Abstract. sizes and the rates of claim

Abstract. We consider the
recursive moments of aggregate discounted claims, where the dependence between
the inter-claim time and the subsequent claim size is 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
equation

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1 INTRODUCTION

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,
2,
3,
4,
5
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,
9,
10
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.

2 CLAIMS MODEL

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

where

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
as:

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. Let

be dependence parameter, the pdfs of the copulas are given
by:

 

                                 

                          (1)

 

                                

                         (2)

 

                                                                                        (3)

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.

3 LAPLACE TRANSFORMS OF THE RECURSIVE MOMENTS

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

follow 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:

 

                         

                  (4)

 

                                                                                        (5)

 

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

is a continuous function in the region

and

is a difference kernel and a continuous function in the
region

(see 16 and 19). VIE is commonly found in the
study of electromagnetic field and viscoelastic materials, as well as
demographic study, insurance mathematics and operational risk modeling 20, 21.
In this study, the VIE of the second kind is given by:

 

                                                            

                                                     (6)

 

The function

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

 

                                     

                              (7)

 

                         

                  (8)

 

 

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

 

                                                             

                                                     (9)

 

Let

and

denote the Laplace transform of

and

respectively. Since

is the convolution integral of

and

then,

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:

4 MOMENTS OF THE AGGREGATE DISCOUNTED CLAIMS

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

and

the values of the first and the second moments under each
copula are summarized in Table 1 and Table 2. In the field of finance and
actuarial studies, ? represents the discount rate applied on future claims. 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 22.
The two marginals are positively
correlated if its dependence parameter, ?Frank ? (0,1) and inversely correlated if ?Frank
? (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).

 

Table 1. First moments of the
aggregate discounted claims with

FGM

Frank

HRT

-0.999

477.658

0.005

400.646

0.005

10.475

-0.950

476.457

0.05

418.704

0.05

249.115

-0.900

475.231

0.5

444.633

0.5

374.047

-0.500

465.428

1

453.173

1

392.109

0.000

453.173

10

478.487

10

443.761

0.500

440.919

500

501.348

500

452.982

0.900

431.115

5000

506.874

5000

453.154

0.950

429.890

100000

510.578

100000

453.172

0.999

428.689

?

516.062

?

453.173

Spread

48.969

 

115.417

 

442.698

 

Table 2. Second moments of the
aggregate discounted claims with

FGM

Frank

HRT

-0.999

334509

0.005

196424

0.005

3814

-0.950

332163

0.05

226374

0.05

129531

-0.900

329774

0.5

272065

0.5

159485

-0.500

310877

1

287786

1

187005

0.000

287786

10

336366

10

271105

0.500

265284

500

382946

500

287442

0.900

247706

5000

394638

5000

287751

0.950

245535

100000

402577

100000

287784

0.999

243413

?

414136

?

287786

Spread

91096

 

217712

 

283972

 

Table 3. Values Var(5) of the aggregate discounted
claims with

FGM

Frank

HRT

-0.999

106351.835

0.005

35906.614

0.005

3704.421

-0.950

105151.727

0.05

51061.432

0.05

67472.040

-0.900

103929.497

0.5

74366.716

0.5

19574.014

-0.500

94253.777

1

82420.025

1

33255.194

0.000

82420.232

10

107416.321

10

74181.637

0.500

70874.435

500

131596.342

500

82248.977

0.900

61845.857

5000

137716.491

5000

82402.872

0.950

60729.588

100000

141886.938

100000

82419.133

0.999

59638.741

?

147815.577

?

82419.931

 

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 287,786
(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. Given the values in Table 1, 2 and 3, 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:

 

 

where ? > 0 is a risk loading factor, which is used to cover additional risk
and expenses, as well as to generate profit for an insurer.

5 CONCLUSIONS

In this study, we represent the m-th order recursive moments of the aggregate discounted claims
using the Volterra integral equation (VIE) of the second kind and solve them
using Laplace transform. We then 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. The results of the
first and second moments computed using the Laplace transform are identical to
results obtained in 16,
whereby the solution to the VIE was derived using Neumann series, as well as
15. 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.