Optimal Dividend Strategy Research Articles

Optimal dividend policy with self-exciting claims in the Gamma–Omega model

In this paper, we consider the optimal dividend policy for an insurance company under a contagious insurance market, where the occurrence of a claim can trigger sequent claims. This clustering effect is modelled by a self-exciting Hawkes process where the intensity of claims depends on its historical path. In addition, we include the concept of bankruptcy to allow the insurance company to operate with a temporary negative surplus. The objective of the management is to obtain the optimal dividend strategy that maximises the expected discounted dividend payments until bankruptcy. The Hamilton–Jacobi–Bellman variational inequalities (HJBVIs) are derived rigorously. When claim sizes follow exponential distributions and the bankruptcy rate is a positive constant, the value function can be obtained based on the Gerber–Shiu penalty function and the optimal dividend barrier can be solved numerically. Finally, numerical examples are demonstrated to show the impact of key parameters on the optimal dividend strategy.

A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks

Our study investigates the optimal dividend strategy for a bank, taking into account the potential for government capital injections. We explore different types of government interventions, such as liberal, transparent, or uncertain strategies, and consider both single and multiple types of interventions. Our approach differs from others as it focuses on interventions that aim to maintain the overall stability of the financial system, rather than just addressing banks that have already sought government assistance or are in dire need of it. Specifically, we focus on situations where the government is more likely to assist banks that have not requested its intervention or that are not too difficult to save. To accomplish this, we conduct a comprehensive examination of all possible scenarios involving a single, one-time capital injection and derive explicit solutions for the associated optimal control problem. Furthermore, we expand the model to include semi-Markov dynamic capital injection processes and show that the optimal control is the unique viscosity solution of a Hamilton–Jacobi–Bellman equation. The government’s strategy also takes into account the bank’s solvency and any past government interventions.

Optimal Investment in a Dual Risk Model

Dual risk models are popular for modeling a venture capital or high-tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrates on the optimal dividend strategies. In this paper, we propose to study the optimal investment strategy on research and development for the dual risk models to minimize the ruin probability of the underlying company. We will also study the optimization problem when, in addition, the investment in a risky asset is allowed.

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterize the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.

Optimal dividend and capital injection strategies in common shock dependence model with time-inconsistent preferences

<p style='text-indent:20px;'>This paper discusses the optimal dividend and capital injection problems when an insurance company has two lines of business with common shock dependence. Suppose that the manager of the company has time-inconsistent preference, which can be described by a quasi-hyperbolic discount function. The value of the company is measured by the expected discounted dividend payments minus the expected discounted costs of capital injection. The goal is to find out the optimal strategies for maximizing the value of the company. By using the stochastic control techniques, we solve the problems in all cases of time-consistent preference manager, naive manager and sophisticated manager, respectively. The closed-form solutions of the value functions are presented. Our results show that the sophisticated manager is inclined to pay out dividends earlier than the naive manager, time-inconsistent preference manager is more likely to pay dividends in advance than time-consistent preference manager. Furthermore, we provide some numerical analysis to reveal the sensitivity of the optimal dividend strategies to the dependence intensity and the cost of capital injection. The results show that, as the cost of capital injection increases, sophisticated manager will give up capital injection first, followed by naive manager. In addition, we also find that: when the cost of capital injection is low, managers think that the value of the company that undertakes two kinds of insurance with high correlation is higher; while, managers have opposite evaluations, when the cost of capital injection is high.</p>

Optimal dividend bands revisited: a gradient-based method and evolutionary algorithms

We reconsider the study of optimal dividend strategies in the Cramér-Lundberg risk model. It is well-known that the solution of the classical dividend problem is in general a band strategy. However, the numerical techniques for the identification of the optimal bands available in the literature are very hard to implement and explicit numerical results are known for very few cases only. In this paper we put a gradient-based method into place which allows to determine optimal bands in more general situations. In addition, we adapt an evolutionary algorithm to this dividend problem, which is not as fast, but applicable in considerable generality, and can serve for providing a competitive benchmark. We illustrate the proposed methods in concrete examples, reproducing earlier results in the literature as well as establishing new ones for claim size distributions that could not be studied before.

On the surplus management of funds with assets and liabilities in presence of solvency requirements

In this paper, we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu [(2003). Geometric Brownian motion models for assets and liabilities: From pension funding to optimal dividends. North American Actuarial Journal 7(3), 37–56]. We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint [as in Paulsen (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 7(4), 457–473] is consequently imposed and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. The existence and uniqueness of the optimal strategies are shown. Results are illustrated.

Optimal Ratcheting of Dividends in a Brownian Risk Model

We study the problem of optimal dividend payout from a surplus process governed by Brownian motion with drift under the additional constraint of ratcheting, i.e., the dividend rate can never decrease. We solve the resulting two-dimensional optimal control problem, identifying the value function to be the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. For finitely many admissible dividend rates we prove that threshold strategies are optimal, and for any closed interval of admissible dividend rates we establish the $\varepsilon$-optimality of curve strategies. This work is a counterpart of [H. Albrecher, P. Azcue, and N. Muler, SIAM J. Control Optim., 58 (2020) pp. 1822--1845], where the ratcheting problem was studied for a compound Poisson surplus process with drift. In the present Brownian setup, calculus of variation techniques allow us to obtain a much more explicit analysis and description of the optimal dividend strategies. We also give some numerical illustrations of the optimality results.

Optimal Dividend Strategies with Reinsurance under Contagious Systemic Risk

This paper studies the multidimensional mixed singular-regular stochastic control problems subject to reduced-form default driven by contagious intensities. The dynamic process of surplus is given by a system of diffusion processes with two controls, and the intensity of the reduced-form model increases when defaults occur. We derive the recursive Hamilton--Jacobi--Bellman variational inequalities by the dynamic programming principle and present analytical and recursive solutions. We prove that the solutions are classical and recursively associated with each other by the default states. The verification theorem is presented.

The policy iteration algorithm for a compound Poisson process applied to optimal dividend strategies under a Cramér–Lundberg risk model

In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér–Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples.

Optimal Dividend Strategy Under Parisian Ruin with Affine Penalty

In this paper, we investigate the optimal dividend problem under Parisian ruin with affine penalty payments at Parisian ruin time. The underlying risk process is assumed to be a spectrally negative Levy risk process. With the help of the dynamic programming principle, we prove that the value function associated to our optimal control problem is the smallest solution with certain characteristics to the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, the form of the performance function under barrier dividend strategy is expressed in terms of various extended scale functions. Then we identify a condition under which the performance function under certain barrier strategy is also a solution to the HJB equation, which in turn illustrates the optimalilty of such barrier dividend strategy among all admissible strategies. Various numerical examples are also given when the underlying risk process is compound Poisson process, Brownian motion with drift and jump-diffusion process.

Optimal dividend strategy for the dual model with surplus-dependent expense

In this paper, we consider the optimal dividend problem for the dual model with surplus-dependent expense. The objective is to find the optimal dividend-payment strategy that maximizes the expected discounted value of dividends until the time of ruin. We give the analytic characterizations of admissible strategy and Markov strategy and show that a dividend strategy is a Markov strategy if and only if it is an additive functional of the controlled surplus process. We use the theory of measure-valued generators to derive a measure-valued dynamic programming equation (DPE). And the verification theorem is proved without additional assumptions on the regularity of the value function. Under the assumption that the value function is locally Lipschitz continuous, the optimal strategy is presented as a Markov strategy and hence a band strategy. Numerical examples are also given.

Optimal risk exposure and dividend payout policies under model uncertainty

This paper investigates a combined optimal risk exposure and dividend distribution decision making problem in presence of model uncertainty. In the context of model uncertainty, the decision maker regards the reference model (fitted by observed information) as an approximation to the true model and believes that the true model exists in a family of alternative models surrounding the reference model. The aim is to find a robust strategy of risk exposure and dividend payments, which maximizes the expected cumulative discounted dividends until ruin plus a “penalty” on the distortion between the reference and alternative models in the worst-case scenario. We provide explicit expressions for the value functions and derive the respective optimal strategies explicitly. We show that the optimal dividend strategy is always of a barrier type and that it is optimal to retain full risk when the surplus is large. We also find that when the insurer is more averse to ambiguity, her optimal strategy on risk retention and dividend payout is more conservative.

Optimal dividend strategy for an insurance group with contagious default risk

This paper studies the optimal dividend for a multi-line insurance group, in which each subsidiary runs a product line and is exposed to some external credit risk. The default contagion is considered such that one default event may increase the default probabilities of all surviving subsidiaries. The total dividend problem for the insurance group is investigated and we find that the optimal dividend strategy is still of the barrier type. Furthermore, we show that the optimal barrier of each subsidiary is modulated by the default state. That is, how many and which subsidiaries have defaulted will determine the dividend threshold of each surviving subsidiary. These conclusions are based on the analysis of the associated recursive system of Hamilton–Jacobi–Bellman variational inequalities (HJBVIs). The existence of the classical solution is established and the verification theorem is proved. In the case of two subsidiaries, the value function and optimal barriers are given in analytical forms, allowing us to conclude that the optimal barrier of one subsidiary decreases if the other subsidiary defaults.

Optimal reinsurance and dividends with transaction costs and taxes under thinning structure

In this paper, we investigate the problem of optimal reinsurance and dividends under the Cramér–Lundberg risk model with the thinning-dependence structure which was first introduced by Wang and Yuen [Wang, G. & Yuen, K. C. (2005). On a correlated aggregate claims model with thinning-dependence structure. Insurance: Mathematics and Economics 36(3), 456–468]. The optimization criterion is to maximize the expected accumulated discounted dividends paid until ruin. To enhance the practical relevance of the optimal dividend and reinsurance problem, non-cheap reinsurance is considered and transaction costs and taxes are imposed on dividends. These realistic features convert our optimization problem into a mixed classical-impulse control problem. For the sake of mathematical tractability, we replace the Cramér–Lundberg risk model by its diffusion approximation. Using the method of quasi-variational inequalities, we show that the optimal reinsurance follows a two-dimensional excess-of-loss reinsurance strategy, and the optimal dividend strategy turns out to be an impulse dividend strategy with an upper and a lower barrier, i.e. everything above the lower barrier is paid as dividends whenever the surplus goes beyond the upper barrier, and no dividends are paid otherwise. Under the diffusion risk model, closed-form expressions for the value function associated with the optimal dividend and reinsurance strategy are derived. In addition, some numerical examples are presented to illustrate the optimality results.

Optimal Dividend Payment in De Finetti Models: Survey and New Results and Strategies

We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value α. This is done in most simple discrete De Finetti models. We characterize the value function V(s,α) for initial surplus s of this problem, characterize the corresponding optimal dividend strategies, and present an algorithm for its computation. In an earlier solution to this problem, a Hamilton-Jacobi-Bellman equation for V(s,α) can be found which leads to its representation as the limit of a monotone iteration scheme. However, this scheme is too complex for numerical computations. Here, we introduce the class of two-barrier dividend strategies with the following property: when dividends are paid above a barrier B, i.e., a dividend of size 1 is paid when reaching B+1 from B, then we repeat this dividend payment until reaching a limit L for some 0≤L≤B. For these strategies we obtain explicit formulas for ruin probabilities and present values of dividend payments, as well as simplifications of the above iteration scheme. The results of numerical experiments show that the values V(s,α) obtained in earlier work can be improved, they are suboptimal.

A consistent estimation of optimal dividend strategy in a risk model with delayed claims

In this paper, we study how to estimate the optimal dividend strategy in a discrete-time risk model with delayed claims. The insurance company pays dividends to shareholders, and each dividend is taxable. The company controls the dividend payments to maximize the expectation of accumulated discounted dividends after tax prior to ruin. This paper aims to develop a method to estimate the optimal dividend strategy by using historical data directly. For the purpose we construct a nonlinear stochastic operator on the space which is proved to have contraction property. Fortunately the stochastic fixed point of the operator converges to a real sequence in probability, from which a consistent estimator of the optimal strategy follows. This is an important advantage of the stochastic operator method because the consistency property is difficult to be obtained by traditional methods with estimated distributions for claims. Finally, the method is applied in an example. The estimation results by using simulation data show that the optimal dividend strategy is a conditional threshold strategy, and the optimal dividend threshold tends to be stable with the sample size increasing.

The Optimal Investment, Liability and Dividends in Insurance

In this paper, we build an optimal control model with the objective to maximize the expected value of the time discount utility by selecting optimal investment, liability and dividend strategies for insurance companies. We then use the techniques from Merton (J Econ Theory 3(4):373–413, 1971) to solve our optimal control problem and deduce the optimal control solutions. Finally, we analyze the economic impacts on the optimal controls of the parameters in insurance market.

Optimality of barrier dividend strategy in a jump-diffusion risk model with debit interest

This paper investigates the optimal dividend problem in a jump-diffusion risk model with debit interest. In this model, the insurer could borrow money at a debit interest when the surplus turns negative. However, when the negative surplus attains a certain critical level, the business stops and absolute ruin happens at this moment. A sufficient condition under which the optimal dividend strategy is of barrier type is given in such a risk model. The main result relies on the smoothness of certain function arising from the dividend problem and we prove that it is twice continuously differentiable by the probability argument. Finally, numerical examples are given to illustrate the effects of the debit interest.

Dividend optimization for jump–diffusion model with solvency constraints

Belhaj (2010) established that a barrier strategy is optimal for the dividend problem under jump–diffusion model. However, if the optimal dividend barrier level is set too low, then the bankruptcy probability may be too high to be acceptable. This paper aims to address this issue by taking the solvency constrain into consideration. Precisely, we consider a dividend payment problem with solvency constraint under a jump–diffusion model. Using stochastic control and PIDE, we derive the optimal dividend strategy of the problem.