Optimal Dividend Strategy Research Articles

Singular Optimal Dividend Control for the Regime-Switching Cramér-Lundberg Model with Credit and Debit Interest

We investigate the dividend optimization problem for a company whose surplus process is modeled by a regime-switching compound Poisson model with credit and debit interest. The surplus process is controlled by subtracting the cumulative dividends. The performance of a dividend distribution strategy which determines the timing and amount of dividend payments, is measured by the expectation of the total discounted dividends until ruin. The objective is to identify an optimal dividend strategy which attains the maximum performance. We show that a regime-switching band strategy is optimal.

Dividend-reinsurance strategy in the Sparre Andersen model

In this paper, we introduce a reinsurance strategy into the Sparre Andersen risk model with a horizon dividend barrier, which is named dividend-reinsurance strategy. It is shown that the value function of the new strategy far exceeds that of the optimal barrier strategy (even that of the optimal dividend strategy). Some results on the advantages of the new strategy are obtained, and the methods for computing the value functions are provided. Numerical illustrations for Erlang (2) and compound Poisson risk models are also given.

On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

On the Optimal Dividend Strategy in a Regime-Switching Diffusion Model

In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

Impulse Stochastic Control for the Optimization of the Dividend Payments of the Compound Poisson Risk Model Perturbed by Diffusion

This article deals with the optimal dividend problem for the classical risk model perturbed by diffusion. With each dividend payment there is a proportional cost and a fixed cost. The objective of the corporation is to maximize the cumulative expected discounted dividends payout until the time of ruin. It controls the timing and the amount of dividends paid out to the shareholders. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes an impulse stochastic control problem. By solving the corresponding quasi-variational inequalities, we obtain the optimal dividend strategy. Furthermore, we solve this problem explicitly in the case of exponential claim amount distributions of this jump-diffusion model. It turns out that there can be essentially two different solutions depending on the model's parameters and the costs. Specifically, (1) whenever the reserve reaches an appropriate level x*, it is reduced to through a dividend payment, and the process continues; (2) whenever the reserve reaches an appropriate level x*, everything is paid out as dividends immediately, and the ruin occurs.

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramér–Lundberg model with arbitrary claim-size distribution. Our objective is to find the dividend payment policy which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy imposing a ceiling on the dividend rates. We characterize the optimal value function as the unique bounded viscosity solution of the associated Hamilton–Jacobi–Bellman equation. We prove that there exists an optimal dividend strategy and that this strategy is stationary with a band structure. We study the regularity of the optimal value function. We find a characterization result to check optimality even in the case where the optimal value function is not differentiable. We construct examples where the claim-size distribution is smooth but the optimal dividend policy is not threshold and the optimal value function is not differentiable. We study the survival probability of the company under the optimal dividend policy. We also present examples where the optimal dividend policy has infinitely many bands even in the case that the claim-size distribution has a bounded density.

Dividends and reinsurance under a penalty for ruin

We find the optimal dividend strategy in a diffusion risk model under a penalty for ruin, as in Thonhauser and Albrecher (2007), although we allow for both a positive and a negative penalty. Furthermore, we determine the optimal proportional reinsurance strategy, when so-called expensive reinsurance is available; that is, the premium loading on reinsurance is greater than the loading on the directly written insurance. One can think of our model as taking the one in Taksar (2000, Section 6) and adding a penalty for ruin.We use the Legendre transform to obtain the optimal dividend and reinsurance strategies. Not surprisingly, the optimal dividend strategy is a barrier strategy. Also, we investigate the effect of the penalty P on the optimal strategies. In particular, we show that the optimal barrier increases with respect to P, while the optimal proportion retained and the value function decrease with respect to P. In the end, we explore the time of ruin, and find that the expected time of ruin increases with respect to P under a net profit condition.

Optimal dividend strategy with transaction costs for an upward jump model

In this paper, we consider the optimal dividend problem with transaction costs when the incomes of a company can be described by an upward jump model. Both fixed and proportional costs are considered in the problem. The value function is defined as the expected total discounted dividends up to the time of ruin. Although the same problem has already been studied in the pure diffusion model and the spectrally negative Lévy process, the optimal dividend problem in an upward jump model has two different aspects in determining the optimal dividends barrier and in the property of the value function. First, the value function is twice continuous differentiable in the diffusion case, but it is not in the jump model. Second, under the spectrally negative Lévy process, downward jumps will not cause any payment actions; however, it might trigger dividend payments when there are upward jumps. In deriving the optimal barriers, we show that the value function is bounded by a linear function. Using this property, we establish the verification theorem for the value function. By solving the quasi-variational inequalities associated with this problem, we obtain the closed-form solution to the value function and hence the optimal dividend strategy when the income sizes follow a common exponential distribution. In the presence of a fixed transaction cost, it is shown that the optimal strategy is a two-barrier policy, and the optimal barriers are only dependent on the fixed cost and not the proportional cost. A numerical example is used to illustrate how the fixed cost plays a significant role in the optimal dividend strategy and also the value function. Moreover, an increased fixed cost results in larger but less frequent dividend payments.

Optimal dividend strategies in a Cramer–Lundberg model with capital injections and administration costs

In this paper, we consider a classical risk model with dividend payments and capital injections in the presence of both fixed and proportionals administration costs. Negative surplus or ruin is not allowed. We measure the value of a strategy by the discounted value of the dividends minus the costs. It turns out, capital injections are only made if the claim process falls below zero. Further, at the time of an injection the company may not only inject the deficit, but inject additional capital C ≥ 0 to prevent future capital injections. We derive the associated Hamilton–Jacobi–Bellman equation and show that the optimal strategy is of band type. By using Gerber–Shiu functions, we derive a method to determine numerically the solution to the integro-differential equation and the unknown value C.

Optimal dividend strategies in a double compound poisson risk process

In this paper, we consider a double compound Poisson risk model involving two independent classes of insurance risks with a threshold dividend strategy. We derived the integro-differential equations (IDE) with certain boundary conditions for the present value of dividends until ruin. When the claims from both classes are exponentially distributed, we show that the threshold dividend strategy is an optimal dividend strategy.

On optimality of the barrier strategy for a general Lévy risk process

We consider the optimal dividend problem for the insurance risk process in a general Lévy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Lévy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. [F. Avram, Z. Palmowski, M.R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability 17 (2007) 156–180], Loeffen [R. Loeffen, On optimality of the barrier strategy in De Finetti’s dividend problem for spectrally negative Lévy processes, The Annals of Applied Probability 18 (2008) 1669–1680] and Kyprianou et al. [A.E. Kyprianou, V. Rivero, R. Song, Convexity and smoothness of scale functions with applications to De Finetti’s control problem, Journal of Theoretical Probability 23 (2010) 547–564] in which the same problem was considered under the spectrally negative Lévy processes setting.

Some Results on Optimal Dividend Problem in Two Risk Models

The compound Poisson risk model and the compound Poisson risk model perturbed by diffusion are considered in the presence of a dividend barrier with solvency constraints. Moreover, it extends the known result due to (1). Ref. (1) finds the optimal dividend policy is of a barrier type for a jump-diffusion model with exponentially distributed jumps. In this paper, it turns out that there can be two different solutions depending on the model's parameters. Furthermore, an interesting result is given: the proportional transaction cost has no effect on the dividend barrier. The objective of the corporation is to maximize the cumulative expected discounted dividends payout with solvency constraints before the time of ruin. It is well known that under some reasonable assumptions, optimal dividend strategy is a barrier strategy, i.e., there is a level so that whenever surplus goes above the level , the excess is paid out as dividends. However, the optimal level may be unacceptably low from a solvency point of view. Therefore, some constraints should imposed on an insurance company such as to pay out dividends unless the surplus has reached a level . We show that in this case a barrier strategy at optimal. 12 () bb 12 () bb

Optimal dividend strategies in discrete risk model with capital injections

In this paper we consider a doubly discrete model used in Dickson and Waters (biASTIN Bulletin 1991; 21:199–221) to approximate the Cramér–Lundberg model. The company controls the amount of dividends paid out to the shareholders as well as the capital injections which make the company never ruin in order to maximize the cumulative expected discounted dividends minus the penalized discounted capital injections. We show that the optimal value function is the unique solution of a discrete Hamilton–Jacobi–Bellman equation by contraction mapping principle. Moreover, with capital injection, we reduce the optimal dividend strategy from band strategy in the discrete classical risk model without external capital injection into barrier strategy, which is consistent with the result in continuous time. We also give the equivalent condition when the optimal dividend barrier is equal to 0. Although there is no explicit solution to the value function and the optimal dividend barrier, we obtain the optimal dividend barrier and the approximating solution of the value function by Bellman's recursive algorithm. From the numerical calculations, we obtain some relevant economical insights. Copyright © 2010 John Wiley & Sons, Ltd.

Optimal dividend strategies in the diffusion model with stochastic return on investments

This paper studies the optimal dividend problem in the diffusion model with stochastic return on investments. The insurance company invests its surplus in a financial market. More specially, the authors consider the case of investment in a Black-Scholes market with risky asset such as stock. The classical problem is to find the optimal dividend payment strategy that maximizes the expectation of discounted dividend payment until ruin. Motivated by the idea of Thonhauser and Albrecher (2007), we take the lifetime of the controlled risk process into account, that is, the value function considers both the expectation of discounted dividend payment and the time value of ruin. The authors conclude that the optimal dividend strategy is a barrier strategy for the unbounded dividend payment case and is of threshold type for the bounded dividend payment case.

Optimal investment policy and dividend payment strategy in an insurance company

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramér–Lundberg process. The firm has the option of investing part of the surplus in a Black–Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton–Jacobi–Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.

Optimal dividend strategies in a dual model with capital injections

We study three types of practical optimization problems faced by a firm that can control its liquid reserves by paying dividends and by issuing new equity. In the first problem, we consider the classical dividend problem without equity issuance. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of issuing new equity over strategies associated with positive reserves at all times. The third problem has the same objective as the second one, but with no constraints on the reserves. Under the assumption of proportional transaction costs, we identify the value functions and the optimal strategies. We also present the relationship between three problems.

Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes

In this paper, we study optimal dividend problem in the classical risk model. Transaction costs and taxes are required when dividends occur. The problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality, we obtain the analytical solutions of the optimal return function and the optimal dividend strategy when claims are exponentially distributed. We also find a formula for the expected time between dividends. The results show that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller but more frequent dividend payments.

Optimal financing and dividend strategies in a dual model with proportional costs

We consider the optimal control problem with dividend payments and issuance of equity in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.

Optimal Dividend Policies with Transaction Costs for a Class of Diffusion Processes

Finding optimal dividend strategies is a classical problem in the financial and actuarial literature. The idea is that the company wants to pay some of its surplus as dividends, and the problem is to find a dividend strategy that maximizes the expected total discounted dividends received by the shareholders until ruin. Here we generalize results in [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669-689] in that the rate of growth of the surplus process is assumed to exceed the discounting factor whenever the surplus process is smaller than a fixed number $x_{\lambda}$. In [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669-689] it was assumed that this rate of growth is always less than or equal to the discounting factor. It turns out that this generalization makes the problem much more complicated, and a simple barrier strategy is no longer always optimal.

An elementary approach to discrete models of dividend strategies

The paper studies a discrete counterpart of Gerber et al. (2006). The surplus of an insurance company (before dividends) is modeled as a time-homogeneous Markov chain with possible changes of size + 1 , 0 , − 1 , − 2 , − 3 , … . If a barrier strategy is applied for paying dividends, it is shown that the dividends-penalty identity holds. The identity expresses the expected present value of a penalty at ruin in terms of the expected discounted dividends until ruin and the expected present value of the penalty at ruin if no dividends are paid. For the problem of maximizing the difference between the expected discounted dividends until ruin and the expected present value of the penalty at ruin, barrier strategies play a prominent role. In some cases an optimal dividend barrier exists. The paper discusses in detail the special case where the distribution of the change in surplus does not depend on the current surplus (so that in the absence of dividends the surplus process has independent increments). A closed-form result for zero initial surplus is given, and it is shown how the relevant quantities can be calculated recursively. Finally, it is shown how optimal dividend strategies can be determined; typically, they are band strategies.