Lluís Bermúdez

A Correlation Sensitivity Analysis of Non-Life Underwriting Risk in Solvency Capital Requirement Estimation

This paper analyses the impact of using different correlation assumptions between lines of business when estimating the risk-based capital reserve, the Solvency Capital Requirement (SCR), under Solvency II regulations. A case study is presented and the SCR is calculated according to the Standard Model approach. Alternatively, the requirement is then calculated using an Internal Model based on a Monte Carlo simulation of the net underwriting result at a one-year horizon, with copulas being used to model the dependence between lines of business. To address the impact of these model assumptions on the SCR we conduct a sensitivity analysis. We examine changes in the correlation matrix between lines of business and address the choice of copulas. Drawing on aggregate historical data from the Spanish non-life insurance market between 2000 and 2009, we conclude that modifications of the correlation and dependence assumptions have a significant impact on SCR estimation.

Modelling Dependence in a Ratemaking Procedure with Multivariate Poisson Regression Models

When actuaries face with the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or homeowners insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce di®erent multivariate Poisson regression models in order to relax the independence assumption, including zero-in°ated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational di±culties. Bayesian inference based on MCMC helps to solve this problem (and also lets us derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claims. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models and their zero-inflated versions.

Mixture of Bivariate Poisson Regression Models with an Application to Insurance

In a recent paper Bermudez [2009] used bivariate Poisson regression models for ratemaking in car insurance, and included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. In the present paper, we revisit this model in order to consider alternatives. We propose a 2-finite mixture of bivariate Poisson regression models to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, we show that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Additionally, we describe a model in which the mixing proportions are dependent on covariates when modelling the way in which each individual belongs to a separate cluster. Finally, an EM algorithm is provided in order to ensure the models’ ease-of-fit. These models are applied to the same automobile insurance claims data set as used in Bermudez [2009] and it is shown that the modelling of the data set can be improved considerably.

Solvency Capital Estimation and Risk Measures

This paper examines why a financial entity’s solvency capital estimation might be underestimated if the total amount required is obtained directly from a risk measurement. Using Monte Carlo simulation we show that, in some instances, a common risk measure such as Value-at-Risk is not subadditive when certain dependence structures are considered. Higher risk evaluations are obtained for independence between random variables than those obtained in the case of comonotonicity. The paper stresses, therefore, the relationship between dependence structures and capital estimation.

A priori ratemaking using bivariate Poisson regression models

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.

Bonus–malus system using an exponential loss function with an Inverse Gaussian distribution

Abstract In most companies, the importance of the auto-insurance has prompted their actuaries to look for different tariff systems that distribute the exact weight of each risk among portfolios. The use of quadratic loss functions in most of classical bonus–malus systems leads to very high maluses. To avoid this problem, we use an exponential loss function and subsequently obtain the result for the Poisson–Inverse Gaussian model. We show, with a numerical example, that this new model fits better than the classical one for our data. Furthermore, it provides maluses that not so high, as well as other advantages for actuaries.

Copula-based regression modeling of bivariate severity of temporary disability and permanent motor injuries.

The analysis of factors influencing the severity of the personal injuries suffered by victims of motor accidents is an issue of major interest. Yet, most of the extant literature has tended to address this question by focusing on either the severity of temporary disability or the severity of permanent injury. In this paper, a bivariate copula-based regression model for temporary disability and permanent injury severities is introduced for the joint analysis of the relationship with the set of factors that might influence both categories of injury. Using a motor insurance database with 21,361 observations, the copula-based regression model is shown to give a better performance than that of a model based on the assumption of independence. The inclusion of the dependence structure in the analysis has a higher impact on the variance estimates of the injury severities than it does on the point estimates. By taking into account the dependence between temporary and permanent severities a more extensive factor analysis can be conducted. We illustrate that the conditional distribution functions of injury severities may be estimated, thus, providing decision makers with valuable information.

A posteriori ratemaking using bivariate Poisson models

Recently, different bivariate Poisson regression models have been used in the actuarial literature to make an a priori ratemaking taking into account the dependence between two types of claims. A natural extension for these models is to consider a posteriori ratemaking (i.e. experience rating models) that also relaxes the independence assumption. We introduce here two bivariate experience rating models that integrate the a priori ratemaking based on the bivariate Poisson regression models, extending the existing literature for the univariate case to the bivariate case. These bivariate experience rating models are applied to an automobile insurance claims data-set to analyse the consequences for posterior premiums when the independence assumption is relaxed. The main finding is that the a posteriori risk factors obtained with the bivariate experience rating models are significantly lower than those factors derived under the independence assumption.

Influence of parties' behavioural features on motor compensation disputes in insurance markets

Disputes between parties involved in motor insurance claims compensations are analysed. The decision to resolve the disagreement by either negotiation or trial may depend on how risk and confrontation adverse or pessimistic the claimant is. The extent to which these behavioural features of the claimant might influence the final compensation amount is examined. An empirical analysis, fitting a switching regression model to a Spanish database, is conducted in order to analyse whether the choice of the conflict resolution procedure is endogenous to the compensation outcomes. The results show that compensations awarded by courts are always higher, although 95% of cases are settled by negotiation. We show that this is because claimants are adverse to risk and confrontation, and are pessimistic about their chances at trial. By contrast, insurers are risk/confrontation neutral and more objective in relation to the expected trial compensation. During the negotiation insurers accept to pay the subjective compensation values of claimants, since these values are lower than their estimates of compensations at trial.

On the use of risk measures in solvency capital estimation

Regulation on the minimum capital that a financial institution or an insurance firm must hold to guarantee its solvency is proportional to a measure of its global risk. Using Monte Carlo simulation we show that, in some instances, risk measures can substantially underestimate risk. So, we address the implications on the choice of the risk measure that determines the economic capital requirement. The paper analyses the relationship between dependence structures, risk measurement and capital estimation.