# Multivariate Credibility in Bonus-Malus Systems Distinguishing between Different Types of Claims

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## Abstract

**:**

## 1. Introduction

## 2. Basic Model

#### Estimation

## 3. Contemplating Heterogeneity

#### The Premiums

- When $t\to 0$, $Z\left(0\right)\to 1$, $\gamma (x,{z}_{1},{z}_{2},0)\to \frac{\alpha}{\beta}$ and therefore ${P}^{*}(x,{z}_{1},{z}_{2},0)\to \frac{\alpha}{\beta}P$. Then, the premium is based only in the prior information about the risk. Therefore, the case is the one in which experience is ignored and external information is used as the sole basis for the process of ratemaking.
- When $t\to \infty $, $Z(\infty )\to 0$, $\gamma (x,{z}_{1},{z}_{2},\infty )\to 0$ and therefore ${P}^{*}(x,{z}_{1},{z}_{2},\infty )\to h(x,{z}_{1},{z}_{2},\infty )$. Then, the premium is based only in the sample information.

## 4. Numerical Applications

`Mathematica`software package. We have taken the integer part of the individual claim amount, this does not seem very relevant in our analysis. It is important to mention that due to

`RandomChoice`function, the partition of the aggregate claim amount is different every time the program is run.

#### The Proposed Premiums

## 5. Final Comments and Future Research

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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1. | As a reviewer has pointed out if this inequality is not sustained, then the likelihood function and posterior distribution that will be defined later are not correct. |

Parameter | Updated Parameter |
---|---|

$\alpha $ | ${\alpha}^{*}=\alpha +t\overline{x}$ |

$\beta $ | ${\beta}^{*}=\beta +t$ |

${\alpha}_{1}$ | ${\alpha}_{1}^{*}={\alpha}_{1}+t{\overline{z}}_{1}$ |

${\beta}_{1}$ | ${\beta}_{1}^{*}={\beta}_{1}+t(\overline{x}-{\overline{z}}_{1})$ |

${\alpha}_{2}$ | ${\alpha}_{2}^{*}={\alpha}_{2}+t{\overline{z}}_{2}$ |

${\beta}_{2}$ | ${\beta}_{2}^{*}={\beta}_{2}+t(\overline{x}-{\overline{z}}_{1}-{\overline{z}}_{2})$ |

$\mathbf{(}\mathit{x}\mathbf{,}{\mathit{z}}_{\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathbf{2}}\mathbf{)}$ | Empirical | Fitted (1) | Fitted (2) | $\mathbf{(}\mathit{x}\mathbf{,}{\mathit{z}}_{\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathbf{2}}\mathbf{)}$ | Empirical | Fitted (1) | Fitted (2) |
---|---|---|---|---|---|---|---|

$(0,0,0)$ | 63,232 | 63,094.30 | 63,233.20 | $(3,0,0)$ | 0 | 0.29 | 1.49 |

$(1,0,0)$ | 1840 | 1921.02 | 1812.24 | $(3,1,0)$ | 5 | 1.05 | 4.75 |

$(1,1,0)$ | 2084 | 2257.62 | 2128.08 | $(3,0,1)$ | 0 | 0.19 | 0.44 |

$(1,0,1)$ | 409 | 411.91 | 387.96 | $(3,0,2)$ | 0 | 0.04 | 0.22 |

$(2,0,0)$ | 31 | 29.24 | 51.83 | $(3,1,1)$ | 3 | 0.45 | 1.28 |

$(2,1,0)$ | 134 | 68.73 | 113.61 | $(3,2,1)$ | 0 | 0.26 | 1.11 |

$(2,0,1)$ | 7 | 12.54 | 13.98 | $(3,1,2)$ | 0 | 0.05 | 0.51 |

$(2,1,1)$ | 16 | 14.74 | 24.32 | $(3,3,0)$ | 3 | 0.48 | 2.04 |

$(2,2,0)$ | 79 | 40.40 | 66.82 | $(3,0,3)$ | 0 | 0.00 | 0.10 |

$(2,0,2)$ | 4 | 1.34 | 5.60 | $(4,\xb7,\xb7)$ | 4 | 0.35 | 0.89 |

**Table 3.**Parameters estimated and standard errors (SE) for the basic and mixture model without including covariates.

Basic Model | Mixture Model | ||||
---|---|---|---|---|---|

Parameter | Estimate | SE | Parameter | Estimate | SE |

$\widehat{\theta}$ | 0.072 | 0.001 | $\widehat{\alpha}$ | 1.157 | 0.121 |

${\widehat{p}}_{1}$ | 0.492 | 0.007 | $\widehat{\beta}$ | 15.903 | 1.681 |

${\widehat{p}}_{2}$ | 0.176 | 0.007 | ${\widehat{\alpha}}_{1}$ | 575.261 | 2.779 |

${\widehat{\beta}}_{1}$ | 594.757 | 2.961 | |||

${\widehat{\alpha}}_{2}$ | 0.365 | 0.268 | |||

${\widehat{\beta}}_{2}$ | 1.705 | 1.257 | |||

${\chi}^{2}$ | 137.06 | 7.18 | |||

p-value | 0.00 | 0.007 | |||

df | 4 | 1 | |||

AIC | 45,129.80 | 45,027.70 | |||

CAIC | 45,160.20 | 45,088.50 |

**Table 4.**BMP’s for claims when there are x claims, ${z}_{1}$ with a claim size between ${\psi}_{1}$ and ${\psi}_{2}$, ${z}_{2}$ claims with a size larger than ${\psi}_{2}$ and $x-{z}_{1}-{z}_{2}$ claims with a claim size smaller than ${\psi}_{1}$ with ${p}_{x}=0.25$, ${p}_{y}=0.50$ and ${p}_{z}=0.75$.

$(\mathit{x},{\mathit{z}}_{1},{\mathit{z}}_{2})$ | t | |||||
---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | |

$(0,0,0)$ | 1.000 | 0.940 | 0.888 | 0.841 | 0.799 | 0.760 |

$(1,0,0)$ | 1.798 | 1.692 | 1.597 | 1.513 | 1.437 | 1.368 |

$(1,1,0)$ | 1.864 | 1.754 | 1.656 | 1.568 | 1.489 | 1.418 |

$(1,0,1)$ | 2.168 | 2.040 | 1.926 | 1.824 | 1.732 | 1.649 |

$(2,0,0)$ | 2.583 | 2.430 | 2.295 | 2.173 | 2.064 | 1.965 |

$(2,1,0)$ | 2.633 | 2.477 | 2.339 | 2.215 | 2.104 | 2.003 |

$(2,1,1)$ | 3.174 | 2.986 | 2.819 | 2.670 | 2.536 | 2.414 |

$(2,2,0)$ | 2.729 | 2.568 | 2.424 | 2.296 | 2.180 | 2.076 |

$(2,2,1)$ | 3.530 | 3.321 | 3.135 | 2.969 | 2.820 | 2.685 |

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**MDPI and ACS Style**

Gómez-Déniz, E.; Calderín-Ojeda, E.
Multivariate Credibility in Bonus-Malus Systems Distinguishing between Different Types of Claims. *Risks* **2018**, *6*, 34.
https://doi.org/10.3390/risks6020034

**AMA Style**

Gómez-Déniz E, Calderín-Ojeda E.
Multivariate Credibility in Bonus-Malus Systems Distinguishing between Different Types of Claims. *Risks*. 2018; 6(2):34.
https://doi.org/10.3390/risks6020034

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, and Enrique Calderín-Ojeda.
2018. "Multivariate Credibility in Bonus-Malus Systems Distinguishing between Different Types of Claims" *Risks* 6, no. 2: 34.
https://doi.org/10.3390/risks6020034