According to Klugman et al. (2012), the aggregate loss in the classical collective risk model is defined as where means the number of claim and denotes individual claim amounts over a fixed period of time with the following assumptions:
, the random variablesare i.i.d. random variables.
Conditional on , the common distribution of the random variables does not depend on .
The distribution of does not depend in any way on the values of .
These assumptions might be convenient in terms of computational ease, however, such simplifying assumptions often lead to bias issues especially when used for risk classification. In relaxing such assumptions, various models have been proposed in the insurance literature. An interesting method to model the dependence in the collective risk model is the so-called two-part dependent frequency-severity model as suggested by Frees et al. (2014a). In this model, the dependence is incorporated by using frequency as an explanatory variable in the severity component. A similar approach has been used by Frees et al. (2011a) in the modeling and prediction of frequency and severity of health care expenditure. Shi et al. (2015) suggested a three-part framework in order to capture the association between frequency and severity components. When generalized linear models (GLMs) are used with the number of claims treated as a covariate in claims severity, Garrido et al. (2016) showed that the pure premium includes a correction term for inducing dependence. When analyzing bonus-malus data, an interesting observation was made by Park et al. (2018) that dependence between claim frequency and severity is driven by the desire to reach a better bonus-malus class.
Applications of copula methods to capture dependence have been recently used in collective risk models. A majority of work in this area focused on modeling the dependence between frequency and average severity with parametric copulas. For example, Czado et al. (2012) used Gaussian copulas to extend traditional compound Poisson-Gamma two-part model and incorporated possible dependence. Krämer et al. (2013) suggested a similar joint copula-based approach and interestingly observed that ignoring dependence causes a severe underestimation of total loss in a portfolio. Frees et al. (2016) extended the copula-based approach to dependent frequency and average severity using claims data with multiple lines of insurance business. While their findings suggested weak association between frequency and average severity, they concluded that there are strong dependencies among the lines of business.
Unlike choosing a suitable family of marginal distributions, it is usually much harder to choose the correct family of copulas when calibrating these dependent models with data. The work of Krämer et al. (2013) investigated test procedures for the selection of a suitable family of copulas in a dependent frequency and average severity model. However, Oh et al. (2020a) illustrated that indeed it is even more difficult to choose the appropriate dependence structure between frequency and average severity that includes the classical collective risk model as a special case. In particular, even under the most naive assumption of independence between frequency and individual severities, choosing the correct parametric copula presents some challenges. Inspired by this phenomenon, Oh et al. (2020a) and Cossette et al. (2019) discussed the construction of single year collective risk models with microlevel data to provide a suitable dependence structure between the frequency and severity components. In part, the extension in this paper that captures dependence of various types of dependence between claim frequency and claim severity over multiple years is motivated by the work of Oh et al. (2020a).
In insurance industry, it is important to model the longitudinal property of the insurance losses to predict the fair premium in the future based on each policyholder’s historical claims information. However, the existing copula methods in the literature cannot be directly applied in prediction of the premium due to at least one of the following difficulties:
Limited to the analysis of data over a single period or cross-sectional data,
The choice of the copula family to provide a suitable dependence structure between claim frequency and average claim severity can be difficult.
Alternatively, the random effect model can be used to model the longitudinal property of the insurance losses. Hernández-Bastida et al. (2009) and Oh et al. (2020b) used the shared random effects model to construct the dependence in a collective risk model, where independence between claim frequency and severity conditional on the random effect is assumed and the dependence structure is naturally derived by the shared random effects. Jeong and Valdez (2020) derived a closed form of credibility premium for compound loss which captures not only the dependence between frequency and severity but also dependence among the multi-year claims of the same policyholder. However, it is known that the overdispersion and serial dependence can be compounded in the random effect model. Such compounded effect of the random effect can possibly result in pseudo or fake dependence structure in the claims, which in turn leads to the poor prediction of the premium (Denuit et al., 2007; Murray et al., 2013; Lee et al., 2020).
In this regard, as a natural extension of shared random effects model and one-year dependent compound risk model, we propose a multi-year framework with microlevel data so that we may incorporate the following dependencies simultaneously:
dependence between a frequency and a severity within a year,
dependence between two distinct severities within a year,
dependence among frequencies across years,
dependence between a frequency and a severity in different years,
dependence between two severities in different years.
Specifically, we use a factor copula representation, which can be viewed as a copula model version of the random effect model (Krupskii and Joe, 2013, 2015), by using 1-year microlevel model as building blocks.
The remainder of this paper is organized as follows. In Section 2, we propose a generalized shared random effects framework for multi-year microlevel collective risk model that incorporates all types of dependencies previously described. We demonstrate that previous methods for dependence modeling can be considered as special cases of our proposed model. In Section 3, we provide a concrete example of our proposed model with elliptical copulas. Because of simplicity, we focus on the family of Gaussian copulas to further explore various correlation structures that satisfy our framework. In Section 4, an empirical analysis with a special case of our proposed model is conducted with a dataset from an automobile insurance company. Concluding remarks are provided in Section 5 with some future directions of research.
2 Construction of the shared random effect parameter model
2.1 A motivating illustration
While copula methods are flexible in modeling the dependence, the “actual” flexibility comes from the proper choice of the parametric copula family. Although one may consider using the nonparametric copula method for the full flexibility in choosing a copula structure, modeling and interpreting dependence based on the non-parametric copula can be difficult as long as the discrete random variables are involved mainly due to the lack of uniquness(Genest and Nešlehová, 2007). While recent study in Yang et al. (2019)
provides the safe copula estimation method for discrete outcomes in a regression context, it is known to suffer from the so-called curse of dimensionality.
Indeed, as shown in Oh et al. (2020a), it is difficult to choose a proper parametric copula family for the frequency and average severity even under the most naive assumption, the case where frequency and individual severities are independent. This subsection summarizes the example in Oh et al. (2020a) to explain such difficult and the necessity to use microlevel claims information.
Consider the classical collective risk model where frequency and the individual severity s are assumed to be independent. Further, assume that is a positive integer valued random variable with
Then, (1) implies
Clearly, and are not independent even though frequency and individual severities are independent. Since is discrete, the visualization and interpretation of the corresponding copula density function for can be difficult. Alternatively, Oh et al. (2020a) provides the density function for the jittered version of as shown in Figure 2 where x-axis and y-axis corresponds to frequency and the average severity , respectively.
be a bivariate random vector sampled from the copula of the jittered version of. As shown in Figure 2, the density of the copula tends to be smaller in the middle part of when is smaller, wheras the density tends to be larger in the middle part of when
is larger. Therefore, it is straightforward to see that conditional variance ofdecreases as increases in Figure 2, which is quite intuitive since in this case.
This example illustrates that we can see that most existing copulas, including Gaussian and Archimedean copulas, are unable to accommodate the dependence between frequency and average severity properly. This is a motivation for the modeling the dependence based on the microlevel claims information rather than summarized claims information. We refer the readers to Oh et al. (2020a) for more details of this example and the detailed construction of the jittered version of .
2.2 Data structure and model specification
For non-life insurance, claims observed are typically a history of frequencies and severities for multiple years. For a policyholder observed for years, we have which stand for frequency for each year, and corresponding individual severities where
We find it convenient to define the following symbols for the description of data.
Define a random vector of length
and the realization of is denoted as
Furthermore, multi-year extension of is defined as
and the realization of is denoted as
In the subsequent, we describe a shared random effect parameter model for modeling the type of claims data we observe that primarily consist of frequencies and severities for multiple years.
Model 1 (The copula linked shared random effect model).
Consider the following random effect model for where the joint distribution between the observed losses and the shared random effect is presented with copulas.
where the joint distribution between the observed losses and the shared random effect is presented with copulas.
Shared random effect
follows a probability distribution with density.
Conditional on , we have that for are independent observations whose distribution function is given by
means marginal cumulative distribution functions ofand , respectively and means joint density function of . As a result, we have the following distribution function of
The parameters and of the copula controls the independence between the frequency and severities and independence among individual severities, respectively, within a year so that we have
and means joint density function of .
for if and only if .
for if and only if .
Figure 3 illustrates the dependence structure of our proposed model. In this figure we show that shared random effect induces the types of dependence that are of interest to us. To illustrate, is linked to the number of claims across years, , through so that is a parameter which captures dependence among claim counts between years. Likewise, is linked to the individual amounts of claims across years, , through so that is a parameter which captures dependence among claim amounts within and across the years. Furthermore, combined with introduces the dependence between the claim counts and individual severities within and across the years.
While, via the shared random effect , the parameters and universally capture dependence among the claims across the years, the other parameters and specifically capture dependence within the claims of the same year. That is, is a parameter which incorporates the dependence between the claim count and claim amounts within a year whereas is a parameter which incorporates the dependence among claim amounts within a year. Similarly, combined with affects the dependence between the claim counts and individual severities within the year. As a result, while dependence among the claims in different years are modeled by only, the dependence among the claims in the same year are modeled by both and . Note that our framework is distinguished from some existing work on dependence modeling with copulas such as Shi and Yang (2018) and Lee and Shi (2019), where average severity in the form of summarized data was used for modeling and implicitly precluded independence among the individual severities within the same year.
The idea of our multi-year microlevel collective risk model is that the observed claim for year , , are independent for given the shared random effect described as follows:
which is straightforward from iii and iv of Model 1.
We note that this construction is similar to the model described by Krupskii and Joe (2013), which develops a factor copula model conditionally on a set of latent variables. In some sense, according to their paper, our approach leads to a one-factor copula model presented in Section 3. The primary difference in our approach is the clear intuitive interpretation of our model to describe the various types of dependence in a dependent collective risk model. The well-definedness of Model 1 will also be discussed in Remark 1 in Section 3.
2.3 Special cases
It is immediate to see that the classical collective risk model of Klugman et al. (2012) is a special case of our proposed model where . This is the case when all frequencies and severities are mutually independent. Baumgartner et al. (2015) proposed shared random effects model to capture association between frequency and the average severity, which is just another special case of our proposed model. This is the case when . Finally, it is also easy to check that single-year microlevel collective risk model, proposed by Oh et al. (2020a), is another special case of our proposed model. This is when . In this regard, our proposed framework is quite comprehensive that allows other dependence models that have appeared in the literature as special cases.
3 Factor copula model based on the elliptical distributions
Copulas generated by elliptical distributions, also called elliptical copulas, have the correlation matrix as the primary parameter describing dependence between the components. The Gaussian and copulas belong to the family of elliptical copulas. We refer to Landsman and Valdez (2003) for other choices of elliptical copulas including the copulas generated from multivariate Cauchy or multivariate logistic distribution. In this section, for simplicity, apparent ease of computations, and steering clear of distractions from the general case, we focus on the case of Gaussian copulas. In B, we illustrate how Gaussian copulas in multi-year microlevel collective risk model can be generalized into the elliptical copulas by providing an example of copula among other choices of elliptical copulas. Specifically, we consider Gaussian copulas with a specific covariance matrix to accommodate the dependence structure of multi-year microlevel collective risk model, and show that such Gaussian copula models can be represented as factor copula models. For the use in elliptical copulas including the Gaussian and copulas in mind, we begin with describing dependence structure via correlation matrices.
3.1 Dependence structure via correlation matrix
We start with definition of symbols. Denote , , , and by the set of positive integer, the set of non-negative integer, the set of real number, and the set of positive real number, respectively.
For a matrix , we denote -th component of as . For a row vector of length , we denote the -th component of as . For , define and as a column vector of with length and a matrix of ones, respectively. We use for to represent the identity matrix.
Suppose , , , and are , , , and matrices, respectively. Define matrix as
If is invertible, the Schur complement of the block of the matrix is the matrix defined by
For and define the partitioned matrix
For , the matrix is a matrix defined as
for . Furthermore, for with , the matrix is a matrix defined as
for and .
Consider the case . Then we can write out by denoting and . As a result, is a defined as
Furthermore, from the above and the following
In the matrix , each component will be used for modeling the correlation between frequencies and severities within and across years. For example, the partitioned matrix is a matrix describing the correlation structure of the random vector . Specifically, in is used for a correlation between a frequency and a severity in the -th year, and in is used for a correlation among the severities in the -th year. On the other hand, the partitioned matrix is a matrix describing the correlation structure between the random vectors and . Specifically, in is used for a correlation between the frequencies in the different years, and in is used for a correlation between a frequency in different years. Finally, in is used for a correlation between a frequency and a severity in different years. The following is summarization for the meaning of each correlation:
: correlation between a frequency and a severity within a year;
: correlation among two distinct severities within a year;
: correlation among frequencies across years;
: correlation between a frequency and a severity in different years;
: correlation between two severities in different years.
We finally note that only depends on while for only depends on . Hence, we find that it is convenient to use with to stand for , and similarly for with to stand for in a clear context.
For , , , define the partitioned matrix as as
where is defined in (4) and is a matrix which can be expressed based on the following partitioned matrix
with being a matrix given by
In Definition 2, we have introduced two parameters and . We impose natural dependence for multiples years of observed claims by using the shared random effect , which will affect all frequency and severities in any calendar year. In this regard, will be served as correlation parameter between the random effect and a frequency, and will be served as correlation parameter between a random effect and each severity, as described in Figure 2.
Consider the case , then one can represent as a partitioned matrix as
where is in (4), and
Hence, we have
Now, for and , we consider reparameterization of a matrix with
The following theorem provides some results related with reparameterization in (6).
For , , consider the Schur Complement of the block of the matrix in (5) denoted as . For convenience, consider the following block matrix representation of as
where is a matrix. Then, we have the following results.
For the proof of part i, it suffices to show that if , then
For the proof of part ii, by Schur decomposition, we have is positive definite if and only if is positive definite. Since is a block diagonal matrix provided (6) is satisfied, checking the positive definiteness of is equivalent to check whether is positive definite or not. Hence, a matrix is positive definite and is a block diagonal matrix for any if and only if is positive definite for any where
Following Corollary 1 in Oh et al. (2020a), we have positive definite for any if and only if
Finally, simple argument shows that (10) with the condition is equivalent with
The proof of part iii immediately follows from part i and ii. ∎
3.2 The special case of Gaussian copulas
Let be non-negative integer-valued distribution functions with the respective probability mass functions for . Let and be non-negative real-valued distribution functions with respective probability densities and for . While it is not necessary but for simplicity, we assume for any .
We use and
to denote the standard normal distribution and the corresponding density function, respectively. For a vectorand a covariance matrix , we use to denote the distribution function of multivariate normal distribution with mean and a covariance matrix , and to denote the corresponding density function. Now, we are ready to present the multi-year microlevel collective risk model where the Gaussian copula is used to model the dependence.
Model 2 (The Gaussian copula model for the multi-year microlevel collective risk model).
Suppose satisfies (6). Then, consider the random vector whose joint distribution function is given by the following copula model representation
where is a Gaussian copula with correlation matrix .
From Lemma 1, the matrix is positive definite for any satisfying (6). Hence, in Model 2 is a valid Gaussian copula. One can see that the estimation of the parameters in (11) is involved with the calculation of multivariate Gaussian density function which depends on the length of the observer years. Let be vertices where each is equal to either or . Then the corresponding density function of the random vector of at in (11) is given by
where the sum is taken over all vertices , and is given by
Here, we note that calculation of the density function in (12) can be difficult due to the following aspects of our model.
Due to the discrete nature of the frequency observations, one can immediately check that the computational complexity in (12) grows exponentially with .
However, here we avoid such difficulty by using the following copula representation which is inspired by factor copula representation in Krupskii and Joe (2013), Nikoloulopoulos and Joe (2015) and Kadhem and Nikoloulopoulos (2019). For defined in (6) satisfying (9), we extend the modeling of by including the random effect so that the joint distribution of is given by
which further implies that the random vector is a natural extension of the random vector . Furthermore, reparameterization in (6) gives us a well-defined and natural dependence structure with the shared random effect so that claims across multiple years would be independent conditional on . For example, if (6) holds and , then one can see that are not only conditionally independent but also marginally independent so that for all . In addition to (6), if , then is not only block-diagonal, but diagonal, which implies that frequency and severity are independent once the shared random effect is controlled. In other words, dependence between frequency and severity are fully explained by the shared random effect . Finally, if (6) holds and , then is diagonal, which implies our model specification includes the traditional model, which assumes independence among the claims in different years and independence between the frequency and severity.
The following theorem shows us the key idea of our copula representation where the observed claim for are independent conditional on the random effect , and can be fitted into the special case of Model 1. In this regard, the copula in (13) has similar spirit as a factor copula. The corresponding copula of the distribution of conditional on is a Gaussian copula which can be represented as 1-factor copula. As a result, the distribution in (13) have 2-factor copula representation. However, such representation of the model increases the complexity of the notation while provides limited benefit in computational complexity, and hence we do not pursue such representation for the simplicity of the paper.
The distribution function of can be obtained as in (11).
The density function of conditional on is given by
where is the conditional density function of conditional on .
for if and only if .
for if and only if .
The proof of part i is trivial from the property of copula function. The proof of part ii, by the invariance property of the copula under the monotone transformation, we have that the corresponding copula of the conditional distribution of random vector conditional on is again a Gaussian copula. Furthermore, knowing that is a Gaussian copula, Theorem 1 shows that are independent conditional on . The proofs of part iii and iv are immediate from the property of Gaussian copulas. ∎
Based on this result in Theorem 2, one can obtain the joint density of just with a single dimensional (numerical) integration as the following manner.
Consider the random vector under the settings in Model 2. Then, the joint density of is given as follows:
with and . Here, is the density function of conditional on , and given by