2. Models and definitions

We first introduce the hierarchical multivariate compound aggregate loss model studied in this paper. Assume that an insurance policy covers $K$ categories of risks, denoted by $\mathcal{K}=\{1,\cdots, K\}$ . An accident can cause different types of claims in combinations, $\mathbf{h}=(h_{1},\ldots,h_{K})^\top$ , where the $k$ th coordinate $h_{k}$ equals to 1 if type $k$ claim occurs and 0 otherwise.

Let $\mathcal{M}=\{1,\cdots, M\}$ , where $M=2^K -1$ be the set of indexes of possible combinations that include at least one claim. For $m\in \mathcal{M}$ , let $N_m$ denote the number of accidents resulting in the $m$ th combination of claims for a portfolio of policies during a time period. The random variables $N_m$ , $m\in \mathcal{M}$ , can be dependent. Let $\mathbf{N}=(N_1, \cdots, N_M)^\top$ and denote its joint probability function by

\begin{equation*} p_{\mathbf {N}}(\mathbf {n})=\Pr [\mathbf {N}=\mathbf {n}]\,, \end{equation*}

where $\mathbf{n}=(n_1,\cdots,n_M)^\top \in \mathbb{N}^M$ .

For a given claim combination $m\in \mathcal{M}$ , let $\mathbf{X}_{(m)}=\left (X_{(m),1},\ldots,X_{(m),K}\right )^\top$ denote the random vector of claim sizes, where $X_{(m),k}$ for $k=1,\ldots, K$ represents the claim size of type $k$ risk in this combination. $X_{(m),k}=0$ if the $m$ th combination does not include a type $k$ claim. Since the claim size vector $\mathbf{X}_{(m)}$ is generated by one accident, its elements are stochastically dependent. However, since the loss size vectors $\mathbf{X}_{(1)},\ldots,\mathbf{X}_{(M)}$ result from different accidents, we assume that they are mutually independent and are independent of $\mathbf{N}$ . A slight extension of the model where $\mathbf{N}$ and $(\mathbf{X}_{(1)},\ldots,\mathbf{X}_{(M)})$ are dependent is considered in Section 4.2 of the paper.

For the portfolio of policies, let the aggregate amount of the $K$ types of claims resulting from the $m$ th combination be represented by the vector

\begin{equation*} \mathbf {S}_{N_m}=\left (S_{N_m,1},\ldots,S_{N_m,K}\right )^\top = \sum _{i=1}^{N_m}\mathbf {X}_{(m)i}= \sum _{i=1}^{N_m} \left (X_{(m)i,1},\ldots,X_{(m)i,K}\right )^\top, \end{equation*}

where $\mathbf{X}_{(m)i}$ , $i\geq 1$ are independent copies of $\mathbf{X}_{(m)}$ .

Let

\begin{equation*} \mathbf {S}_{\mathbf {N}}=\left (\mathbf {S}_{N_1},\ldots,\mathbf {S}_{N_M}\right ) \end{equation*}

be the $K\times M$ dimensional compound loss matrix. Equivalently,

(2.1) \begin{equation} \mathbf{S}_{\mathbf{N}} = \left [{\begin{array}{cccc} S_{N_1,1} & S_{N_2,1} & \cdots & S_{N_M,1}\\ S_{N_1,2} & S_{N_2,2} & \cdots & S_{N_M,2}\\ \vdots & \vdots & \ddots & \vdots \\ S_{N_1,K} & S_{N_2,K} & \cdots & S_{N_M,K}\\ \end{array} } \right ], \end{equation}

where the element $S_{N_m,k}$ represents the aggregate amount of the type $k$ claims that result from the claim combination $m$ . Then, the total amount of claims of all types for the portfolio of business is given by

\begin{equation*} S_{\bullet }=\sum _{m=1}^M \sum _{k=1}^K S_{N_m,k}. \end{equation*}

Example 2.1. Suppose that an auto insurance policy covers two types of risks: PD and bodily injury (BI). An accident can cause a PD claim only, a BI claim only, or a claim that combines both PD and BI. Then we may denote the type of risk by $\mathcal{K}= \{1,2\}$ ; and the possible combinations of claims caused by an accident can be represented by three two-dimensional vectors $\mathbf{h}_1=(1,0)^\top$ , $\mathbf{h}_2=(0,1)^\top$ , and $\mathbf{h}_3=(1,1)^\top$ . Then we have $\mathcal{M}= \{1,2,3\}$ , and the numbers of claims of the three combinations of risk types, i.e., PD only, BI only, and both PD and BI, incurred in a time period is given by $\mathbf{N} = (N_1, N_2, N_3)^\top$ . The claim sizes are given by $\mathbf{X}_{(1)}=(X_{(1),1},0)^\top$ , $\mathbf{X}_{(2)}=(0,X_{(2),2})^\top$ and $\mathbf{X}_{(3)}=(X_{(3),1},X_{(3),2})^\top$ , respectively.

For this case, the multivariate aggregate loss matrix $\mathbf{S}_{\mathbf{N}}$ is given by

\begin{equation*} \mathbf {S}_{\mathbf {N}} = \left [ {\begin {array}{ccc} S_{N_1,1} & 0 & S_{N_3,1}\\ 0 & S_{N_2,2} & S_{N_3,2} \end {array} } \right ]= \left [ {\begin {array}{ccc} \sum _{i=1}^{N_1} X_{(1)i,1} & 0 & \sum _{i=1}^{N_3} X_{(3)i, 1}\\ 0 & \sum _{i=1}^{N_2} X_{(2)i,2} & \sum _{i=1}^{N_3} X_{(3)i,2} \end {array} } \right ]. \end{equation*}

For $q\in (0, 1)$ , let $s_q$ denote the Value-at-Risk (VaR) of $S_{\bullet }$ at the $100q\%$ confidence level.

The TCE at level $q$ of $S_{\bullet }$ is defined by

\begin{equation*} \text {TCE}_{S_{\bullet }}( {q})={\mathbb {E}}[S_{\bullet }|S_{\bullet }\gt s_q]\,. \end{equation*}

According to the TCE-based capital allocation rule (Dhaene et al., Reference Dhaene, Henrard, Landsman, Vandendorpe and Vanduffel2008), the total capital requirement in a portfolio of business is $\text{TCE}_{S_{\bullet }}({q})$ for some $q$ and the part allocated to the type $k$ risk in the portfolio is

(2.2) \begin{equation} \text{TCE}_{S_{\bullet,k}}({q})=\sum _{m=1}^M{\mathbb{E}}[S_{N_m,k}|S_{\bullet }\gt s_q],\quad k\in \mathcal{K}\,. \end{equation}

The capital required for the $m$ th combination of risk type is given by

(2.3) \begin{equation} \text{TCE}_{S_{N_m,\bullet }}({q})=\sum _{k=1}^K{\mathbb{E}}[S_{N_m,k}|S_{\bullet }\gt s_q],\quad m\in \mathcal{M}\,. \end{equation}

Note that

\begin{equation*} \sum _{k=1}^K \text {TCE}_{S_{\bullet,k}}( {q}) = \sum _{m=1}^M \text {TCE}_{S_{N_m,\bullet }}( {q})=\text {TCE}_{S_{\bullet }}( {q})\,. \end{equation*}

Likewise, if the total required capital is determined by the TV of $S_{\bullet }$ at probability level $q$ , which is defined by (Furman and Landsman, Reference Furman and Landsman2006)

\begin{equation*} \text {TV}_{S_{\bullet }}( {q}) =\text {Var}[S_{\bullet }|S_{\bullet }\gt s_q]\,, \end{equation*}

then, according to the TV-based capital allocation rule, the capital allocated to the type $k$ risk is given by

(2.4) \begin{equation} \text{TV}_{S_{\bullet,k}}({q}) = \sum _{m=1}^M \text{Cov}[(S_{N_m,k},S_{\bullet })|S_{\bullet }\gt s_q]\,, \end{equation}

and the capital required for the $m$ th combination of risk type is given by

(2.5) \begin{equation} \text{TV}_{S_{N_m,\bullet }}({q})=\sum _{k=1}^K \text{Cov}[(S_{N_m,k},S_{\bullet })|S_{\bullet }\gt s_q]\,. \end{equation}

Notably,

\begin{equation*} \sum _{k=1}^K \text {TV}_{S_{\bullet,k}}( {q}) =\sum _{m=1}^M \text {TV}_{S_{N_m,\bullet }}( {q})= \text {Var}[S_{\bullet }|S_{\bullet }\gt s_q]\,. \end{equation*}

For more details about the TCE- and TV-based capital allocation, one can refer to, for example, Cummins (Reference Cummins2000), Dhaene et al. (Reference Dhaene, Henrard, Landsman, Vandendorpe and Vanduffel2008), Furman and Zitikis (Reference Furman and Zitikis2008), and references therein. We note that other frameworks of risk measures exist in the literature. As an example, Furman et al. (Reference Furman, Wang and Zitikis2017) proposed Gini-type risk measures and developed the corresponding capital allocation rules. Notably, this framework requires only finiteness of the first moment of the underlying random variable.

In the next section, we derive formulas for computing the TCE and TV risk measures of the proposed multivariate compound loss model and performing the associated capital allocation. The main tool we use is the concept of moment (size-biased) transforms, which is widely used in statistics. For detailed studies of the moment transforms, one is referred to, for example, Patil and Ord (Reference Patil and Ord1976), Arratia and Goldstein (Reference Arratia and Goldstein2010), Furman and Landsman (Reference Furman and Landsman2005), Denuit (Reference Denuit2020), Denuit and Robert (Reference Denuit and Robert2022), Mohammed et al. (Reference Mohammed, Furman and Su2021), and Furman et al. (Reference Furman, Kye and Su2021).

For completeness of this paper, we provide definitions for the size-biased transform of univariate and multivariate random variables in the following.

Definition 2.1. Consider a non-negative random variable $X$ with the distribution function $F_X$ and moments $\mathbb{E}[X^{\alpha }]\lt \infty$ for some positive integer $\alpha$ . A random variable $\widetilde{X^{\alpha }}$ is said to be a copy of the $\alpha$ th moment transform of $X$ if its cumulative distribution function (c.d.f.) is given by

\begin{equation*} F_{\widetilde {X^{\alpha }}}(x)=\frac {\int _0^xt^{\alpha }dF_X(t)}{\mathbb {E}[X^{\alpha }]}=\frac {\mathbb {E}[X^{\alpha }I(X\leq x)]}{\mathbb {E}[X^{\alpha }]},\quad x\ge 0. \end{equation*}

The first-moment transform of $X$ is commonly referred to as the size-biased transform. It is simply denoted by $\widetilde{X}$ .

Definition 2.2. Consider a random vector $\mathbf{X}=(X_1,\ldots,X_K)^\top$ with the c.d.f. $F_{\mathbf{X}}$ and moments $\mathbb{E}[X^{\alpha }_k]\lt \infty$ and $\mathbb{E}[X^{\alpha _1}_{k_1}X^{\alpha _2}_{k_2}]\lt \infty$ for some $k, k_1, k_2 \in \{1,\ldots,K\}$ and non-negative integers, $\alpha$ , $\alpha _1$ , and $\alpha _2$ .

The $k$ th component $\alpha$ th moment transform of $\mathbf{X}$ is any random vector $\widetilde{\mathbf{X}^{\alpha }}^{[k]}$ with the c.d.f.

\begin{equation*} F_{\widetilde {\mathbf {X}^{\alpha }}^{[k]}}(\mathbf {x})=\frac {\int _0^{x_1}\ldots \int _0^{x_K}t_k^{\alpha }d F_{\mathbf {X}}(t_1,\ldots,t_K)}{\mathbb {E}[X^{\alpha }_k]}=\frac {\mathbb {E}[X^{\alpha }_k I(\mathbf {X}\leq \mathbf {x})]}{\mathbb {E}[X^{\alpha }_k]}\,,\quad \mathbf {x}\ge \mathbf {0}\,, \end{equation*}

where $\mathbf{x}=(x_1,\ldots, x_K)^\top$ .

The $(k_1,k_2)$ th component, $(\alpha _1,\alpha _2)$ th moment transform of $\mathbf{X}$ is any random vector $\widetilde{\mathbf{X}^{\alpha _1,\alpha _2}}^{[k_1,k_2]}$ with the c.d.f.

\begin{equation*} F_{\widetilde {\mathbf {X}^{\alpha _1,\alpha _2}}^{[k_1,k_2]}}(\mathbf {x})=\frac {\int _0^{x_1}\ldots \int _0^{x_K}t_{k_1}^{\alpha _1}t_{k_2}^{\alpha _2}d F_{\mathbf {X}}(t_1,\ldots,t_K)}{\mathbb {E}[X^{\alpha _1}_{k_1}X^{\alpha _2}_{k_2}]}=\frac {\mathbb {E}[X^{\alpha _1}_{k_1}X^{\alpha _2}_{k_2}I(\mathbf {X}\leq \mathbf {x})]}{\mathbb {E}[X^{\alpha _1}_{k_1}X^{\alpha _2}_{k_2}]}\,,\quad \mathbf {x}\ge \mathbf {0}\,. \end{equation*}

The $k$ th component first-moment transform of $\mathbf{X}$ is denoted as $\widetilde{\mathbf{X}}^{[k]}$ , and the $(k_1,k_2)$ th component $(1,1)$ th moment transform of $\mathbf{X}$ is denoted as $\widetilde{\mathbf{X}}^{[k_1,k_2]}$ .

For discrete distributions, following Patil and Ord (Reference Patil and Ord1976), we consider the factorial moment transform. For a positive integer $I$ , define

\begin{align*} I^{(\alpha )}=\left \{\begin{array}{cc} I(I-1)\ldots (I-\alpha +1), & \text{if}\ \alpha \leq I \\ 0, & otherwise \end{array}\right. .\end{align*}

Then we have

Definition 2.3. Consider a non-negative discrete random variable $N$ with the probability mass function (p.m.f.) $p_N$ . A random variable $\widetilde{N^{(\alpha )}}$ is said to be a copy of the $\alpha$ th factorial moment transform of $N$ if its p.m.f. is given by

\begin{equation*} p_{\widetilde {N^{(\alpha )}}}(n)=\frac {\mathbb {E}[N^{(\alpha )}I(N= n)]}{\mathbb {E}[N^{(\alpha )}]}=\frac {n^{(\alpha )} p_N(n)}{\mathbb {E}[N^{(\alpha )}]}\,,\quad n\geq 0\,. \end{equation*}

The first factorial moment transform of $N$ is denoted by $\widetilde{N}$ .

Definition 2.4. Consider a vector of discrete random variables $\mathbf{N}=(N_1,\ldots,N_K)^\top$ having the p.m.f. $p_{\mathbf{N}}(\mathbf{n})$ .

The $k$ th component, $\alpha$ th moment transform of $\mathbf{N}$ is any random vector $\widetilde{\mathbf{N}^{(\alpha )}}^{[k]}$ with p.m.f.

\begin{equation*} p_{\widetilde {\mathbf {N}^{(\alpha )}}^{[k]}}(\mathbf {n})=\frac {n_k^{(\alpha )}p_{\mathbf {N}}(\mathbf {n})}{\mathbb {E}[N^{\alpha }_k]}=\frac {\mathbb {E}[N^{(\alpha )}_k I(\mathbf {N}= \mathbf {n})]}{\mathbb {E}[N^{\alpha }_k]}\,,\quad \mathbf {n}\geq \mathbf {0}\,. \end{equation*}

The $(k_1,k_2)$ th component, $(\alpha _1,\alpha _2)$ th moment transform of $\mathbf{N}$ is any random vector $\widetilde{\mathbf{N}^{(\alpha _1),(\alpha _2)}}^{[k_1,k_2]}$ with the p.m.f.

\begin{equation*} p_{\widetilde {\mathbf {N}^{(\alpha _1),(\alpha _2)}}^{[k_1,k_2]}}(\mathbf {n})=\frac {n_{k_1}^{(\alpha _1)}n_{k_2}^{(\alpha _2)} p_{\mathbf {N}}(\mathbf {n})}{\mathbb {E}[N^{(\alpha _1)}_{k_1}N^{(\alpha _2)}_{k_2}]}=\frac {\mathbb {E}[N^{(\alpha _1)}_{k_1}N^{(\alpha _2)}_{k_2}I(\mathbf {N}= \mathbf {n})]}{\mathbb {E}[N^{(\alpha _1)}_{k_1}N^{(\alpha _2)}_{k_2}]}\,,\quad \mathbf {n}\geq \mathbf {0}\,. \end{equation*}

The $k$ th component, first-moment transform of $\mathbf{N}$ is denoted as $\widetilde{\mathbf{N}}^{[k]}$ , and the $(k_1,k_2)$ th component $(1,1)$ th moment transform of $\mathbf{N}$ is denoted as $\widetilde{\mathbf{N}}^{[k_1,k_2]}$ .

3. Risk measures and capital allocation for the multivariate compound loss model

In this section, we present results for calculating the risk measures and capital allocation for the multivariate compound loss model represented by the matrix $\mathbf{S}_{\mathbf{N}}$ defined in equation (2.1). To this purpose, we define the $(k,m)$ th ( $k$ th row, $m$ th column) component, first-moment transform of $\mathbf{S}_{\mathbf{N}}$ to be a matrix of random variables $\widetilde{\mathbf{S}_{\mathbf{N}^{[m]}}}^{[k]}$ , which has the same size as $\mathbf{S}_{\mathbf{N}}$ and the c.d.f.

\begin{equation*} F_{\widetilde {\mathbf {S}_{\mathbf {N}^{[m]}}}^{[k]}}(\mathbf {s})=\frac {\mathbb {E}[S_{N_m,k} I(\mathbf {S}_{\mathbf {N}}\leq \mathbf {s})]}{\mathbb {E}[S_{N_m,k}]}\,, \end{equation*}

where $\mathbf{s}$ is a matrix of non-negative constants

\begin{equation*} \mathbf {s} = \left [ {\begin {array}{cccc} {s}_{1,1} &\cdots & {s}_{M,1}\\ \vdots & \ddots & \vdots \\ {s}_{1,K} &\cdots & {s}_{M,K} \end {array} } \right ], \end{equation*}

and the $\le$ operation is defined piecewisely.

Then, following Proposition 1 of Furman and Landsman (Reference Furman and Landsman2005), Proposition 3.1 of Denuit and Robert (Reference Denuit and Robert2022), or Lemma 2.1 in Jiang and Ren (Reference Jiang and Ren2022), we have, for $m\in \mathcal{M}$ and $k\in \mathcal{K}$ ,

(3.1) \begin{equation} {\mathbb{E}}[S_{N_m,k}|S_{\bullet }\gt s_q]={\mathbb{E}}[S_{N_m,k}]\frac{\Pr (\widetilde{S_{\bullet }}^{m[k]}\gt s_q)}{\Pr (S_{\bullet }\gt s_q)}\,, \end{equation}

where

\begin{equation*}\widetilde {S_{\bullet }}^{m[k]}=\sum _{i=1}^M\sum _{j=1}^K \widetilde {S_{\mathbf {N}^{[m]}}}^{[k]}_{i,j}\end{equation*}

and $\widetilde{S_{\mathbf{N}^{[m]}}}^{[k]}_{i,j}$ is the $i,j$ th element of $\widetilde{\mathbf{S}_{\mathbf{N}^{[m]}}}^{[k]}$ .

In addition, define the $[(k_1, m_1), (k_2, m_2)]$ th components joint moment transform $\mathbf{S}_{\mathbf{N}}$ to be a random matrix $\widetilde{\mathbf{S}_{\mathbf{N}^{[m_1,m_2]}}}^{[k_1,k_2]}$ with c.d.f.

\begin{equation*} F_{\widetilde {\mathbf {S}_{\mathbf {N}^{[m_1,m_2]}}}^{[k_1,k_2]}}(\mathbf {s})=\frac {\mathbb {E}[S_{N_{m_1},k_1} S_{N_{m_2},k_2}I(\mathbf {S}_{\mathbf {N}}\leq \mathbf {s})]}{\mathbb {E}[S_{N_{m_1},k_1} S_{N_{m_2},k_2}]}\,. \end{equation*}

Then, for $m_1,m_2\in \mathcal{M}$ and $k_1,k_2\in \mathcal{K}$ ,

(3.2) \begin{equation} {\mathbb{E}}[S_{N_{m_1},k_1}S_{N_{m_2},k_2}|S_{\bullet }\gt s_q]={\mathbb{E}}[S_{N_{m_1},k_1}S_{N_{m_2},k_2}]\frac{\Pr (\widetilde{S_{\bullet }}^{m_1,m_2[k_1,k_2]}\gt s_q)}{\Pr (S_{\bullet }\gt s_q)}\,, \end{equation}

where

\begin{equation*}\widetilde {S_{\bullet }}^{m_1,m_2[k_1,k_2]}=\sum _{i=1}^M\sum _{j=1}^K \widetilde {S_{\mathbf {N}^{[m_1,m_2]}}}^{[k_1,k_2]}_{i,j}\end{equation*}

and $\widetilde{S_{\mathbf{N}^{[m_1,m_2]}}}^{[k_1,k_2]}_{i,j}$ is the $(i,j)$ th element of $\widetilde{\mathbf{S}_{\mathbf{N}^{[m_1,m_2]}}}^{[k_1,k_2]}$ .

Therefore, together with Equations (2.2) to (2.5), it is seen that if we can compute the distribution function of $S_{\bullet }$ , $\widetilde{S_{\bullet }}^{m[k]}$ and $\widetilde{S_{\bullet }}^{m_1,m_2[k_1,k_2]}$ , then the TCE and TV of $S_{\bullet }$ can be determined and the associated capital allocation can be performed.

To determine the distribution of $S_{\bullet }$ , $\widetilde{S_{\bullet }}^{m[k]}$ , and $\widetilde{S_{\bullet }}^{m_1,m_2[k_1,k_2]}$ , we need the distribution functions of ${\mathbf{S}}_{\mathbf{N}}$ , $\widetilde{\mathbf{S}_{\mathbf{N}^{[m]}}}^{[k]}$ , and $\widetilde{\mathbf{S}_{\mathbf{N}^{[m_1,m_2]}}}^{[k_1,k_2]}$ , for which we have the following results.

Theorem 3.1. For $m\in \mathcal{M}$ , let $\mathbf{1}^{[m]}$ denote an $M$ dimensional vector with the $m$ th element being one and all others zero. Let

\begin{equation*} \mathbf {L}^{[m]}=\widetilde {\mathbf {N}}^{[m]}-\mathbf {1}^{[m]}\,, \end{equation*}

and

\begin{equation*} \mathbf {S}_{\mathbf {L}^{[m]}}=\left (\mathbf {S}_{L_1^{[m]}},\ldots,\mathbf {S}_{L_M^{[m]}}\right ), \end{equation*}

where

\begin{equation*} \mathbf {S}_{L_i^{[m]}}=\sum _{j=1}^{L_i^{[m]}}\mathbf {X}_{(m)j}= \sum _{j=1}^{L_i^{[m]}} \left (X_{(m)j,1},\ldots,X_{(m)j,K}\right )^\top =\left (S_{L_i^{[m]},1},\ldots,S_{L_i^{[m]},K}\right )^\top \end{equation*}

and $L_i^{[m]}$ , $i\in \mathcal{M}$ , is the $i$ th element of $\mathbf{L}^{[m]}$ . Then, for $k\in \mathcal{K}$ ,

(3.3) \begin{equation} \widetilde{\mathbf{S}_{\mathbf{N}^{[m]}}}^{[k]}\stackrel{d}{=}\mathbf{S}_{\mathbf{L}^{[m]}}+\widetilde{\mathbf{X}_{(m)1}}^{[k]}\times{\mathbf{1}^{[m]}}^\top \,, \end{equation}

where $\widetilde{\mathbf{X}_{(m)1}}^{[k]}$ is an independent copy of the $k$ th component, first-moment transform of $\mathbf{X}_{(m)}$ .

Further, let

\begin{equation*} {\mathbf {L}^{(2)}}^{[m]}=\widetilde {\mathbf {N}^{(2)}}^{[m]}-2\times \mathbf {1}^{[m]}\,, \end{equation*}

then, for $k_1,k_2\in \mathcal{K}$ ,

(3.4) \begin{align} &\nonumber \Pr \left (\widetilde{\mathbf{S}_{\mathbf{N}^{[m,m]}}}^{[k_1,k_2]}\leq{\mathbf{s}}\right )\\ =&\,\frac{{\mathbb{E}}[N_m]{\mathbb{E}}[X_{(m)1,k_1}X_{(m)1,k_2}]}{{\mathbb{E}}[S_{N_m,k_1}S_{N_m,k_2}]} \textrm{Pr}\left (\mathbf{S}_{\mathbf{L}^{[m]}}+\widetilde{\mathbf{X}_{(m)1}}^{[k_1,k_2]}\times{\mathbf{1}^{[m]}}^\top \leq{\mathbf{s}}\right )\\ \nonumber &+\frac{{\mathbb{E}}[N_m^{(2)}]{\mathbb{E}}[X_{(m)1,k_1}]{\mathbb{E}}[X_{(m)2,k_2}]}{{\mathbb{E}}[S_{N_m,k_1}S_{N_m,k_2}]}\textrm{Pr}\left (\mathbf{S}_{{\mathbf{L}^{(2)}}^{[m]}}+(\widetilde{\mathbf{X}_{(m)1}}^{[k_1]}+\widetilde{\mathbf{X}_{(m)2}}^{[k_2]})\times{\mathbf{1}^{[m]}}^\top \leq{\mathbf{s}}\right ), \end{align}

where $\widetilde{\mathbf{X}_{(m)1}}^{[k_1]}$ and $\widetilde{\mathbf{X}_{(m)2}}^{[k_2]}$ are copies of the first-moment transform of $\mathbf{X}_{(m)}$ , and $\widetilde{\mathbf{X}_{(m)1}}^{[k_1,k_2]}$ is a copy of the $(k_1,k_2)$ th component (1,1)th moment transform of $\mathbf{X}_{(m)}$ . The random variables $\widetilde{\mathbf{X}_{(m)1}}^{[k_1]}$ , $\widetilde{\mathbf{X}_{(m)2}}^{[k_2]}$ , $\widetilde{\mathbf{X}_{(m)1}}^{[k_1,k_2]}$ , and $\mathbf{X}_{(m)}$ are mutually independent.

In addition, for $m_1,m_2\in \mathcal{M}$ and $m_1\neq m_2$ , let

\begin{equation*} \mathbf {L}^{[m_1,m_2]}=\widetilde {\mathbf {N}}^{[m_1,m_2]}-\mathbf {1}^{[m_1]}-\mathbf {1}^{[m_2]}, \end{equation*}

then

(3.5) \begin{equation}{} \widetilde{\mathbf{S}_{\mathbf{N}^{[m_1,m_2]}}}^{[k_1,k_2]}\stackrel{d}{=}\mathbf{S}_{\mathbf{L}^{[m_1,m_2]}}+\widetilde{\mathbf{X}_{(m_1)1}}^{[k_1]}\times{\mathbf{1}^{[m_1]}}^\top +\widetilde{\mathbf{X}_{(m_2)1}}^{[k_2]}\times{\mathbf{1}^{[m_2]}}^\top . \end{equation}

The proof of this theorem is provided in the appendix of the paper.

Remark 3.1. Equation (3.4) shows that the distribution of $\widetilde{\mathbf{S}_{\mathbf{N}^{[m,m]}}}^{[k_1,k_2]}$ is a mixture of

\begin{equation*} \mathbf {S}_{\mathbf {L}^{[m]}}+\widetilde {\mathbf {X}_{(m)1}}^{[k_1,k_2]}\times {\mathbf {1}^{[m]}}^\top \end{equation*}

and

\begin{equation*} \mathbf {S}_{{\mathbf {L}^{(2)}}^{[m]}}+(\widetilde {\mathbf {X}_{(m)1}}^{[k_1]}+\widetilde {\mathbf {X}_{(m)2}}^{[k_2]})\times {\mathbf {1}^{[m]}}^\top, \end{equation*}

with weights

\begin{equation*} \frac {{\mathbb {E}}[N_m]{\mathbb {E}}[X_{(m)1,k_1}X_{(m)1,k_2}]}{{\mathbb {E}}[S_{N_m,k_1}S_{N_m,k_2}]} \end{equation*}

and

\begin{equation*} \frac {{\mathbb {E}}[N_m^{(2)}]{\mathbb {E}}[X_{(m)1,k_1}]{\mathbb {E}}[X_{(m)2,k_2}]}{{\mathbb {E}}[S_{N_m,k_1}S_{N_m,k_2}]}\,, \end{equation*}

respectively.

We summarize the procedures for performing the capital allocation computation as follows:

4. Numerical examples

In this section, we illustrate how to apply the formulas derived in last section to compute the risk measures and to perform the capital allocations for the proposed multivariate aggregate loss models.

4.1 The multivariate aggregate claim model

In this subsection, we provide the general structure of the model that will be used in the numerical examples. Let $W$ be a counting random variable that follows $\text{NB}(r, \beta )$ distribution with probability mass function

\begin{equation*} p_W(w)=\left (\begin {array}{c} r+w-1 \\ w \end {array}\right )\left (\frac {\beta }{1+\beta }\right )^w\left (\frac {1}{1+\beta }\right )^r,\ \ w\geq 0\,. \end{equation*}

Conditional on $W=w$ , let the claim number vector $\mathbf{N}=(N_1, \ldots, N_M)^\top$ follow a multinomial distribution with parameters $(w, q_1, \ldots, q_M)$ . That is,

\begin{equation*} \Pr (\mathbf {N}=\mathbf {n}|W=w)=\frac {w!}{n_1!n_2!\ldots n_M!}q_1^{n_1}q_2^{n_2}\ldots q_M^{n_M},\ \ n_1+n_2+\ldots +n_M=w\,. \end{equation*}

This model was introduced by Hesselager (Reference Hesselager1996), and Jiang and Ren (Reference Jiang and Ren2022) denoted the unconditional distribution of $\mathbf{N}$ by $\text{HMN}(W, q_1, \ldots, q_M)$ . Regression analysis of this model was provided by Frees et al. (Reference Frees, Shi and Valdez2009).

We could easily obtain that

\begin{equation*} {\mathbb {E}}[N_m]={\mathbb {E}}[W]q_m = r\beta q_m,\ \ m\in \mathcal {M}\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[N^{(2)}_m]={\mathbb {E}}[W^{(2)}]q^2_m=r(r+1)\beta ^2 q^2_m,\ \ m\in \mathcal {M}\,, \end{equation*}

and

\begin{equation*} {\mathbb {E}}[N_{m_1}N_{m_2}]={\mathbb {E}}[W^{(2)}]q_{m_1}q_{m_2}=r(r+1)\beta ^2 q_{m_1}q_{m_2},\ \ m_1,m_2\in \mathcal {M},\ m_1\neq m_2\,. \end{equation*}

The joint p.g.f. of $\mathbf{N}$ is

\begin{equation*} \mathcal {P}_{\mathbf {N}}(z_1,\ldots,z_M)=\left [1-\beta (q_1z_1+\ldots +q_Mz_M-1)\right ]^{-r}. \end{equation*}

As shown in Theorem 4.1 of Jiang and Ren (Reference Jiang and Ren2022), for $m,m_1,m_2\in \mathcal{M}$ and $m_1\neq m_2$ , we have

\begin{equation*} \mathbf {L}^{[m]}=\widetilde {\mathbf {N}}^{[m]}-\mathbf {1}^{[m]}\sim \text {HMN}(\widetilde {W}-1,q_1,\ldots,q_M)\,, \end{equation*}

\begin{equation*} {\mathbf {L}^{(2)}}^{[m]}={\widetilde {\mathbf {N}^{(2)}}}^{[m]}-2\times \mathbf {1}^{[m]}\sim \text {HMN}(\widetilde {W}^{(2)}-2,q_1,\ldots,q_M)\,, \end{equation*}

and

\begin{equation*} \mathbf {L}^{[m_1,m_2]}=\widetilde {\mathbf {N}}^{[m_1,m_2]}-\mathbf {1}^{[m_1]}-\mathbf {1}^{[m_2]}\sim \text {HMN}(\widetilde {W}^{(2)}-2,q_1,\ldots,q_M)\,. \end{equation*}

In addition, $\widetilde{W}-1$ follows $\text{NB}(r + 1, \beta )$ distribution, and $\widetilde{W}^{(2)}-2$ follows $\text{NB}(r + 2, \beta )$ distribution. Therefore, the distribution of $\mathbf{L}^{[m]}$ , ${\mathbf{L}^{(2)}}^{[m]}$ , and $\mathbf{L}^{[m_1,m_2]}$ is all in the same family as $\mathbf{N}$ .

For claim severity, we assume that if a claim combination $m$ only consists of a type $k\in \mathcal{K}$ claim, then its size follows a Poisson distribution with mean $a_k$ . Then we have

\begin{equation*} {\mathbb {E}}[X_{(m),k}]=a_k, \end{equation*}

and

\begin{equation*} {\mathbb {E}}[X^2_{(m),k}]=a_k+a_k^2. \end{equation*}

If a claim combination $m$ consists of non-zero claims of types $\{k_1,\ldots, k_h\}$ , then the joint distribution of the claim sizes is assumed to be a common Poisson mixture. That is, conditional on a mixing variable $\Lambda =\lambda$ , for $j = 1, \ldots, h$ , the size of type $k_j$ claim follows a Poisson distribution with mean $b_{k_j}\lambda$ . Further, we assume $\Lambda$ follows a gamma distribution with shape parameter $\alpha$ and p.d.f.

\begin{equation*} f_{\Lambda }(\lambda )=\frac {\alpha ^{\alpha }\lambda ^{\alpha -1}e^{-\alpha \lambda }}{\Gamma (\alpha )}. \end{equation*}

Consequently, ${\mathbb{E}}[\Lambda ]=1$ and for $i,j \in \{1,\ldots,h\}$ and $i\neq j$ , we have

\begin{equation*} {\mathbb {E}}[X_{(m),k_j}]=b_{k_j}\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X^2_{(m),k_j}]=b_{k_j}+\frac {\alpha +1}{\alpha }b_{k_j}^2\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X_{(m),k_i}X_{(m),k_j}]=\frac {\alpha +1}{\alpha }b_{k_i}b_{k_j}\,. \end{equation*}

In addition, for claim combinations $m, m_1, m_2\in \mathcal{M}$ , $m_1\neq m_2$ , and $k_1, k_2\in \mathcal{K}$ , we have

\begin{equation*} {\mathbb {E}}[S_{N_{m},{k_1}}S_{N_{m},k_2}]={\mathbb {E}}[N_m]{\mathbb {E}}[X_{(m),k_1}X_{(m),k_2}]+{\mathbb {E}}[N^{(2)}_m]{\mathbb {E}}[X_{(m),k_1}]{\mathbb {E}}[X_{(m),k_2}]\,, \end{equation*}

and

\begin{equation*} {\mathbb {E}}[S_{N_{m_1},{k_1}}S_{N_{m_2},k_2}]={\mathbb {E}}[N_{m_1}N_{m_2}]{\mathbb {E}}[X_{(m_1),k_1}]{\mathbb {E}}[X_{(m_2),k_2}]\,. \end{equation*}

Let

\begin{equation*}\psi _{\mathcal {S}_{\mathbf {N}}}(\mathbf {t})={\mathbb {E}}\left [exp(i\cdot tr(\mathbf {t}^\top \mathbf {S}_{\mathbf {N}}))\right ],\end{equation*}

where

\begin{align*} \mathbf{t}=\left ((t_{1,1},\ldots, t_{K,1})^\top,\ldots,(t_{1,M},\ldots, t_{K,M})^\top \right )\,, \end{align*}

denote the characteristic function (c.f.) of $\mathbf{S}_{\mathbf{N}}$ . Let $\mathcal{P}_{\mathbf{N}}(\cdot )$ denote the probability generating function (p.g.f.) of $\mathbf{N}$ and $\phi _{\mathbf{X_{(m)}}} (\cdot )$ the c.f. of $\mathbf{X_{(m)}}$ . Then

\begin{align*} &\psi _{\mathcal{S}_{\mathbf{N}}}(\mathbf{t}) ={\mathbb{E}}\left [exp\left (i\sum _{m=1}^M\sum _{k=1}^K t_{k,m}\sum _{j=1}^{N_m}X_{(m)j,k}\right )\right ]\\=&{\mathbb{E}}\left [\prod _{m=1}^M\prod _{j=1}^{N_m} exp\left (it_{k,m}\sum _{k=1}^K X_{(m)j,k}\right )\right ]={\mathbb{E}}\left [{\mathbb{E}}\left [\prod _{m=1}^M\prod _{j=1}^{N_m} exp\left (it_{k,m}\sum _{k=1}^K X_{(m)j,k}\right )\Biggl |\mathbf{N}\right ]\right ]\\=&{\mathbb{E}}\left [\prod _{m=1}^M\prod _{j=1}^{N_m}{\mathbb{E}}\left [exp\left (it_{k,m}\sum _{k=1}^K X_{(m)j,k}\right )\right ]\right ]={\mathbb{E}}\left [\prod _{m=1}^M\left ({\mathbb{E}}\left [ exp\left (it_{k,m}\sum _{k=1}^K X_{(m),k}\right )\right ]\right )^{N_m}\right ] \\=&\mathcal{P}_{\mathbf{N}}\left (\phi _{\mathbf{X_{(1)}}}(t_{1,1},\ldots, t_{K,1}),\ldots,\phi _{\mathbf{X_{(M)}}}(t_{1,M},\ldots, t_{K,M})\right )\\ =&\left [1-\beta \left (q_1\phi _{\mathbf{X_{(1)}}}(t_{1,1},\ldots, t_{K,1})+\ldots +q_M\phi _{\mathbf{X_{(M)}}}(t_{1,M},\ldots, t_{K,M})-1\right )\right ]^{-r}. \end{align*}

The c.f. of $\mathbf{S}_{\mathbf{L}^{[m]}}$ , $\mathcal{S}_{{\mathbf{L}^{(2)}}^{[m]}}$ , and $\mathcal{S}_{\mathbf{L}^{[m_1,m_2]}}$ can be derived similarly. Specifically,

\begin{align*} \psi _{\mathcal{S}_{\mathbf{L}^{[m]}}}(\mathbf{t})=\left [1-\beta \left (q_1\phi _{\mathbf{X_{(1)}}}(t_{1,1},\ldots, t_{K,1})+\ldots +q_M\phi _{\mathbf{X_{(M)}}}(t_{1,M},\ldots, t_{K,M})-1\right )\right ]^{-(r+1)}, \end{align*}

and

\begin{align*} &\psi _{\mathcal{S}_{{\mathbf{L}^{(2)}}^{[m]}}}(\mathbf{t})=\psi _{\mathcal{S}_{\mathbf{L}^{[m_1,m_2]}}}(\mathbf{t})\\&=\left [1-\beta \left (q_1\phi _{\mathbf{X_{(1)}}}(t_{1,1},\ldots, t_{K,1})+\ldots +q_M\phi _{\mathbf{X_{(M)}}}(t_{1,M},\ldots, t_{K,M})-1\right )\right ]^{-(r+2)}. \end{align*}

With the above, all steps in computation procedure 3.1 can be carried out, and the risk analysis of $\mathbf{S}_{\mathbf{N}}$ can be performed.

We remark that the selection of the distribution of $\mathbf{N}$ and $\mathbf{X}_{(m)}$ is arbitrary in this section. Other distributions of $\mathbf{N}$ and $\mathbf{X}_{(m)}$ can be used as long as their moments, characteristic function, and moment transforms can be evaluated.

4.1.1 An example with two types of risks

We apply the setting in Example 2.1 where two types of claims, PD and BI, are considered. Recall that the claim number vector is $\mathbf{N} = (N_1, N_2, N_3)^\top$ and the claim sizes be $\mathbf{X}_{(1)}=(X_{(1),1},0)^\top$ , $\mathbf{X}_{(2)}=(0,X_{(2),2})^\top$ , and $\mathbf{X}_{(3)}=(X_{(3),1},X_{(3),2})^\top$ , respectively.

We assume that $\mathbf{N} \sim \text{HMN}(W, q_1=0.9, q_2 = 0.02, q_3= 0.08)$ and $W \sim \text{NB}(r=10, \beta =1)$ .

Let ${X_{(1),1}}\sim \text{Poi}(a_1=1)$ , ${X_{(2),2}}\sim \text{Poi}(a_2=5)$ , and $\mathbf{X}_{(3)}$ follow a common Poisson mixture, where conditional on $\Lambda =\lambda$ , ${X_{(3),1}}\sim \text{Poi}(1.2\lambda )$ and ${X_{(3),2}}\sim \text{Poi}(6\lambda )$ , and $\Lambda$ follows a gamma distribution with shape parameter $\alpha =2$ and mean one.

These parameter values are selected hypothetically to reflect the fact that accidents that cause only PDs usually have high frequency and low severity; it is unlikely ( $q_2 = 0.02$ ) that an accident causes BI but no PDs; accidents that cause both BI and PDs have low frequency and high severity. Note that we assume discrete distributions for the claim sizes for simplicity. If continuous distributions are assumed, they need to be discretized to apply the FFT or recursive methods.

The proportions of capital allocated to the two types of risks according to TCE with selected values of $q$ in $(0, 1)$ are plotted in Figure 1, panel (a). It shows that the proportion of risk capital allocated to PD (BI) claims decreases (increases) with $q$ . When $q$ is small, more capital is allocated to PD claims, whereas more risk capital is allocated to BI claims when $q$ is large.

The proportions of capital allocated according to TV are shown in panel (b) of Figure 1. We observe that when $q$ is small, the proportion allocated to BI claims is a decreasing function of $q$ , and when $q$ is large, the proportion allocated to BI claims increases with $q$ . The opposite pattern is observed for PD claims.

The amounts of capital allocated to the two types of risks according to both TCE and TV criteria increase with $q$ , as shown in panels (c) and (d) of Figure 1.

Figure 1 The proportions and amounts of capital allocated to the two types of risks according to TCE and TV criteria.

Figure 2 compares the proportions of capital allocated to the two types of risks according to TCE and TV criteria obtained by using the moment transform method proposed in this paper and those by using the Monte Carlo simulation (with $10^7$ runs). It shows that the results based on the moment transform are accurate. We note that the moment transform method takes much less computation time than the Monte Carlo simulation.

The capital allocation to the three combinations of risk types according to TCE and TV criteria can be performed following the same procedure. To avoid redundancy, we omit the analysis here.

Tables 1 and 2 show the numerical values of the amounts and proportions of capital allocated to the two types of risk according to TCE and TV criteria for the HMN models under different values of $\alpha$ , which is the parameter of the mixing random variable $\Lambda$ for $\mathbf{X}_{(3)}=(X_{(3),1}, X_{(3),2}$ ). Since $\text{Var}({X_{(3),1}})=b_1+b_1^2/\alpha$ , $\text{Var}({X_{(3),2}})=b_2+b_2^2/\alpha$ , and

\begin{equation*} r(X_3,X_4)=\frac {\text {Cov}(X_3,X_4)}{\sqrt {\text {Var}(X_3)}\sqrt {\text {Var}(X_4)}}=\frac {b_1b_2/\alpha }{\sqrt {b_1+b_1^2/\alpha }\sqrt {b_2+b_2^2/\alpha }}=\sqrt {\frac {b_1b_2}{\alpha ^2+(b_1+b_2)\alpha +b_1b_2}}, \end{equation*}

we see that the variance of ${X_{(3),1}}$ and ${X_{(3),2}}$ and their correlation increases as $\alpha$ decreases. In particular, when $\alpha \rightarrow \infty$ , ${X_{(3),1}}$ , and ${X_{(3),2}}$ are uncorrelated; when $\alpha \rightarrow 0$ , the corelation coefficient approaches to one.

Table 1. Comparison of the amounts of capital allocated to the two risk types under Tail Conditional Expectation (TCE) criterion

Table 2. Comparison of the amounts and proportions of capital allocated to the two risk types under Tail Variance (TV) criterion

Figure 2 Comparison of the simulated and theoretical results.

From Tables 1 and 2, we observe that larger variance of and stronger dependence between ${X_{(3),1}}$ and ${X_{(3),2}}$ lead to greater values of VaR and TCE of the total losses and higher proportion of capital allocated to BI risks.

4.1.2 An example with three types of risks

In this subsection, we consider the automobile insurance claim model discussed by Frees and Valdez (Reference Frees and Valdez2008), in which three types of claims, own damage (OD), third-party property (TPP), and third-party injury (TPI), are considered. An accident can cause any combination of the three types of claims with proportions shown in Table 3. Frees and Valdez (Reference Frees and Valdez2008) proposed a hierarchical, three-component (loss frequency, severity, and dependence) regression model to analyze this highly complex data structure.

Table 3. Possible combinations and their occurrence frequencies

In this example, we study the risk measure and capital allocation problem for the model. We assume that the vector of the number of claim combinations is given by

\begin{equation*}\mathbf {N} = (N_1, N_2, \ldots, N_7)^\top \sim \text {HMN}(W,q_1,\ldots,q_7),\end{equation*}

where $W\sim \text{NB}(r=10,\beta =1)$ , with $q_1=0.004$ , $q_2=0.732$ , $q_3=0.123$ , $q_4=0.003$ , $q_5=0.001$ , $q_6=0.135$ , $q_7=0.002$ . The claim size vectors are denoted by $\mathbf{X}_{(1)}=(X_{(1),1},0,0)^\top$ , $\mathbf{X}_{(2)}=(0,X_{(2),2},0)^\top$ , $\mathbf{X}_{(3)}=(0,0,X_{(3),3})^\top$ , $\mathbf{X}_{(4)}=(X_{(4),1},X_{(4),2},0)^\top$ , $\mathbf{X}_{(5)}=(X_{(5),1},0,X_{(5),3})^\top$ , $\mathbf{X}_{(6)}=(0,X_{(6),2},X_{(6),3})^\top$ , and $\mathbf{X}_{(7)}=(X_{(7),1},X_{(7),2},X_{(7),3})^\top$ .

Frees and Valdez (Reference Frees and Valdez2008) fitted the claim sizes by the generalized beta of the second kind (GB2) distribution and modelled their dependence by multivariate t-copula. Here, for illustration of our method, we simply assume that ${X_{(1),1}}\sim \text{Poi}(a_1)$ , ${X_{(2),2}}\sim \text{Poi}(a_2)$ , ${X_{(3),1}}\sim \text{Poi}(a_3)$ , and the non-zero elements of $\mathbf{X}_{(4)}$ , $\mathbf{X}_{(5)}$ , $\mathbf{X}_{(6)}$ , $\mathbf{X}_{(7)}$ follow common Poisson mixtures. Specifically, let $\Lambda _i$ for $i=1,2,3,4$ assumed to be independent; all follow a gamma distribution with shape parameter $\alpha =2$ and mean one. Conditional on $\Lambda _1=\lambda _1$ , ${X_{(4),1}}$ and ${X_{(4),2}}$ are independent Poisson variables $\text{Poi}(b_1\lambda _1)$ and $\text{Poi}(b_2\lambda _1)$ ; conditional on $\Lambda _2=\lambda _2$ , ${X_{(5),1}}$ and ${X_{(5),3}}$ are independent Poisson variables $\text{Poi}(b_1\lambda _2)$ and $\text{Poi}(b_3\lambda _2)$ ; conditional on $\Lambda _3=\lambda _3$ , ${X_{(6),2}}$ and ${X_{(6),3}}$ are independent Poisson variables $\text{Poi}(b_2\lambda _3)$ and $\text{Poi}(b_3\lambda _3)$ ; and conditional on $\Lambda _4=\lambda _4$ , ${X_{(7),1}}$ , ${X_{(7),2}}$ , and ${X_{(7),3}}$ are independent Poisson variables $\text{Poi}(b_1\lambda _4)$ , $\text{Poi}(b_2\lambda _4)$ , and $\text{Poi}(b_3\lambda _4)$ . The parameter values are set to be $a_1=5$ , $a_2=1$ , $a_3=0.8$ , $b_1=6$ , $b_2=1.2$ , and $b_3=0.96$ .

The proportions and amounts of capital allocated to the three types of risks according to TCE and TV with selected values of ${q}$ are plotted in Figure 3. As we can see, the pattern for TPI (OD) is similar to BI (PD) in the last example, and the pattern for TPP is somewhat in the middle.

Figure 3 The proportions and amounts of capital allocated to the three types of risks under TCE and TV criteria.

Remark 4.1. Applying simulation methods to estimate risk measures and compute capital allocations for this complex hierarchical risk model can be time-consuming and/or inaccurate. Our proposed method, based on moment transform and FFT, can solve the problem efficiently. This is especially true because, as pointed out in Remark 3.3, we do not need to apply multivariate IFFT to obtain the joint distribution of the seven possible combinations of the three types of losses. Instead, we only need to perform one-dimensional IFFT to the diagonal terms to get the distribution of the total.

4.2 A model with dependent claim frequency and size

In this subsection, we study a model in which the claim frequency and size are dependent through a common mixing variable, $\Xi$ defined on $(0,\ \infty )$ . Similar to the example in Section 4.1.1, we suppose that insurance policies cover two types of claims, PD and BI. The claim frequency vector $\mathbf{N} = (N_1, N_2, N_3)^\top$ follows the $\text{HMN}(W, q_1, q_2, q_3)$ distribution defined in Section 4.1, where $W\sim \text{NB}(r\xi, \beta )$ . The claim sizes are denoted by $\mathbf{X}_{(1)}=(X_{(1),1}, 0)^\top$ , $\mathbf{X}_{(2)}=(0, X_{(2),2})^\top$ , and $\mathbf{X}_{(3)}=(X_{(3),1}, X_{(3),2})^\top$ , respectively. We assume that, conditional on $\Xi =\xi$ , ${X_{(1),1}}\sim \text{Poi}(a_1\xi )$ , ${X_{(2),2}} \sim \text{Poi}(a_2\xi )$ , and $\mathbf{X}_3$ follows a common Poisson mixture, where conditional on $\Lambda =\lambda$ , ${X_{(3),1}}\sim \text{Poi}(b_1\lambda \xi )$ and ${X_{(3),2}}\sim \text{Poi}(b_2\lambda \xi )$ . Finally, we assume that $\Xi$ and $\Lambda$ are independent and follow gamma distribution with shape parameters $\alpha _1$ and $\alpha _2$ , respectively. Both have unit mean.

The characteristic function of $\mathbf{S}_{\mathbf{N}}$ is given by

(4.1) \begin{align} \nonumber &\psi _{\mathcal{S}_{\mathbf{N}}}(\mathbf{t})={\mathbb{E}}\left [{\mathbb{E}}\left [exp\left (i\sum _{m=1}^3\sum _{k=1}^2 t_{k,l}\sum _{j=1}^{N_m(\Xi )}X_{(m)j,k}(\Xi )\right )\Biggl |\Xi \right ]\right ]\\ \nonumber =&{\mathbb{E}}\left [{\mathbb{E}}\left [\prod _{m=1}^3\left ({\mathbb{E}}\left [ exp\left (it_{k,m}\sum _{k=1}^2 X_{(m),k}(\Xi )\right )\right ]\right )^{N_m(\Xi )}\Biggl |\Xi \right ]\right ] \\ \nonumber =&{\mathbb{E}}\left [{\mathbb{E}}\left [\mathcal{P}_{\mathbf{N}(\Xi )}\left (\phi _{\mathbf{X_{(1)}}(\Xi )}(t_{1,1},t_{2,1}),\phi _{\mathbf{X_{(2)}}(\Xi )}(t_{1,2},t_{2,2}),\phi _{\mathbf{X_{(3)}}(\Xi )}(t_{1,3}, t_{2,3})\right )\Biggl |\Xi \right ]\right ]\\ =&{\mathbb{E}}\left [\mathcal{P}_{\mathbf{N}(\Xi )}\left (\phi _{\mathbf{X_{(1)}}(\Xi )}(t_{1,1},t_{2,1}),\phi _{\mathbf{X_{(2)}}(\Xi )}(t_{1,2},t_{2,2}),\phi _{\mathbf{X_{(3)}}(\Xi )}(t_{1,3}, t_{2,3})\right )\right ]. \end{align}

Since it is difficult to calculate the explicate expression of integral in Equation (4.1), even with the simple assumptions for the distributions of claim frequency and severity, in computation, we discretize the distribution of $\Xi$ and compute the expectation in Equation (4.1) numerically.

The computation for capital allocation can be performed by using the following equations. For $m,m_1,m_2=1,2,3$ and $k,k_1,k_2=1,2$

\begin{align*}{\mathbb{E}}[S_{N_m,k}|S_{\bullet }\gt s_q]&=\frac{{\mathbb{E}}[S_{N_m,k}I(S_{\bullet }\gt s_q)]}{\Pr (S_{\bullet }\gt s_q)}=\frac{{\mathbb{E}}\left [{\mathbb{E}}[S_{N_m,k}I(S_{\bullet }\gt s_q)|\Xi ]\right ]}{\Pr (S_{\bullet }\gt s_q)}\\ &=\frac{{\mathbb{E}}\left [{\mathbb{E}}[S_{N_m,k}(\Xi )|\Xi ]\Pr \left (\widetilde{S_{\bullet }}^{m[k]}(\Xi )\gt s_q|\Xi \right )\right ]}{\Pr (S_{\bullet }\gt s_q)}\,, \end{align*}

and

\begin{align*}{\mathbb{E}}[S_{N_{m_1},k_1}S_{N_{m_2},k_2}|S_{\bullet }\gt s_q]&=\frac{{\mathbb{E}}[S_{N_{m_1},k_1}S_{N_{m_2},k_2}I(S_{\bullet }\gt s_q)]}{\Pr (S_{\bullet }\gt s_q)}\\&=\frac{{\mathbb{E}}\left [{\mathbb{E}}[S_{N_{m_1},k_1}(\Xi )S_{N_{m_2},k_2}(\Xi )|\Xi ]\Pr \left (\widetilde{S_{\bullet }}^{m_1,m_2[k_1,k_2]}(\Xi )\gt s_q|\Xi \right )\right ]}{\Pr (S_{\bullet }\gt s_q)}\,. \end{align*}

Conditional on $\Xi =\xi$ , the above quantities can be calculated by applying Theorem 3.1. Specifically, for $m=1,2,3$ and $k_1,k_2=1,2$ , we have the following:

\begin{equation*} {\mathbb {E}}[S_{N_{m},{k_1}}(\xi )S_{N_{m},k_2}(\xi )]={\mathbb {E}}[N_m(\xi )]{\mathbb {E}}[X_{(m),k_1}(\xi )X_{(m),k_2}(\xi )]+{\mathbb {E}}[N^{(2)}_m(\xi )]{\mathbb {E}}[X_{(m),k_1}(\xi )]{\mathbb {E}}[X_{(m),k_2}(\xi )], \end{equation*}

where

\begin{equation*} {\mathbb {E}}[N_m(\xi )]=r\xi \beta q_m\,, \ \ m=1,2,3\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[N^{(2)}_m(\xi )]=r\xi (r\xi +1)\beta ^2 q_m^2\,, \ \ m=1,2,3\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X_{(m),k}(\xi )]=a_k\xi \,, \ \ m=1,2,3, \ k=1,2, \ \text {and}\ X_{(m),k}\neq 0\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X^2_{(m),k}(\xi )]=a_k\xi +a_k^2\xi ^2, \ \ m=1,2,\ k=1,2, \ \text {and}\ X_{(m),k}\neq 0\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X^2_{(3),k}(\xi )]=b_k\xi +\frac {\alpha _2+1}{\alpha _2}b_k^2\xi ^2, \ \ k=1,2\,, \end{equation*}

\begin{equation*} {\mathbb {E}}[X_{(3),1}(\xi )X_{(3),2}(\xi )]=\frac {\alpha _2+1}{\alpha _2}b_1b_2\xi ^2\,. \end{equation*}

For $m_1,m_2=1,2,3$ , $k_1,k_2=1,2$ , and $m_1\neq m_2$ , we have

\begin{equation*} {\mathbb {E}}[S_{N_{m_1},{k_1}}(\xi )S_{N_{m_2},k_2}(\xi )]={\mathbb {E}}[N_{m_1}(\xi )N_{m_2}(\xi )]{\mathbb {E}}[X_{(m_1),k_1}(\xi )]{\mathbb {E}}[X_{(m_2),k_2}(\xi )], \end{equation*}

where

\begin{equation*} {\mathbb {E}}[N_{m_1}(\xi )N_{m_2}(\xi )]=r\xi (r\xi +1)\beta ^2 q_{m_1} q_{m_2}\,, \ \ m=1,2,3\,. \end{equation*}

Then, the expectation with regard to $\Xi$ can be computed numerically by discretizing the distribution of $\Xi$ .

In the following, we set the value of the shape parameter of the gamma distributed variable $\Xi$ to $\alpha _1=10$ and assume that all other parameters are the same as those in Section 4.1.1.

The proportions and amounts of capital allocated to the two types of risks according to TCE and TV are shown in Figure 4. It shows a similar pattern to that in Figure 1 in Section 4.1.1. That is, according to TCE, the proportion of risk capital allocated to PD (BI) claims decreases (increases) with ${q}$ ; and according to TV, the proportion allocated to BI(PD) claims decreases (increases) with ${q}$ when ${q}$ is small and increases (decreases) when ${q}$ is large. Also, the amounts of capital allocated to the two types of risks increase with ${q}$ in all cases.

Tables 4 and 5 show the numerical values of the amounts of capital allocated to the two types of risk according to TCE and TV criteria for different values of $\alpha _1$ , the coefficient of the mixing random variable $\Xi$ . Recall that a smaller value of $\alpha _1$ indicates a larger variance of $\Xi$ , and a stronger dependence between the claim frequency and severity. When $\alpha _1\rightarrow \infty$ , the loss frequency and severities are independent.

Table 4. Comparison of the amounts and proportions of capital allocated to the two types of risks according to Tail Conditional Expectation (TCE) criterion when the dependence between loss frequency and sizes changes

Figure 4 The proportions and amounts of capital allocated to the two types of risks according to TCE and TV criteria for the model with dependent frequency and severity.

Table 5. Comparison of the amounts and proportions of capital allocated to the two types of risks according to Tail Variance (TV) criterion when the dependence between loss frequency and sizes changes

From Tables 4 and 5, we conclude that larger variance of the claim frequency and size, and stronger dependence between them, leads to greater values of VaR, TCE and TV of the total losses. The total capital allocated to each type of risk also increases.