Open access peer-reviewed chapter - ONLINE FIRST

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

By Franck Adékambi

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

DOI: 10.5772/intechopen.88699

Downloaded: 33

Abstract

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.

Keywords

  • 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:

Z0t=k=1NoteδTkXk,Zdt=k=1NdteδTkXk,E1

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

FXi,τixv=CFXixFτiv=FXixFτiv+θFXixFτiv1FXix1Fτiv,E2

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

cθFGMuv=1+θ12u12v,E3

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

fXi,Tixv=cθFGMFXixFTivfXixfTiv=fXixfTiv+θfXixfTiv12FXix12FTiv,E4

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:

π˜Zdmr=1+λ2r+mδ+λ1λ2r+mδ+λ1u˜mr=λ11+λ2r+mδ+λ1λ2r+mδ+λ1×j=0m1mjμmjθμmjμmjλ1+mδ+r+2θμmjμmj2λ1+mδ+rπ˜ZojrE5

where

π˜Zdmr=u˜mr+λ2mδu˜mr×Lτ1mδr+λ1λ2mδ+λ1u˜m×Lτ1mδ+λ1r.E6

Proof

Conditioning on the arrival of the first claim leads to

πZdmt=EZmt=EEeδsX1+eδsZotsmτ1=s=j=0m1mj0t fτ1semδsEXmjτ1=sπZojtsds+0t fτ1semδsπZomtsds.E7

We have

EXmjτ1=s=0xmjfXτ1=sxdx=0xmj1+θ12FXx12Fτ1sfXxdx=EXmj+θ0xmj12FXx12Fτ1sfXxdx=EXmj+θ0xmj22FXx12Fτ1sfXxdxθ0xmj12Fτ1sfXxdx=EXmj1θ12Fτ1s+θ12Fτ1s0mjxmj1Fτ1sdx.E8

We let,

μmj=EXmj=0mjxmj11FXx2dx<0mjxmj11FXxdx=EXmj<E9

such that the above equation becomes

EXmjτ1=s=μmj+θ12Fτ1sμmjμmj.E10
πZdmt=EZmt=EEeδsX1+eδsZotsmτ1=s=j=0m1mj0t fτ1semδsμmj+θ12Fτ1sμmjμmjπZojtsds+0t fτ1semδsπZomtsds.

Let us 0tfτ1semδsds=Hδt, 0tfτ2semδsds=Iδtthen

πZdmt=j=0m1mj0t fτ1semδsμmj+θ12Fτ1sμmjμmjπZojtsds+HmδπZo.m=um+Hmδum+ImδπZo.m=um+Hmδum+HmδImδπZo.m=um+Hmδum+HmδImδum+ImδπZo.m=um+Hmδum+umk=1HmδImδkt=um+umk=0HmδImδkt=um+0tumtsemδsdmds,E11

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:

fτ1t=λ1eλ1t,fτ2t=λ2eλ2t,Lτ1λ1s=0esvfτ1vdv=λ1λ1+s,Lτ2λ2s=λ2λ2+s.
mdt=λ2t+λ1λ2λ11eλ1tE12

The mth moment of Zdtis then given by,

πZdmt=um+0tumtsemδsdmds=um+λ20tumtsemδsds+λ1λ20tumtsemδ+λ1sds=um+λ2mδ0tumtsmδemδsds+λ1λ2mδ+λ10tumtsmδ+λ1emδ+λ1sdsE13

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

π˜Zdmr=u˜mr+λ2mδu˜mr×Lτ1mδr+λ1λ2mδ+λ1u˜m×Lτ1mδ+λ1rE14

But,

umt=j=0m1mj0t fτ1semδsμj+θ12Fτ1sμjμjπZomjtsds=j=0m1mj0tλ1eλ1semδsμj+θ2eλ1s1μjμjπZomjtsds=λ1μjθμjμjλ1+mδj=0m1mj0tλ1+mδeλ1+mδsπZomjtsds+2θλ1μjμj2λ1+mδj=0m1mj0t2λ1+mδe2λ1+mδsπZomjtsdsE15

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

u˜mr=λ1j=0m1mjμjθμjμjλ1+mδ+r+2θμjμj2λ1+mδ+rπ˜ZomjrE16

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

π˜Zdmr=1+λ2r+mδ+λ1λ2r+mδ+λ1u˜mr=λ11+λ2r+mδ+λ1λ2r+mδ+λ1j=0m1mjμjθμjμjλ1+mδ+r+2θμjμj2λ1+mδ+rπ˜ZomjrE17

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

π˜Zomr=λ2μmrr+δm+λ2+λ2r+δm+λ2k=1m1Cmkμkπ˜Zomkr+θμmμmλ2r+δmrr+δm+λ2r+δm+2λ2+θλ2r+δmr+δm+λ2r+δm+2λ2k=1m1Cmkμkμkπ˜Zomkr+λ2r+δm+λ2π˜Z0mrE18

Rearranging the above equation, we will get

π˜Zomr=λ2μmrr+δm+λ2r+δmk=1m1Cmkμkπ˜Zomkr+θλ2μmμmrr+δm+2λ2+θλ2r+δm+2λ2k=1m1Cmkμkμkπ˜ZomkrE19

Corollary 3.1

The first moment of Zdtt0is given by:

πZdt=θλ1μ1μ1λ2+δδ+λ1δ+2λ2+λ1λ2+δδδ+λ1μ1+θλ1μ1μ1λ1λ2λ12λ2λ1λ2μ11δ+λ1eδ+λ1tθλ11δ+2λ2λ2λ12λ2μ1μ1eδ+2λ2t2θλ11δ+2λ1μ1μ1eδ+2λ1tλ2δμ1eδtE20

Proof:

From Theorem 3.1, we have:

π˜Zdr=λ1μ1rr+δ+λ1+λ1r+δ+λ1π˜Z0r+θμ1μ1λ1r+δrr+δ+λ1r+δ+2λ1E21

From Bargès et al. [8], we have

π˜Zor=λ2μ1rr+δ+θλ2μ1μ1rr+δ+2λ2E22

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

π˜Zdr=λ1μ1rr+δ+λ1+λ1r+δ+λ1λ2μ1rr+δ+θλ2μ1μ1rr+δ+2λ2+θλ1μ1μ1r+δrr+δ+λ1r+δ+2λ1=λ1λ2rr+δr+δ+λ1+λ1rr+δ+λ1μ1+θλ1μ1μ1λ2rr+δ+λ1r+δ+2λ2+r+δrr+δ+λ1r+δ+2λ1E23

with

λ1rr+δ+λ1=λ1δ+λ1.1rλ1δ+λ1.1r+δ+λ1E24
λ1λ2rr+δr+δ+λ1=λ1λ2δδ+λ1.1r+λ2δ+λ1.1r+δ+λ1λ2δ.1r+δE25
λ2rr+δ+λ1r+δ+2λ2=λ2δ+λ1δ+2λ21r+λ2λ12λ21δ+λ11r+δ+λ11δ+2λ21r+δ+2λ2E26
r+δrr+δ+λ1r+δ+2λ1=δδ+λ1δ+2λ11r+1δ+λ11r+δ+λ12δ+2λ11r+δ+2λ1E27

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

π˜Zdr=μ1λ1δ+λ1.1rλ1δ+λ1.1r+δ+λ1+θμ1μ1λ2δ+λ1δ+2λ21r+λ2λ12λ21δ+λ11r+δ+λ11δ+2λ21r+δ+2λ2+θλ1μ1μ1δδ+λ1δ+2λ11r+1δ+λ11r+δ+λ12δ+2λ11r+δ+2λ1+μ1λ1λ2δδ+λ1.1r+λ2δ+λ1.1r+δ+λ1λ2δ.1r+δE28

Rearranging the above equation, will give

πZdt=θλ1μ1μ1λ2+δδ+λ1δ+2λ2+λ1λ2+δδδ+λ1μ11r+θλ1μ1μ1λ1λ2λ12λ2λ1λ2μ11δ+λ11r+δ+λ1θλ11δ+2λ2λ2λ12λ2μ1μ11r+δ+2λ22θλ11δ+2λ1μ1μ11r+δ+2λ1λ2δμ11r+δE29

Remark 1

If λ1=λ2then Eq. (29) becomes

π˜Zdr=θλμ1μ11δ+2λ+λδμ11r1δ+2λθλμ1μ11r+δ+2λλδμ11r+δ=λδμ11r1r+δ+θλμ1μ11δ+2λ1r1r+δ+2λ=λμ1rr+δ+λθμ1μ1rr+δ+2λ,E30

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

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

πZdt=θλ1μ1μ1λ2+δδ+λ1δ+2λ2+λ1λ2+λ1δδδ+λ1μ1+θλ1μ1μ1λ2λ12λ2+1+λ2λ1μ11δ+λ1eδ+λ1tθλ11δ+2λ2λ2λ12λ2μ1μ1eδ+2λ2t2θλ11δ+2λ1μ1μ1eδ+2λ1tλ2δμ1eδtE31

Remarks 2

If θ=0and λ1λ2then

πZdt=λ1δλ2+δδ+λ1λ2δeδtμ1+λ2λ1δ+λ1μ1eδ+λ1t=λ21eδtδμ1+λ1λ21eδ+λ1tδ+λ1μ1=λ2a¯tδ+λ1λ2a¯tδ+λ1μ1E32

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

If λ1=λ2and θ0then

πZot=θλμ1μ11δ+2λ2+λδμ1θλ1δ+2λμ1μ1eδ+2λtλδμ1eδt=λδ1eδtμ1+θλ1eδ+2λtδ+2λμ1μ1E33

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

If λ1=λ2and θ=0then

πZot=λδ1eδtμ1=λa¯tδμ1,E34

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:

π˜Zd2r=2λ1λ1λ2rr+2δ+λ1r+2δ+2λ11rμ2+2μ1π˜Zor+λ1r+2δ+λ1r+2δr+2δ+2λ2λ2r+2δ+2λ1+1π˜Z02r,E35

From Bargès et al. [8], we have.

π˜Zor=λ2μ1rr+δ+θλ2μ1μ1rr+δ+2λ2E36

and

π˜Zo2r=λ2μ2rr+2δ+θλ2μ2μ2rr+2δ+2λ2+2λ22μ12rr+δr+2δ+2θλ22μ1μ1μ1rr+δ+2λ2r+2δ+2θλ22μ1μ1μ1rr+2δ+2λ2r+δ+2θ2λ22μ1μ12rr+δ+2λ2r+2δ+2λ2E37

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

π˜Zd2r=2λ1λ1λ2rr+2δ+λ1r+2δ+2λ11rμ2+2μ1λ2μ1rr+δ+θλ2μ1μ1rr+δ+2λ2+λ1r+2δ+λ1r+2δr+2δ+2λ2λ2r+2δ+2λ1+1×λ2μ2rr+2δ+θλ2μ2μ2rr+2δ+2λ2+2λ22μ12rr+δr+2δ+2θλ22μ1μ1μ1rr+δ+2λ2r+2δ+2θλ22μ1μ1μ1rr+2δ+2λ2r+δ+2θ2λ22μ1μ12rr+δ+2λ2r+2δ+2λ2,E38

and rearranging Eq. (38), will give:

π˜Zd2r=λ1μ2rr+2δ+λ1+2λ1λ2μ12rr+δr+2δ+λ1+2θλ1λ2μ1μ1μ1rr+δ+2λ2r+2δ+λ1+θλ1μ2μ2r+2δrr+2δ+λ1r+2δ+2λ1+2θλ1λ2μ1μ1μ1r+2δrr+δr+2δ+λ1r+2δ+2λ1+2θ2λ1λ2μ1μ12r+2δrr+δ+2λ2r+2δ+λ1r+2δ+2λ1+λ1λ2μ21rr+2δr+2δ+λ1+θλ1λ2μ2μ21rr+2δ+2λ2r+2δ+λ1+2λ1λ22μ121rr+δr+2δr+2δ+λ1+2θλ1λ22μ1μ1μ11rr+δ+2λ2r+2δr+2δ+λ1+2θλ22λ1μ1μ1μ11rr+2δ+2λ2r+δr+2δ+λ1+2θ2λ22λ1μ1μ121rr+δ+2λ2r+2δ+2λ2r+2δ+λ1,E39

which can be simplified to

π˜Zd2r=γ0r+γ1r+δ+γ2r+2δ+γ3r+2δ+λ1+γ4r+δ+2λ2+γ5r+2δ+2λ1+γ6r+2δ+2λ2,E40

with,

γ0=λ1μ22δ+λ1+2λ1λ2μ12δ2δ+λ1+2θλ1λ2μ1μ1μ12δ+λ1δ+2λ2+θδλ1μ2μ22δ+λ1δ+λ1+2θλ1λ2μ1μ1μ12δ+λ1δ+λ1+2δθ2λ1λ2μ1μ122δ+λ1δ+λ1δ+2λ2+λ1λ2μ22δ2δ+λ1+θλ1λ2μ2μ22δ+λ22δ+λ1+λ1λ22μ12δ22δ+λ1+θλ1λ22μ1μ1μ1δδ+2λ22δ+λ1+θλ1λ22μ1μ1μ1δδ+λ22δ+λ1+θ2λ1λ22μ1μ122δ+2λ2δ+λ22δ+λ1E41
γ1=2λ1λ2μ12δδ+λ12θλ1λ2μ1μ1μ1δ+λ1δ+2λ12λ1λ22μ12δ2δ+λ12θλ1λ22μ1μ1μ1δδ+2λ2δ+λ1E42
γ2=λ2μ22δ+λ22μ12δ2+θλ22μ1μ1μ1δδ2λ2E43
γ3=λ1μ22δ+λ1+2λ1λ2μ122δ+λ1δ+λ1+2θλ1λ2μ1μ1μ12δ+λ1λ1+δ2λ2+θλ1μ2μ22δ+λ12θλ1λ2μ1μ1μ12δ+λ1δ+λ12θ2λ1λ2μ1μ122δ+λ1λ1+δ2λ2+λ2μ22δ+λ1+θλ1λ2μ2μ22δ+λ1λ12λ22λ22μ122δ+λ1δ+λ12θλ22μ1μ1μ12δ+λ1λ1+δ2λ2+2θλ1λ22μ1μ1μ12λ2λ1δ+λ12δ+λ1+2θ2λ1λ22μ1μ122δ+λ1λ1+δ2λ22λ2λ1E44
γ4=2θλ1λ2μ1μ1μ1δ+2λ2λ1+δ2λ2+2θ2λ1λ2μ1μ122λ2δδ+2λ2λ1+δ2λ22λ12λ2+δ2θλ1λ22μ1μ1μ1δ+2λ2δ2λ2λ1+δ2λ22θ2λ1λ22μ1μ12δδ+2λ2λ1+δ2λ2E45
γ5=θλ1μ2μ2δ+λ1+2θλ1λ2μ1μ1μ1δ+λ1δ+2λ12θ2λ1λ2μ1μ12δ+λ12λ1δ2λ2E46
γ6=θλ1λ2μ2μ22δ+λ2λ12λ2+θλ1λ22μ1μ1μ1δ+λ2δ+2λ2λ12λ2+θ2λ1λ22μ1μ12δδ+λ2λ12λ2E47

Remark 2

When

λ1=λ2
π˜Zd2r=2λ2μ2rr+δr+2δ+2θλ2μ1μ1μ1rr+δ+2λr+2δ+λμ2rr+2δ+θλμ2μ2rr+2δ+2λ2θλ2μ1μ1μ1rr+δr+2δ+2λ+2θ2λ2μ1μ12rr+δ+2λr+2δ+2λ,E48

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

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

g˜r=1rα1+rα2+rαn+r,E49

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

gt=γ0+γ1eα1t+γ2eα2t++γneαnt.E50

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

πZd2t=γ0+γ1eδt+γ2e2δt+γ3e2δ+λ1t+γ4eδ+2λ2t+γ5e2δ+2λ1t+γ6e2δ+2λ2t,t0,E51

where γii0126are given by equation Eq. (50).

Remarks

If θ=0then

γ0θ=0=λ1μ22δ+λ1+2λ1λ2μ12δ2δ+λ1+λ1λ2μ22δ2δ+λ1+λ1λ22μ12δ22δ+λ1=λ12δ+λ12δ+λ22δμ2+λ1λ2δ2δ+λ12δ+λ2δμ12=λ12δ+λ22δ2δ+λ1μ2+2λ2μ12δ,E52

γ1θ=0=2λ1λ2μ12δδ+λ12λ1λ22μ12δ2δ+λ1=2λ1λ2μ12δδ+λ11+λ2δ,γ2θ=0=λ22μ12δ2λ2μ22δE53

γ3θ=0=λ1μ22δ+λ1+2λ1λ2μ122δ+λ1δ+λ1+λ2μ22δ+λ12λ22μ122δ+λ1δ+λ1=λ2λ1μ22δ+λ1+2λ2λ1λ2μ122δ+λ1δ+λ1,E54
γ4θ=0=γ5θ=0=γ6θ=0.E55

Then,

πZd2t=λ12δ+λ22δ2δ+λ1μ2+2λ2μ12δ2λ1λ2μ12δδ+λ11+λ2δeδt+λ22μ12δ2λ2μ22δe2δt+12δ+λ1λ2λ1μ2+2λ2λ1λ2μ122δ+λ1δ+λ1e2δ+λ1t=λ12δ+λ22δ2δ+λ1λ22δe2δt+λ2λ12δ+λ1e2δ+λ1tμ2+λ2λ12δ+λ2δ22δ+λ12λ1λ2δδ+λ11+λ2δeδt+λ22δ2e2δt+2λ2λ1λ22δ+λ1δ+λ1e2δ+λ1tμ12=λ12δ+λ22δ2δ+λ1λ22δ12δa¯t2δ+λ2λ12δ+λ112δ+λ1a¯t2δ+λ1μ2+λ2λ12δ+λ2δ22δ+λ12λ1λ2δδ+λ11+λ2δeδt+2λ22δ2e2δtλ22δ2e2δtμ12+2λ2λ1λ2δδ+λ12λ2λ1λ2δ2δ+λ1μ12e2δ+λ1t=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+λ2λ12δ+λ2δ22δ+λ12λ1λ2δδ+λ11+λ2δeδt+2λ22δ2e2δtλ22δ212δa¯t2δμ12+2λ2λ1λ2δδ+λ1e2δ+λ1tμ122λ2λ1λ212δ+λ1a¯t2δ+λ1μ12δ2δ+λ1=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1μ12eδt2λ1λ2δδ+λ11+λ2δ2λ22δ2eδt+2λ2λ1λ2δδ+λ1eδ+λ1tE56
πZd2t=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1μ12eδt2λ1λ2δδ+λ11+λ2δ2λ22δ2eδt+2λ2λ1λ2δδ+λ1eδ+λ1t=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1μ12eδt2λ1λ2δδ+λ11+λ2δ2λ22δ21δa¯t2δ+2λ2λ1λ2δδ+λ11δ+λ1a¯tδ+λ1=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1μ122λ2δeδtλ2a¯tδ+λ1λ2a¯tδ+λ1=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1eδtλ2a¯tδ+λ1λ2a¯tδ+λ1μ12.E57

To finally have:

πZd2t=λ2a¯t2δ+λ1λ2a¯t2δ+λ1μ2+2λ2δλ2a¯t2δ+λ1λ2a¯t2δ+λ1eδtλ2a¯tδ+λ1λ2a¯tδ+λ1μ12,E58

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

If λ1=λ2then

γ0λ1=λ2=λμ22δ+λ2μ12δ2+θλ2δδ+λ+θλ2δδ+2λμ1μ1μ1+θλμ2μ22δ+λ+θ2λ2μ1μ12δ+λδ+2λ,E59

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

γ4λ1=λ2=2θλ2μ1μ1μ1δ+2λδ2λ2θ2λ2μ1μ12δδ+2λ,E61
γ5λ1=λ2=θλμ2μ2δ+λ+2θλ2μ1μ1μ1δ+λδ+2λ+2θ2λ2μ1μ12δδ+λ,E62
γ6λ1=λ2=θλμ2μ22δ+λθλ2μ1μ1μ1δ+λδ+2λθ2λ2μ1μ12δδ+λ.E63

Then,

πZd2t=λμ21δeδt2δ+θλμ2μ212λ+2δe2λ+2δt2λ+2δ+2λ2μ1212δ2eδtδ2+e2δt2δ2+2θλ2μ1μ1μ112δ2λ+δe2λ+δt2λ+δ2λ+δ+e2δt2δ2λ+δ+2θλ2μ1μ1μ11δ2λ+2δeδtδ2λ+δ+e2λ+2δt2λ+2δ2λ+δ+2θ2λ2μ1μ1212λ+δ2λ+2δe2λ+δtδ2λ+δ+e2λ+2δtδ2λ+2δE64

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

If λ1=λ2and θ=0then

πZo2t=λa¯t2δμ2+λa¯t2δμ12,E65

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

ηkijp=ijθpλkEXj1pEXjEXjpr+p×2λ+k.E66

We can rewrite π˜Zorand π˜Zo2ras

π˜Zor=1rηk110+ηk111,E67
π˜Zo2r=1rηk220+ηk221+ηk210+ηk211ηk110+ηk111=1rηk220+ηk221+ηk210ηk110+ηk210ηk111+ηk211ηk110+ηk211ηk111E68

The term π˜Zomrcan also be expressed using

π˜Zomr=1ri=1mi1j1p1injnpnm,nηkinjnpn××ηki1j1p1,E69

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

and

π˜Zdmr=2λ1λ1λ2rr+mδ+λ1r+mδ+2λ11rμm+k=1m1CmkEX1kπ˜Zomkr+λ1r+mδ+λ1r+mδr+mδ+2λ2λ2r+mδ+2λ1+1π˜Z0mr,E71

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

θt=1t=10t=100
−1482.3375445016,428
0438.10574407.116,385
1393.8744364.216,342

Table 1.

EZdtfor λ1=1,λ2=10,δ=3%.

θt=1t=10t=100
−1597.16334578.716,557
0554.52374535.516,513
1511.88414492.316,470

Table 2.

EZdtfor λ1=5,λ2=10,δ=3%.

θt=1t=10t=100
−133900.8529646.717928.481
023972.6520976.125711.548
1359.87951036.2231542.412

Table 3.

StdZdtfor λ1=1,λ2=10,δ=3%.

θt=1t=10t=100
−134027.129756.447954.606
024061.8521053.725731.564
1366.14841035.4911545.671

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

Gt=EZdt+MtE73

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:

Gt=EZdt+θEZdt,E74

where Mt=θEZdt.

4.2.2 The variance principle

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

Gt=EZdt+θVarZdt,E75

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:

Gt=EZdt+θVarZdt,E76

where Mt=θVarZdt.

4.2.4 The quantile principle

The standard deviation principle defines the loaded premium as:

Gt=FZdt11ε,E77

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

In this case, the safety loading Mtis given by

Mt=FZdt11εEZdtE78

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.

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Franck Adékambi (September 3rd 2019). Moments of the Discounted Aggregate Claims with Delay Inter-Occurrence Distribution and Dependence Introduced by a FGM Copula [Online First], IntechOpen, DOI: 10.5772/intechopen.88699. Available from:

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