#### Abstract

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company taking into account the time value of ruin. We assume the preference of the insurer is of the CRRA form. The discounting factor is modeled as a geometric Brownian motion. We introduce the VaR control levels for the insurer to control its loss in reinsurance strategies. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategy. Finally, we provide some numerical examples to illustrate the results and analyze the VaR control levels on the optimal strategy.

#### 1. Introduction

In recent years, dividend optimization problems for insurance company have attracted extensive attention. The problem of optimal dividend was proposed by De Finetti in 1957. He suggested that a company would seek to find a strategy in order to maximize the accumulated value of expected discounted dividends up to the ruin time. Since then, many results on optimal dividend problems have been obtained. Some literatures on the optimal dividend include Avanzi et al. [1], Avanzi et al. [2], Avram et al. [3], Bayraktar et al. [4], Dai et al. [5], Dai et al. [6], Hubalek and Schachermayer [7], Li and Wu [8], Loeffen [9], Ng [10], Thonhauser and Albrecher [11], Yao et al. [12], and Yin and Wen [13].

On the other hand, reinsurance can effectively reduce the risk, so reinsurance is an important way for an insurer to control its risk exposure. Many interesting results have been obtained under reinsurance strategy in recent years. See, for example, Azcue and Muler [14], who studied the optimal reinsurance in the framework of Cramér-Lundberg model. Zhou and Yuen [15] considered reinsurance for a diffusion model with capital injection under variance premium principle. For some related discussions, among others, we refer the reader to Li and Shen [16], Liang and Palmowski [17], Wen and Yin [18], Wu [19], and Yao et al. [20].

In practice, Value-at-Risk (VaR) is often used to measure and control the risk of an insurance company. Recently, Chen et al. [21] investigated the optimal reinsurance strategies and the minimum probability of ruin with VaR constraints. Zhang et al. [22] discussed the optimal reinsurance strategies and the survival probability under dynamic VaR constraints. Bi and Cai [23] studied the equilibrium strategies under the mean-variance criterion with VaR constraints. Other works about optimal reinsurance problems with VaR constraints can be found in Cai and Tan [24], Cai et al. [25], Cheung et al. [26], and Yiu [27].

The interest rate is the main factor of an insurance company. And, it is affected by many factors. So, we must pay attention to the current situation of interest rate and its changing trend. It is more reasonable to assume that the interest rate is a function of time. Eisenberg [28] solved the optimal dividends under a stochastic interest rate for the case of the Brownian motion. In this paper, we are going to study the problem of maximizing the expected discounted utility of dividend, taking into account both reinsurance under VaR constraints and a stochastic interest rate. The insurer’s surplus process is approximated by a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategies with and without VaR constraints.

This paper is organized as follows. In Section 2, we provide a general formulation of the optimal reinsurance-dividend problem. Then, we investigate the problem of the expected discounted utility of dividend maximization under noncheap reinsurance assumption. We give the explicit expressions for the expected discounted utility of dividend payments and the optimal strategies in Section 3. In Section 4, we consider the optimization problem with VaR constraints and solve the optimization problem using the results derived in Section 3. We also present some numerical examples to illustrate the results in Section 5.

#### 2. Model Settings and Problem Formulations

Let be a probability space equipped with a filtration .

##### 2.1. Reserve Process of an Insurer

The reserve process of an insurer is modeled by for the aggregate cumulative amount of claims counted up to time : where and are positive constants, is the initial surplus, is the premium rate with the safety loading , and is a standard Brownian motion.

In the following, the insurer is allowed to pay dividends. The cumulated dividends are described by a process , which is an adapted and nondecreasing process. represents the total dividends up to time . In this paper, we assume that is absolutely continuous with respect to the Lebesgue measure. We suppose that the process admits almost surely a density process denoted by modeling the intensity of the dividend payments in continuous time. To manage the risk, the insurer could choose a reinsurance policy at time . When , it means the insurer purchases proportional reinsurance. In this case, the insurer only pays its , while the reinsurer pays the rest . A strategy is said to be admissible, if the ruin cannot be caused by the dividend payment, and is -progressively measurable and satisfies , for all . We denote the set of all admissible strategies by . The set of admissible strategies consists of nonnegative, nondecreasing, adapted, cádlág process. Let denote the insurers total wealth at time under the strategy . Then, the dynamic of is given by where is the safety loading of the reinsurer and the condition of is required for avoiding the insurers arbitrage.

As a risk measure, we consider that dividends are discounted by the geometric Brownian motion where , , and . Let further be a standard Brownian motion independent of . We suppose that the underlying filtration is generated by . The time of ruin for this process is defined by . We define the return function corresponding to to be where is some fixed utility function, , means expectation with respect to . Usually, it is assumed that is differentiable and nonnegative. can be interpreted as the present value of an amount which the insurer earns as long as the company is alive. In this way, the lifetime of the portfolio becomes part of the value function and is weighted according to the choice of . We seek to find an optimal strategy so that

In the present paper, we consider the Constant Relative Risk Aversion (CRRA) utility function:

##### 2.2. Value-at-Risk Constraints for the Proportional Reinsurance and Dividend Strategy

Under the reinsurance-dividend strategy , the insurer’s wealth process is a risk process. The insurer may or has to use the risk measure of VaR to control its wealth for avoiding huge loss. For is small enough and for in the interval , we approximate by , . This is a reasonable approximation since the insurance strategy can only be adjusted at discrete time, and the decision is made based on the surplus at time . Thus, the loss of the insurer in time interval can be expressed as . Then, the SDE (3) admits a solution

Thus,

For a given risk level and a given horizon , we denote the conditional VaR of conditioning on by , namely,

Proposition 1. *Given risk level and time length . We have
where is the inverse function of the cumulative standard normal distribution function .*

*Proof. *We have
where the last equality follows from the fact that conditional on the filtration is a standard normal random variable. Then, we have
Thus, we obtain (11).

In this paper, we will derive the optimal strategy under the constraint

#### 3. Solution to the Optimization Problem without VaR Constraints

In this section, we will solve the optimal reinsurance-dividend problem without VaR constraints. The HJB equation corresponding to the problem (6) is given by with the boundary condition

We conjecture that the value function can be written in the form . In this case, the HJB equation becomes

Note that supremum in (17) without constraints is realized for

Denote . Let

We will solve our optimization problem under the assumption:

*Assumption 2. *For noncheap reinsurance, assume that , and .

For the optimal value function , we can derive the following property.

Lemma 3. *The optimal value function satisfies , , and .*

*Proof. *It is easy to see that is increasing in and decreasing in . We prove the property of . For , denote . From the dynamic programming principle, we claim that satisfies the following equality:
We first show the part. Let . There exists such that . We choose the strategy where is an admissible strategy. Then,
and therefore, we have
For the other direction, since
we have
Thus, (21) is proved.

For , let be the set of strategies such that
on the set , where is a stopping time defined by
By putting from (21), we obtain that
Then,
Since and is a nondecreasing function of , the right-hand side of (29) is a decreasing function of . Thus, is decreasing in . Hence, is also decreasing and . This finishes the proof.

Proposition 4. *If Assumption 2 is true, then on , the function solving (15) is given by
where and are given in (19) and (20) and , where is the inverse of function :
for and satisfies
*

Above where

The maximizers and are given by

*Proof. *From the expression of and , we have that the HJB equation of (17) becomes
Letting and , then we have . Plugging it into (36) and taking derivatives with produce
We take derivatives with , and we get
Denote , we have
where . In view of , the value of should be nonnegative. Then, . If is strictly positive, from (37) and (38), we get , where is some fixed constant. Note that
where is some fixed constant. Therefore, by Assumption 2, we must choose to be equal zero. Recall that and (37), we obtain the representation of the function
Thus, we have given in (30). Since and , we get (31) and (32).

Since
we get . Since , to satisfy the required condition , we need to show that the inequality is true. It is easy to verify that given in (34) is the unique solution of the equation . In view of Assumption 2 and , we have , so we have for , i.e., on .

It follows from the facts and that is increasing and strictly concave. Substituting the values of and into the left of (17), we have
Since is an increasing and concave function and for , therefore, the maximizers and are given by and with . Hence, Equation (43) can be rewritten as
It follows from (30) and (31) that (44) is true. This finishes the proof.

Since on the interval , it is reasonable to let ; then, the HJB equation is equivalent to the following equation:

We also conjecture that the value function can be written in the form . In this case, the HJB equation of (45) becomes with the boundary conditions

If we set , then the HJB equation of (46) becomes with the boundary conditions

Similar to the proof of Proposition 5 in Liang and Palmowski [17], we have the following results whose proof is omitted.Summarizing Propositions 4 and 5 gives the following result.

Proposition 5. *Assume that Assumption 2 holds. Suppose also that on solves (46)–(48). Then , on , and the the maximum in
is attained when in this case.*

Theorem 6. *Assume that Assumption 2 holds. Then the value function is given by
with and given in (19) and (20) and , where is defined in (31). Moreover, is given in (33), and solves (46). The corresponding optimal dividend strategy is
and the optimal reinsurance proportion is
*

*Proof. *If , denote , where
Since is a continuously differentiable solution of Equation (17), it is easy to obtain that is a solution of (15). If , denote , where is a continuously differentiable solution of Equation (36); it is easy to obtain that is a solution of (15).

From now on, we prove the optimality of strategy
Let be an admissible strategy with dividend and reinsurance strategy. Select . Define
It follows from the Itô’s formula that we have
Taking expectations on the both sides of the above inequality yields
Note that the equality holds. We claim that is uniformly integrable. Then, letting , , and by applying the dominated convergence theorem and the monotone convergence theorem, we have
Thus,
And, we can conclude that . By the concavity of , we have
Thus, for each ,
We introduce now a new stochastic process having the following dynamics:
Then, one can observe that is a submartingale and for any . We denote . Then, by the Cauchy-Swartz inequality and Doob’s maximal inequality for submartingale, we have
where and are some fixed constants. Therefore, in view of (64) and (65), we can conclude that is uniformly integrable. This completes our proof.

*Remark 7. *Letting and and in Theorem 6, the result reduces to the result of Theorem 7 in Liang and Palmowski [17].

#### 4. Solution to the Optimization Problem under VaR Constraints

In this section, we will use the results in Section 3 to solve the optimal problem with VaR constraints (14). To do so, we make the following assumption.

*Assumption 8. *We assume that and or equivalently .

Under Assumption 8, the VaR constraint is equivalent to

Note that , so there exists at least one strategy that satisfies (66). Then, the control space defined in Assumption 8 is not empty. The strategy under VaR constraints should satisfy

If the strategy defined in Theorem 6 satisfies the last inequality in (67), then the strategy is also a solution with the constraint, namely . Otherwise, if the strategy defined in the first and second equations in (67) is outside the control space at the initial point , then the optimal strategy is just the boundary of the control space. If the strategy defined in the first and second equations in (67) is inside the control space at the initial point , but it leaves the control space at sometime, we define the first exit point of the control space as where and . Then, under the VaR constraints, the optimal strategy is with

#### 5. Numerical Examples

In this section, we illustrate the results obtained in Sections 3 and 4 by numerical examples. We set the model parameters of the insurers reserve process and the financial market in Table 1.

First, we plot the optimal dividend and optimal reinsurance proportion for surplus ranging from to without VaR constraint. The strategies are shown in Figure 1. From Figure 1, we find the optimal threshold equals to 10.3337 according to Theorem 6, which implies that the insurer will take all the claims without buying any reinsurance when the surplus is larger than 10.3337. From this graph, we can also observe that the optimal dividend rate is nearly linear increasing with the surplus , while the optimal reinsurance proportion is decreasing with . When its surplus is sufficiently small, the insurer will divert most of its risk incurred by claims to the reinsurer, which seems intuitively reasonable. It is a reasonable result because when the insurer has a bigger initial wealth , the insurer would like to retain a bigger proportion of its insurance risks and increase the dividend rate.

Second, we plot the optimal dividend and optimal reinsurance proportion as the functions of for fixed without VaR constraint. The strategies are shown in Figure 2. From Figure 2, we find that the optimal dividend rate increases with increasing , while the optimal reinsurance proportion decreases. This phenomenon can be explained by noting that the parameter represents the risk aversion of the insurer, so the insurer will retain more proportion of the insurance risks and give more dividend payment.

Third, we use the model parameters of the insurers reserve process market as in Table 2 and consider the VaR constraints for three different cases as in Table 2, where the time interval is equal to , and the VaR control levels of are set so that (66) holds. Cases 1 of Table 2: for the VaR control level and risk level given in Case 1 of Table 2, time interval , and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:

Cases 2 of Table 2: for the VaR control level and risk level given in Case 2 of Table 2, time interval , and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:

Cases 3 of Table 2: for the VaR control level and risk level given in Case 3 of Table 2, time interval , and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:

Under the VaR constraints, both the reinsurance strategy and the investment strategy are increasing while the initial capital increases, which are presented in Figures 3 and 4. It is a reasonable result because when the insurer has a bigger initial wealth , the insurer would like to retain a bigger proportion of its insurance risks and a larger dividend rate. The upper bounds represent the effect of the VaR constraints on the strategies, which implies that to limit the loss of the insurer at the VaR control level , the insurer has to limit both the amounts of dividends and the retained insurance risks within certain scope. And, we can see that the upper bounds of the reinsurance strategy and the rate of dividend in case 2 are larger than those in case 1, which means that the insurer can retain more insurance risks and pay more dividend payment in Case 2 than in Case 1. This finding is consistent with the fact that the insurer has a tougher VaR control level of 0.08 in case 1 than that of the VaR control level 0.1 in case 2.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflict of interests.

#### Authors’ Contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11501319, No. 11571198).