On the independence between risk profiles in the compound collective risk actuarial model

Abstract

This paper examines a compound collective risk model in which the primary distribution comprised the Poisson–Lindley distribution with a λ parameter, and where the secondary distribution is an exponential one with a θ parameter. We consider the case of dependence between risk profiles (i.e., the parameters λ and θ), where the dependence is modelled by a Farlie–Gumbel–Morgenstern family. We analyze the consequences of the dependence on the Bayes premium. We conclude that the consequences of the dependence on the Bayes premium may vary considerably.

Introduction

In actuarial risk theory, the collective risk model is described by a frequency distribution for the number of claims N and a sequence of independent and identically distributed random variables representing the size of the single claims X_i. Frequency N and Severity X_i are assumed to be independent, conditional on distribution parameters. There is an extensive body of literature on the risk modelling process, see e.g. McNeil et al. [28].

For each likelihood assessment and for each probabilistic modelling of the prior information, a different model is derived. The most commonly used models are

• N has a Poisson distribution [16], Negative Binomial ([17], [47]; among others).

• The claim severity distribution is Exponential [35], Gamma [44], Lognormal [23], [3], Pareto and Weibull [9], among others.

Our interest is focussed on S = X₁ + ⋯ + X_N which denotes the aggregate losses or the total cost over a period.

A set of interesting results on the sum of random variables with an exponential distribution and a random number of summands is presented in Kozubowskiu and Panoska [22]. Furthermore, a comprehensive collection of approximate forms for the compound mixed Poisson distribution is discussed in Nadarajah and Kotz [32], [33].

Estimation of the annual loss distribution by modelling the frequency and severity of losses is a well known actuarial technique. It is also used for model solvency requirements in the insurance industry [37], [48]. Under the Basel II accords [5] banks are required to quantify the capital charge for operational risk. Several aspects of operational risk modelling are examined in Chavez-Demoulin et al. [8] and Cruz [11], among others. Many banks have adopted the Loss Distribution Approach (LDA) as the Advanced Measurement Approach (AMA), where the frequency and severity of operational losses for each risk cell are estimated over a one year period.

Notwithstanding the above, it is well known that a considerable difficulty arises in many data sets in which the variance of the number of claims is appreciably larger than the mean (a phenomenon known as overdispersion). This has led researchers to examine distributions other than the Poisson for modelling the random variable N, and in particular the mixed Poisson distributions, in which the phenomenon of overdispersion always occurs (see for instance Nikoloulopoulos and Karlis [34], among others).

On the other hand, the computations required to obtain S under the different models above cited are difficult to perform without the independence hypothesis.

Carriere [50], using a bootstrap procedure, investigated how to test the hypothesis about the independence of claim frequency and severity. The need to relax this sometimes restrictive hypothesis has recently led to some interesting studies [1], [6], [10]. Peters et al. [51] proposed that this kind of independence assumption in operational risk models should be investigated further.

Fundamentally, this paper presents two novel aspects: (1) the good behaviour of the Poisson–Lindley (PL) distribution for fitting certain actuarial data presenting certain properties in addition to the overdispersion: the fact that they are “zero-inflated” and the presence of “deflation of ones”; (2) a simple and easily implementable approach to quantify, in actuarial terms, departures from independence between risk profiles. In actuarial practice, the following properties are normally present in sample observations.

• Overdispersion: the variance is greater than the mean.

• Zero-inflated: the sample data present a higher frequency of occurrence of zero claims than would be expected if the sample had been generated by a Poisson distribution, for example.

• Deflation of ones: fewer policyholders make only one claim.

PL distributions also have properties in common with other useful count-distributions such as unimodality, infinite divisibility, controlled skewness and kurtosis [14].

It can be considered that the ultimate goal of the actuary is to provide a good estimate of the premium to be charged. On the basis of this assumption, associations (correlations) between risk profiles may be drawn. In our opinion, this paper could be considered a preliminary step towards the ultimate goal. Thus, our second aim is to investigate the importance of the independence assumption, in the light of its importance in the literature. With respect to previous studies, we address the problem of independence from a different standpoint. First, we focus on the hypothesis of the independence of risk profiles as an indirect way of analyzing the independence between claim frequency and claim severity. Subsequently, we propose a model of (weak) dependence between the prior densities of these risk profiles, including the case of independence as a particular case (for this purpose, using the Farlie–Gumbel–Morgenstern (FGM) family of distributions [30]). By means of these tools, it is a straightforward matter to study how the independence hypothesis affects actuarial decisions. By setting a measure of comparison (for example, the Bayes premium), it suffices to compare this measure over the entire class under consideration with the one that would be obtained under independence.

The FGM family used presents two significant advantages: (a) it means the problem can be approached mathematically, and solutions are readily achieved by intensive computation; (b) we can control the form or structure of the dependence arising between the parameters within the class and we know that it involves a distancing from independence (regardless of the path being taken).

The results obtained reveal that the Bayes premium is quite sensitive to this hypothesis of independence.

Obviously, one could suggest alternatives such as the Sarmanov family of bivariate distributions [39]. Focusing on modelling dependence between the risk profiles, Hernández-Bastida et al. [18] proposed the use of a Sarmanov family developed by Ting Lee [45] under the composed collective model using Poisson and Exponential distributions.

The article is organized as follows. Section 2 provides details of the statistical model proposed containing the likelihood of a Poisson–Lindley count distribution and exponential severities. Section 3 contains the scenario of prior information considered in the paper: the FGM family of priors elicited on the risk profiles. Section 3 also presents the derivation of the (prior) joint moments of parameters, including the covariance and correlation coefficients, while the variation of the uncertainty between risk profiles as measured by the relative entropy or Kullback–Leibler divergence over the FGM family is presented in Section 3.1. In Section 4 we describe how the models react to variations in the independence of the risk profile priors with respect to the Bayes premium, and how the results obtained can be used in practice. Some conclusions and comments are made in Section 5.