1. Introduction
The classical compound Poisson risk model has been extensively analyzed in the actuarial literature. One of the key assumptions of this model is that the inter-claim times and the claim amounts are independent. This assumption can be rather restrictive in applications. For example, in the case of earthquake damages, it is usually believed that the longer the period between earthquakes, the greater the damages expected.
In this paper, we consider a compound Poisson risk model in which the inter-claim time and the subsequent claim size are statistically dependent. Specifically, we assume that the claim sizes
,
are non-negative independent and identically distributed (i.i.d.) random variables (rvs) with a common distribution function (df)
. The claim arrival process
is modelled as a homogeneous Poisson process with intensity
. Let
,
denote the
ith inter-claim waiting time. Then, the following i.i.d. exponential distribution has rate
. Crucially, we assume that the bivariate random vectors
are mutually independent, but that the rvs
and
are no longer independent. As usual, the aggregate claim process
over a finite time horizon
is defined as
Risk models that consider the dependence between the waiting time
and the claim size
have been studied extensively in the literature. For example, Boudreault et al. [
1] introduced a dependence structure where the conditional density of
is defined through a mixture of functions. They provided explicit expressions for quantities of interest, such as the ruin probability and the Gerber-Shiu function for a large class of claim size distributions. Asimit and Badescu [
2] proposed a general dependence structure for
via the conditional tail probability of
. As stated in [
3], this dependence structure is satisfied by several commonly used bivariate copulas and allows for both positive and negative dependencies. It is also very useful for analyzing the tail behavior of the sum or product of two dependent random variables. Under this dependence structure and assuming that the distribution of the claim amounts has a heavy tail, Asimit and Badescu [
2] derived the asymptotic finite-time ruin probabilities and asymptotic results for Value at Risk (VaR) and Tail Conditional Expectation (TCE) of the aggregate losses. For other applications of this dependence structure in risk analysis and probability theory, one may refer to, for example, [
3,
4,
5], among others. Bargès et al. [
6] studied the moments of the compound Poisson sums when the dependence between the inter-claim time and the subsequent claim size is modelled by a Farlie-Gumbel-Morgenstern copula. Zhang and Chen [
7] provided closed-form formulas for the densities of the discounted aggregate claims by assuming that the dependence is through mixing.
The moment (size-biased) transform of distributions, studied in [
8], is a useful statistical tool, which has been exploited in many research areas. In risk management, for example, Furman and Landsman [
9] applied moment transforms to compute the TCE. Further, Furman and Landsman [
10] showed that the Tail Variance (TV) and other weighted risk measures can also be determined by moment transforms. More recently, Denuit [
11] obtained the size-biased transform of compound sums and illustrated their applications in determining the TCE. Ren [
12] studied the moment transform of both univariate and multivariate compound sums, and derived formulas to efficiently compute TCE, TV and higher tail moments.
In this paper, as detailed in
Section 2, we assume that the dependence between the waiting time
and the claim size
is as proposed in [
2]. We apply moment transforms to analyze TCE and TV of the risk process with dependence. Our approach generalizes that proposed in [
2], which is based on extreme value theory. It allows us to derive the asymptotic results for the TCE, TV, and even higher tail moments. In addition, our numerical examples show that our asymptotic results provide more accurate values of TCE than those computed using the method in Asimit and Badescu [
2].
The remainder of this paper is organized as follows.
Section 2 provides some preliminary results and definitions needed.
Section 3 presents asymptotic results for the first two tail moments of the aggregate claims with heavy-tailed claim amounts.
Section 4 provides numerical examples with detailed computations to illustrate the results we obtained and compares with the existing results.
Section 5 concludes.
4. Numerical Results
In this section, we present some numerical examples to examine the accuracy of the asymptotic results obtained in
Section 3.
Asimit and Badescu [
2] obtained results for
with distributions of claim sizes belonging to the maximum domain of attraction of Gumbel and Fréchet types, respectively. To facilitate comparison with their results, we next present one example for each scenario.
4.1. Weibull Distributed Claim Size
Assume that
X follows the Weibull distribution with df
for
and
. As indicated in [
18], this distribution is sub-exponential with a non-regularly varying tail, which is in the maximum domain of attraction of Gumbel type, i.e.,
.
In this example, we select , the Poisson intensity = 3, and the time horizon t = 100. We assume that the inter-claim times and claim sizes are dependent through an FGM copula, with the parameter equal to , representing negative dependence, independence, and positive dependence, respectively, between the claim waiting time and claim sizes.
For
, we apply Theorem 1 in [
2]. For
and
, we utilize formulas derived in Theorems 1 and 2 in
Section 3, respectively. Due to the complexity of obtaining exact expressions for these asymptotic results, we use R software Version 4.4.2 to compute the numerical results.
Table 1 presents the asymptotic results of
,
and
under different choices of
q and
.
We next present simulation studies to validate our asymptotic results. For each scenario, rounds of simulations of the risk process are used.
To calculate the tail moment
, we simulated
and used the relationship
where the expression of
is provided in Equation (
3) in
Section 2.1. A similar approach was applied when simulating
.
The simulated results for
,
and
are shown in
Table 2.
Remark 6. The results in Table 1 and Table 2 indicate that when the parameter θ in the FGM copula changes from negative to positive values (the correlation between inter-claim waiting time and subsequent claim size shifts from negative to positive), the tail risk measures , , and slightly decrease. Intuitively, this may be because a negative dependence between claim waiting time and claim size leads to more chance of large claims occurring within a short period of time. To evaluate the accuracy of the asymptotic results, we report in
Table 3 and
Table 4 their relative errors with respect to the simulation results, which is defined as the absolute value of the ratio of asymptotic and simulated results minus 1.
As a comparison, we calculate the relative errors of the asymptotic results of
in [
2], which states that when
,
i.e., asymptotically, the values of
and
are the same. The relative error values are reported in
Table 5.
It is evident from
Table 3 and
Table 4 that our asymptotic results for
and
obtained using Theorems 1 and 2 are fairly accurate. Furthermore, the accuracy of the asymptotic results for
improves gradually when the confidence level
q is high. In addition, by comparing
Table 3 and
Table 5, we see that the accuracy of our results is notably better than that obtained using the formulas in [
2] for every level of
q.
To facilitate a clearer comparison, we present the asymptotic results obtained in our study (“This paper”), the asymptotic results calculated through [
2] (“Asimit and Badescu [
2]”), and simulated results (“Simulation”) of
,
, and
in
Table 6. Specifically, we choose
,
, and consider
, 6, and 9.
From
Table 6, it can be seen that accuracy of the asymptotic results for all quantities of interests
,
and
improve as the parameter
increases. This is because the tail of a Weibull distribution becomes heavier as the parameter
increases and the asymptotic formulas work better for heavier-tailed distributions. For smaller values of
, our results are more accurate than those obtained using [
2].
4.2. Pareto Distributed Claim Size
Assume that
X follows the Pareto (Type 1) distribution with df
for
and
. As mentioned in [
18], this distribution is sub-exponential with a regularly varying tail, i.e.,
. It belongs to the maximum domain of attraction of Fréchet type, i.e.,
.
In this example, we set
. The Poisson parameter and the dependence structure between the inter-claim times and claim sizes are set to the same values as in
Section 4.1. Exact expressions and numerical evaluation of the asymptotic results of
and
in Theorems 1 and 2 can be efficiently carried out, for example, by using the software Mathematica 14.2.
Table 7 presents the asymptotic results of
and
under different choices of
q and
. Please note that the variance and
are infinite for a Pareto distribution with parameter
. Thus, the values of
are not reported for this case.
Similar to
Table 1,
Table 7 shows that a transition from negative to positive dependence between inter-claim waiting time and subsequent claim size leads to slightly smaller values of risk measures.
We simulated the values of
and
based on a sample size of
. Their values under different choices of
q and
are shown in
Table 8.
In
Table 9, we report the relative errors of the asymptotic results of
presented in
Table 7 with respect to the simulated results in
Table 8.
Next, we compare our asymptotic results for
with those in [
2]. As mentioned in Lemma 4, when
,
The calculated values of
using Equation (
8) are presented in
Table 10.
Table 11 reports the relative errors of the asymptotic results of
in
Table 10 with respect to the simulated results in
Table 8.
Table 9 and
Table 11 demonstrate that both our approach and the one presented in [
2] yield rather accurate asymptotic results for
. However, our approach results in smaller relative errors.
We next provide a summary of the asymptotic results obtained in our study, those calculated using [
2], and simulated results of
,
and
(if applicable) in
Table 12. Specifically, we select
,
, and consider
.
Table 12 shows that the asymptotic results in both this paper and those in [
2] are more accurate when the tail of the claim size distribution becomes heavier. For larger values of
(lighter tail cases), the results in this paper provide more accurate values of
.
Remark 7. Combining the results from the two numerical examples we conducted, we claim that the moment transform technique serves as an efficient method for calculating the asymptotic tail moments for aggregate losses. Our approach extends the existing results presented in [2] in two key aspects. Firstly, it enables the derivation of asymptotic results for not only the first tail moment but also the second and higher tail moments of the aggregate claims. Secondly, our methodology enhances the accuracy of existing results.