.46	186.35	176.42	167.50	159.44
	$M=$ 1		229.73	215.16	202.34	190.95	180.78	171.64	163.37
	$M=$ 2		235.26	220.35	207.21	195.55	185.13	175.77	167.31
	$M=$ 3		240.80	225.53	212.08	200.15	189.49	179.90	171.24
$\mathit{N}=$ 4	$M=$ 0		224.84	210.59	198.03	186.89	176.93	167.99	159.90
	$M=$ 1		229.90	215.32	202.48	191.09	180.91	171.76	163.49
	$M=$ 2		234.95	220.05	206.93	195.29	184.88	175.53	167.08
	$M=$ 3		240.00	224.78	211.38	199.49	188.86	179.31	170.68
	$M=$ 4		245.05	229.51	215.83	203.69	192.83	183.08	174.27

.

The base premiums and premiums with a claim-free year for low-risk policyholders are slightly different between the bonus-malus premiums proposed in Table 5, Table 6 and Table 7 and the traditional bonus-malus premiums in Table 8, Table 9 and Table 10. Moreover, the traditional bonus-malus premiums in Table 8, Table 9 and Table 10 are stricter for high-risk policyholders than premiums presented in Table 5, Table 6 and Table 7 when the total number of claims with claim size over the limit value is high. However, the traditional bonus-malus premiums in Table 8, Table 9 and Table 10 are more generous for high-risk policyholders than premiums presented in Table 5, Table 6 and Table 7 when the total number of claims with the claim size over the limit value is small.

Table 11. Comparison of premiums for various number of claims and claim sizes in the first year.

Claim Size	Number of Claims
	N = 1		N = 2			N = 3				N = 4
	M = 0	M = 1	M = 0	M = 1	M = 2	M = 0	M = 1	M = 2	M = 3	M = 0	M = 1	M = 2	M = 3	M = 4
100	87	90	89	91	94	90	92	94	96	90	92	94	95	97
1000	136	140	139	143	146	140	143	147	150	140	143	146	149	152
5000	354	364	362	371	380	364	373	381	389	365	372	380	387	395
10,000	627	644	641	657	672	645	659	674	689	645	659	672	686	699
20,000	1172	1204	1198	1227	1257	1205	1233	1260	1287	1207	1232	1257	1282	1307
30,000	1717	1764	1755	1798	1842	1766	1806	1846	1886	1768	1804	1841	1878	1914

shows the premiums (malus) with Poisson binomial-Lindley beta for claim frequency distribution and gamma-Lindley for claim severity distribution that must be paid by policyholders during the first year of the policy, with the number of accidents ranging from 1 to 4, number of claims corresponding to the limit value and the aggregate claim amount of accidents ranging from AUD 100 to AUD 30,000. The results shown in Table 11 are depicted in Figure 6. Premiums were dependent on aggregate claim amount, number of claims and number of claims with claim size corresponding to the limit value.

4. Conclusions

A model was proposed to determine optimal bonus-malus premiums that considered both claim frequency and claim severity components by distinguishing two types of claims based on claim amounts beneath and above a threshold value. Two mixed distributions, the Poisson binomial with the Lindley beta distribution and a mixed gamma with Lindley distribution, were investigated and employed in the model as frequency distribution and severity distribution, respectively. The Bayesian method was applied in the model as a frequency component and a severity component. Both models were then integrated. The new model rewarded and penalized all policyholders in a fairer manner.

An example of real automobile insurance data was used to illustrate our model. Fitted results of claim frequency were better than the traditional Poisson binomial-exponential beta model. When claims occurred, high-risk policyholders were liable to more severe penalties under the traditional model. Our proposed model presents a reasonable alternative method for rewarding and penalizing both low- and high-risk policyholders. The obtained premiums could be used by insurers to gain market share in the highly competitive insurance industry.

To increase impartiality of assigning premium to all policyholders, further different types of claims could be distinguished as a topic for further research.