Journal of Operational Risk
ISSN:
17446740 (print)
17552710 (online)
Editorinchief: Marcelo Cruz
Measuring tail operational risk in univariate and multivariate models with extreme losses
Need to know
 The authors consider operational risk models with weakly tail dependent, heavytailed loss severities and general loss frequency processes.
 Based on capital approximation within the Basel II/III regulatory capital accords, the Loss Distribution Approach is used to analyse operational risks.
 Some limit behaviors for the ValueatRisk and Conditional Tail Expectation of aggregate operational risks are derived.
Abstract
This paper considers some univariate and multivariate operational risk models, in which the loss severities are modeled by some weakly tail dependent and heavytailed positive random variables, and the loss frequency processes are some general counting processes. We derive some limit behaviors for the valueatrisk and conditional tail expectation of aggregate operational risks in such models. The methodology is based on capital approximation within the Basel II/III framework (the socalled loss distribution approach). We also conduct some simulation studies to check the accuracy of our approximations and the (in)sensitivity due to different dependence structures or to the heavytailedness of the severities.
Introduction
1 Introduction
Operational risk is defined by Basel Committee on Banking Supervision (2003) as “the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events”. The qualitative and quantitative modeling of operational risk in the banking and insurance system started to attract more research attention as lessons were learned following the severe loss (£827 million) by the UK merchant bank Barings in 1995 (see, for example, Böcker and Klüppelberg 2005, 2010; ChavezDemoulin et al 2006; Abbate et al 2009; Hernández et al 2013; Peters et al 2016). Under the advanced measurement approach of Basel II (Basel Committee on Banking Supervision 2019), calculating the regulatory capital of operational risk is equivalent to finding the quantile (ie, valueatrisk (VaR)) of aggregate operational loss at a 99.9% confidence level with a holding period of one year. Hence, the tail risk of the aggregate operational loss should be investigated carefully.
Chernobai et al (2007) stated that operational loss is highly rightskewed and can be classified in terms of its severity and frequency as follows:
 •
lowfrequency–lowseverity,
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highfrequency–lowseverity,
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highfrequency–highseverity or
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lowfrequency–highseverity.
The first two types do not have severe consequences for a financial institution and can often be prevented. Losses of the third type are considered implausible and thus are not expected to occur. Operational losses with lowfrequency and highseverity, however, will have serious consequences, including potential bankruptcy. In longterm empirical studies, researchers have found that high severities show characteristics such as high peaks and fat tails (see, for example, Moscadelli 2004; de Fontnouvelle et al 2007). It is natural to consider the use of some typical heavytailed distributions, such as the Pareto and lognormal distributions, to model severities instead of the Gaussian.
There are three general approaches to operational risk management proposed by the Basel Committee on Banking Supervision (BCBS). The first, simplest approach, the basic indicator approach (BIA), is suitable for some smaller firms with simplex transactions. A more sophisticated approach, the standardized approach, is suitable for banks that are in the process of developing their internal loss data. If the internal losses are available, the advanced measurement approach (AMA) is better than the standardized and basic indicator approaches because it captures the data characteristics (see Basel Committee on Banking Supervision 2019).^{1}^{1} 1 The BCBS has simplified the above three methods into a single one, called the standardized measurement approach (Basel Committee on Banking Supervision 2017). The AMA is more risksensitive than these two approaches, and it is a better fit to the collected data than banks’ internally developed models, which include internal or relevant external losses. In addition, both frequency and severity distributions are taken into account to measure the capital requirement.
One of the most popular approaches satisfying the AMA is the loss distribution approach (LDA), which employs banks’ internal and external data to model the probability distribution of their operational losses. As well as being adopted by larger banks, the LDA is a typical technique in actuarial modeling, and hence it is commonly used for solvency requirements in insurance companies (see, for example, Wüthrich and Merz 2008; Peters et al 2011). Under the LDA, operational losses are divided into eight business lines based on the quantitative impact studies of the BCBS, and each line contains seven nonoverlapping loss types (see McNeil et al 2015). Hence, there are 56 categories of operational losses. This inspired us to consider a generalized multivariate operational risk model that applies the LDA.
Under Basel III, conditional tail expectation (CTE) has gradually replaced VaR as a new risk measure to determine the capital requirements under the internal models approach (Basel Committee on Banking Supervision 2013). The latter is used as the minimum capital requirement, whereas the former is used as the average minimum capital charge. In this paper, we derive some approximations for the VaR and CTE based on operational risks (which we call OpVaR and OpCTE, respectively) under different dependence structures. A simple closedform expression for OpVaR in a single operational risk cell was provided by Böcker and Klüppelberg (2005), who describe the severities by some independent and identically distributed (iid) heavytailed random variables. Böcker and Klüppelberg (2010) further derived some asymptotic results for OpVaR in a multivariate model, in which a Lévy copula dependence is used to model different cells, but the severities in each cell are assumed to be iid and the frequency process is a homogeneous Poisson process.
Motivated by Böcker and Klüppelberg (2010), the goal of our paper is to look for some more transparent asymptotic estimates for both the OpVaR and the OpCTE of total aggregate loss with heavytailed severities. First, we consider a single operational risk cell model, in which there is some specific weak tail dependence (ie, asymptotic independence) between the severities. Second, the multivariate operational risk cell model is constructed based on the above dependent singlecell model (that is, in which the severities in the same cell are dependent on each other). In addition, we also allow the frequency processes of different cells to be dependent in three situations. In the first situation, the frequency processes are arbitrarily dependent and not necessarily Poisson processes. In the other two (special) situations, a common frequency process is shared by all cells and the frequency processes are mutually independent. In fact, in reality it is difficult to model the dependence between different frequency processes. For example, a common shock may result in the dependence between the numbers of losses from different cells (see Lindskog and McNeil 2003). Frachot et al (2004) and ChavezDemoulin et al (2006) studied specific dependent frequency processes.
We perform some numerical simulations via the Monte Carlo method to check our main results. The resulting twodimensional plots show that approximations for OpVaR and OpCTE in each case converge reasonably well in a given fixed time interval when the confidence level is relatively high (say, 99.9% or above). The corresponding threedimensional plots involving time $t$ indicate that the planes of simulated and asymptotic estimates on OpVaR or OpCTE appear to almost overlap in the extreme regions. In addition, to illustrate how the simulation parameterization is related to the datagenerating processes of individual risk cells, the (mean) relative errors are further calculated with respect to different tail indexes of loss severities and different dependence parameters between losses. The simulation results show that our theoretical formulas are reasonable and accurate.
The rest of this paper is organized as follows. After some preliminaries in Section 2, in Section 3 we introduce a pairwise asymptotically independent singlecell model in relation to the asymptotic behavior of OpVaR and OpCTE. In Section 4, we extend the singlecell model to a multivariate one and establish some corresponding approximations in three situations. In Section 5 we show the findings of our simulation studies. Section 6 offers some concluding remarks. The proofs of all the main results are given in an online appendix.
2 Preliminaries
2.1 Notational conventions
Throughout the paper, all limit relationships are according to $x\to \mathrm{\infty}$ unless otherwise stated. For two positive functions ${g}_{1}(\cdot )$ and ${g}_{2}(\cdot )$, we write
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${g}_{1}(x)\lesssim {g}_{2}(x)$ or ${g}_{2}(x)\gtrsim {g}_{1}(x)$ if $lim\; sup{g}_{1}(x)/{g}_{2}(x)\le 1$,
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${g}_{1}(x)\sim {g}_{2}(x)$ if $lim{g}_{1}(x)/{g}_{2}(x)=1$,
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${g}_{1}(x)=o({g}_{2}(x))$ if $lim{g}_{1}(x)/{g}_{2}(x)=0$,
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${g}_{1}(x)=O({g}_{2}(x))$ if $$.
For a nondecreasing function $g:\mathbb{R}\mapsto \mathbb{R}$, we denote by ${g}^{\leftarrow}$ the general inverse (that is, for any $y\in \mathbb{R}$, ${g}^{\leftarrow}(y)=inf\{x\in \mathbb{R}:g(x)\ge y\}$, where $inf\mathrm{\varnothing}=\mathrm{\infty}$ by convention).
2.2 Regular variation
Recall that a positive measurable function $h$ on ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$ is said to be regularly varying at $\mathrm{\infty}$ with index $\alpha \in \mathbb{R}$, written as $h\in {\mathrm{RV}}_{\alpha}$, if
$$\underset{x\to \mathrm{\infty}}{lim}\frac{h(xy)}{h(x)}={y}^{\alpha},y>0.$$ 
For a comprehensive study of regular variation, we refer the reader to Resnick (1987) or Embrechts et al (1997). A positive random variable $\xi $ (or its distribution $V$) is said to be regularly varying at the tails if its tail distribution $\overline{V}=1V\in {\mathrm{RV}}_{\alpha}$ for some $\alpha >0$. A typical example is the Pareto distribution of type II,
$$V(x)=1{\left(\frac{1}{x+1}\right)}^{\alpha},x>0,$$  (2.1) 
with parameter $\alpha >0$, which clearly satisfies $\overline{V}\in {\mathrm{RV}}_{\alpha}$.
3 The singlecell case
Motivated by the early work of Böcker and Klüppelberg (2005), we first study the LDA model with pairwise asymptotically independent and regularly varying tail losses in the singlecell case and we aim to establish the asymptotic formulas for both the OpVaR and OpCTE of the aggregate loss.
We note that some weak tail dependence structure between losses may exist when the international monetary and financial system is stable, while some strong tail dependence could be used to model losses if the financial system is fragile or experiencing economic downturns (see, for example, Hartmann et al 2010). The asymptotic independence is a weak tail dependence structure commonly used for modeling losses. Indeed, many empirical studies have indicated that it is an efficient tool for handling catastrophic losses. Venter (2002) discovered that the Frank and Gaussian copulas (both asymptotically independent) have a good fit to a data set of hurricane losses in Delaware and Maryland. Dupuis and Jones (2006) showed by fitting the exceedances over high thresholds that observations above the 80% and 85% thresholds fit the Frank and Clayton copulas (also asymptotically independent) well. See Das et al (2013) and Yuan (2017) for detailed discussions on asymptotically independent losses. In this paper, we use the asymptotic independence to model losses when the economy is stable.
Two random variables ${\xi}_{1}$ and ${\xi}_{2}$ are said to be asymptotically independent if
$$\underset{x\to \mathrm{\infty}}{lim}P({\xi}_{j}>x\mid {\xi}_{i}>x)=0,i\ne j=1,2,$$  (3.1) 
while they are said to be asymptotically dependent if the limit in (3.1) is positive. According to Resnick (1987), asymptotic independence suggests that the probability of one component becoming large is more significant than the probability of two components being simultaneously large, so the latter can be negligible when measuring them both on the same scale. Section 5.1 shows some examples of pairwise asymptotically independent and asymptotically dependent copulas that can be found in Joe (2014).
In this section, we start with the singlecell case, which constitutes the basis of the investigation for the multivariate cell model.
Definition 3.1 (The asymptotically independent singlecell model).
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The severity process: the severity ${({X}_{k})}_{k\in \mathbb{N}}$ is a sequence of pairwise asymptotically iid positive random variables with common distribution ${F}_{1}$ describing the magnitude of each loss event.
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The frequency process: the counting process ${N}_{1}(t)$ represents the number of loss events within the time interval $[0,t]$ for some $t\ge 0$. We denote the mean function by ${\lambda}_{1}(t)=E[{N}_{1}(t)]$.
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The severity process ${({X}_{k})}_{k\in \mathbb{N}}$ and the frequency process ${({N}_{1}(t))}_{t\ge 0}$ are mutually independent.
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The aggregate loss process: the aggregate loss ${S}_{1}(t)$ in $[0,t]$ constitutes a compound process
${S}_{1}(t)={\displaystyle \sum _{k=1}^{{N}_{1}(t)}}{X}_{k},t\ge 0,$ (3.2) with distribution ${G}_{1t}(x)=P({S}_{1}(t)\le x)$.
In such a singlecell model, the OpVaR can be viewed as the quantile of ${G}_{1t}$, and the OpCTE captures the expectation of the aggregate loss beyond the OpVaR.
Definition 3.2 (OpVaR and OpCTE).
Suppose ${G}_{1t}$ is the distribution of the aggregate loss ${S}_{1}(t)$. Then, for some $q\in (0,1)$, the operational VaR up to time $t$ at confidence level $q$ is defined at the $q$quantile as
$${\mathrm{OpVaR}}_{q}({S}_{1}(t))={G}_{1t}^{\leftarrow}(q)=inf\{x\in \mathbb{R}:P({S}_{1}(t)\le x)\ge q\},$$ 
and the operational CTE is defined as
$${\mathrm{OpCTE}}_{q}({S}_{1}(t))=E[{S}_{1}(t)\mid {S}_{1}(t)>{\mathrm{OpVaR}}_{q}({S}_{1}(t))].$$ 
A closedform expression for ${G}_{1t}(x)$ is not available except in a few cases under ideal distributional assumptions. Hence, in general, ${G}_{1t}^{\leftarrow}(q)$ (and therefore ${\mathrm{OpVaR}}_{q}({S}_{1}(t))$ and ${\mathrm{OpCTE}}_{q}({S}_{1}(t))$) cannot be calculated analytically. Our first main result establishes the asymptotic formulas for both ${\mathrm{OpVaR}}_{q}({S}_{1}(t))$ and ${\mathrm{OpCTE}}_{q}({S}_{1}(t))$ as $q\uparrow 1$.
Theorem 3.3.
Consider the asymptotically independent singlecell model with ${\overline{F}}_{\mathrm{1}}\mathrm{\in}{\mathrm{RV}}_{\mathrm{}{\alpha}_{\mathrm{1}}}$ for some ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{0}$. If $$ for some $p\mathrm{>}{\alpha}_{\mathrm{1}}$, then it holds that, as $q\mathrm{\uparrow}\mathrm{1}$,
$${\mathrm{OpVaR}}_{q}({S}_{1}(t))\sim {({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q).$$  (3.3) 
Further, if ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{1}$, then it holds that, as $q\mathrm{\uparrow}\mathrm{1}$,
$${\mathrm{OpCTE}}_{q}({S}_{1}(t))\sim \frac{{\alpha}_{1}}{{\alpha}_{1}1}{({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q).$$  (3.4) 
4 The multivariate cell case
In a multivariate subexponential compound Poisson model, Böcker and Klüppelberg (2010) gave an approximation for the total OpVaR with the dependence between different cells modeled by a Lévy copula, but where the severities in each cell are assumed to be iid.
In this section, we continue to study the total OpVaR and OpCTE of the total aggregate loss. The simplest way to obtain the total OpVaR is just to sum all the OpVaRs of different cells. However, the simplesum OpVaR may under or overestimate the actual total OpVaR due to the heavytailedness of the severity data or the dependence between severity losses, as shown by Böcker and Klüppelberg (2008). As suggested by Böcker and Klüppelberg (2010), we adopt the following multivariate model with asymptotically independent severities to investigate the total OpVaR and OpCTE.
Definition 4.1.
(The asymptotically independent multivariate loss distribution model)
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The cell process: all the operational risk cells, indexed by $i=1,\mathrm{\dots},d$, are described by some asymptotically independent singlecell models in which, for each $i=1,\mathrm{\dots},d$, the severity ${({X}_{k}^{(i)})}_{k\in \mathbb{N}}$ is a sequence of asymptotically independent positive random variables with common distribution ${F}_{i}$, and the frequency process ${({N}_{i}(t))}_{t\ge 0}$ is a general counting process with mean function ${\lambda}_{i}(t)=E[{N}_{i}(t)]$. The aggregate loss process can thus be described by
$${S}_{i}(t)=\sum _{k=1}^{{N}_{i}(t)}{X}_{k}^{(i)}.$$  •
Assume that the severity processes $\{{({X}_{k}^{(i)})}_{k\in \mathbb{N}},i=1,\mathrm{\dots},d\}$ and the frequency processes $\{{({N}_{i}(t))}_{t\ge 0},i=1,\mathrm{\dots},d\}$ are mutually independent.
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The total aggregate loss process: the bank’s total aggregate loss process can be defined as
$$S(t)=\sum _{i=1}^{d}{S}_{i}(t),t\ge 0,$$ with distribution ${G}_{t}(x)=P(S(t)\le x)$.
Similarly to Definition 3.2, in a multivariate cell model, the total OpVaR and OpCTE can respectively be regarded as the quantile of ${G}_{t}$ and the conditional expectation of $S(t)$ beyond $\mathrm{OpVaR}(S(t))$:
${\mathrm{OpVaR}}_{q}(S(t))$  $={G}_{t}^{\leftarrow}(q)$  
and  
${\mathrm{OpCTE}}_{q}(S(t))$  $=E[S(t)\mid S(t)>{\mathrm{OpVaR}}_{q}(S(t))]$ 
for confidence level $q\in (0,1)$ and time $t\ge 0$.
We consider three situations (ie, Theorems 4.2–4.4) in which to investigate the asymptotic behavior of the total OpVaR and OpCTE in the asymptotically independent multivariate loss distribution model. The first situation requires that the first severity process dominates all the others, but allows for arbitrary dependence between the severity processes ${({X}_{k}^{(1)})}_{k\in \mathbb{N}},\mathrm{\dots},{({X}_{k}^{(d)})}_{k\in \mathbb{N}}$. The frequencycounting processes ${({N}_{1}(t))}_{t\ge 0},\mathrm{\dots},{({N}_{d}(t))}_{t\ge 0}$ may also be arbitrarily dependent.
Theorem 4.2.
Consider the asymptotically independent multivariate loss distribution model with ${\overline{F}}_{\mathrm{1}}\mathrm{\in}{\mathrm{RV}}_{\mathrm{}{\alpha}_{\mathrm{1}}}$ for some ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{0}$. If ${\overline{F}}_{i}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}o\mathit{}\mathrm{(}{\overline{F}}_{\mathrm{1}}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$, $i\mathrm{=}\mathrm{2}\mathrm{,}\mathrm{\dots}\mathrm{,}d$, and $$ for some $p\mathrm{>}{\alpha}_{\mathrm{1}}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}d$, then, regardless of the arbitrary dependence between $\mathrm{(}{S}_{\mathrm{1}}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{,}\mathrm{\dots}\mathrm{,}{S}_{d}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{)}$, it holds that, as $q\mathrm{\uparrow}\mathrm{1}$,
$${\mathrm{OpVaR}}_{q}(S(t))\sim {({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q),$$  (4.1) 
and further, if ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{1}$, then
$${\mathrm{OpCTE}}_{q}(S(t))\sim \frac{{\alpha}_{1}}{{\alpha}_{1}1}{({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q).$$  (4.2) 
We note that in Theorem 4.2, for each $i=2,\mathrm{\dots},d$, ${({X}_{k}^{(i)})}_{k\in \mathbb{N}}$ can be arbitrarily dependent, whereas the losses ${({X}_{k}^{(i)})}_{k\in \mathbb{N}}$ in the $i$th cell should be dominated by ${({X}_{k}^{(1)})}_{k\in \mathbb{N}}$ in the first cell. For example, if ${\overline{F}}_{i}\in {\mathrm{RV}}_{{\alpha}_{i}}$ for some ${\alpha}_{i}>0$, $i=1,\mathrm{\dots},d$, and $$, then ${\overline{F}}_{i}(x)=o({\overline{F}}_{1}(x))$, $i=2,\mathrm{\dots},d$.
The following two theorems consider two special multivariate models. The first model requires that the loss events from different cells are affected by a common economic shock, and thus it shares a common frequencycounting process (see similar discussions in Lindskog and McNeil (2003)). The second model deals with independent business lines, which leads to mutually independent aggregate loss processes ${S}_{1}(t),\mathrm{\dots},{S}_{d}(t)$.
Theorem 4.3.
Consider the asymptotically independent multivariate loss distribution model with ${\overline{F}}_{\mathrm{1}}\mathrm{\in}{\mathrm{RV}}_{\mathrm{}{\alpha}_{\mathrm{1}}}$ for some ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{0}$. Assume that the severity processes ${\mathrm{(}{X}_{k}^{\mathrm{(}\mathrm{1}\mathrm{)}}\mathrm{)}}_{k\mathrm{\in}\mathrm{N}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\mathrm{(}{X}_{k}^{\mathrm{(}d\mathrm{)}}\mathrm{)}}_{k\mathrm{\in}\mathrm{N}}$ are mutually independent, and the frequency processes ${N}_{\mathrm{1}}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{=}\mathrm{\cdots}\mathrm{=}{N}_{d}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{=}N\mathit{}\mathrm{(}t\mathrm{)}$ with $\lambda \mathit{}\mathrm{(}t\mathrm{)}\mathrm{=}E\mathit{}\mathrm{[}N\mathit{}\mathrm{(}t\mathrm{)}\mathrm{]}$. If ${\overline{F}}_{i}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{\sim}{c}_{i}\mathit{}{\overline{F}}_{\mathrm{1}}\mathit{}\mathrm{(}x\mathrm{)}$ for some ${c}_{i}\mathrm{\in}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{\infty}\mathrm{)}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}d$, and $$ for some $p\mathrm{>}{\alpha}_{\mathrm{1}}$, then it holds that, as $q\mathrm{\uparrow}\mathrm{1}$,
$${\mathrm{OpVaR}}_{q}(S(t))\sim {\left(\lambda (t)\sum _{i=1}^{d}{c}_{i}\right)}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q),$$  (4.3) 
and further, if ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{1}$, then
$${\mathrm{OpCTE}}_{q}(S(t))\sim \frac{{\alpha}_{1}}{{\alpha}_{1}1}{\left(\lambda (t)\sum _{i=1}^{d}{c}_{i}\right)}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q).$$  (4.4) 
Theorem 4.4.
Consider the asymptotically independent multivariate loss distribution model with ${\overline{F}}_{\mathrm{1}}\mathrm{\in}{\mathrm{RV}}_{\mathrm{}{\alpha}_{\mathrm{1}}}$ for some ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{0}$. Assume that the severity processes ${\mathrm{(}{X}_{k}^{\mathrm{(}\mathrm{1}\mathrm{)}}\mathrm{)}}_{k\mathrm{\in}\mathrm{N}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\mathrm{(}{X}_{k}^{\mathrm{(}d\mathrm{)}}\mathrm{)}}_{k\mathrm{\in}\mathrm{N}}$ are mutually independent, and the frequency processes ${\mathrm{(}{N}_{\mathrm{1}}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{)}}_{t\mathrm{\ge}\mathrm{0}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\mathrm{(}{N}_{d}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{)}}_{t\mathrm{\ge}\mathrm{0}}$ are also mutually independent with intensities ${\lambda}_{i}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{=}E\mathit{}\mathrm{[}{N}_{i}\mathit{}\mathrm{(}t\mathrm{)}\mathrm{]}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}d$. If ${\overline{F}}_{i}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{\sim}{c}_{i}\mathit{}{\overline{F}}_{\mathrm{1}}\mathit{}\mathrm{(}x\mathrm{)}$ for some ${c}_{i}\mathrm{\in}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{\infty}\mathrm{)}$, and $$ for some $p\mathrm{>}{\alpha}_{\mathrm{1}}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}d$, then it holds that, as $q\mathrm{\uparrow}\mathrm{1}$,
$${\mathrm{OpVaR}}_{q}(S(t))\sim {\left(\sum _{i=1}^{d}{c}_{i}{\lambda}_{i}(t)\right)}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q),$$  (4.5) 
and further, if ${\alpha}_{\mathrm{1}}\mathrm{>}\mathrm{1}$, then
$${\mathrm{OpCTE}}_{q}(S(t))\sim \frac{{\alpha}_{1}}{{\alpha}_{1}1}{\left(\sum _{i=1}^{d}{c}_{i}{\lambda}_{i}(t)\right)}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q).$$  (4.6) 
Note that it is complicated to calculate ${\mathrm{OpVaR}}_{q}(S(t))$ and ${\mathrm{OpCTE}}_{q}(S(t))$ directly due to the complexity of the dependence between the severities and between the frequency processes as well as to the heavytailedness of the severities. However, the asymptotic expressions on the righthand sides of (4.1)–(4.6) can be calculated in a less computationally costly way, because we only need to find the general inverse function ${F}_{1}^{\leftarrow}(q)$, and all the others are constants. The details of the calculations are presented in Section 5.
5 Numerical studies
In this section, we first present some typical examples for calculating asymptotic independence (AI) and asymptotic dependence (AD) structures via copulas, and we state some specifications of our proposed models. Then, we perform some Monte Carlo simulation studies by using R software (R Core Team 2021) to verify that approximations (3.3) and (3.4) are reasonable in the singlecell case. Later, we present the numerical studies on the multivariate cell models using a similar method, to demonstrate the asymptotic estimations given by our theorems, as well as some sensitivity analysis for the OpVaR and OpCTE due to different dependence structures or the different heavytailednesses of the severities.
5.1 Copulas
For simplicity, assume that $H$, the joint distribution of $({\xi}_{1},\mathrm{\dots},{\xi}_{n})$, has continuous marginal distributions ${V}_{1},\mathrm{\dots},{V}_{n}$. Then, by Sklar’s theorem, there exists a unique copula $C({u}_{1},\mathrm{\dots},{u}_{n}):{[0,1]}^{n}\mapsto [0,1]$ such that
$$H({x}_{1},\mathrm{\dots},{x}_{n})=C({V}_{1}({x}_{1}),\mathrm{\dots},{V}_{n}({x}_{n})),n\ge 2.$$ 
Example 5.1.
The Frank copula is of the form
$$C({u}_{1},\mathrm{\dots},{u}_{n})=\frac{1}{\theta}\mathrm{log}\left(1+\frac{({\mathrm{e}}^{\theta {u}_{1}}1)\mathrm{\cdots}({\mathrm{e}}^{\theta {u}_{n}}1)}{{({\mathrm{e}}^{\theta}1)}^{n1}}\right),\theta >0.$$  (5.1) 
Example 5.2.
The Clayton copula is of the form
$$C({u}_{1},\mathrm{\dots},{u}_{n})={[\mathrm{max}\{({u}_{1}^{\vartheta}+\mathrm{\cdots}+{u}_{n}^{\vartheta}n+1),0\}]}^{1/\vartheta},\vartheta >0.$$  (5.2) 
It can be verified that if $n$ identically distributed ${\xi}_{1},\mathrm{\dots},{\xi}_{n}$ are dependent through the Frank copula or the Clayton copula, then they are pairwise asymptotically independent, but the following Gumbel copula is a typical asymptotically dependent case.
Example 5.3.
The Gumbel copula is of the form
$$C({u}_{1},\mathrm{\dots},{u}_{n})=\mathrm{exp}\{{({(\mathrm{log}{u}_{1})}^{\rho}+\mathrm{\cdots}+{(\mathrm{log}{u}_{n})}^{\rho})}^{1/\rho}\},\rho \ge 1.$$  (5.3) 
5.2 Model specifications
The model specifications for the numerical studies throughout this section are as follows.
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The severities follow a type II Pareto distribution of the form (2.1) or a lognormal distribution with density function
$$f(x)=\frac{1}{\sqrt{2\pi}\sigma x}\mathrm{exp}\left\{\frac{{(\mathrm{ln}x\mu )}^{2}}{2{\sigma}^{2}}\right\},x>0,$$ (5.4) for parameters $\mu \in \mathbb{R}$ and $\sigma >0$, which will be used to analyze the sensitivity.
 •
The frequency processes are described by ${(N(t))}_{t\ge 0}$, a negative binomial process with probability mass function
$$P(N(t)=k)=\left(\genfrac{}{}{0pt}{}{t+k1}{k}\right){p}^{t}{(1p)}^{k},k\in \mathbb{N},$$ (5.5) for parameter $p\in (0,1)$.
 •
All the numerical studies are conducted with a sample size $m={10}^{6}$.
5.3 The singlecell case
For the simulated estimation ${\widehat{\mathrm{OpVaR}}}_{q}({S}_{1}(t))$, we first generate $m$ samples ${N}_{1,l}(t)$, $l=1,\mathrm{\dots},m$, where ${N}_{1,l}(t)$ is a negative binomial process with parameter $p\in (0,1)$. Then, for each $l=1,\mathrm{\dots},m$, we generate ${N}_{1,l}(t)$ dependent and identically distributed severities ${X}_{k,l}$, $k=1,\mathrm{\dots},{N}_{1,l}(t)$, using a Frank copula of the form (5.1) with parameter $\theta >0$, and with a common Pareto distribution ${F}_{1}$ of the form (2.1) with ${\alpha}_{1}>0$. Then, all the conditions in Theorem 3.3 are satisfied and the aggregate loss can be calculated according to (3.2):
${S}_{1,l}(t)={\displaystyle \sum _{k=1}^{{N}_{1,l}(t)}}{X}_{k,l},l=1,\mathrm{\dots},m.$ 
Thus, the ${\mathrm{OpVaR}}_{q}({S}_{1}(t))$ can be estimated from
${\widehat{\mathrm{OpVaR}}}_{q}({S}_{1}(t))=inf\{x:{\displaystyle \frac{1}{m}}{\displaystyle \sum _{l=1}^{m}}{\mathrm{\U0001d7cf}}_{({S}_{1,l}(t)\le x)}\ge q\}$  (5.6) 
for confidence level $q\in (0,1)$, where ${\mathrm{\U0001d7cf}}_{A}$ denotes the indicator function of a set $A$. The asymptotic value on the righthand side of (3.3) can be calculated as
$${\left(t\frac{1p}{p}\right)}^{1/{\alpha}_{1}}({(1q)}^{1/{\alpha}_{1}}1),$$ 
while ${\mathrm{OpCTE}}_{q}({S}_{1}(t))$ can be estimated from
${\widehat{\mathrm{OpCTE}}}_{q}({S}_{1}(t))={\displaystyle \frac{{\sum}_{l=1}^{m}{S}_{1,l}(t){\mathrm{\U0001d7cf}}_{({S}_{1,l}(t)>{\widehat{\mathrm{OpVaR}}}_{q}({S}_{1}(t)))}}{{\sum}_{l=1}^{m}{\mathrm{\U0001d7cf}}_{({S}_{1,l}(t)>{\widehat{\mathrm{OpVaR}}}_{q}({S}_{1}(t)))}}}.$ 
We are now ready to perform some Monte Carlo simulations for Theorem 3.3. The various parameters are set to $p=0.5$, ${\alpha}_{1}=0.24$ (or ${\alpha}_{1}=1.66$), $\theta =1$, $t\in [4.2,9.8]$. Part (a) of Figure 1 shows the simulated OpVaR as a bivariate function in terms of time $t$ and confidence level $q$ that obviously increases with respect to either $q$ or $t$. Part (c) compares the simulated and asymptotic estimates for the OpVaR with different $t$. It can be seen that both estimates get closer as $q\to 1$. Then, choosing a fixed time $t=7$ and taking the cross profile, from the twodimensional plot in Figure 1(e), we can observe the limiting behavior more clearly for the OpVaR. The analysis of OpCTE is similar to that of OpVaR but with another parameter, ${\alpha}_{1}=1.66>1$, of the Pareto severities, and the graphs are shown in parts (b), (d) and (f) of Figure 1. Here, threedimensional graphs are provided only for the singlecell case, but they can be plotted for the multivariate model in the same way.
Figure 1 illustrates the accuracy of approximations visually, but it is difficult to see the error existing between the asymptotic and simulated estimates. Hence, to illustrate the accuracy precisely, we further calculate the mean relative error (MRE) statistical measure for the estimated OpVaR and OpCTE after repeating the Monte Carlo simulation procedure $n=100$ times. The MRE for the simulated OpVaR can be defined as
$\frac{1}{n}}{\displaystyle \sum _{j=1}^{n}}{\displaystyle \frac{{\widehat{\mathrm{OpVaR}}}_{q}^{j}({S}_{1}(t))}{{({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q)}}1,$ 
where the denominator ${({\lambda}_{1}(t))}^{1/{\alpha}_{1}}{F}_{1}^{\leftarrow}(q)$ in the summand is the asymptotic value on the righthand side of (3.3), and the numerator ${\widehat{\mathrm{OpVaR}}}_{q}^{j}({S}_{1}(t))$ is the estimated value defined by (5.6) in the $j$th repeating procedure. The MRE for the estimated OpCTE can be defined in a similar way to that for OpVaR.
$\bm{q}$  

0.995  0.996  0.997  0.998  0.999  
MRE  1.188  2.137  3.514  3.437  4.274 
$\bm{q}$  

0.995  0.996  0.997  0.998  0.999  
MRE  9.948  8.513  6.894  5.056  2.763 
Now we calculate the MREs for estimated OpVaR and OpCTE with $t=7$, which correspond to those appearing in parts (e) and (f) of Figures 1, respectively, and we summarize them in Tables 1 and 2.
From Tables 1 and 2, it is obvious that all the MREs for OpVaR and OpCTE do not exceed 10%, indicating that our theoretical result is reasonable. More experimental findings are given in Tables 3 and 4. These show that a change in tail index ${\alpha}_{1}$ would not lead to a significant change in the MRE, since all the MREs for both OpVaR and OpCTE are less than 10% when the tail index ${\alpha}_{1}$ changes. This indicates that our asymptotic formulas can be adapted for different distribution parameters.
MREs ($\mathbf{\times}\text{\U0001d7cf\U0001d7ce\U0001d7ce}$) for OpVaR  

Change in  
${\bm{\alpha}}_{\text{\U0001d7cf}}$ (%)  0.995  0.996  0.997  0.998  0.999 
$+$2  0.462  1.375  1.455  1.438  3.974 
$+$1  2.414  2.116  2.936  3.267  1.203 
0  1.188  2.137  3.514  3.437  4.274 
$$1  4.309  3.923  5.621  5.613  0.315 
$$2  0.807  2.022  1.485  0.548  1.477 
MREs ($\mathbf{\times}\text{\U0001d7cf\U0001d7ce\U0001d7ce}$) for OpVaR  

Change in  
${\bm{\alpha}}_{\text{\U0001d7cf}}$ (%)  0.995  0.996  0.997  0.998  0.999 
$+$2  8.402  6.725  4.811  2.470  0.901 
$+$1  8.439  6.827  5.013  2.843  0.161 
0  9.948  8.513  6.894  5.056  2.763 
$$1  8.497  7.021  5.381  3.441  0.936 
$$2  6.629  5.017  3.167  0.891  2.477 
In addition to the simulations for tail index ${\alpha}_{1}$, we investigate the influence of the dependence between loss severities by adjusting the parameter of the corresponding copula. Using the model with independent severities (ie, $\theta =0$) as the benchmark model, we compute the relative errors with respect to the different dependence parameters $\theta $ of the Frank copula of the form (5.1), where the relative error can be computed as
$$\left\frac{{\widehat{\mathrm{OpVaR}}}_{\mathrm{AI}}}{{\widehat{\mathrm{OpVaR}}}_{\mathrm{Independent}}}1\right.$$ 
It can be seen from Tables 5 and 6 that, based on Theorem 3.3, the relative errors are less than 10% regardless of the value of $\theta $. This also shows that the change in dependence between loss severities would not affect our asymptotic formulas.
${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{A}\mathbf{I}}}$ versus ${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{I}\mathbf{N}\mathbf{D}}}$  
$\bm{\theta}$  0.995  0.996  0.997  0.998  0.999 
0  ($\text{1.236}\times {\text{10}}^{\text{13}}$)  ($\text{3.156}\times {\text{10}}^{\text{13}}$)  ($\text{1.029}\times {\text{10}}^{\text{14}}$)  ($\text{5.453}\times {\text{10}}^{\text{14}}$)  ($\text{1.028}\times {\text{10}}^{\text{16}}$) 
0.5  1.279  3.789  7.420  5.961  5.981 
1.0  2.293  5.069  0.423  7.199  8.372 
1.5  2.998  1.900  5.328  5.664  8.413 
${\text{\mathbf{O}\mathbf{p}\mathbf{C}\mathbf{T}\mathbf{E}}}_{\text{\mathbf{A}\mathbf{I}}}$ versus ${\text{\mathbf{O}\mathbf{p}\mathbf{C}\mathbf{T}\mathbf{E}}}_{\text{\mathbf{I}\mathbf{N}\mathbf{D}}}$  

$\bm{\theta}$  0.995  0.996  0.997  0.998  0.999 
0  (206.889)  (234.819)  (276.770)  (349.787)  (524.667) 
0.5  1.064  1.536  2.125  3.014  4.748 
1.0  0.733  1.668  2.846  4.515  7.059 
1.5  0.022  1.135  2.529  4.475  7.202 
5.4 The multivariate cell case
To simplify the simulation studies, we consider the multivariate cell model consisting of $d=3$ nonoverlapping cells in all the cases in Section 4.
We first consider the results based on Theorem 4.2 via situations (1)–(5).
Situation (1).
Under a simplex setting, we assume all severities are Pareto distributed with the form (2.1) and the weak tail dependence within the same cell is constructed through the Frank copula (5.1) with $\theta =1$. For different cells, the three severity sequences ${({X}_{k}^{(1)})}_{k\in \mathbb{N}}$, ${({X}_{k}^{(2)})}_{k\in \mathbb{N}}$, ${({X}_{k}^{(3)})}_{k\in \mathbb{N}}$ are independent. The three frequency processes with fixed time $t$ are generated through the Clayton copula (5.2). The various parameters are set to ${p}_{1}=0.3$, ${p}_{2}=0.4$, ${p}_{3}=0.5$ in (5.5) for the negative binomial frequency processes $\{{({N}_{i}(t))}_{t\ge 0},i=1,2,3\}$ with $t=4$, and ${\alpha}_{1}=0.42$, ${\alpha}_{2}=0.7$, ${\alpha}_{3}=0.62$ (or ${\alpha}_{1}=1.2$, ${\alpha}_{2}=1.6$, ${\alpha}_{3}=1.9$) in (2.1) for the Pareto severities $\{{({X}_{k}^{(i)})}_{k\in \mathbb{N}},i=1,2,3\}$. We note that ${\alpha}_{1}$ is chosen as the smallest parameter, which leads to the first severity sequence dominating all the others. Clearly, $$ for any $v>0$ and $t\ge 0$, $i=1,2,3$. Then all the conditions in Theorem 4.2 are satisfied.
Parts (a) and (b) of Figure 2 plot the simulated and asymptotic estimates for OpVaR for severities with different Pareto indexes. They show that the two estimates increase quickly with increasing confidence level due to the heavytailedness of the total aggregate loss $S(t)$, and the two lines get closer. Comparing parts (a) and (b), it can be seen that the smaller the parameter ${\alpha}_{1}$, the greater are the simulated and asymptotic estimates for OpVaR, showing that OpVaR is very sensitive to ${\alpha}_{1}$. This is because the total aggregate loss $S(t)$ is regularly varying tailed with index ${\alpha}_{1}$. The approximations for OpCTE in (2.1) are shown in part (c) of Figure 2 with parameters ${\alpha}_{1}=1.2$, ${\alpha}_{2}=1.6$ and ${\alpha}_{3}=1.9$.
$\bm{q}$  

0.995  0.996  0.997  0.998  0.999  
MRE  3.217  2.862  2.284  2.028  0.532 
$\bm{q}$  

0.995  0.996  0.997  0.998  0.999  
MRE  17.908  14.887  11.514  7.386  1.361 
$\bm{q}$  

0.995  0.996  0.997  0.998  0.999  
MRE  3.091  2.487  1.828  1.075  0.077 
MREs ($\mathbf{\times}\text{\U0001d7cf\U0001d7ce\U0001d7ce}$) for OpVaR  

Change in  
${\bm{\alpha}}_{\text{\U0001d7cf}}$ (%)  0.995  0.996  0.997  0.998  0.999 
$+$2  6.667  5.254  3.101  2.219  1.861 
$+$1  5.291  3.752  3.095  4.217  3.333 
0  3.217  2.862  2.284  2.028  0.532 
$$1  2.635  2.914  1.382  0.427  2.755 
$$2  1.765  1.218  0.706  1.595  0.291 
MREs ($\mathbf{\times}\text{\U0001d7cf\U0001d7ce\U0001d7ce}$) for OpVaR  

Change in  
${\bm{\alpha}}_{\text{\U0001d7cf}}$ (%)  0.995  0.996  0.997  0.998  0.999 
$+$2  20.256  16.358  14.073  9.703  2.083 
$+$1  18.185  14.270  11.754  7.019  1.159 
0  17.908  14.887  11.514  7.386  1.361 
$$1  16.576  13.329  10.395  6.804  2.032 
$$2  15.167  12.420  9.413  5.882  1.201 
MREs ($\mathbf{\times}\text{\U0001d7cf\U0001d7ce\U0001d7ce}$) for OpVaR  

Change in  
${\bm{\alpha}}_{\text{\U0001d7cf}}$ (%)  0.995  0.996  0.997  0.998  0.999 
$+$2  1.670  1.119  0.522  0.166  1.481 
$+$1  3.296  2.890  2.494  2.180  1.873 
0  3.091  2.487  1.828  1.075  0.077 
$$1  2.545  2.994  3.560  4.269  5.587 
$$2  0.844  0.576  2.034  2.879  4.308 
The MREs for OpVaR and OpCTE are shown in Tables 7–9. Although the MREs sometimes exceed 10% in Table 8, they show a decreasing trend as the confidence level increases, and the MRE for OpVaR reaches 1.361% when the confidence level is 99.9%. All the other MREs in Tables 7 and 9 are less than 5%. This shows our approximations are reasonable. For further illustrations of the parameterization in our proposed method, the results for the OpVaR and OpCTE sensitivity analysis are shown in Tables 10–12.
${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{A}\mathbf{I}}}$ versus ${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{I}\mathbf{N}\mathbf{D}}}$  
$\bm{\theta}$  0.995  0.996  0.997  0.998  0.999 
0  ($\text{6.115}\times {\text{10}}^{\text{7}}$)  ($\text{1.045}\times {\text{10}}^{\text{8}}$)  ($\text{2.079}\times {\text{10}}^{\text{8}}$)  ($\text{5.493}\times {\text{10}}^{\text{8}}$)  ($\text{2.783}\times {\text{10}}^{\text{9}}$) 
0.5  0.270  8.239  0.303  7.258  2.112 
1.0  2.755  5.839  2.588  9.937  2.555 
1.5  0.536  1.276  1.553  6.239  1.685 
${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{A}\mathbf{I}}}$ versus ${\text{\mathbf{O}\mathbf{p}\mathbf{V}\mathbf{a}\mathbf{R}}}_{\text{\mathbf{I}\mathbf{N}\mathbf{D}}}$  

$\bm{\theta}$  0.995  0.996  0.997  0.998  0.999 
0  (614.367)  (726.310)  (898.444)  (1254.647)  (2178.641) 
0.5  1.197  2.013  2.746  2.786  4.356 
1.0  2.854  1.539  2.776  0.280  0.690 
1.5  3.317  2.219  5.013  1.011  4.516 
${\text{\mathbf{O}\mathbf{p}\mathbf{C}\mathbf{T}\mathbf{E}}}_{\text{\mathbf{A}\mathbf{I}}}$ versus ${\text{\mathbf{O}\mathbf{p}\mathbf{C}\mathbf{T}\mathbf{E}}}_{\text{\mathbf{I}\mathbf{N}\mathbf{D}}}$  

$\bm{\theta}$  0.995  0.996  0.997  0.998  0.999 
0  (3275.212)  (3924.804)  (4960.422)  (6916.185)  (12244.778) 
0.5  1.209  1.271  1.375  1.213  1.370 
1.0  2.881  3.067  3.215  3.252  3.391 
1.5  6.281  6.453  6.635  7.079  7.782 
It is obvious that the MREs in Tables 10 and 12 do not yet exceed 10% for different values of ${\alpha}_{1}$, and the MREs in Table 11 show a decreasing trend as the confidence level increases, although there are some values greater than 10%. Tables 13–15 conclude the simulation analysis for relative errors with respect to different dependence parameter $\theta $ based on Theorem 4.2, from which it can be seen that all the relative errors are less than 10%, and this ensures the rationality of our theoretical results.
In the following situations, the MREs and relative errors will not be calculated repeatedly, because the same conclusions can be obtained. Further, we note that the conditions in Theorem 4.2 are relatively general. For example, the severities from different cells are allowed to be arbitrarily dependent. In the following, we consider all kinds of situations to demonstrate the influence of different dependence structures and severity distributions.
Situation (2).
In this situation we use a Clayton copula of the form (5.2) to model the dependence between the severities within the same cell and then we compare the simulated estimates with those obtained via the Frank copula. We set the parameter $\vartheta =1$ in (5.2), and all the other assumptions are the same as those in situation (1). As shown in Figure 3, the curves of the two simulated estimates from the Frank and Clayton copulas almost overlap the asymptotic estimates. This indicates that the simulated OpVaR is insensitive to different asymptotically independent structures, which coincides with the theoretical Theorem 4.2.
Situation (3).
In this situation we analyze the influence of the different dependencies between ${S}_{1}(t)$, ${S}_{2}(t)$ and ${S}_{3}(t)$ and we consider two further cases.
 Situation (3a).

In this situation we generate a long sequence of pairwise asymptotically independent severities $\{{X}_{k}^{(1)},{X}_{k}^{(2)},{X}_{k}^{(3)};k\in \mathbb{N}\}$. This implies that within the same cell the severities are asymptotically independent, and the severities from different cells are weakly tail dependent. More precisely, as in situation (1), we first generate the dependent ${N}_{1}(t)$, ${N}_{2}(t)$ and ${N}_{3}(t)$, then simulate the $({N}_{1}(t)+{N}_{2}(t)+{N}_{3}(t))$dimensional severity matrix $({X}_{1}^{(1)},\mathrm{\dots},{X}_{{N}_{1}(t)}^{(1)},{X}_{1}^{(2)},\mathrm{\dots},{X}_{{N}_{2}(t)}^{(2)},{X}_{1}^{(3)},\mathrm{\dots},{X}_{{N}_{3}(t)}^{(3)})$ via a multivariate Frank copula of the form (5.1). The parameters are set to $t=4$, ${p}_{1}=0.3$, ${p}_{2}=0.4$, ${p}_{3}=0.5$ in (5.5), ${\alpha}_{1}=0.42$, ${\alpha}_{2}=0.58$, ${\alpha}_{3}=0.62$ in (2.1) and $\theta =1$ in (5.1).
 Situation (3b).

In this situation the tail dependence between different cells is changed from weak to strong. Specifically, the random vector $({X}^{(1)},{X}^{(2)},{X}^{(3)})$ is assumed to be dependent via the threedimensional Gumbel copula (5.3), and ${({X}_{k}^{(1)},{X}_{k}^{(2)},{X}_{k}^{(3)})}_{k\in \mathbb{N}}$ are the independent copies of $({X}^{(1)},{X}^{(2)},{X}^{(3)})$. All the other assumptions are the same as those in situation (1). Then, all conditions in Theorem 4.2 are satisfied, but the severities from different cells are strongly tail dependent due to the fact that the Gumbel copula is an asymptotically dependent case. The related parameters are set the same as those in situation (3a), except that $\rho =1.4$ for the Gumbel copula (5.3). Comparing parts (a) and (b) of Figure 4, it can be seen that different dependencies (either weak or strong tail dependencies) between ${S}_{1}(t)$, ${S}_{2}(t)$ and ${S}_{3}(t)$ affect the asymptotic behavior for OpVaR only a little. In other words, arbitrary dependence between ${S}_{1}(t),\mathrm{\dots},{S}_{d}(t)$ is allowed, which is ensured by Theorem 4.2.
Situation (4).
In this situation we show that if the first severity sequence dominates all the others, then the dependence structures in the second or third cell can be chosen arbitrarily. In this situation, we use the Gumbel copula to model the dependence between severities in the second or third cell, and we retain the Frank copula for the first cell, as in situation (1). The parameter for the Gumbel copula (5.3) is set to $\rho =1.2$. As shown in Figure 5, the three estimates almost overlap as $q\to 1$. This indicates that the simulated estimates are insensitive to different dependencies (either weak or strong tail dependencies) in all cells except the first one, which verifies Theorem 4.2.
Situation (5).
In this situation we keep the severities in the first cell to be Pareto distributed, but model the severities in the second or third cell by the lognormal distribution of the form (5.4), then we compare the simulations with the case of all Pareto distributed severities. We set the parameters $\mu =0$ and $\sigma =1$ in (5.4), and all the other assumptions are the same as those in situation (1). In Figure 6, the three curves coincide, which shows that there is no significant influence on the asymptotic behavior, even if the severities from the second or third cell are moderately heavytailed.
Finally, we check the accuracy of the approximations based on Theorems 4.3 and 4.4 by using Figures 7 and 8. In the setting of Theorem 4.3, the common frequency process is assumed to be the negative binomial process (5.5), and the severities from different cells are tail equivalent and Pareto distributed in the form (2.1). All the other assumptions are the same as those in situation (1). The parameters are set to $t=3$, $p=0.5$ in (5.5), ${c}_{1}={c}_{2}={c}_{3}=1$ in (4.5), $\theta =1$ in (5.1) and $\alpha =0.3$ for OpVaR and $\alpha =1.57$ for OpCTE in (2.1).
In the setting of Theorem 4.4, the three frequency processes are assumed to be iid negative binomial processes (5.5), and all the other assumptions remain the same as the settings of Theorem 4.3. The related parameters are set to $t=4$, ${p}_{1}={p}_{2}={p}_{3}=0.5$ in (5.5), ${c}_{1}={c}_{2}={c}_{3}=1$ in (4.5), $\theta =1$ in (5.1) and $\alpha =0.35$ for OpVaR and $\alpha =1.6$ for OpCTE in (2.1).
6 Concluding remarks
In this paper, we considered some dependent univariate and multivariate operational risk models. A major contribution of this study is that in the multivariate cell model the frequency processes can be arbitrarily dependent, and the severities from the same cell are pairwise asymptotically independent. This is suitable for modeling a stable economic environment. Another significant contribution of the paper is the demonstration that the severities can be modeled by some extremely or moderately heavytailed distributions that show the highseverity characteristics of empirical data from the banking and insurance industries. Our models used VaR and CTE to measure operational risks, in line with Basel III. By employing the LDA approach, we established some asymptotic formulas for the OpCTE and OpVaR of aggregate loss as the confidence level increases. Our results could serve as good approximations of capital requirement estimates for operational risks. In future studies, we will consider strong tail dependence between the severities, when the financial system experiences economic downturns. In addition, the more general heavytailed (subexponential) distributions could be candidates for modeling the severities. In our future studies, we will adopt the standardized measurement approach to update our results.
Declaration of interest
The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.
Acknowledgements
We are very grateful to the referees for their constructive suggestions. The research of Yang Yang was supported by the National Social Science Fund of China (grant 22BTJ060), the Humanities and Social Sciences Foundation of the Ministry of Education of China (grant 20YJA910006), the Natural Science Foundation of Jiangsu Province (grant BK20201396) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (grant 19KJA180003). The research of Jiajun Liu was supported by the National Natural Science Foundation of China (grants 72171055, 12201507), the Natural Science Foundation of Jiangsu Higher Education Institutions (grant 21KJB110019), the Xi’an Jiaotong Liverpool University (XJTLU) Postgraduate Research Scholarship (grants PGRS2012012, FOSA200701, FOSA200702) and the XJTLU University Research Fund Project (grant RDF170121).
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