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

Emilio Gómez-Déniz,Enrique Calderín-Ojeda

doi:10.3390/risks6020034

Emilio Gómez-Déniz, Enrique Calderín-Ojeda

Open Access

https://doi.org/10.3390/risks6020034

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Abstract

In the classical bonus-malus system the premium assigned to each policyholder is based only on the number of claims made without having into account the claims size. Thus, a policyholder who has declared a claim that results in a relatively small loss is penalised to the same extent as one who has declared a more expensive claim. Of course, this is seen unfair by many policyholders. In this paper, we study the factors that affect the number of claims in car insurance by using a trivariate discrete distribution. This approach allows us to discern between three types of claims depending wether the claims are above, between or below certain thresholds. Therefore, this model implements the two fundamental random variables in this scenario, the number of claims as well as the amount associated with them. In addition, we introduce a trivariate prior distribution conjugated with this discrete distribution that produce credibility bonus-malus premiums that satisfy appropriate traditional transition rules. A practical example based on real data is shown to examine the differences with respect to the premiums obtained under the traditional system of tarification.

Highlights

In an attempt of reducing the economic and casualty losses, the bonus-malus systems (BMS)have been introduced in the actuarial community
Our findings reveal that the bounus-malus premium (BMP)’s computed by using the methodology proposed in this work does not modify the discounts make in the absence of claims
It generally consists of accepting that each policy or insured is represented by a risk parameter that is unknown but random with a certain probability distribution, called a priori distribution or structure function. This way of proceeding is even more useful in the BMS scenario since the premium obeys certain transition rules that classify the policyholders as a bonus or malus

Summary

Introduction

In an attempt of reducing the economic and casualty losses, the bonus-malus systems (BMS). Used a multivariate credibility model that allows the practitioner to consider the positive correlation in customer behaviour between different financial products and estimate the customer specific risk profiles for a specific product not owned by the customer This approach uses only two quantities, the a priori expected number of events and the observed number of events. It generally consists of accepting that each policy or insured is represented by a risk parameter that is unknown but random with a certain probability distribution (in the insurance portfolio), called a priori distribution or structure function This way of proceeding is even more useful in the BMS scenario since the premium obeys certain transition rules that classify the policyholders as a bonus or malus.

Basic Model

Contemplating Heterogeneity

The Premiums

Numerical Applications

The Proposed Premiums

Final Comments and Future Research