# Multiple Bonus–Malus Scale Models for Insureds of Different Sizes

## Abstract

**:**

## 1. Introduction

- Models allowing a dynamic time dependence between contracts of the same insured: models based on series of correlated random effects (Bolancé et al. 2007; Abdallah et al. 2016), jitter models (Shi and Valdez 2014), time series for count data (Gourieroux and Jasiak 2004; Bermúdez et al. 2018; Bermúdez and Karlis 2021 or Pinquet 2020).
- Longitudinal models allowing for dependence structures between many type of claims (Abdallah et al. 2016; Pechon et al. 2019, 2021; Gómez-Déniz and Calderín-Ojeda 2018).
- Longitudinal models allowing dependence between claim frequency and claim severity (Shi et al. 2016; Oh et al. 2020; Jeong and Valdez 2020).

## 2. Review of the BMS Model

#### 2.1. Summary of the Past-Claims-Rating Model

- The variable to model, named the
**target**variable; - The information used to define what we consider the past claim experience, named the
**scope**variable.

#### 2.2. The Bonus–Malus Scale Models

**Relativity**parameter, and $\mathsf{\Psi}=\frac{{\gamma}_{1}}{{\gamma}_{0}}$ is the

**Jump**parameter. The new variable ${\ell}_{i,T}$, based on ${\kappa}_{i,\u2022}$ and ${n}_{i,\u2022}$, summarizes all past claim experience and is called a

**claim score**. This approach is called the Kappa-N model (Boucher 2022), and because it is defined by the mean parameter, it can easily be used with many count distributions. The Kappa-N model can be interpreted as follows:

- For an insured i without insurance experience, we have ${n}_{i,\u2022}=0$ and ${\kappa}_{i,\u2022}=0$, which means that a new insured without experience has a claim score of 100, i.e., an
**entry level**of 100. - Each year without a claim decreases the claim score by 1;
- Each claim increases the claim score by $\mathsf{\Psi}$.
- The impact of a single claim on the premium is equal to $\mathsf{\Psi}$ years without claims.
- The penalty for a claim is $(exp(\mathsf{\Psi}{\gamma}_{0})-1)$%.
- Each year without a claim decreases the premium by $(1-exp(-{\gamma}_{0}))$%.

#### Numerical Example

**Bonus–Malus Scale**system (BMS), where the claim score can be seen as a

**BMS level**, or simply a BMS score. More formally, the BMS is a class system with a finite number of levels (when $\mathsf{\Psi}$ is an integer), where a relativity is assigned to each level. For this BMS model, the transition rule is simple: an insured moves down by one level if there is no claim over the course of the contract and moves up by $\mathsf{\Psi}$ levels for each claim.

#### 2.3. Summary of the Numerical Illustration

#### 2.3.1. Data Used

#### 2.3.2. Estimated Parameters of the BMS Model

- The jump parameter $\mathsf{\Psi}$ is equal to 6, meaning that each claim increases the BMS level by 6. After a claim, an insured would need 6 years without a claim to return to the original premium (before the claim);
- The value of ${\gamma}_{0}$ is $0.0312$. That means that the penalty for a claim is equal to $exp(0.0312\times 6)-1=20.6\%$, and each year without claim decreases the premium by $1-exp(-0.0312)=3.07\%$;
- The maximum BMS level is ${\ell}_{max}=116$, meaning that the maximum surcharge, compared with level 100, is $exp(0.0312\times 16)-1=64.7\%$;
- The minimum BMS level is ${\ell}_{min}=85$, meaning that the minimum surcharge, compared with level 100, is $1-exp(-0.0312\times 15)=37.3\%$.

#### 2.3.3. Problems with the Size of Farms

## 3. Partitioning the Portfolio

- The first iteration divides the insureds into two groups: insured with four or fewer than four insured items, which represents 62.3% of all insureds, and insureds with more than four items, which represents 37.7% of the insureds.
- The first group of insureds is then divided again into two groups: farms with one insured item (26.9%) and farms with between two and four insured items (35.4%). The second group is also divided into two more groups: between 5 and 12 items (31.0%) and insureds with more than 12 items (6.7%).
- Finally, only the group with between 5 and 12 items can be divided again: groups with between 5 and 8, and between 9 and 12 items, are created.

#### 3.1. Analyzing Each Group of Farms

#### 3.2. Different BMS Models versus the Original BMS Model

- Group 3A, which is composed of farms with only one insured item, seems very different than the other farms. Perhaps these small insureds are family farms, while the farms in the other groups are more industrial.
- Group 3A generates extreme surcharge values for insureds who claim too much: they are more than 53 times the basic premium for insureds at the top of the Bonus–Malus Scale. Even if this BMS model seems to be better than the original BMS model for group 3A (when looking at the logarithmic score), this is not realistic and cannot be applied in practice. On the other hand, groups 3B, 3C and 3D have similar maximum surcharges ranging from 0.780 to 0.920, which can also be seen as an indication that small farms with one insured item seem very different than the other farms. The maximum surcharge for larger farms is limited to an increase of 35.2% compared with the basic level (100).
- The maximum discounts are similar for each group, except group 3A. Compared with a new insured at level 100, insureds from group 3A can expect a maximum discount of only around 7%, but other insureds can obtain discounts ranging from 31.1% to 47.1%.
- The surcharge for each claim is very different from one group to another. We see that for group 3A and 3B the impact of a claim is similar, while the impacts are similar for groups 3C, 3D and 3E.

#### 3.3. Limits of the Approach

#### A Study of Transition Rules

## 4. Conclusions

## Funding

## Conflicts of Interest

## Note

1 | Farms are sometimes passed from generation to generation. Insurance experience would not be reset in such a case. |

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**Figure 3.**Average number of insured items by BMS level (

**left**) and average BMS level by number of insured items (

**right**).

**Figure 7.**Distribution of the BMS relativities for each group (

**left**: original BMS,

**right**: BMS by group).

Insured | Years | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

i | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | $-{\mathit{\kappa}}_{\mathit{i},\u2022}$ | ${\mathit{n}}_{\mathit{i},\u2022}$ |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −10 | 0 |

2 | 2 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | −6 | 6 |

3 | 4 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −7 | 7 |

Policy Number | Number of Items | Effective Date | First Insurance | Coverage | … | Province | Number of Claims | Costs of Claims |
---|---|---|---|---|---|---|---|---|

… | … | … | … | … | ⋯ | … | … | … |

125721 | 2 | 15 January 2017 | 15 January 1995 | MACHINERY | ⋯ | Ontario | 2 | 186,592 |

125722 | 15 | 22 March 2017 | 22 March 2013 | MACHINERY | ⋯ | Quebec | 0 | 0 |

125723 | 1 | 11 January 2016 | 5 November 1993 | MACHINERY | ⋯ | Manitoba | 1 | 18,889 |

125724 | 27 | 17 February 2018 | 17 February 2018 | MACHINERY | ⋯ | Nova Scotia | 1 | 7444 |

… | … | … | … | … | ⋯ | … | … | … |

BMS Parameters | Log-Likelihood | Log. Score | ||||
---|---|---|---|---|---|---|

Distributions | ${\mathit{\ell}}_{\mathit{max}}$ | ${\mathit{\ell}}_{\mathit{min}}$ | $\widehat{\mathsf{\Psi}}$ | ${\widehat{\mathit{\gamma}}}_{0}$ | (Train) | (Test) |

Poisson | 116 | 85 | 6 | 0.0312 | −8490.026 | 2857.029 |

Division | Number of | Proportion | Average | Claims | Past Claims History | |||
---|---|---|---|---|---|---|---|---|

Depth | Group | Items | of Contracts | Nb. of Items | Frequency | ${\overline{\mathit{n}}}_{\u2022}$ | ${\overline{\mathit{\kappa}}}_{\u2022}$ | $\overline{\mathit{\tau}}$ |

0 | 0A | $[1,212]$ | 100.0% | 5.85 | 2.36% | 0.83 | 10.30 | 11.08 |

1 | 1A | $[1,4]$ | 62.3% | 2.00 | 0.71% | 0.58 | 9.95 | 10.50 |

1B | $[5,212]$ | 37.7% | 12.22 | 5.08% | 1.24 | 10.88 | 12.03 | |

2 | 2A | $\left\{1\right\}$ | 26.9% | 1.00 | 0.36% | 0.49 | 9.53 | 10.00 |

2B | $[2,4]$ | 35.4% | 2.75 | 0.98% | 0.64 | 10.26 | 10.87 | |

2C | $[5,12]$ | 25.5% | 7.57 | 3.25% | 1.04 | 10.99 | 11.96 | |

2D | $[13,212]$ | 12.2% | 21.94 | 8.91% | 1.68 | 10.64 | 12.17 | |

3 | 3A | $\left\{1\right\}$ | 26.9% | 1.00 | 0.36% | 0.49 | 9.53 | 10.00 |

3B | $[2,4]$ | 35.4% | 2.75 | 0.98% | 0.64 | 10.26 | 10.87 | |

3C | $[5,8]$ | 17.1% | 6.24 | 2.63% | 0.94 | 10.94 | 11.83 | |

3D | $[9,12]$ | 8.4% | 10.29 | 4.52% | 1.23 | 11.09 | 12.24 | |

3E | $[13,212]$ | 12.2% | 21.94 | 8.91% | 1.68 | 10.64 | 12.17 |

Original BMS | BMS by Group | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Group | ${\widehat{\mathit{\ell}}}_{\mathit{max}}$ | ${\widehat{\mathit{\ell}}}_{\mathit{min}}$ | $\widehat{\mathsf{\Psi}}$ | ${\widehat{\mathit{\gamma}}}_{0}$ | Loglik. | Log.Score | ${\widehat{\mathit{\ell}}}_{\mathit{max}}$ | ${\widehat{\mathit{\ell}}}_{\mathit{min}}$ | $\widehat{\mathsf{\Psi}}$ | ${\widehat{\mathit{\gamma}}}_{0}$ | Loglik. | Log.Score |

3A | 116 | 85 | 6 | 0.0312 | −554.37 | 114.57 | 156 | 99 | 7 | 0.0713 | −539.06 | 107.70 |

3B | 116 | 85 | 6 | 0.0312 | −1655.34 | 550.50 | 114 | 85 | 11 | 0.0425 | −1633.46 | 553.19 |

3C | 116 | 85 | 6 | 0.0312 | −1778.24 | 563.72 | 118 | 85 | 6 | 0.0320 | −1768.57 | 558.16 |

3D | 116 | 85 | 6 | 0.0312 | −1336.60 | 459.63 | 120 | 88 | 6 | 0.0311 | −1320.47 | 457.57 |

3E | 116 | 85 | 6 | 0.0312 | −3165.47 | 1168.61 | 107 | 85 | 4 | 0.0431 | −3148.91 | 1166.47 |

Total | 116 | 85 | 6 | 0.0312 | −8490.03 | 2857.03 | . | . | . | −8410.47 | 2843.08 |

Group | Discount by Year without Claim | Surcharge by Claim | Claim Impact | Maximum Surcharge | Maximum Discount |
---|---|---|---|---|---|

0A | 3.08% | 20.6% | 24.5% | 65.0% | 37.5% |

3A | 6.88% | 64.7% | 76.9% | 5321% | 6.90% |

3B | 4.16% | 59.6% | 66.5% | 81.3% | 47.1% |

3C | 3.15% | 17.4% | 21.2% | 78.0% | 38.2% |

3D | 3.06% | 20.5% | 24.3% | 92.0% | 31.1% |

3E | 4.22% | 18.8% | 24.1% | 35.2% | 47.6% |

${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | ${\widehat{\mathit{\beta}}}_{3}$ | ${\widehat{\mathit{\beta}}}_{4}$ | ${\widehat{\mathit{\beta}}}_{5}$ | ${\widehat{\mathit{\beta}}}_{6}$ | |
---|---|---|---|---|---|---|

0A | 0.931 | −0.173 | −0.176 | 0.091 | 0.052 | 0.008 |

3A | 0.976 | −0.953 | −0.719 | 0.141 | 0.157 | −0.133 |

3B | 0.888 | −0.799 | 0.170 | 0.009 | 0.247 | 0.350 |

3C | 0.892 | −0.864 | −0.268 | 0.026 | 0.014 | −0.031 |

3D | 0.828 | −1.099 | −0.045 | −0.010 | 0.159 | −0.012 |

3E | 0.877 | −0.445 | −0.252 | 0.167 | −0.022 | −0.045 |

Nb. of Items | Actual Group | New Group If an Item Is Added | Nb. of Items | Actual Group | New Group If an Item Is Removed |
---|---|---|---|---|---|

1 | 3A | 3B | 2 | 3B | 3A |

4 | 3B | 3C | 5 | 3C | 3B |

8 | 3C | 3D | 9 | 3D | 3C |

12 | 3D | 3E | 13 | 3E | 3D |

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**MDPI and ACS Style**

Boucher, J.-P.
Multiple Bonus–Malus Scale Models for Insureds of Different Sizes. *Risks* **2022**, *10*, 152.
https://doi.org/10.3390/risks10080152

**AMA Style**

Boucher J-P.
Multiple Bonus–Malus Scale Models for Insureds of Different Sizes. *Risks*. 2022; 10(8):152.
https://doi.org/10.3390/risks10080152

**Chicago/Turabian Style**

Boucher, Jean-Philippe.
2022. "Multiple Bonus–Malus Scale Models for Insureds of Different Sizes" *Risks* 10, no. 8: 152.
https://doi.org/10.3390/risks10080152