Abstract. by [7] in which insurance risks distribution and

Abstract. We consider the recursive moments of aggregate
discounted claims, where the dependence between the inter-claim time and the
subsequent claim size are captured by a copula distribution. The equations of
the recursive moments, which take the form of the Volterra integral equation
(VIE), are then solved using the Laplace transform. We then compute its mean
and variance, and compare with the results obtained in previous literature.

Keywords: aggregate
discounted claim; copula;
Laplace transform; Volterra integral

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The classical risk model of discounted claims which assumed
claim amounts and inter-claim time to be independent has been studied for
decades in the literature by 1,
and 6.
The independence assumption is no longer appropriate in modeling insurance risk
portfolio which could give rise to reserving and solvency issues, especially
due to the increased frequency of catastrophic event. This is supported by 7
in which insurance risks distribution and level of dependence has significant
impact on the estimated risk measure and is important in assessing the risk
based capital. Dependence between claim occurrences and claim sizes has already
been explored in previous studies such as in 8,
and 11.

The first two moments of aggregate discounted claim was obtained using Martingale approach in 6 and the author then considered jump diffusion process to relax the classical
independence assumption in 12. The explicit expression on the first two moments of discounted claims was derived in 13 in which claim arrivals and claims sizes are correlated whereby the claims sizes and the rates
of claim occurrence are dependent under Markovian environment known as circumstance process. This was then extended in 14 whereby
the higher moments of discounted claims were
obtained using the Laplace transform through a martingale argument.

The dependency between claim amount and inter-claim time in the recursive moments of aggregate discounted claims has been addressed in 15 based on the idea in 9. The general expression for the m-th recursive moments was obtained in 15
assuming dependency between claim severity and inter-claim time conditioned on the first arrival and using a renewal theory argument, which was then solved using the Laplace transform approach. Farlie- Gumbel-Morgenstern (FGM) copula was utilized for its simplicity and mathematically tractable properties to describe the dependency between the variables. The recursive moments obtained in 15 was then extended in 16 using the Volterra integral equation which was then solved by Neumann series expression. The dependency was represented using three copulas which are FGM copula, the Gaussian copula and Gumbel copula in which Gumbel copula showed the most sensitive value on its first and second moments, as well as the premium charged,
with respect to changes in dependence parameter.
The results in 16
are applicable to any claim sizes following continuous distribution so long as
the claim arrival process follows a Poisson distribution. In this paper, the
Volterra integral equation used in 16
will be solved using Laplace transform method and applied to two other copulas
to define the dependency between two exponential marginals.

This paper is organized as follows: Section 2 introduces continuous time renewal risk model in which claims occur according to a Poisson counting process with exponentially distributed inter-claim time. The dependency between claim size and inter-claim time is described by the FGM copula, the Frank copula and the heavy right tail (HRT) copula. Section 3 shows the derivation of the first and second recursive moments of the aggregate discounted claims which are then expressed in terms of Volterra
integral equation. Laplace
transform of the recursive moments will be used to solve this and to allow for the general form of the m-th moments. Moments of the aggregate discounted claims under each copula are summarized in Section 4
with assumption of exponentially distributed claims amount. Section 5 will conclude the article.


Consider a continuous time
renewal risk model,

 as discussed in 15 and 16 here:

The independent and identically
distributed (i.i.d) random variables (r.v.’s)

 represent non-negative
claims amount occurring at time


is a homogenous Poisson counting process. The parameter

represents a deterministic instantaneous rate of net
interest. Meanwhile, the inter-claim arriving time continuous r.v.’s is defined

A sequence of i.i.d random vectors,

is formed by relaxing the independent assumption between the
claim size and the inter-claim time. The components of the vectors,

are dependent. In this study, three copulas are used to
describe the dependency between an inter-claim time and its claim amount,
namely: the FGM copula, the Frank copula and the heavy right tail (HRT) copula.

be dependence parameter, the pdfs of the copulas are given

While the FGM copula allows for simplicity and analytically
tractable structure, the dependence is rather weak. It is included in this
study to ensure that the resulting Laplace transform will result in the same
result as obtained by the Neumann series in 15
and 16.
The Frank copula is chosen can be constructed easily in addition to its ability
to model a wide range of dependency 1.
The HRT copula was selected since it is suitable to model upper tail dependence
which is relevant in studying dependency between extreme values. It was used to
model seismic loss in 17
and was considered in modeling claim size and the delay between its occurrence
time until its reporting time by 18.


In this study, the inter-claim arriving time is
exponentially distributed because it is assumed that the jump occurrences
follow a Poisson distribution 16. Let

follows an exponential distribution with mean

 as in 16 and by using
substitution, the first and the second recursive moments of the aggregate
discounted claims are given by:

The moments can be expressed in
terms of the Volterra integral equation (IE) of the second kind with

is a continuous function in the region


is a difference kernel and a continuous function in the

(see 16 and 19). In this case, the Volterra IE
of the second kind is given by:

The function

for the first and second moments are as follow, respectively:

Meanwhile, the kernel function for
the moments is given by:



denote the Laplace transform of


respectively. Since

is the convolution integral of



Therefore, by replacing Eqn. 5 and Eqn.6 into Eqn. 7, the
value of

can be found using the following:

The solution of

is the inverse Laplace transform of

which is:



Assuming that the claims amount, X, is exponentially distributed with mean


the values of the first and the second moments under each
copula are summarized in Table 1 and Table 2. The values for the first and
second moments under FGM copula in Table 1 and Table 2 at ? = -0.9, 0 and 0.9 are similar to results obtained in Table 1 of 16 under the Neumann series method. We also note that the
values for the second moment under the BCLM method in Table 1 of 16 contain programming error which has been
corrected and the resulting values are the same as in 15.
Therefore, the expression of moments derived in Section 3 under the Laplace
transform method is correct as it yields similar results as in 15
and 16.

The Frank copula is a two-sided copula capturing both
positive and negative dependencies 20.
The two marginals are positively correlated
if its dependence parameter, ? ? (0,1) and
inversely correlated if ? ? (1,?) (i.e. a large claim size
following a short inter-waiting time, vice versa). Meanwhile, the value of ?Frank?1 indicates an independence between the
marginals. The spread, which is the difference of moments under the two extreme
ends of ?Frank, i.e. ?Frank = 0.005 and ?Frank?? is 115.417 (first moment) and 217,712 (second moment),
which is larger than values under the FGM copula. Moments under the HRT copula
are comparable with the moments values under the FGM copula when ?FGM
? 0. As ?HRT??, the moment values converge towards the
values under the independent case of FGM copula (?FGM = 0) and Frank copula (?Frank = 1).

We can see that among the three copulas, HRT copula produces
the highest spread for both moments. Nevertheless, the moment values are capped
at 453.173 (first moment) and 287786
(second moment) which are values when the marginals are independent.
Additionally, although the spreads under Frank and FGM copulas are smaller than
the HRT, the moments values take into consideration of frequent severe claims.
Insurers would have to charge a higher
premium to policyholders should they expect frequent severe claims coming in
the future. This effect can be illustrated easily using the expected value
premium principle, the variance premium principle or the standard deviation
premium principle, as the following:


In this study, we chose Farlie-Gumbel-Morgenstern copula, Frank copula and heavy right
tail copula to describe the dependence structure between inter-claim arrival
time and claim sizes, in which both marginals are represented by exponential
distributions. We then derived the Laplace transform to solve the m-th order recursive moments of the
aggregate discounted claims which can be expressed in the form of the Volterra
integral equation (VIE) of the second kind. The results of the first and second
moments calculated using the Laplace transform are identical to results
obtained in 15
and 16.
As we vary the dependence parameters from one end to the other, our computation also shows that FGM gives the narrowest range of first and second moments, followed by Frank and HRT. Insurers however, must make their own judgment in
choosing the best copula that can capture the dependency of their inter-claim
time and claim amount data. Future research may consider using different
marginal distributions for claim sizes including Pareto or Gamma distribution
and Weibull distribution for inter-claim time.


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