Open access peer-reviewed chapter

Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution and Dependence Introduced by a FGM Copula

Written By

Franck Adékambi

Submitted: May 3rd, 2019 Reviewed: July 19th, 2019 Published: September 3rd, 2019

DOI: 10.5772/intechopen.88699

Chapter metrics overview

565 Chapter Downloads

View Full Metrics


In this chapter, with renewal argument, we derive higher simple moments of the Discounted Compound Delay Renewal Risk Process (DCDRRP) when introducing dependence between the inter-occurrence time and the subsequent claim size. To illustrate our results, we assume that the inter-occurrence time is following a delay-Poisson process and the claim amounts is following a mixture of Exponential distribution, we then provide numerical results for the first two moments. The dependence structure between the inter-occurrence time and the subsequent claim size is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we obtain a general formula for all the moments of the DCDRRP process.


  • compound delay-Poisson process
  • discounted aggregate claims
  • moments
  • FGM copula
  • constant interest rate

1. Introduction

The classical Poisson model is attractive in the sense that the memoryless property of the exponential distribution makes calculations easy. Then the research was extended to ordinary Sparre-Andersen renewal risk models where the inter-claim times have other distributions than the exponential distribution. Dickson and Hipp [1, 2] considered the Erlang-2 distribution, Li and Garrido [3] the Erlang-n distribution, Gerber and Shiu [4] the generalized Erlang-n distribution (a sum of n independent exponential distributions with different scale parameters) and Li and Garrido [5] looked into the Coxian class distributions. One difficulty with these models is that we have to assume that a claim occurs at time 0, which is not the case in usual setting.

Albrecher and Teugels [6] considered modeling dependence with the use of an arbitrary copula. In a similar dependence model to Albrecher and Teugels as well, Asimit and Badescu [7] considered a constant force of interest and heavy tailed claim amounts.

Barges et al. [8] followed the idea of Albrecher and Teugels [6] and supposed that the dependence is introduced by a copula, the Farlie-Gumbel-Morgenstern (GGM) copula, between a claim inter-arrival time and its subsequent claim amount.

Adékambi and Dziwa [9] and Adékambi [10] provide a direct point of extension but assuming that the claim counting process to follow an unknown general distribution in a framework of dependence with random force of interest to calculate the first two moments of the present value of aggregate random cash flows or random dividends.

The discounted aggregate sum has also been applied in many other fields. For example, it can be used in health cost modeling, see Govorun and Latouche [11], Adékambi [12], or in reliability, in civil engineering, see Van Noortwijk and Frangopol [13].

The delayed or modified renewal risk model solves this problem by assuming that the time until the first claim has a different distribution than the rest of the inter-claim times. Not much research has been done for this model at this stage. Among the first works was Willmot [14] where a mixture of a “generalized equilibrium” distribution and an exponential distribution is considered for the distribution of the time until the first claim. Special cases of the model include the stationary renewal risk model and the delayed renewal risk model with the time until the first claim exponentially distributed. Our focus is to extend the work of Bargès et al. [8], Adékambi and Dziwa [9] and Adékambi [10] by allowing the counting process to follow a delay renewal risk process and thus derive a recursive formula of the moments of this subsequent Discounted Compound Delay Poisson Risk Value (DCDPRV).

For example, young performer companies typically have a high growth rate at the beginning, but as they mature their growth rate may decrease with the increasing scarcity of investment opportunities. That makes dividends dependent on the economic climate at the dividend occurrence time. Obviously the distribution of inter-dividends time in times of economic expansion and in times of economic crisis cannot be identically distributed. So it would be appropriate to use a delayed renewal model to model the distribution of the inter-dividend time. A delayed renewal process is just like an ordinary renewal process, except that the first arrival time is allowed to have a different distribution than the other inter-dividends times.

The chapter is organized as follows: In the second section, we present the model of the continuous time discounted compound delay-Poisson risk process that we use and give some notation. In Section 3, we present a general formula for all the moments of the DCDPRV process. A numerical example of the first two moments will then follow in Section 4.


2. The model

We use the same model as the one in Bargès et al. [8], where the instantaneous interest rate δis constant.

Define our risk model as follows:

  1. The number of claims Ntt0and Ndtt0form, respectively, an ordinary and a delayed renewal process and, for kN=123:

  • the positive claim occurrence times are given by Tk,

  • the positive claim inter-arrival times are given by τk=TkTk1,kN, and T0=0.

  • τkk2τ2are independent and identically distributed (i.i.d),

  1. The kthrandom claim is given by Xk, and

  • XkkNare independent and identically distributed (i.i.d),

  • XkτkkNare mutually independent; and the higher moments, μk=EX1kof X1exist.

  1. The discounted aggregate value at time t=0of the claims recorded over the period 0tyields, respectively, for the ordinary and the delayed renewal case:


where Z0t=Zdt=0if N0t=Ndt=0.

2.1 The dependence

We introduce a specific structure of dependence based on the Farlie-Gumbel-Morgenstern (FGM) copula. The advantage of using the FGM copula and its generalizations lies in its mathematical manageability. The joint cumulative distribution function (c.d.f.) of Xiτi, the ith claim and its occurrence time is


for xvR+R+and where FXixand Fτivare the marginals of Xiand τirespectively. Recall that the density of the FGM copula is


for uv0101so that the joint probability density function (p.d.f.) of Xiτiis


where fXiand fτiare the p.d.f.’s of Xiand τirespectively.

With these hypotheses, we present in Section 3 recursive formula of the higher moments of this present value risk process, for a constant instantaneous interest rate.


3. Recursive expression for higher moments

It is often easier to calculate the moments of the random variable Zdtt0than finding its distribution. If the probability generation function of Zdtt0or its moment generating function (mgf) exists, it is possible to obtain the corresponding distribution by inversion of its mgf. Since, there is relatively little research devoted to the study of the distribution of the discounted compound renewal sums. We could then think about another technique other than the one proposed by the above authors by studying the moments of Zdtt0.

3.1 Delay renewal case

The mathematical expectation of total claims plays an important role in the determination of the pure premium, in addition to giving a measure of the central tendency of its distribution. The moments centered at the average of order 2, 3 and 4 are the other moments usually considered because they usually give a good indication of the pace of distribution, and these give us respectively a measure of the dispersion of the distribution around its mean, a measure of the asymmetry and flattening of the distribution considered.

Moments, whether simple, joined or conditional, may eventually be used to construct approximations of the distribution of the DCDPRV.

Theorem 3.1

The Laplace transform of the mth moment of Zdtt0is given by:





Conditioning on the arrival of the first claim leads to


We have


We let,


such that the above equation becomes


Let us 0tfτ1semδsds=Hδt, 0tfτ2semδsds=Iδtthen


where umt=j=0m1mj0tfτ1semδsμj+θ12Fτ1sμjμjπZomjtsds.

We consider the case where the canonical random variable τ2has an Exponential distribution with parameter λ2>0and τ1has an Exponential distribution with parameter λ1>0.

That is, we have:


The mth moment of Zdtis then given by,


Taking the Laplace transform of the above equation, we get:




Then the Laplace transform of umt, at r, will give:


Substituting Eq. (14) into Eq. (13), we have:


Solving the above equation for the ordinary case, where τ2k2τ2, we have:


Rearranging the above equation, we will get


Corollary 3.1

The first moment of Zdtt0is given by:



From Theorem 3.1, we have:


From Bargès et al. [8], we have


Substituting Eq. (22) into Eq. (21), yields




Substituting Eqs. (24), (25), (26) and (27) into Eq. (23), yields:


Rearranging the above equation, will give


Remark 1

If λ1=λ2then Eq. (29) becomes


which is exactly the result of Bargès et al. [8].

The inverse of the Laplace transform in Eq. (29) will give


Remarks 2

If θ=0and λ1λ2then


which is exactly the result of Léveillé et al. [15].

If λ1=λ2and θ0then


which is exactly the result of Bargès et al. [8].

If λ1=λ2and θ=0then


which is exactly the result of Léveillé et al. [15].

Corollary 3.2

The second moment of Zdtt0is given by the following development:

The result in Theorem 3.1 when n=2gives:


From Bargès et al. [8], we have.




Substituting Eqs. (39) and (40) into Eq. (38), yields:


and rearranging Eq. (38), will give:


which can be simplified to




Remark 2



which is exactly the result of Bargès et al. [8].

The Laplace transform in Eq. (49), is a combination of terms of the form:


with ga function defined for all non-negative real numbers. As described in the proof of Theorem 1.1 in Baeumer [16], each of these terms can be expressed as a combinations of partial fraction such as g˜r=γ01r+γ11α1+r++γn1αn+rwhere.

γ0=1α1αn, for i=1,,n, γi=1αij=1,ji1αjαi.

Since the inverse Laplace transform of 1αi+ris eαit, it is easy to invert g˜and obtain


Using Eq. (49) in Eq. (53), it results that


where γii0126are given by equation Eq. (50).


If θ=0then






To finally have:


which is exactly the result of Léveillé et al.[15].

If λ1=λ2then


γ1λ1=λ2=2λ2μ12δ22θλ2μ1μ1μ1δδ+2λ, γ2λ1=λ2=λμ22δ+λ2μ12δ2+θλ2μ1μ1μ1δδ2λ,γ3λ1=λ2=0(60)




which is exactly the result of Bargès et al. [8].

If λ1=λ2and θ=0then


which is exactly the result of Léveillé et al. [15].

Remark 3.1

Noting for i=1,2,,m,j=1,2,,m, p=0,1and kN0


We can rewrite π˜Zorand π˜Zo2ras


The term π˜Zomrcan also be expressed using


where m,n=i1j1p1,,injnpn:i1=m,i1++in=m1+n,i1>>in,j1=m1+n,j1++jn=m,j1>>jn,p01, (70)



4. Application

4.1 First two moments

For the numerical illustration, suppose that XpExpβ1=180+1pExpβ2=1200, the inter-claim time distribution parameters λ1=2;4and λ2=1, the interest rate δ=3%(Tables 14). We use three different values for the copula parameter θ=1;0;1, p=13and fix the time t=1;10;100. The mth moment of Xis


Table 1.

EZdtfor λ1=1,λ2=10,δ=3%.


Table 2.

EZdtfor λ1=5,λ2=10,δ=3%.


Table 3.

StdZdtfor λ1=1,λ2=10,δ=3%.


Table 4.

StdZdtfor λ1=5,λ2=10,δ=3%.

μm=pm!β1m+1pm!β2mand μm=0mxm11FXx2dx=μm=pm!2β1m+1pm!2β2m.(72)

4.2 Premium calculation

From the results in Section 4.1, we can compute the premium related to the risk of an insurance portfolio represented by Gt, depending on the premium calculation principles adopted by the insurance company. The loaded premium Zdtconsists in the sum of the pure premium EZdt, the expected value of the costs related to the portfolio, and a loading for the risk Mtas


The loading for the risk differs according to the premium calculation principles.

4.2.1 The expected value principle

Denote by θ>0the safety loading. The expected value principle defines the loaded premium as:


where Mt=θEZdt.

4.2.2 The variance principle

Denote by θ>0the safety loading. The variance principle defines the loaded premium as:


where Mt=θVarZdt.

4.2.3 The standard deviation principle

Denote by θ>0the safety loading. The standard deviation principle defines the loaded premium as:


where Mt=θVarZdt.

4.2.4 The quantile principle

The standard deviation principle defines the loaded premium as:


where εis smallest (for example: ε=0.5%,1%,2.5%,5%).

In this case, the safety loading Mtis given by


The principles of standard deviation and variance only require partial information on the distribution of the random variable, Zdt, i.e., its expectation and its variance.

Often, the actuary only has this information for different reasons (time constraints, information …).

If the actuary has more information about the random variable, Zdti.e., he knows the form of FZdt, then he can apply the quantile principle.

But he does not know much about FZdt, then he can approximate the distribution of Zdtusing the matching moments technique.


5. Conclusion

We have derived exact expressions for all the moments of the DCDPRV process using renewal arguments, again disproving the popular belief that renewal techniques cannot be applied in the presence of economic factors. Our results, for the DCDPRV process, are consistent: (i) with the results of Léveillé et al. [15] for θ=0,λ1λ2and for θ=0,λ1=λ2, (ii) with the results of Bargès et al. [8] for θ0,λ1=λ2.

Within this framework, further research is needed to get exact expressions (or approximations) of certain functional of the Zdtt0process, as stop-loss premiums and ruin probabilities.

Our models have applications in reinsurance, house insurance and car insurance. They can also be used in evaluation of health programs, finance, and other areas.

For example, consider the case of a male currently aged 25 who is starting a defined contribution (DC) pension plan and is planning to retire in, say, 40 years at the age of 65. He anticipates that when he reaches that age he will convert his accumulated pension fund into a life annuity in order to hedge his own longevity risk and avoid outliving his own financial resources. The value of his retirement income will depend not only on the value of his pension fund, but also on the price of annuities at the time. Other things being equal, this means that his retirement income prospects will be affected by the distribution on future annuity value: the greater the dispersion of that distribution, the riskier his retirement income will be. For the assessment of the accumulated pension fund and its variability our models can be used. We can suppose that this man makes a deposit to a bank account, and that the time between successive deposits follows a renewal process and the force of interest is stochastic. Our model allows us to calculate the accumulated pension fund and its variability at the age of 65.

Another possible application is in reliability, to model the net present value of aggregate equipment failures costs until its total breakdown. A piece of equipment is deemed to be beyond repair when the repair time exceeds a predetermined gap. Of course, another possible definition of total breakdown is when the cost of repair exceeds a predetermined gap. But, since the cost of repair is defined per unit time, the two definitions are somewhat equivalent.


  1. 1. Dickson DCM, Hipp C. Ruin probabilities for Erlang(2) risk process. Insurance: Mathematics & Economics. 1998;22:251-262
  2. 2. Dickson DCM, Hipp C. On the time to ruin for Erlang(2) risk process. Insurance: Mathematics & Economics. 2001;29:333-344
  3. 3. Li S, Garrido J. On ruin for the Erlang(n) risk process. Insurance: Mathematics & Economics. 2004;34:391-408
  4. 4. Gerber H, Shiu E. The time value of ruin in a sparre Andersen model. North American Actuarial Journal. 2005;9:49-69
  5. 5. Li S, Garrido J. On a general class of renewal risk process: Analysis of the Gerber-Shiu function. Advances in Applied Probability. 2005;37:836-856
  6. 6. Albrecher Hö, Teugels JL. Exponential behaviour in the presence of dependence in risk theory. Journal of Applied Probability. March 2006;43(1):257-273
  7. 7. Asimit AV, Badescu AL. Extremes on the discounted aggregate claims in a time dependent risk model. Scandinavian Actuarial Journal. 2010;2:93-104
  8. 8. Bargès M, Cossette H, Loisel S, Marceau E. On the moments of aggregate discounted claims with dependence introduced by a FGM copula. ASTIN Bulletin: The Journal of the IAA. 2011;41(1):215-238
  9. 9. Adékambi F, Dziwa S. Moment of the discounted compound renewal cash flows with dependence: The use of Farlie-Gumbel-Morgenstern copula. In: Proceedings of the 58th Annual Conference of the South African Statistical Association for 2016; Cape Town, South Africa: University of Cape Town. 2016. pp. 1-8
  10. 10. Adékambi F. Second moment of the discounted aggregate renewal cash flow with dependence. In: Proceedings of the 58th Annual Conference of the South African Statistical Association for 2016; Cape Town, South Africa: University of Cape Town. 2017. pp. 1-8
  11. 11. Govorun M, Latouche G. Modeling the effect of health: Phase-type approach. European Actuarial Journal. 2014;4(1):197-218
  12. 12. Adékambi F. Moments of phase-type aging modeling for health dependent costs. Advances in Decision Sciences. 2019;23(2):1-28
  13. 13. Van Noortwijk J, Frangopol D. Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures. Probabilistic Engineering Mechanics. 2004;19(4):345-359
  14. 14. Willmot GE. A note on a class of delayed renewal risk processes. Insurance: Mathematics & Economics. 2004;34:251-257
  15. 15. Léveillé G, Garrido J. Moments of compound renewal sums with discounted claims. Insurance: Mathematics & Economics. 2001;28:217-231
  16. 16. Baeumer B. On the inversion of the convolution and Laplace transform. Transactions of the American Mathematical Society. 2003;355(3):1201-1212

Written By

Franck Adékambi

Submitted: May 3rd, 2019 Reviewed: July 19th, 2019 Published: September 3rd, 2019