On a Multi-Year Microlevel Collective Risk Model

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 $N$ and the individual severity $Y_{j}$s are assumed to be independent. Further, assume that $N$ is a positive integer valued random variable with

and

Then, (1) implies

Clearly, $N$ and $M$ are not independent even though frequency and individual severities are independent. Since $N$ is discrete, the visualization and interpretation of the corresponding copula density function for $(N,M)$ can be difficult. Alternatively, Oh et al., 2020a provides the density function for the jittered version of $(N,M)$ as shown in Figure 1 where x-axis and y-axis corresponds to frequency $N$ and the average severity $M$, respectively.

Let $(U_{1},U_{2})$ be a bivariate random vector sampled from the copula of the jittered version of $(N,M)$. As shown in Figure 1, the density of the copula tends to be smaller in the middle part of $U_{2}$ when $U_{1}$ is smaller, wheras the density tends to be larger in the middle part of $U_{2}$ when $U_{1}$ is larger. Therefore, it is straightforward to see that conditional variance of $M$ decreases as $N$ increases in Figure 1, which is quite intuitive since ${\mathrm{V}ar}\left(M|N\right)=\xi^{2}\psi/N$ 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 $(N,M)$.

For non-life insurance, claims observed are typically a history of frequencies and severities for multiple years. For a policyholder observed for $\tau$ years, we have $n_{1},\cdots,n_{\tau}$ which stand for frequency for each year, and corresponding individual severities $(\boldsymbol{y}_{1},\cdots,\boldsymbol{y}_{\tau})$ where

We find it convenient to define the following symbols for the description of data.

Define a random vector of length $N_{t}+1$

and the realization of $\boldsymbol{Z}_{t}$ is denoted as

Furthermore, multi-year extension of $\boldsymbol{Z}_{t}$ is defined as

and the realization of $\boldsymbol{Z}_{(\tau)}$ 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.

Figure 2 illustrates the dependence structure of our proposed model. In this figure we show that shared random effect $R$ induces the types of dependence that are of interest to us. To illustrate, $R$ is linked to the number of claims across years, $(N_{1},\ldots,N_{\tau})$, through $C_{\theta_{1}}$ so that $\theta_{1}$ is a parameter which captures dependence among claim counts between years. Likewise, $R$ is linked to the individual amounts of claims across years, $(\boldsymbol{Y}_{1},\ldots,\boldsymbol{Y}_{\tau})$, through $C_{\theta_{2}}$ so that $\theta_{2}$ is a parameter which captures dependence among claim amounts within and across the years. Furthermore, $C_{\theta_{1}}$ combined with $C_{\theta_{2}}$ introduces the dependence between the claim counts and individual severities within and across the years.

While, via the shared random effect $R$, the parameters $\theta_{1}$ and $\theta_{2}$ universally capture dependence among the claims across the years, the other parameters $\theta_{3}$ and $\theta_{4}$ specifically capture dependence within the claims of the same year. That is, $\theta_{3}$ is a parameter which incorporates the dependence between the claim count and claim amounts within a year whereas $\theta_{4}$ is a parameter which incorporates the dependence among claim amounts within a year. Similarly, $\theta_{3}$ combined with $\theta_{4}$ 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 $(\theta_{1},\theta_{2})$ only, the dependence among the claims in the same year are modeled by both $(\theta_{1},\theta_{2})$ and $(\theta_{3},\theta_{4})$. 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 $t$, $Z_{t}$, are independent for $t=1,\ldots,\tau$ given the shared random effect $R=r$ described as follows:

and

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.

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 $\theta_{1}=\theta_{2}=\theta_{3}=\theta_{4}=0$. 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 $\theta_{3}=\theta_{4}=0$. 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 $\theta_{1}=\theta_{2}=0$. In this regard, our proposed framework is quite comprehensive that allows other dependence models that have appeared in the literature as special cases.

We start with definition of symbols. Denote $\mathbb{N}$, $\mathbb{N}_{0}$, $\mathbb{R}$, and $\mathbb{R}^{+}$ 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 $n\times m$ matrix $\boldsymbol{M}$, we denote $(i,j)$-th component of $\boldsymbol{M}$ as $\left[\boldsymbol{M}\right]_{ij}$. For a row vector $\boldsymbol{v}$ of length $n$, we denote the $i$-th component of $\boldsymbol{v}$ as $\left[\boldsymbol{v}\right]_{i}$. For $n\in\mathbb{N}$, define ${\bf 1}_{n}$ and $\boldsymbol{J}_{n\times n}$ as a column vector of $1$ with length $n$ and a $n\times n$ matrix of ones, respectively. We use $\boldsymbol{I}_{n}$ for $n\in\mathbb{N}$ to represent the $n\times n$ identity matrix.

Suppose $\boldsymbol{\Sigma}_{1,1}$, $\boldsymbol{\Sigma}_{1,2}$, $\boldsymbol{\Sigma}_{2,1}$, and $\boldsymbol{\Sigma}_{2,2}$ are $\ell\times\ell$, $\ell\times m$, $m\times\ell$, and $m\times m$ matrices, respectively. Define $(\ell+m)\times(\ell+m)$ matrix $\boldsymbol{\Sigma}$ as

If $\boldsymbol{\Sigma}_{2,2}$ is invertible, the Schur complement of the block $\boldsymbol{\Sigma}_{2,2}$ of the matrix $\boldsymbol{\Sigma}$ is the $\ell\times\ell$ matrix defined by

In the matrix $\boldsymbol{\Sigma}_{(\boldsymbol{n})}^{(\boldsymbol{\rho})}$, each component will be used for modeling the correlation between frequencies and severities within and across years. For example, the partitioned matrix $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho})}$ is a $(n_{t}+1)\times(n_{t}+1)$ matrix describing the correlation structure of the random vector $(N_{t},Y_{t,1},\cdots,Y_{t,n_{t}})$. Specifically, $\rho_{1}$ in $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho})}$ is used for a correlation between a frequency and a severity in the $t$-th year, and $\rho_{2}$ in $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho})}$ is used for a correlation among the severities in the $t$-th year. On the other hand, the partitioned matrix $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}$ is a $(n_{t}+1)\times(n_{j}+1)$ matrix describing the correlation structure between the random vectors $(N_{t},Y_{t,1},\cdots,Y_{t,n_{t}})$ and $(N_{j},Y_{j,1},\cdots,Y_{j,n_{j}})$. Specifically, $\rho_{3}$ in $\left[\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}\right]_{11}$ is used for a correlation between the frequencies in the different years, and $\rho_{4}$ in $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}$ is used for a correlation between a frequency in different years. Finally, $\rho_{5}$ in $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}$ is used for a correlation between a frequency and a severity in different years. The following is summarization for the meaning of each correlation:

• 1.

  $\rho_{1}$: correlation between a frequency and a severity within a year;

• 2.

  $\rho_{2}$: correlation among two distinct severities within a year;

• 3.

  $\rho_{3}$: correlation among frequencies across years;

• 4.

  $\rho_{4}$: correlation between a frequency and a severity in different years;

• 5.

  $\rho_{5}$: correlation between two severities in different years.

We finally note that $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho})}$ only depends on $(\rho_{1},\rho_{2})$ while $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}$ for $t\neq j$ only depends on $(\rho_{3},\rho_{4},\rho_{5})$. Hence, we find that it is convenient to use $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho}^{*})}$ with $\boldsymbol{\rho}^{*}=(\rho_{1},\rho_{2})$ to stand for $\boldsymbol{\Sigma}_{tt}^{(\boldsymbol{\rho})}$, and similarly $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho}^{*})}$ for $t\neq j$ with $\boldsymbol{\rho}^{*}=(\rho_{3},\rho_{4},\rho_{5})$ to stand for $\boldsymbol{\Sigma}_{tj}^{(\boldsymbol{\rho})}$ in a clear context.

In Definition 2, we have introduced two parameters $\theta_{1}$ and $\theta_{2}$. We impose natural dependence for multiples years of observed claims by using the shared random effect $R$, which will affect all frequency and severities in any calendar year. In this regard, $\theta_{1}$ will be served as correlation parameter between the random effect $R$ and a frequency, and $\theta_{2}$ will be served as correlation parameter between a random effect $R$ and each severity, as described in Figure 1.

Now, for $\boldsymbol{n}=(n_{1},\cdots,n_{\tau})\in\left(\mathbb{N}_{0}\right)^{\tau}$ and $\boldsymbol{\rho}=(\rho_{1},\cdots,\rho_{5})\in\mathbb{R}^{5}$, we consider reparameterization of a matrix $\boldsymbol{\Sigma}_{(\boldsymbol{n})}^{(\boldsymbol{\rho})}$ with

for

The following theorem provides some results related with reparameterization in (6).

Let $F_{t}$ be non-negative integer-valued distribution functions with the respective probability mass functions $f_{t}$ for $t\in\mathbb{N}$. Let $G_{t}$ and $G_{t,j}$ be non-negative real-valued distribution functions with respective probability densities $g_{t}$ and $g_{t,j}$ for $t,j\in\mathbb{N}$. While it is not necessary but for simplicity, we assume $G_{tj}=G_{t}$ for any $t,j\in\mathbb{N}$.

We use $\Phi$ and $\phi$ to denote the standard normal distribution and the corresponding density function, respectively. For a vector $\boldsymbol{\mu}\in\mathbb{R}^{n}$ and a $n\times n$ covariance matrix $\boldsymbol{\Sigma}$, we use $\Phi_{\boldsymbol{\mu},\boldsymbol{\Sigma}}$ to denote the distribution function of multivariate normal distribution with mean $\boldsymbol{\mu}$ and a covariance matrix $\boldsymbol{\Sigma}$, and $\phi_{\boldsymbol{\mu},\boldsymbol{\Sigma}}$ 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.

From Lemma 1, the matrix $\boldsymbol{\Sigma}_{(\boldsymbol{n})}^{(\boldsymbol{\rho})}$ is positive definite for any $\boldsymbol{\rho}$ satisfying (6). Hence, $C_{(\boldsymbol{n})}^{(\boldsymbol{\rho})}$ 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 $\boldsymbol{b}=\left(b_{1},\cdots,b_{\tau}\right)$ be vertices where each $b_{j}$ is equal to either $n_{j}$ or $n_{j}-1$. Then the corresponding density function of the random vector of $\boldsymbol{Z}_{(\boldsymbol{\tau})}$ at $\boldsymbol{Z}_{(\boldsymbol{\tau})}=\boldsymbol{z}_{(\boldsymbol{\tau})}$ in (11) is given by

where the sum is taken over all vertices $\boldsymbol{b}$, and ${\rm sgn}(\boldsymbol{b})$ 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.

• 1.

  Due to the discrete nature of the frequency observations, one can immediately check that the computational complexity in (12) grows exponentially with $\tau$.

• 2.

  The calculation of each summation in (12), which requires a numerical multivariate integration due to the nature of multivariate Gaussian function, can be even difficult especially in high dimensions (Genz and Bretz,, 2009)

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 $\boldsymbol{\rho}$ defined in (6) satisfying (9), we extend the modeling of $\boldsymbol{Z}_{(\boldsymbol{\tau})}$ by including the random effect $R$ so that the joint distribution of $\left(R,\boldsymbol{Z}_{(\boldsymbol{\tau})}\right)$ is given by

Naturally, by the property of the copula $C_{(\boldsymbol{n})}^{(\boldsymbol{\rho},\boldsymbol{\theta})}$, the joint distribution in (13) implies the joint distribution function in (11) in the following sense

which further implies that the random vector $\left(R,\boldsymbol{Z}_{(\boldsymbol{\tau})}\right)$ is a natural extension of the random vector $\boldsymbol{Z}_{(\boldsymbol{\tau})}$. Furthermore, reparameterization in (6) gives us a well-defined and natural dependence structure with the shared random effect ${R}$ so that claims across multiple years would be independent conditional on ${R}$. For example, if (6) holds and $\theta_{1}=\theta_{2}=0$, then one can see that $\boldsymbol{Z}_{(\boldsymbol{\tau})}$ are not only conditionally independent but also marginally independent so that $\boldsymbol{Z}_{t}\perp\boldsymbol{Z}_{t^{\prime}}$ for all $t\neq t^{\prime}$. In addition to (6), if $\theta_{3}=\theta_{4}=0$, then $\boldsymbol{M}$ is not only block-diagonal, but diagonal, which implies that frequency and severity are independent once the shared random effect $R$ is controlled. In other words, dependence between frequency and severity are fully explained by the shared random effect ${R}$. Finally, if (6) holds and $\theta_{1}=\theta_{2}=\theta_{3}=\theta_{4}=0$, then $\boldsymbol{\Sigma}_{(\boldsymbol{n})}^{(\boldsymbol{\rho})}$ 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 $\boldsymbol{Z}_{t}$ for $t=1,\ldots,\tau$ are independent conditional on the random effect $R$, 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 $\boldsymbol{Z}_{t}$ conditional on $R$ 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.

Based on this result in Theorem 2, one can obtain the joint density of $(\boldsymbol{Z}_{1},\cdots,\boldsymbol{Z}_{\tau})$ just with a single dimensional (numerical) integration as the following manner.