Abstract
In this paper, we consider the optimization problem of dividends for the terminal bankruptcy model, in which some money would be returned to shareholders at the state of terminal bankruptcy, while accounting for the tax rate and transaction cost for dividend payout. Maximization of both expected total discounted dividends before bankruptcy and expected discounted returned money at the state of terminal bankruptcy becomes a mixed classical-impulse stochastic control problem. In order to solve this problem, we reduce it to quasi-variational inequalities with a nonzero boundary condition. We explicitly construct and verify solutions of these inequalities and present the value function together with the optimal policy.
1. Introduction
Corporate dividend policy has long engaged the attention of financial economists. The classical paper [1] by Miller and Modigliani provides the valuation formula for an infinite horizon firm under perfect certainty. In a world of perfect capital markets, they show that the dividend policy is irrelevant as a firm can always raise funds to meet the need for continuing operation. However, under a more realistic condition with imperfections such as the presence of financial constraints, information asymmetry, agency costs, taxes, risk exposure under uncertainty, transaction costs, and other frictions, it has been shown that there exists an optimal dividend policy (see [2–5]). Thus, determining the optimal dividend payouts becomes an important issue as it affects firm value. More recent models have focused on the issue of how to set the optimal dividend policy in a dynamic uncertain environment.
The valuation model used by Miller-Modigliani in 1961 can be extended to the situation of controllable business activities in a stochastic environment. During the recent decades, there have been increasing interests in applying diffusion models to financial decision problems, especially in (re)insurance modelling (see [6–17]). For most of these models, the liquid assets processes of the corporation contain a Brownian motion with drift and diffusion terms. The drift term corresponds to the expected profit per unit time, and the diffusion term represents risk exposure. By using diffusion models, many kinds of optimal dividend problems, such as in [7, 14–16], are discussed and optimal policies are presented in these papers. Particularly, in some papers (see [14–16, 18]), authors discuss much more practical problems by considering a fixed transaction cost for each dividend payout. In [14], the optimal dividend problem without bankruptcy for insurance firms is considered under the assumptions of constant tax rate and fixed cost for dividend payout. In [15, 16], the author considers the general income process with drift term and diffusion term and the case of bankruptcy. Moreover, numerical methods, such as the Runge-Kutta method, are implemented to simulate the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear differential equation. In [19], the author studies the dual risk model with a barrier strategy under the concept of bankruptcy, in which one has a positive probability to continue business despite temporary negative surplus. In [20], the author considers an insurance entity endowed with an initial capital and an income, modelled as a Brownian motion with drift and finds an explicit expression for the value function and for the optimal strategy in the first but not in the second case, where one has to switch to the viscosity ansatz. In [21], the authors suppose that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals and obtain the explicit solutions for value function and optimal strategy.
In these papers, the value function is typically assumed to be zero when there is a bankruptcy. But in the real world, some shareholders, especially for preferred shareholders, can get some money back when a terminal bankruptcy occurs. That means for this case the value function is not zero at bankruptcy. Thus, it is very useful and necessary for us to consider this kind of problem. In this paper, we postulate that the amount of money, shareholders can obtain for the terminal bankruptcy, is a positive constant, . Moreover, we assume that the liquid assets follows a process with constant drift and diffusion coefficients.
In the model of this paper, as that in [14], the dividend distribution policy is given by a purely discontinuous increasing functional. The net amount of money received by shareholders is for the -th dividends, where is the amount of the dividend payments, is the tax rate the shareholder pays, and is the fixed cost whenever the dividends are paid out. Further, represents the moments of dividend payments and is the discount rate. Based on these assumptions, we transform the value function into quasi-variational inequalities (QVI) and list out a candidate solution to QVI with a positive boundary condition . Subsequently, we show that the value function can be given by and the optimal policy can be presented based on the solution . A natural question is how to point out whether there is a bankruptcy or not in order to obtain the optimal policy under some conditions. To answer this question, some criteria are provided.
A major difficulty in this paper is that the structure of the candidate solution is uncertain since the existing interval of it has unfixed endpoints, which depends on some unknown parameters. This phenomenon does not appear in [14] and other related papers. Enlightened by the derivatives of candidate solutions, we construct the integral and then discuss it by several cases for , , , , and . For the model mentioned above, which is restricted to stay at the bankruptcy state, it is denoted by terminal bankruptcy model as in [22].
The structure of this paper is as follows. In the next section, we provide a rigorous mathematical model for the optimal dividend problem. Then the stochastic control problem is transformed to a QVI. Moreover, some definitions and an important verification are presented. Following this, the detailed structure of candidate solutions is given under different situations in Section 3. In Section 4, the uniqueness of some unfixed parameters is verified; some formulas to calculate these parameters are proposed and some numerical examples are shown to support our theoretical results. In Section 5, we demonstrate that the candidate solutions satisfy QVI and obtain the optimal dividend policy. In the last section, we summarize our results and suggest a direction for future research.
2. The Mathematical Model
The proposed model considers the dividend optimization problem for a firm which can control its business activities that affect its risk and potential profit. It extends the classical Miller-Modigliani model of firm valuation to the situation of controllable business activities in a stochastic environment with a possibility of bankruptcy and a positive residual value to shareholders upon bankruptcy. The model is quite general and can be applied to any firm in which management has control on the dividend stream as well as the risk exposure. Without loss of generality, the model is cast in the framework of a large insurance company, as it possesses many nice features to best illustrate the model.
2.1. Value Function
Let be a probability space with a filtration and be a standard Brownian motion adapted to that filtration. Moreover, the reserve process is a state variable, which denotes the liquid assets of the company. For an insurance company, in order to reduce risk, the risk control takes up the form of proportional reinsurance, which mathematically corresponds to decreasing the drift and diffusion coefficient by multiplying both quantities by the same factor . The time of dividends is described by a sequence of increasing stopping times and the amounts of the dividends paid out to the shareholders, associated with the times, are represented by a sequence of random variables . Then the controlled state process before bankruptcy is given bywhere is the initial reserve and is an indicator function.
Let the time of bankruptcy be given by
Definition 1. Let be an -adapted process; let , , be a stopping time with respect to , and let the random variable , , be measurable with ; thenis called an admissible control or an admissible policy. The class of all admissible controls is denoted by .
In addition, we denote the net amount of money that shareholders receive by a function aswhere the constant is a fixed setup cost incurred each time that a dividend is paid out, and the constant is the tax rate at which the dividends are taxed, and is a real value variable with respect to the amount of liquid assets withdrawn.
A performance functional with each admissible control is defined bywhich represents the total expected discounted value received by shareholders until the time of bankruptcy, where is the known amount paid out to shareholders when the terminal bankruptcy happens.
Define the value function byThen the optimal control is a policy for which the following equality can be satisfied
2.2. Properties of the Value Function
In this section, the QVI associated with the stochastic control problem is provided. Moreover, we derive some properties of the value function.
Proposition 2. For every , the value function in (6) satisfies
Proof. By the same method as in [14] and letting instead of , then the result can be obtained.
Let be given by (4); then, for a function , define the maximum utility operator of it by
Suppose that the payment of dividends occurs at time and the amount of it equals ; then the reserve decreases from initial position to . After that, if the optimal policy is followed, the total expected utility is . Consequently, under such a policy, the total maximal expected utility would be equal to . On the other hand, for each initial position, suppose that there exists an optimal policy, which is optimal for the whole domain. Then the expected utility associated with this optimal policy is , which is greater or equal to any expected utility associated with another different policy. So, we have
Now, defineBy the dynamic programming principle, we know that in the continuation region, satisfies
The arguments in (10) and (12) give us an intuition for the following two definitions and one theorem.
Definition 3. Assume that function . For every and , if we havethen we claim that satisfies the quasi-variational inequalities of the control problem.
Definition 4. The control is called the QVI control associated with ifand, for every ,
As in Cadenillas et al. [14], we also have the following theorem.
Theorem 5. Let be a solution of QVI. Suppose there exists such that is twice continuously differentiable on and is linear on . Then, for any ,Furthermore, if the QVI control associated with is admissible, then coincides with the value function and the QVI control associated with is the optimal policy; that is,
Proof. The idea of this proof is very similar to that of Theorem in Cadenillas et al. [14]. So, we do not show it in this paper.
3. Smooth Solutions to the QVI Properties
In this section, we first recall the zero boundary (no recovery) problem in Cadenillas et al. [14], and then by the similar method of this solved problem, we obtain the smooth solutions of QVI properties.
3.1. Solution for the Problem with Zero Boundary Condition
Let us consider the similar problem of QVI as follows:
Letand defineThen from Cadenillas et al. [14], we can obtain the structure of the solution of :where is a free constant and , , , and are given byand is the unique solution of the following equation:In Cadenillas et al. [14], it has been shown that and . Define a function by
Letthen is a decreasing function of with the range , and there exists unique , , and , such that .
Further, Cadenillas et al. [14] show that the solution of can be given by (21) with , , and .
3.2. Smooth Solutions of QVI Properties on
The difference between QVI and is just the boundary condition. Thus, we conjecture that solutions of them may have some similarities.
3.2.1. Smooth Solutions of QVI Properties on
First, as in (20), we defineThen, on the interval , from QVI we have that
Let be the maximizer of the expression on the left-hand side of (28); then
Putting (29) into (28), we haveA general solution of (30) under the boundary condition iswhere is a free constant and is presented by (19). From (29) and (31), it follows that
Since above is an increasing linear function, if and only if , whereThus, if and , it follows that . But since the range of is , then we must have for ; consequently (28) becomes
One general solution to (34) can be written bywhere and are free constants and and are given by (22). Continuity of the function and its derivative at the point implies that and , where is a free constant and and are defined by (23).
On the other hand, if , then , for any , and (28) becomesSolving (36), we can obtain the general solution of (36) as follows:with .
Now, let us summarize the possible structure for the solution of (28) on . If , thenwhere is a free constant. If , then the structure of on is given by (37).
Remark 6. From (33), we notice that depends on the uncertain parameter , which will be estimated later. For different , the sign of may be different. Moreover, it is easy to show that the assumption, , is equivalent to and is equivalent to . In addition, for (37) and (38), they are consistent at . That is, for any ,At , from the consistency of two solutions, we have that . In addition, the condition of existence for (37) implies . Moreover, there is no conflict to denote by when . So, we let for .
Remark 7. For in (38), from and , it is easy to show that on , which implies that obtained from (38) have convexity on .
3.2.2. Smooth Solution of QVI Properties at
From the definition of , we have that . Then, byit follows that the maximizing sequence for cannot have zero as a limiting point. So, at , the supremum on the right-hand side of (9) can be taken over for some . Therefore, there exists , such thatLet ; then andFrom (42), it follows thatthen we have
Remark 8. If , then , which means the dividend happens once and then a bankruptcy follows. This is an important phenomenon that deserves to be discussed.
4. Uniqueness for the Unfixed Parameters
In Section 3, some parameters, such as , , and , are unfixed numbers. In this section, we discuss the uniqueness of their corresponding parameters by two useful integral functions, whose integrands can be used to obtain solutions of QVI. As in Remark 6, we claim that can be used to denote for .
4.1. Definitions and Properties of Two Integral Functions
From Remark 6, it is known that different will lead to two possible cases for . The first case is and the second is . In the following, we discuss these two cases by two constructed integral functions, respectively.
Case 1 (). Let be a function, with constant , constructed bywhere is also defined by (33) with .
Definewhere and are two nonnegative roots of the equation with and denotes . If does not exist on , then let .
From the definitions of and , obviously is a continuous function of . Further, and are also continuous functions of if the existence condition of them is satisfied. So, is a continuous function.
Proposition 9. Let be defined by (45); then it has convexity on by . Moreover, is an increasing function with respect to by . For , if the condition of its existence is satisfied on some subintervals of , then it is a strictly decreasing function on these subintervals.
Case 2 (). Letwhere .
Definewhere and are two nonnegative roots of the equation with . If does not exist on , then let .
For (48), taking the derivative of with respect to givesConsequently, for any positive , due to the fact that and . So, we have the following result.
Proposition 10. For , if its existence condition is satisfied on some subintervals of , then it is a continuous and strictly decreasing function on these subintervals. Moreover, we can show that and
Remark 11. For small enough and positive , it is possible that . Under this situation, may have no convexity since may not be satisfied.
In the following, we discuss the property of two integral functions at .
At there are some common properties of and . Firstly, it follows that and . Then fromwe obtain that at the integrands of and are the same. Consequently, we can conclude that if exists at , then To judge whether , we have the following result.
Proposition 12. One equivalent condition of is , where is given by
Proof. Let Notice that , and so has convexity. One equivalent condition of is thatSolving , we have thatIt can check that , so in (53).
Putting (53) into , we haveFrom (52) and (54), we can complete the proof.
Remark 13. An important inequality with respect to given by (51) is .
Proof. For given by (51), taking its logarithm, we can getLet us check the derivative of on as follows:Therefore, is an increasing function of . From , we can take limits as and to get the infimum and the supremum of . For (55), it follows thatSince the infimum of is and the supremum of is , we can get
4.2. Integral Functions under Different Conditions
By the convexity of on , it shows that , that is, , is equivalent to . From the definition of , it is known that . We compare to as follows.
Case 1 (). By simplicity, this condition is equivalent to In addition, at point , from (45) we have that and . Combining with the convexity of , it follows that and on .
Proposition 14. If , then is satisfied. Consequently, from the monotonicity of , we have .
From the above, it shows that can be the left domain endpoint for . On the other hand, from as and is a continuous function, then there exists , such that . Therefore, can be the domain of . Moreover, we have that can be the domain of from Section 4.1.
Remark 15. If , we have , for and , for , where .
Therefore, we can defineas a useful integral function under . Moreover, is a strictly decreasing function.
Case 2 (). By simplicity, this assumption is equivalent to In addition, in the inequality is equivalent to From (22), we can obtain that is equivalent to . Then, for , it can be easily seen that at . By the similar discussion as above, we can show that as follows.
Proposition 16. If , then . If , then and for any .
Remark 17. If , we have for , where . If , we have for , where . If , we have , for , and , for , where .
Therefore, we can also use and to define three types of similar to (58) as useful integral functions under . Moreover, these types of are strictly decreasing functions.
4.3. Compute Parameter and Numerical Examples
In the process of calculating , we can follow the same steps for these two cases. By the following steps, the uncertain parameter can be obtained. Further, we can get other parameters, such as , , , and .
Step 1. Compare with .
Step 2. If , Compute , and then compare with .(i)If , get on , such that .(ii)If , get on , such that .
Step 3. If , then compare with and solve .
In practical problems, , but the restriction for is just . In order to satisfy , for convenience, we just choose different values for and never change the values of , , , and in the following numerical examples. The values of these parameters are selected as , , , and . For the value of , we choose to satisfy four different conditions. In Figure 1, the graphs of are shown, and both forms of , , and are listed out in their corresponding domains. Moreover, the right domain endpoint of is also presented. Figure 1 shows that is always a decreasing function.
(a)
(b)
(c)
(d)
In Figure 1(a), , which satisfies . We let and . We can see is composed of for and for with and . This phenomenon is consistent with Proposition 14. In Figure 1(b), , which satisfies , and is given by . Since is very close to , we just draw the position of in the graph. In fact, under , this phenomenon happens almost on any other trial, for which we select different , , , and . Figure 1(b) shows , which supports Proposition 12 correctly. In Figure 1(c), we choose , which satisfies . is just composed of with , which matches Proposition 16 very well. Since the value , compared with the numbers of -axis, is so small, it is not obvious in the graph. In Figure 1(d), , which satisfies . We can obtain by calculation. From numerical simulation we get , so is satisfied. Therefore, the conclusion in Proposition 16 is shown.
Table 1 summarizes the results for different parameter values. As shown in Table 1 (starting from row 4), for each case in Figure 1 where , we choose three different values of to calculate parameters such as , , , and from . For Figure 1(a) with , from and , three potential values of are selected as , , and . By the same selection method, for Figure 1(b) with , from and , three trials for are . For Figure 1(c) with , from , three comparable values of are given by and the same is applied in Figure 1(d) with . This table shows that the results, whether the values of and are zeroes or not, are totally consistent with the theoretical results in Sections 4.1 and 4.2. In addition, , , and are decreasing and is increasing when is increasing under the same . On the other hand, under the same , is increasing and is decreasing as is increasing.
5. Solutions to the QVI and the Optimal Policy
Notice that all possible structures to calculate are and . Moreover, all integrands of these structures can be used to construct the candidate solutions to the QVI. In the following, we first demonstrate that all constructed candidate solutions are solutions to the QVI under different conditions and give several corresponding numerical examples by using the results of Section 4.3. Then, we provide and verify the optimal policy based on the solutions to the QVI.
5.1. Solutions to the QVI and Numerical Examples
Assume that we have got , such that , by those steps provided in Section 4.3. Then, other parameters, such as , , , and , can also be obtained. For simplicity, we use the same denotations as before, such as , , , , , and , to represent all calculated parameters.
For , such that , we define a function byThen and satisfy .
Define ; then we can obtain thatwhere .
In addition, for , such that , definewhere . Then and satisfy .
Define ; it follows that
In the following, for and , we use the same notations, , to denote both and and to denote both and . From the construction of and , we have the following theorem by verifying the system QVI.
Theorem 18. Both functions and given by (60) and (62) are continuously differentiable on , and for each of and , there exists such that this function is twice continuously differentiable on . Moreover, and provide a solution to QVI.
In addition, from the discussion of Section 4, we have an important result as follows to make a decision with respect to whether there is a bankruptcy or not.(1)Under the assumption , if , then and there would be a bankruptcy. If , then and the optimal policy is continuation strategy.(2)Under the assumption , if , then and there would be a bankruptcy. If , then and the optimal policy is continuation strategy.(3)If , then and there would be a bankruptcy.
Based on the calculated parameters, , , and of Section 4.3, several numerical examples are provided here to support our theoretical results. In Figure 2, the graphs of are shown under , , , and , respectively. These figures show that at the same value of , the graphs of with greater are below those of with smaller . This phenomenon means that the optimal dividends would be greater if the fixed costs become smaller, which accords with the actual situation in the real world. Moreover, given , the value of the firm increases with the recovery rate .
(a)
(b)
(c)
(d)
In Figure 3, the corresponding derivatives of in Figure 2 are plotted. For Figure 3(a) at , Figure 3(b) at and , and Figure 3(c) at , it shows that they have in corresponding graphs. In addition, for Figure 3(b) at , the graph of it has convexity. Moreover, for Figure 3(b) at , Figure 3(c) at and , and Figure 3(d) at , , and , all graphs of them are below the level line of .
(a)
(b)
(c)
(d)
For comparative purposes, we also compute the results for the cases when , which are considered by Cadenillas et al. [14]. The first three rows in Table 1 present the cases for . As shown, the results for are quite different from those for (see rows 4–15 of Table 1). An important difference is that, for , and are all greater than zero, which are critical factors to reduce the complexity of verification for the uniqueness of solutions. The value of and its derivatives under are plotted in Figure 4. In fact, in Figure 4(a) is the lower limit lines of those corresponding lines with the same parameters except for in Figure 2 as . Comparing Figure 4(b) with Figure 3, it shows that the case has much simpler graphs than the case for the derivative function . The former only has a convex shape, whereas the latter has both convex and concave shapes, which are consistent with the complicated properties of in Section 4.2. More importantly, for any given , the value of the firm is always lower in the case with no recovery () than that with a recovery value (). Results show that the model with underestimates the firm value when there is a residual value at bankruptcy.
(a)
(b)
5.2. Optimal Policy
In the following theorem, the optimal policy is presented and verified, and it is shown that the solution to QVI constructed above is the value function.
Theorem 19. Let ; then the control defined byif , then let and ; if , then every ,where is the solution to the stochastic differential equation.is the QVI control associated with the functions defined by (60) and defined by (62). This control is optimal and the functions and coincide with the value function.
Proof. In Section 5.1, for any functions defined by (60) and defined by (62), we can see that and satisfy all conditions of Theorem 5. From Definition 3 and the discussion in Sections 3 and 4, we show that the control defined in (63)–(65) is the control associated with and . In addition, the control is admissible from Definition 1. So, by Theorem 18, the claim that and are the value functions and is the optimal policy is proved.
6. Conclusion
In this paper, we consider the dividend optimization problem that some fixed money would be returned to shareholders at the state of terminal bankruptcy. For this problem, the optimal timing and the optimal amount of dividends paid out to the shareholders and the resulting firm values are provided. Moreover, we identify some conditions that there would be a terminal bankruptcy under the optimal policy. A challenge in this paper is that the structure of the candidate solution is not explicit since the existing interval of it has unfixed endpoints, which depends on some unknown parameters. We overcome this difficulty by using the integral, , and dividing it into several cases. The importance of this paper is that it generalizes the terminal bankruptcy model and provides several new kinds of optimal dividend strategies, especially at the state of bankruptcy.
In the future, we can address some problems with general drift and diffusion terms under the assumption that shareholders can get some fixed money when there is terminal bankruptcy. For this kind of problem, sometimes we cannot write out the explicit expression for the solutions. Then we need to use some suitable numerical methods to simulate some nonlinear equations in the quasi-variational inequalities. Moreover, we can compare two optimal policies derived from two choices, investing new money or terminating the business, and, based on the obtained result, the best corporate decision can be determined.
Disclosure
The authors declare that this paper is revised and based on Dr. Peimin Chen’s dissertation presented to University of Missouri.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work on this paper was supported by the Funds for “1000 Talents Plan” of Sichuan Province (Fund number listed in Southwestern University of Finance and Economics is 221410001003040002).