Abstract
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 , the excess of the reserve over 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.
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. [1] 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 [2] 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 [3] 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 [4] 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 [5] investigate the expected present value of total dividends under stochastic interest rates. Wu and Li [6] 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. [7]. Zhang and Liu [8] 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 [9], Yang and Zhang [10], Labbé and Sendova [11], Karnaukh [12], Bao and Liu [13], and Zhou et al. [14].
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. [14]. However, we analyze the model with a general premium rate , while Zhou et al. [14] investigate the model with time-delayed claims and unit premium assumption. As stated in Landriault [4], 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 .
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 . Similar to Bao and Liu [13], we assume that the premium income is a binomial process with parameter . More precisely, we denote by , the probability that a premium of size is received, and , the probability that no premium is received in each period. The number of claims is assumed to be another binomial process with parameter . It means that the probability of having a claim is and the probability of no claim is in each period. The claims are independent and identically distributed (i.i.d.) positive and integer-valued random variables with the same distribution as the generic random variable . The probability mass function (p.m.f.) and mean of are denoted by and , respectively. Let be the total amount of settled claims up to the end of the th time period with . The independence among , , and 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 is higher than a constant dividend barrier . 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 of the insurer at the end of the th time period can be described aswhere is the initial surplus and is the sum of the total dividend payments in the first period with . The amount of dividends paid out in the th time period is defined aswhere is a sequence of i.i.d. Bernoulli random variables with .
Note that the positive safety loading condition holds if . The time of ultimate ruin for model (1) is defined as . Let be a constant annual discount rate for each period. Then, the expected present value of the dividend payments due till ruin is
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 , we can obtain the following result by using the total probability formula:where holds for the convolution product of and .
Now, we show that (4) is a homogeneous difference equation of order . The (forward) difference operator is defined as
Using the property of the forward difference operator , we obtain
Substituting (6) into (4) implies thator, equivalentlywhere
Letting and , which are two polynomials of degree in . Then, (8) can be reformulated as
Equation (10) implies that satisfies a homogeneous difference equation of order . From the general theory of difference equations, the solution to (10) can be expressed aswhere are linearly independent and satisfy
Multiplying (12) by and then summing over from 0 to lead towhich can be reformulated as
To find a set of fundamental solutions to (14), we choose for . An application of Theorem 3.4 in [15] implies that , for , are linearly independent. From (14), the generating functions associated with satisfywhere
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 [16].
Lemma 1. When , the denominator in (15) has exactly solutions, say , inside the unit circle .
Lemma 2. When , the denominator in (15) has exactly solutions, say , inside the unit circle and another trivial root .
In what follows, we assume these zeros are distinct for simplicity, as the analysis of the multiple ’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 . Since the numerator in (15) is analytic for all , the zeros of the denominator in (15) have to be zeros of the numerator.
Let and . Since is a polynomial of degree in , using the Lagrange interpolating theorem, one knows
It is easy to see that , and (17) can be simplified aswhich yields
Now, we use a discrete operator defined as , and the corresponding properties of can be found in Li [17]. Then, can be rewritten aswhere is a polynomial of degree in . By imitating the same steps discussed above, we can prove that
Therefore, substituting (20) into (19) yields
Note that (15) can be modified as
For the numerator in (23), partial fractions lead to an equivalent representation
By inserting (22) and (24) into (23), we obtain
The direct inversion of the generating functions in (25) gives the following theorem immediately.
Theorem 1. For , satisfies the following defective renewal equation:where
Proof. To complete the proof, it remains to show that . In the case of , it follows from (22) thatIn the case of , since is a root of the denominator in (15), we haveDifferentiating (29) with respect to and taking the limit , from the positive safety loading condition, we eventually findThus, taking the limit in (28), one deduceswhich completes the proof.
Starting from for , (26) allows for a recursive calculation of the fundamental solutions to the homogeneous difference (14). Regarding the constants in the structural form of of (12), they are determined such that the remaining equations are satisfied with initial surplus , namely,
4. Numerical Example
In this section, we explain the solution procedure to the difference equation through a concrete example. In what follows, it is assumed that is a geometrical distribution with , and the corresponding probability generating function is . Thus, equation (15) becomeswhere . Since is a polynomial of degree with the leading coefficient , it can be factored aswhere is the zero of on the complex plane. Note that has a module larger than 1 and has a module less than 1. Moreover, routine calculations lead to
By substituting (24) and (35) into (33), we obtain
Upon inversion, we obtain from (36) thatwhere is calculated through
As the explicit expression for is given by (37), the constants can be achieved such that (32) holds for , which completes the determination of . We provide a numerical example to illustrate the theoretical results.
Example 1. Suppose , and from (33), we haveLet , , and , and the relative safety loading condition holds for all . By solving , we obtain the values of ’s and , see Table 1.
Explicit expressions for are determined by (37). For example, one has, for ,Then, we obtain the values of , see Table 2.
To finish the calculation of , we solve the system of linear equations (32) satisfied by , see Tables 3 and 4, respectively.
We depict in Figure 1, and it is not surprising that is an increasing function of . When is fixed, the first part of Figure 1 shows that a larger value of corresponds to a smaller amount of dividend payments. On the contrary, for a fixed , the second part of Figure 1 illustrates that a smaller leads to a larger number of dividend payments.
(a)
(b)
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 . 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.