# A priori ratemaking using bivariate Poisson regression models

Abstract: In automobile insurance, it is useful to achieve a priori ratemaking by resorting to generalized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper examines an a priori ratemaking procedure when including two different types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tariff system might be affected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.

## Summary (2 min read)

### 1 Introduction

- Designing a tariff structure for insurance is one of the main tasks for actuaries.
- When this assumption is relaxed, it is interesting to see how the tariff system might be affected.
- In the next section, the model used here is defined.

### 2 Bivariate Poisson regression models

- The usual methodology to obtain the a priori premium under the assumption of independence between types of claims can be described as follows.
- This principle builds on the net premium by including a risk loading that is proportional to the variance of the risk.
- This is the so-called trivariate reduction method that leads to the bivariate Poisson distribution.
- Here the authors follow the zero-inflated bivariate Poisson model proposed by Karlis and Ntzoufras (2005).
- Standard errors for the parameter estimates are calculated using standard bootstrap methods (boot package in R).

### 3 The database

- The original sample comprised a ten percent sample of the automobile portfolio of a major insurance company operating in Spain in 1995.
- Only cars categorised as being for private use were considered.
- The sample is not representative of the actual portfolio as it was drawn from a larger panel of policyholders who had been customers of the company for at least seven years; however, it will be helpful for illustrative purposes.
- The meaning of those variables referring to the policyholders’ coverage should also be clarified.
- The simplest policy only includes third-party liability (claimed and counted as N1 type) and a set of basic guarantees such as emergency roadside assistance, legal assistance or insurance covering medical costs (claimed and counted as N2 type).

### 4.1 Fitting bivariate Poisson models

- First, parameters related to the type of coverage (v10 and v11 ) were always significant and their presence increased the expected number o claims markedly.
- In order to model the covariance term (λ3 ), the covariates were introduced in the bivariate Poisson model with the result that only the parameter for v10 was significant.
- A profile with a mean lying very close to this average was chosen for the third profile.
- In Table 7, it can be observed that the zero-inflated bivariate models did not present any noticeable differences with the non zero-inflated models in terms of the mean scores, but they were present in the case of the variance.

### 5 Conclusions

- This paper has tested the independence assumption between claim types given a set of known risk factors and it has shown that independence should be rejected.
- The interpretation of a number of bivariate Poisson models has been illustrated in the context of automobile insurance claims and the conclusion is that using a bivariate Poisson model leads to an a priori ratemaking that presents larger variances and, hence, larger loadings than those obtained under the independence assumption.
- For the five models analysed here there seems to be a relationship between the goodness of fit and the level of overdispersion considered in each model.
- In short, the main finding is that the independence assumption that is implicitely used when pricing automobile insurance by adding the pure premium for each guarantee (which are obtained using count data regression models) is insufficient because correlations (conditional on the covariates) are ignored.
- 3In Frees and Valdez (2008) a hierarchical model allows to capture possible dependencies of claims among the various types through a t-copula specification.

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