JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/55751875575187Research ArticleOn a Discrete-Time Risk Model with Random Income and a Constant Dividend Barrierhttps://orcid.org/0000-0002-1722-6407BaoZhenhuaHuangJunqinghttps://orcid.org/0000-0001-8346-6815WangJingLuXuewenSchool of MathematicsLiaoning Normal UniversityDalian 116029Chinalnnu.edu.cn2021255202120213012021115202125520212021Copyright © 2021 Zhenhua Bao et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a discrete-time risk model with random income and a constant dividend barrier is considered. Under such a dividend policy, once the insurer’s reserve hits the level bb>0, the excess of the reserve over b is paid off as dividends. We derive a homogeneous difference equation for the expected present value of dividend payments. Corresponding solution procedures for the difference equation are invested. Finally, we give a numerical example to illustrate the applicability of the results obtained.

Ministry of Education of the People's Republic of China20YJA910001Foundation of Educational DepartmentW201783664Science and Technology Department20180550196
1. Introduction

In the actuarial literature, many authors focus their research interests on discrete-time risk models, which can be used as an approximation to continuous time models. Li et al.  present a review of results for discrete-time risk models, including the compound binomial risk model and some of its extensions. In recent years, discrete-time risk models with dependent structure have received increasing attention, and the readers are referred to Liu and Bao  for the related studies on different kinds of dependent models.

Additionally, problems related to dividends have been considered extensively in the discrete-time setting. Because of the certainty of ruin for a risk model with a constant dividend barrier, the calculation of the expected discounted dividend payment is a major problem of interest in the context. Among the class of discrete-time risk models, Tan and Yang  derive a recursive algorithm to compute a particular class of Gerber–Shiu penalty functions in the framework of the compound binomial model with randomized dividend payments. Landriault  then generalize Tan and Yang’s model to consider the compound binomial model with a multithreshold dividend structure and randomized dividend payments. For the compound binomial risk model with possible delay of claims, Xie and Zou  investigate the expected present value of total dividends under stochastic interest rates. Wu and Li  further consider the discrete-time risk model with time-delayed claims and a constant dividend barrier, but the main claim is dependent on its associated by-claim. In the framework of a discrete semi-Markov risk model, a randomized dividend policy is studied by Yuen et al. . Zhang and Liu  consider a discrete-time risk model with a mathematically tractable dependence structure between interclaim time and claim size in the presence of an impulsive dividend strategy.

All the risk models discussed above are based on a common assumption that the premium is collected with a positive deterministic constant. However, this assumption can be unrealistic and inappropriate in practical contexts because the insurance company may have lump sums of income. Therefore, many authors consider the risk models with stochastic income to capture the uncertainty of the customers’ arrivals, for example, Boikov , Yang and Zhang , Labbé and Sendova , Karnaukh , Bao and Liu , and Zhou et al. .

In the present paper, we propose a discrete-time risk model with random income and a constant dividend barrier. A similar model has been discussed by Zhou et al. . However, we analyze the model with a general premium rate c, while Zhou et al.  investigate the model with time-delayed claims and unit premium assumption. As stated in Landriault , unlike the classical compound Poisson risk model for which a unit premium rate can be assumed without loss of generality, such reasoning does not hold for the discrete-time risk model. On the contrary, the technique used in the present paper is to solve a high-order difference equation that often arises in the discrete-time risk model with general premium c.

The rest of the paper is organized as follows. A brief description of the discrete-time model and the introduction of the expected present value of total dividends are considered in Section 2. In Section 3, we obtain and solve a homogeneous difference equation satisfied by the expected present value of dividend payments. When the claim sizes follow the geometrical distribution, a closed-form solution to the expected present value of dividends and the corresponding numerical example are provided in Section 4. Finally, we end our works with conclusion in Section 5.

2. The Model

Throughout, denote by the set of nature numbers and +=\0. Similar to Bao and Liu , we assume that the premium income is a binomial process Mk,k with parameter p00<p01. More precisely, we denote by p0, the probability that a premium of size cc+ is received, and q0=1p0, the probability that no premium is received in each period. The number of claims is assumed to be another binomial process Nk,k with parameter p0<p<1. It means that the probability of having a claim is p and the probability of no claim is q=1p in each period. The claims Xj,k+ are independent and identically distributed (i.i.d.) positive and integer-valued random variables with the same distribution as the generic random variable X. The probability mass function (p.m.f.) and mean of X are denoted by fXx and μ, respectively. Let Sk=i=1NkXi be the total amount of settled claims up to the end of the kth time period with S0=0. The independence among Nk,k, Mk,k, and Xj,k+ are also assumed.

Suppose that premiums are received at the beginning of each period, and claims are paid out at the end of each period. We introduce a dividend policy to the company that a certain amount of dividends will be paid to the policyholder instantly, as long as the surplus of the company at time k is higher than a constant dividend barrier bb>0. It implies that the dividend payments will only possibly occur at the beginning of each period, right after receiving the premium payment. Then, the corresponding surplus Ubk of the insurer at the end of the kth time period can be described as(1)Ubk=u+cMkSkDk,u=0,1,2,,b,where Ub0=u is the initial surplus and Dk=D1+D2++Dk is the sum of the total dividend payments in the first k period with D0=0. The amount of dividends Dk paid out in the kth time period is defined as(2)Dk=maxUbk1+cηkb,0,k+,where ηk,k+ is a sequence of i.i.d. Bernoulli random variables with ηk=1=p0and ηk=0=q0.

Note that the positive safety loading condition holds if cp0>pμ. The time of ultimate ruin for model (1) is defined as τb=mink,Ubk<0. Let v0<v1 be a constant annual discount rate for each period. Then, the expected present value of the dividend payments due till ruin is(3)Vu;b=Ek=1τbDkvk1|U0=u.

3. Expected Present Value of Dividend Payments

By considering the occurrence (or not) of the premium income and claims in the next period, we separate the four possible cases as follows: no premium arrival and no claim occurs, a random premium arrival and no claim occurs, a random premium arrival and a claim occurs, and no premium arrival and a claim occurs. For u=0,1,,bc, we can obtain the following result by using the total probability formula:(4)Vu;b=vq0qVu;b+vp0qVu+c;b+vp0pxu+cVu+cx;bfx+vq0px=1uVux;bfx=vq0qVu;b+vp0qVu+c;b+vp0pVfu+c+vq0pVfu,where Vf holds for the convolution product of V and f.

Now, we show that (4) is a homogeneous difference equation of order c. The (forward) difference operator Δ is defined as(5)ΔVu;b=Vu+1;bVu;b.

Using the property of the forward difference operator Δ, we obtain(6)Vu+c;b=j=0ccjΔjVu;b.

Substituting (6) into (4) implies that(7)Vu;b=vq0qVu;b+vp0qj=0ccjΔjVu;b+vp0pj=0ccjΔjVfu+vq0pVfu,or, equivalently(8)j=0cajcjΔjVu;b=j=0cbjcjΔjVfu,where(9)aj=1j=0vq0q1j=0vp0q,bj=vq0p1j=0+vp0p.

Letting Az=j=0cajzj and Bz=j=0cbjzj, which are two polynomials of degree c in z. Then, (8) can be reformulated as(10)AΔVu;b=BΔVfu.

Equation (10) implies that Vu;b satisfies a homogeneous difference equation of order c. From the general theory of difference equations, the solution to (10) can be expressed as(11)Vu;b=j=0c1αjyju;b,where yju;bu=0 are linearly independent and satisfy(12)AΔyu;b=BΔyfu.

Multiplying (12) by zu+c and then summing over u from 0 to lead to(13)u=0zu+cyu;b=vq0qu=0zu+cyu;b+vp0qu=0zu+cyu+c;b+vp0pu=0zu+cyfu+c+vq0pu=0zu+cyfu,which can be reformulated as(14)y˜z;b=vp0qu=0c1zuyu;b+pu=0c1zuyfu1vq0qzcvp0q+pg˜zf˜z.

To find a set of fundamental solutions to (14), we choose yju;b=1j=u for j,u0,1,,c1. An application of Theorem 3.4 in  implies that yju;bu=0, for j=0,1,,c1, are linearly independent. From (14), the generating functions associated with yju;bu=0 satisfy(15)y˜jz;b=Rjzh˜1zh˜2z,where(16)h˜1z=1vq0qzc,h˜2z=vp0q+pg˜zf˜z,g˜zk=0gzzk=p0+q0zc,Rjz=vp0qzj+pu=j+1c1zufuj.

We can easily identify the number of zeros on the denominator in (15) by using Rouché’s theorem and its generalization. The proof is omitted, and similar discussions can be found in Bao and Liu .

Lemma 1.

When 0<v<1, the denominator in (15) has exactly c solutions, say zii=1c, inside the unit circle C=z:z=1.

Lemma 2.

When v=1, the denominator in (15) has exactly c1 solutions, say zii=1c1, inside the unit circle C=z:z=1 and another trivial root zc=1.

In what follows, we assume these zeros zii=1c are distinct for simplicity, as the analysis of the multiple zi’s leads to tedious derivations. We are ready to give alternative expressions for both the numerator and the denominator in (15) with the help of zii=1c. Since the numerator in (15) is analytic for all Rez1,1, the zeros of the denominator in (15) have to be zeros of the numerator.

Let πcz=j=1czzj and πczk=j=1,jkczkzj. Since h˜1z is a polynomial of degree c in z, using the Lagrange interpolating theorem, one knows(17)h˜1z=πczh˜10πc0+i=1ch˜2zizi1πczi+i=1ch˜2zizzi1πczi.

It is easy to see that limzh˜1z/πcz=1vq0q, and (17) can be simplified as(18)h˜1zπcz=1vq0q+i=1ch˜2zizzi1πczi,which yields(19)h˜1zh˜2zπcz=1vq0qh˜2zπczi=1ch˜2zizzi1πczi.

Now, we use a discrete operator Tz defined as Tzyc=u=0zuyu+c, and the corresponding properties of Tz can be found in Li . Then, h˜2z can be rewritten as(20)h˜2z=vpzcTzgfc+ηz,where ηz=vp0q+vpu=0c1zugfu is a polynomial of degree c1 in z. By imitating the same steps discussed above, we can prove that(21)ηzπczi=1cηzizzi1πczi=0.

Therefore, substituting (20) into (19) yields(22)h˜1zh˜2zπcz=1vq0qvpzcTzgfcπczi=1czicTzigfczzi1πczi=1vq0qvpTzTzcTz2Tz1gfc.

Note that (15) can be modified as(23)y˜jz;b=Rjz/πczh˜1zh˜2z/πcz.

For the numerator in (23), partial fractions lead to an equivalent representation(24)Rjzπcz=k=1cRjzkπczk1zkz.

By inserting (22) and (24) into (23), we obtain(25)y˜jz;b=11vq0qvpy˜jz;bTzTzcTz2Tz1gfc+k=1cRjzkπczk1zkz.

The direct inversion of the generating functions in (25) gives the following theorem immediately.

Theorem 1.

For j=0,1,,c1, yju,b satisfies the following defective renewal equation:(26)yju;b=ςn=0uyjun;bχn+ζu,u,where(27)ζu=11vq0qk=1cRjzkπczk1zku+1,ς=vp1vq0qT1TzcTz2Tz1gfc,χn=TzcTz2Tz1gfc+nT1TzcTz2Tz1gfc.

Proof.

To complete the proof, it remains to show that ς<1. In the case of v0,1, it follows from (22) that(28)ς=11vq0q1vq0qh˜11h˜21πc1=11v1vq0qπc1<1.

In the case of v=1, since zc is a root of the denominator in (15), we have(29)h˜1zch˜2zc=0.

Differentiating (29) with respect to v and taking the limit v1, from the positive safety loading condition, we eventually find(30)limv1zc=1cp0pμ>0.

Thus, taking the limit v1 in (28), one deduces(31)limv1ς=111q0qπc1limv11v1zc<1,which completes the proof.

Starting from yju;b=1j=u for u=0,1,,bc, (26) allows for a recursive calculation of the fundamental solutions to the homogeneous difference (14). Regarding the constants αjj=0,1,,c1 in the structural form of of (12), they are determined such that the remaining c equations are satisfied with initial surplus u=bc+1,,b, namely,(32)Vu;b=vq0qVu;b+vp0qu+cb+Vb;b+vp0p1xu+cbfxu+cbx+Vb;b+x=u+cbu+cVu+cx;bfx+vq0px=1uVux;bfx.

4. Numerical Example

In this section, we explain the solution procedure to the difference equation Vu;b through a concrete example. In what follows, it is assumed that f is a geometrical distribution with fn=1θθn1,n+, and the corresponding probability generating function is f˜z=1θz/1θz. Thus, equation (15) becomes(33)y˜jz;b=Rjz1θzΛz,where Λz=zc1θzvq0qzc1θzvp0q1θzvp0p1θzvq0p1θzc+1. Since Λz is a polynomial of degree c+1 with the leading coefficient a=θ+vq0qθvq0p1θ, it can be factored as(34)Λz=aπczzξ,where ξ is the zero of Λz on the complex plane. Note that ξ has a module larger than 1 and zii=1,2,,c has a module less than 1. Moreover, routine calculations lead to(35)πcz1θzΛz=πcz1θzaπczzξ=θa1+1/θξξz.

By substituting (24) and (35) into (33), we obtain(36)y˜jz;b=1θzΛzk=1cRjzkπczkπczzkz=θak=1cRjzkπczk1+1/θξξz1zkz.

Upon inversion, we obtain from (36) that(37)yju;b=θak=1cRjzkπczkzku+1+1θξl=0uξu+1lzkl+1=θak=1cRjzkπczk11/θξzkξzku+1θak=1cRjzkπczk1/θξzkξξu+1,where Rjzk is calculated through(38)Rjz=vp0qzj+p1θzj+11θz1θzcj11c1j+1.

As the explicit expression for yju;b is given by (37), the constants αjj=0c1 can be achieved such that (32) holds for u=bc+1,,b, which completes the determination of Vu;b. We provide a numerical example to illustrate the theoretical results.

Example 1.

Suppose c=2, and from (33), we have(39)Λz=z21θzvq0qz21θzvp0q1θzvp0p1θzvq0p1θz3.

Let p=0.6, p0=0.7, and v=0.85, and the relative safety loading condition cp0>pμ holds for all θ0,4/7. By solving Λz=0, we obtain the values of zi’s and ξ, see Table 1.

Explicit expressions for yju;b are determined by (37). For example, one has, for θ=0.3,(40)y0u;b=0.36125×2.50685u+0.77878×1.25337u0.14003×0.50348u,y1u;b=0.24797×2.50685u+0.33810×1.25337u0.09014×0.50348u.

Then, we obtain the values of yju;b, see Table 2.

To finish the calculation of Vu;b, we solve the system of linear equations (32) satisfied by αi, see Tables 3 and 4, respectively.

We depict Vu;b in Figure 1, and it is not surprising that Vu;b is an increasing function of u. When θ=0.3 is fixed, the first part of Figure 1 shows that a larger value of b corresponds to a smaller amount of dividend payments. On the contrary, for a fixed b=15, the second part of Figure 1 illustrates that a smaller θ leads to a larger number of dividend payments.

Numerical results of zi and ξ for p=0.6, p0=0.7, θ=0.3, and v=0.85.

ξz1z2
θ=0.13.486045347954590.3637662755053890.824973674803551
θ=0.22.541394025421210.3812513100605230.813367218414144
θ=0.31.986184913410060.398907210868630.797848459477162

Numerical results of yju;b for b=15.

u0123
y0u;b15.52900×10173.458114.17551
y1u;b3.31740×101711.054.56061

u4567

y0u;b16.179533.359892.6721220.969
y1u;b8.9641525.591860.2312155.915

u891011

y0u;b568.1611406.463548.148866.63
y1u;b384.679972.0752427.146096.64

u12131415

y0u;b22262.455764.5139848350509
y1u;b15268.138293.995973.1240620

Numerical results of αi for θ=0.3.

b1516171819
α00.071300.056890.045390.036210.02889
α10.103870.082870.066120.052750.04209

Numerical results of αi for b=15.

θ0.10.150.20.250.3
α00.197160.162880.129680.098780.07130
α10.275820.229610.184530.142120.10387

Impact of θ and b for Vu;b. (a) Numerical results for Vu;b when θ=0.3. (b) Numerical results for Vu;b when b=15.

5. Concluding Remarks

In this paper, we consider the compound binomial model with random income and a constant dividend barrier. Furthermore, we analyze the model with a general premium rate c. Using the roots of a generalization of Lundberg’s fundamental equation and the general theory on difference equations, we derive an explicit expression for the expected present value of dividend payments up to the time of ruin. In particular, a numerical example is provided to show that the formulae are readily programmable in practice. From the numerical example given above, we can see that the barrier level has a negative effect on the total expected present value of dividends.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have equal contributions. All authors have read and approved the final manuscript.

Acknowledgments

This research was supported by the Ministry of Education of Humanities and Social Science Project (20YJA910001), Foundation of Educational Department (W201783664), and Science and Technology Department (20180550196) of Liaoning Province.

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