The Compound Binomial Risk Model with Randomly Charging Premiums and Paying Dividends to Shareholders

Based on characteristics of the nonlife joint-stock insurance company, this paper presents a compound binomial risk model that randomizes the premium incomeonunit time and sets the thresholdx for paying dividends to shareholders. In thismodel, the insurance company obtains the insurance policy in unit time with probability p 0 and pays dividends to shareholders with probability p 1 when the surplus is no less thanx.We thenderive the recursive formulas of the expected discounted penalty function and the asymptotic estimate for it. And we will derive the recursive formulas and asymptotic estimates for the ruin probability and the distribution function of the deficit at ruin. The numerical examples have been shown to illustrate the accuracy of the asymptotic estimations.


Introduction
The compound binomial risk model is one of the classical actuarial models that have been studied extensively.The classic literatures about the compound risk model primarily include [1][2][3][4][5][6][7][8][9].With the emergence and development of dividend insurance, compound binomial risk models considering the case of paying dividends to policyholder are attracting more and more attention of actuarial scholars; see [10][11][12][13][14]. Reference [10] builds the compound binomial risk model with randomly dividends payment and derives the ruin problem by recursive algorithm for cases where the company pays dividends to its policyholders with a certain probability.Reference [11] obtains and solves two defective renewal equations for the Gerber-Shiu penalty function under the compound binomial model proposed in [10].Reference [12] generalizes the model of [10] and derives the discounted penalty function under the compound binomial model with a multithreshold dividend structure and randomized dividend payments.Considering the fact that the joint-stock company may pay dividends to the policyholders and shareholders, [13] builds the compound binomial risk model with random dividends payment to the policyholders and shareholders and studies the ruin problem with the model.furthermore, [14] derives the arbitrary moments of discounted dividend payments under the compound binomial risk model with interest on the surplus and periodically paying dividend.
The previously mentioned models are compound binomial risk models with a constant premium rate suited for depicting the surplus of the life insurance companies which collect installment premiums.However, nonlife insurance companies (e.g., property insurance companies) charge premiums immediately, and insurance policies are obtained randomly in unit time.Thus the model with a constant premium rate cannot reasonably describe the surplus of the nonlife insurance companies.Moreover, joint-stock nonlife insurance companies need to pay dividends to the shareholders randomly (see [10]).However, these characteristics have not been considered in [10] together.Thus, [10] cannot be suitable for describing the surplus of the joint-stock nonlife insurance companies.In order to address the deficiencies of the models in [10], a compound binomial risk model has been developed with random premiums and dividends payment to shareholders, and it derives the recursive formulas and asymptotic estimates of the ruin probability and the distribution of the deficit at ruin.This paper is organized as follows.In Section 2, we build the compound binomial risk model with randomly charging premiums and paying dividends to shareholders.In Section 3, we derive the recursive formulas of discounted penalty

The Model and Preliminaries
Consider the compound binomial model with randomized decisions on paying dividends, which is described by with the initial surplus of the insurance  (≥0).  is the aggregate claim up to time ; that is, where   denotes whether the claim occurs or not in ( − 1, ]; the event in which the claim occurs is denoted by  +1 = 1; the event in which no claim occurs is denoted by  +1 = 0.The probability of a claim is  and the probability of no claim is  = 1 −  in any period (,  + 1]. = {  ,  = 1, 2, . ..} is independent and identically distributed random series. = {  ,  = 1, 2, . ..} is independent and identically distributed as  = {() = Pr( = );  = 1, 2, . ..}. () is the aggregate dividends payment; that is, where  (>0) is the threshold such that the insurance company may pay dividends to the shareholders, and () is the indicator function of a set .   denotes whether dividend is paid or not in ( − 1, ].When the surplus is no less than , the company pays one dividend to the shareholders with probability  1 (denoted by   = 1) and not with the probability  1 = 1 −  1 (denoted by   = 0);  = {  ,  = 1, 2, . ..} is independent and identically distributed as ( 1 ) (0 <  1 < 1).
According to the feature of the variety of the surplus in joint-stock nonlife insurance companies, we further assume that premium is charged randomly in unit period ( − 1, ].Then the aggregate premium is where   is variable with distribution ( 0 ) (0 <  0 ≤ 1),  0 = 1 −  0 , and denotes obtaining an insurance policy by   = 1 in (−1, ], as well as denotes not obtaining an insurance policy by   = 0.  = {  ,  = 1, 2, . ..} is independent and identically distributed.Furthermore, the random series , , ,  are assumed to be mutually independent.Then, the surplus of the nonlife insurance joint-stock company is The model is called as the compound binomial risk model with random premiums and dividends payment to shareholders. Remark.(1) Model ( 5) is the general form of the model in [10] and is the model in [10] if  0 = 1.And also, model ( 5) can be regarded as general form of the classic risk model and exactly the classic risk model if  0 = 1,  1 = 0.
(2) In this model, the initial time  = 0 is some time in the past at which point we begin studying the surplus of the jointstock nonlife insurance company, but not the time when the company is created.

Lemma 3.
If  is a strictly diagonally dominant matrix, then  is a nonsingular matrix.
Theorem 4.Under the assumption that the security loading  > 0, the set of linear equations (8) and ( 9) have a unique solution.
Proof.Let  = ((0), (1), (2), . . ., ()), Δ = (, (0), then, the set of linear equations ( 8) and ( 9) can be rewritten as The coefficient matrix  will be carried out by a series of elementary row operations as follows: the ( + 1) row is replaced by itself plus the first  rows; the  row is replaced by itself plus the first  − 1 rows, and so on; the second row is replaced by itself plus the first row.The matrix  is changed into For the first row, because  > 0 and  = ∑ +∞ =0 () > 1, For the second row, because  0  +  0 (1) − 1 < 0, then For the  row (2 <  ≤  − 1), For the  row, owing to For  − 1 row, owing to  0 (1) −  −  1 < 0, Thus, the matrix  is a strictly diagonally dominant matrix.According to Lemma 3, matrix  is a nonsingular matrix.So the set of linear equations have a unique solution.The theorem has been proved.

The Asymptotic Estimate of the Penalty Function
Let  =  1  1 +  1 −  1 denote the generating function of ; then where   is the generating function of .
Assumption 5.There exists a  ∞ such that   () → +∞ This assumption is similar to the one in [11].
Let   () = 1, and then = 1, () is a convex and increasing function in [0,  ∞ ), and thus (38) has two real nonnegative roots at most, and one of them is 1.
Therefore, there exist two real roots in (38).Denote the other root by , and then  > 1.
Note that if  0 = 0, (38) becomes which is just the adjustment coefficient equation of the compound binomial model with randomized decisions on dividends payment (see [10]).The following lemma will be used to derive the asymptotic estimates of ().
Theorem 7. The asymptotic estimate for the penalty () is where Proof.When  > , from (12), we can obtain Equation ( 27) is equivalent to Combining ( 44) and (45), we can obtain the renewal equation Denoting that () = ()  , Multiplying ( 46) by   , we can obtain We will prove that (48) satisfies the conditions of Lemma 6.Consider where the last equation is valid because  is the root of (38).
The following steps will prove that (52) According to Lemma 6, we can derive lim Equation ( 53) is equivalent to (42).The theorem has been proved.

The Application of the Penalty Function
We will give some examples of ruin quantities such as the ultimate ruin probability, the distribution of the surplus of the deficit at ruin to illustrate the application of the recursive formulas, and asymptotic estimates for the penalty function (). where (2) for  ≥ , the penalty functions () satisfy By Theorem 7, the asymptotic estimates of the ultimate ruin probability are where It suggests that the asymptotic values are more close to the exact values with the surplus increasing under the cases  = (0.75, 0.015), (0.65, 0.055).

Conclusions
In order to describe the surplus of the nonlife insurance companies reasonably, the compound binomial risk model with randomly charging premiums and paying dividends to shareholders is proposed in this paper.Further, we derive the recursive formulas and asymptotic estimation of penalty function using classical method.The results about penalty function are applied to obtain the recursive formulas and asymptotic estimations of the ruin probability and the distribution of the deficit at ruin.The numerical examples show that the actual penalty function can be approached by asymptotic estimation.The results about the model are meaningful to analyze the ruin problem about the joint-stock nonlife insurance companies.It may provide the reference for decision-making of the joint-stock nonlife insurance companies about risk management.
Theorem 7, we can obtain the asymptotic estimates of the distribution function of deficit at ruin.Consider  (, ) ∼   ()  − ,

Table 2 :
Exact values and asymptotic values for the ruin probability.