Abstract
For some firms with large nonliquid assets, preferred shareholders can still get back a little bit of money when the firms finish disbursement of loans at the status of bankruptcy. For such a situation, to investigate the optimal dividend policy, a stochastic dynamic dividend model with nonzero terminal bankruptcy values is put forward in this paper. Moreover, an analytic solution for the optimal objective function of the discounted dividends is provided and verified. An important application of this result is that it can be employed to construct the solution for the optimal value function on the dividend problem with bailouts at bankruptcy. Further, the relationship for the solutions of these two different problems is demonstrated. In the end, some numerical examples are provided to support our theoretical results and the corresponding economic interpretations are illustrated.
1. Introduction
In the past decades, optimal dividend problems have been an important issue in financial and actuarial sciences. Its origin can be traced as early as the work of de Finetti [1], in which a discrete time model for optimal dividend was introduced. Recently, diffusion models for firms with controllable risk exposure and dividends payout have drawn increasing attention of researchers. We refer the readers to Jeanblanc-Picqué and Shiryaev [2], Radner and Shepp [3], Taksar and Zhou [4], Højgaard and Taksar [5, 6], Hubalek and Schachermayer [7], Cadenillas et al. [8], Paulsen [9, 10], Paulsen and Gjessing [11], Avanzi and Wong [12], Hunting and Paulsen [13], Chen et al. [14], Eisenberg [15], Vierkötter and Schmidli [16], and so forth. In all of those works, the terminal value of a company is assumed to be equal to zero when there is a status of bankruptcy, where the bankruptcy is defined as the time when the liquid assets of the company vanish. But in the real world, sometimes preferred shareholders can get some money back at bankruptcy if the value of the nonliquid assets (such as real estate or the rights to conduct business or the trade name) is large enough to pay for loans. This means that for this case the value function is not zero at the time of ruin. For such firms, how to manage the liquid assets and how to distribute dividends become a necessary problem for managers to consider. Unfortunately, there are very few results concerned with the terminal values (see, e.g., Taksar and Hunderup [17], Taksar [18], Xu and Zhou [19], and Chen and Li [20]). In addition, for firms at status of bankruptcy, sometimes they can get bailouts from governments or other firms with abundant cash flows. In such a situation, the optimal dividend policy is also an urgent problem for managers to analyze. In this paper, we put forward a dividend model with nonzero terminal values and then provide optimal dividend strategies for these two problems.
In the real financial market, capital injections are an important approach for insurance company to maintain the business when cash flow is insufficient. Recently, the optimal dividend problem with capital injections becomes hot issue in the research field of insurance. In works of Kulenko and Schimidli [21], Løkka and Zervos [22], He and Liang [23, 24], Yao et al. [25], Li and Liu [26], and so on, they assume that firms can raise capitals by issuing new shares or bonds. That implies the firms are healthy in management and financial situation and they can attract much more capitals from externals to scale up their business. In this case, the event and amounts of capital injections can be controlled by firms, so the capital injections can be a controllable variable in the objective function of optimal dividends (see, e.g., Løkka and Zervos [22] and He and Liang [23, 24]). However, nearby bankruptcy investors are in panic for prospects and firms have lots of difficulties to persuade shareholders to pay for new shares so that shareholders seldom inject capitals. In addition, at bankruptcy bailouts are mainly composed of loans or purchasing agreements from governments. For instance, in the financial crisis of 2008, the bailouts are mainly from the purchasing programs of US governments (see https://projects.propublica.org/bailout/list). Also, bailouts from governments are determined by a complicated political process, which is beyond firms’ control. Therefore, in this situation it is unrealistic to treat bailouts as a controllable and internal variable for shareholders in the value function although bailouts need to be refunded in the future. In all, it is reasonable for us to view bailouts as an exogenous variable, which does not appear in objective function.
Following the classical model, we posit that the manager maximizes the expected discounted value of cumulative dividends payout. The value function is defined from shareholder’s perspective; meanwhile, a residual terminal value is permitted when the firm goes bankrupt. Very closely related to the above is the problem with nonterminal bankruptcy, where a company reaches the bankruptcy state, and it does not go out of business but rather stays in this state a random amount of time and then resumes “business as usual.” As in Sethi and Taksar [27], a diffusion limit description of such behavior would be via Brownian motion with delayed reflection at zero. With this setup, we then transform the optimal cumulative dividend problem to be an equivalent Hamilton-Jacobi-Bellman equation with mixed Dirichlet-Neumann boundary condition. Further, a closed form solution and optimal dividend policy for the original problem are obtained.
The main contribution of this paper is to develop a new dividend model with terminal values or bailouts, in which bankruptcy is not terminal. A novel result obtained is the concise necessary and sufficient condition for immediate dividend events after bailouts. This result presents a guild way for regulators. To deter dividend distributions in embarrassing situation, a policy maker can set the rate of capital injection lower than the maximum dividend rate. Numerical examples are provided to support the theoretical results of the model.
The rest of this paper is organized as follows. In Section 2, we propose a mathematical model with nonzero terminal bankruptcy values for the optimal dividend problem. In Section 3, the HJB equation corresponding to this problem is derived and the detailed structure of candidate solutions is given under different situations. In Section 4, a nonterminal bankruptcy model with bailouts is presented. Then based on the results in Section 3, its smooth solution and the optimal control strategies are obtained. Meanwhile, some numerical examples are provided to verify the theoretical model and the corresponding economic interpretations are illustrated. Finally, in Section 5, we summarize our main findings and suggest a direction for future research.
2. The Mathematical Model
Let be a probability space endowed with a filtration , and let be a standard Brownian motion adapted to that filtration. To understand motivation for our diffusion control problem, one can start with the classical Cramér-Lundberg model of an insurance company. Assume that claims arrive according to a Poisson process with rate and the size of th claim is , where are i.i.d. with mean and variance . The risk process representing the liquid assets of the company, also called reserve or surplus, is governed by where is the amount of premium per unit time received by the insurance company and the initial reserve is supposed to be -measurable. As in Taksar [28], this process can be approximated by a diffusion process with a constant drift and diffusion coefficient . Thus, in the absence of control the reserve process can be modelled as
For an insurance company, it often considers to do reinsurance to reduce its risk and to pay out dividends to show its bright prospect. Proportional reinsurance, which we will consider, consists of paying a certain fraction of the premiums to the reinsurance company in exchange for an obligation from the latter to pick up the same fraction of each claim. If is the fraction of each claim picked up by the reinsurance company, then we call the risk exposure of the cedent. When is fixed, the reserve of the insurance company evolves as follows: A diffusion approximation for the above process yields a Brownian motion with drift and diffusion coefficient , where and are the same as before. We describe a control policy by a two-dimensional stochastic process , where corresponds to the risk exposure and is the restricted dividend rate paid out to the shareholders at time . Consequently, the dynamics of the reserve process under this policy are given byThe set of all admissible policies is denoted by . For a given admissible policy , the corresponding value function is defined aswhere is the time of bankruptcy, denotes a discount rate, and is the residual values left to shareholders when the company goes bankrupt. Generally, once bankruptcy occurs, shareholders have very few opportunities to get imbursements. However, in the case of some companies with large nonliquid assets, sometimes shareholders with preferred rights can still have a finger in the pie and obtain some returned money. So in this paper, we assume that .
For given , the objective is to findThat is, we wish to find an admissible policy so as to maximize the expected present value of the cumulative dividend payouts and bankruptcy amounts shareholders achieve. The function defined by (7) is called the optimal value function, and the policy , which satisfies , is termed the optimal policy.
The following gives a characterization of the value function .
Proposition 1. The function defined by (7) is concave in .
Proof. Let be an admissible policy for the initial reserve and for the initial reserve . Take and define by Then, by linearity of (4), is an admissible policy for the initial reserve and with . The linearity of (6) and (9) implies For any , we can choose and such that As is suboptimal, it follows that From the arbitrariness of , we conclude that is concave.
3. The Hamilton-Jacobi-Bellman Equation and Its Solution
In this section, we firstly derive the Hamilton-Jacobi-Bellman equation satisfied by , and then discuss how to find its analytic solutions.
For any function , we define a differential operator on it by
Proposition 2. Assume that the function defined by (7) is twice continuous differential on ; then it satisfies the Hamilton-Jacobi-Bellman (HJB) equationwith the boundary condition
Proof. By the similar arguments of Højgaard and Taksar [29], the HJB equation (14) can be derived. To avoid tedious repetition, here we omit it. In addition, at , it means that a firm is at the state of bankruptcy. Under no arbitrage theorem in markets, the summation of future dividends should be equal to the residual values of the firm at bankruptcy. Thus, it follows that .
3.1. Constructing a Solution to the HJB Equation
In this section, we construct a solution of (14) and (15), where for convenience is replaced by . Let . Then, by concavity of , we have Therefore, for all , the HJB equation (14) becomesLet be the maximizer of the expression on the left-hand side of (17); thenPut (18) into (17); we obtain
A general solution of (19) with the boundary condition is given bywhere is a free constant and is given byThis solution, however, is valid only when . From (18) and (20), it follows thatSince above is an increasing linear function of , then if and only if , where
Comparing and , there are two possible cases to be considered: and , whose necessary and sufficient conditions will be given later.
3.1.1. The Case:
From (23), we notice that depends on the uncertain parameter , which would lead to two different cases for . So, in the following we consider two cases, and , and discuss them, respectively.
If , then for . But as , we must have for . Consequently, (17) becomes
A general solution to (24) is given by where and are defined byand , are free constants.
The continuity of the function and its derivative at implies that whereand .
For , we have and . Thus, satisfies the following equation:Concavity of implies that when with . Thereby, a solution of (29) is given by where
Summarizing the above, we get the structure of the solution to (14) and (15) for where , , and are unknown constants and are determined by (23).
By continuity of and at , it implies that where we have used (33) in (35).
In view of (33), we getDividing (35) by (34), we eliminate and solve for , thereby obtaining
To simplify this expression, let . Then , , and . This yields By the same argument as above, we haveSubstituting (39) and (40) into (37), we getBy (34), we easily obtain
Consequently, all free constants have been determined. In addition, as in Lemma of Højgaard and Taksar [29], it shows that if and only if .
Remark 3. From (23), it easily shows that the assumption, , is equivalent to that is,
On the other hand, for the case of , namely, we also need to discuss the following two cases by and , respectively.
If and , it implies that for all . Then, the HJB equation (14) becomesUsing the similar argument as above, we obtain a general solution of (46) as follows: where is given by (26), by (31), and , , , and need to be determined.
The continuity of and at , together with , implies thatBy (49), we getCombining (50) and (51), for given we can solve for and Put (53) into (48); we obtain the following equation, where must satisfyBy continuity of at , (52) and (53), it follows that
Remark 4. Let . Under the condition of , it is easily verified that on . Thus, owing to (55). Suppose that satisfiesthen by intermediate value theorem, it shows that (54) has a unique positive root . Thereby, the values of and can be determined certainly.
Secondly, if and , that is,we get (29) and find as before a solution where .
Summarizing the above, in the case of , the solutions of (14) and (15) are presented by the following theorem.
Theorem 5. Assume that . Let , , , and be defined as above.
If (44) holds, then the solution to (14) and (15) is given bywhere is given by (42), by (23), and by (41).
If (56) is true, thenwhere is given by (53) and is determined by (54).
If (57) is true, thenMoreover, in any of the cases described above, is a concave solution of (14) and (15).
Proof. From the construction above, it is easily to verify that the expressions (59)–(61) are solutions to (14) and (15). In addition, the proof on concavity of is similar to that of Theorem in Højgaard and Taksar [29], so we omit it here.
3.1.2. The Case:
Now we turn to the case , which is equivalent to . From Højgaard and Taksar [29], it shows that there exists only one “switching point” because of . Therefore, the solution of (14) and (15) depends on whether or .
If , for , we obtain (17) and find as before a solution as follows: where is given by (21) and is a free constant.
For , we have . Thus, the HJB equation (14) becomesLet be a solution of the following homogeneous equation:Then, a solution of (63) can be expressed byDifferentiating (64) with respect to yields the maximizerSubstituting (66) into (64), we obtain
A general solution of (67) can be given bywhere and are free constants.
Input (68) into (67); we obtain Then, it is easy to derive thatwhere is defined by (21).
Thus, combining (65), (68), and (70), a solution of (63) is given by
Consequently, for the case of , the following solution to (14) and (15) is suggested where , , and need to be specified.
Making “smooth fit” at , we have the following equations:where we have used (73) in (75).
From (73), we obtain Dividing (75) by (74) yields thatPut (77) into (74); we getCombining (77) with (78), it follows thatIn addition, by (79), it shows that is equivalent to
On the other hand, if , that is,we get (63) and find as above a solution as follows: By the boundary condition , we have
Since for all , the following theorem follows easily from construction.
Theorem 6. Assume that . Let be defined by (21).
If (80) is true, then the solution to (14) and (15) is given bywhere is given by (78) and by (79).
On the other hand, if (81) holds, thenMoreover, the functions defined as above are concave solutions of (14) and (15).
3.2. Optimal Policy
Based on sections above, in the following, we present our optimal policy. The admissible policy for all is defined as follows.
Case 1 (). If (44) holds, then where is given by (23) and by (41).
If (56) is true, then where is given by (54).
If (57) holds, then
Case 2 (). If (80) is true, then where is given by (79).
If (81) holds, then Moreover, in the case of , we have for all .
In addition, is the solution of
The following theorem shows that the solution of (14) and (15) constructed above is the optimal value function.
Theorem 7. Let be given by (7), and let be given by (59)–(61) for and by (84)-(85) for . Then
Proof. The proof of this theorem can be carried out in the same way as that of Theorem in Højgaard and Taksar [29]. To avoid tedious repetition, we omit it here.
4. Nonterminal Bankruptcy Model with Bailouts
4.1. Nonterminal Bankruptcy Model and Its HJB Equation
In the real world, firms may receive financial bailouts from the government or other corporations when they are at the edge of bankruptcy. Therefore, it is necessary for us to consider the optimal dividend problem with capital injections when a firm may encounter bankruptcy. In this section, based on the solutions of (14) and (15), the nonterminal bankruptcy model with bailouts is developed and further its solution is obtained under an appropriate boundary condition.
In order to specify the development of the reserve process in our model when , we consider the following discrete time model of bankruptcy. Assume that at time the company is at the stage of bankruptcy, and meanwhile the probability it receives an amount of external capital inflow is , while it remains in bankruptcy with probability . Thus, the capital injections of the reserve process at time are , where are i.i.d. random variables with the distributionThus, Again, we adopt the continuous analogCombining (4), (5), and (95), we obtainEquation (96) shows that the recovery rate can be viewed as a rate at which the company can receive new capital at the time when . As in Sethi and Taksar [27], for the model considered in this part, we denote it by nonterminal bankruptcy (or bankruptcy with recovery). On the other hand, for the model, which is restricted to stay at the bankruptcy state, it is termed as terminal bankruptcy. Consequently, the optimal value function becomes as follows:Then, the HJB equation satisfied by (98) can be also given by (14). To find the function , we need to specify the behavior of at the origin. Moreover, its boundary condition should be related to the capital injection rate . Now, let us verify the following theorem, which may give us some enlightenment about its boundary condition.
Theorem 8. Suppose that is a nonnegative -function on with bounded and satisfies the HJB equationwith the mixed Dirichlet-Neumann boundary conditionThen, it follows that
Proof. Let and be an admissible policy. For , using the generalized Itô’s formula, we have As is bounded, the last term on the right-hand side is a zero-mean square integrable martingale. Then from (99) and (100), we obtain For , it implies that . Thus,Let ; using Fatou’s lemma and taking the supremum over all policies in (104), we obtain
4.2. Solution of the Nonterminal Bankruptcy Model
Although the HJB equation (14) with the mixed boundary condition (100) cannot be solved in a straightforward manner, we can establish a relation between and , in which the boundary condition (100) is replaced by (15), by comparing it with the nonzero terminal bankruptcy problem.
Theorem 9. Let be given by (98). Suppose that is the optimal value function for problem (7) and that is the corresponding optimal policy. Supposeand let be a solution to the stochastic different equation (96) with given by (106), and . Then, the optimal value function can be expressed by . Moreover, the optimal policy in this model is given by
Proof. According to Gihman and Skorohod [30], there exists a solution to (96) with and . Thus, defined by (107) is admissible.
Since is a solution of the HJB (14), it follows thatEquation (106) is equivalent toRepeating the similar argument as the proof of Theorem 8, then by (108) and (109), we obtainBy boundedness of , for some . Therefore,Furthermore, Thus, the term on the left-hand side of (111) is majorized by a positive rand variable with a finite expectation.
Letting in (110) and using dominated and monotone convergence theorems, we haveOn the other hand, let ; then according to Theorem 8, it implies thatCombining (113) and (114), it shows that .
Next, we will construct an optimal value function of nonterminal bankruptcy problem with bailouts in the following way. Given a capital injection rate , we wish to find a function depending on such that the function satisfies Then, according to Theorem 9, we obtain
For any given , a specific analysis of the behavior of is given as follows.
Case 1 (). From (59), it is easy to derive thatwhere , which implies that and .
In view that implies , then for the case , by (117) we obtainSimilarly, from (61), we havewhere . It is easily verified that is a strictly increasing function of , which implies .
Thus, for , (119) implies that By continuity of , for , then applying (60) we get Thus,where is given by (53) and .
Substituting (122) into (54) yields thatAccording to (123), is given byCombining (53), (122), and (124), it shows that is uniquely determined by .
Remark 10. To ensure that in (124), we need to verify for . Let , then is a strictly decreasing function with respect to on . Taking and applying (55), it follows that Therefore, is surely true for .
Case 2 (). In view of (84), we obtainwhere is given by (78). The validity of follows from .
Thus, for the case , by (126) we get On the other hand, for , equivalent to , (85) implies that Then, Consequently, it shows that for any there exists a unique , satisfying . Then, based on Theorems 5, 6, and 9, we derive the following theorem, which presents an explicit expression for the optimal value function .
Theorem 11. Let be given by (98).
Case 1 (). (1) If , then is given bywhere and and are given by (41) and (42), respectively.
(2) If , thenwhere is given by (54) and is determined by (53).
(3) If , thenCase 2 (). (1) If , then is given bywhere is given by (79).
(2) If , then
Proof. From Theorem 9, (59) and (118), for , can be given by with , that is, (130). By the same method, as the proof of the case , we can verify that is given by (131)–(134), respectively.
4.3. Optimal Policy
According to Section 3.2 and (107), the corresponding admissible policy for the nonterminal bankruptcy model with bailouts is given as follows.
Case 1 (). (1) If , then where and is given by (41).
(2) If , then where is given by (54).
(3) If , then
Case 2 (). (1) If , where is given by (79).
(2) If , then In the case of , it is easily seen that for all .
In addition, is the solution of
4.4. Numerical Examples
Based on the calculated parameters , , , and , several numerical examples are provided here to support our theoretical results. It is interesting to look at the relationship between the optimal control and the capital injection rate . In Figures 1 and 2, the graphs of the optimal value function are shown for different under and , respectively. In these plots, parameters used are given by , , and . From these graphs, it shows that, for any given , the value of increases with the capital injection rate .
Figures 3 and 4 present the optimal risk exposure for different at cases and , respectively. In these graphs, the used parameters are also given by , , and as in Figures 1 and 2. In the case of (), for , the optimal risk exposure is a linear function of the current reserve process as long as this process is less than , after which it is a constant function equal to one. For , the optimal risk exposure is constantly equal to one; that is, it is optimal for the company to take the maximum risk all the time and in essence to optimize only the dividend payments. Under the condition (), for the optimal risk exposure monotonically increases from zero to maximum as a linear function of the reserve process and when this value is attained for the risk exposure stays here for all beyond . For , the optimal risk exposure is always equal to the maximum, but less than one. Moreover, it also shows that for any fixed the optimal risk exposure is a nondecreasing function of .
In Figures 5 and 6, parameters , , , and are the same as in Figures 3 and 4, respectively. From the two graphs, it is easily seen that the dividend threshold level is a decreasing function with respect to .
All of this has the following economic interpretation. In an attempt to pay dividend to the shareholders, at least the company is interested in extending the time before bankruptcy. For an extension of time before bankruptcy, a reduction of risk is needed which simultaneously reduces potential profit. However, with increasing capital injection rate, the company can be less sensitive to bankruptcy, since an increasing rate of capital injection hedges against potential losses of the future profits. The same explanation holds for the decrease of , when increases. The company can start distributing dividends earlier by lowering the threshold not being concerned that such a policy can bring the bankruptcy faster.
If is large enough and the reserve or the capital injection rate is large enough, it is optimal for the company to take the maximal risk exposure all the time. However, if is small enough, it will never be optimal to take the maximal risk exposure, no matter how large the reserve or the capital injection rate is. In particular, if the rate of capital injection is larger than the upper of the dividend rate, it is the best for a company not to do business at all but to “take money and run.”
5. Conclusions
In this paper, we develop a stochastic dynamic dividend model with terminal values at bankruptcy status and achieve the analytic solution for the optimal value function. Based on this result, the dividend problem with bailouts at bankruptcy can also be solved and its solution has been constructed. A wonderful result obtained is the concise necessary and sufficient condition for immediate dividend actions after bailouts. That is, if the rate of bailouts is greater than the upper of the dividend rate, then the dividend event would happen immediately after bailouts. It demonstrates that to prohibit dividend payout after bailouts the manager or governor should make the rate of bailouts less than the maximum dividend rate.
In the future, following the researches of Zhu [31, 32], Zhu and Zhang [33], Wang and Zhu [34], and so on, we can address some dividend optimization problems for general diffusion models with the drift and diffusion coefficients being general functions of the surplus under the assumption that terminal bankruptcy values are nonzero. For this kind of problem, sometimes we cannot derive the explicit solutions of optimal value function. Then we need to use some suitable numerical methods as in Zhou et al. [35] and Zhou [36] to simulate.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was jointly supported by the Scientific Research Fund of Sichuan Provincial Department of Education [Grant no. 11ZA220] and the Funds for “1000 Talents Plan” of Sichuan Province [Grant no. 221410001003040002].