A two-dimensional dividend problem for collaborating companies and an optimal stopping problem

We consider two insurance companies with wealth processes described by two independent Brownian motions with drift. The goal of the companies is to maximize their expected aggregated discounted dividend payments until ruin. The companies are allowed to help each other by means of transfer payments. But in contrast to Gu et al. [(2018). Optimal dividend strategies of two collaborating businesses in the diffusion approximation model. Mathematics of Operations Research 43(2), 377–398], they are not obliged to do so, if one company faces ruin. We show that the problem is equivalent to a mixture of a one-dimensional singular control problem and an optimal stopping problem. The value function is explicitly constructed and a verification result is proved. Moreover, the optimal strategy is provided as well.     

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 distribution under Markov regime switching

We investigate the problem of optimal dividend distribution for a company in the presence of regime shifts. We consider a company whose cumulative net revenues evolve as a Brownian motion with positive drift that is modulated by a finite state Markov chain, and model the discount rate as a deterministic function of the current state of the chain. In this setting, the objective of the company is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, which is taken to be the first time that the cash reserves (the cumulative net revenues minus cumulative dividend payments) are zero. We show that if the drift is positive in each state, it is optimal to adopt a barrier strategy at certain positive regime-dependent levels, and provide an explicit characterization of the value function as the fixed point of a contraction. In the case that the drift is small and negative in one state, the optimal strategy takes a different form, which we explicitly identify if there are two regimes. We also provide a numerical illustration of the sensitivities of the optimal barriers and the influence of regime switching.     

Optimal dividend policies for piecewise-deterministic compound Poisson risk models

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.     

Optimal dividend control for a generalized risk model with investment incomes and debit interest

This paper investigates dividend optimization of an insurance corporation under a more realistic model, which takes into consideration refinancing or capital injections. The model follows the compound Poisson framework with credit interest for positive reserve and debit interest for negative reserve. Ruin occurs when the reserve drops below the critical value. The company controls the dividend pay-out dynamically with the objective to maximize the expected total discounted dividends until ruin. We show that the optimal strategy, is a band strategy and it is optimal to pay no dividends when the reserve is negative.     

Semiparametric estimation in the optimal dividend barrier for the classical risk model

In the context of an insurance portfolio which provides dividend income for the insurance company’s shareholders, an important problem in risk theory is how the premium income will be paid to the shareholders as dividends according to a barrier strategy until the next claim occurs whenever the surplus attains the level of ‘barrier’. In this paper, we are concerned with the estimation of optimal dividend barrier, defined as the level of the barrier that maximizes the expected discounted dividends until ruin, under the widely used compound Poisson model as the aggregate claims process. We propose a semi-parametric statistical procedure for estimation of the optimal dividend barrier, which is critically needed in applications. We first construct a consistent estimator of the objective function that is complexly related to the expected discounted dividends and then the estimated optimal dividend barrier as the minimizer of the estimated objective function. In theory, we show that the constructed estimator of the optimal dividend barrier is statistically consistent. Numerical experiments by both simulated and real data analyses demonstrate that the proposed estimators work reasonably well with an appropriate size of samples.     

A note on optimal expected utility of dividend payments with proportional reinsurance

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company that controls risk exposure by purchasing proportional reinsurance. We assume the preference of the insurer is of CRRA form. By solving the corresponding Hamilton–Jacobi–Bellman equation, we identify the value function and the corresponding optimal strategy. We also analyze the asymptotic behavior of the value function for large initial reserves. Finally, we provide some numerical examples to illustrate the results and analyze the sensitivity of the parameters.     

A multivariate Markov chain stock model

ABSTRACT

We propose a dividend stock valuation model where multiple dividend growth series and their dependencies are modelled using a multivariate Markov chain. Our model advances existing Markov chain stock models. First, we determine assumptions that guarantee the finiteness of the price and risk as well as the fulfilment of transversality conditions. Then, we compute the first- and second-order price-dividend ratios by solving corresponding linear systems of equations and show that a different price-dividend ratio is attached to each combination of states of the dividend growth process of each stock. Subsequently, we provide a formula for the computation of the variances and covariances between stocks in a portfolio. Finally, we apply the theoretical model to the dividend series of three US stocks and perform comparisons with existing models. The results could also be applied for actuarial purposes as a general stochastic investment model and for calculating the initial endowment to fund a portfolio of dependent perpetuities.     

Dividend Barrier Strategies in A Renewal Risk Model with Generalized Erlang Interarrival Times

Abstract

We consider a renewal risk model with generalized Erlang distributed interarrival times. We assume that the phases of the interarrival time can be observed. In order to solve de Finetti's dividend problem, we first consider phasewise barrier strategies and look for the optimal barriers when the initial capital is 0. For exponentially distributed claim sizes, we show that the barrier strategy is optimal among all admissible strategies. For the special case of Erlang(2) interarrival times, we calculate the value function and the optimal barriers.     

Periodic threshold-type dividend strategy in the compound Poisson risk model

In this paper, the compound Poisson risk model is considered. Inspired by Albrecher, Cheung, & Thonhauser. [(2011b). Randomized observation periods for the compound Poisson risk model: dividend. ASTIN Bulletin 41(2), 645–672], it is assumed that the insurer observes its surplus level periodically to decide on dividend payments at the arrival times of an Erlang(n) renewal process. If the observed surplus is larger than the maximum of a threshold b and the last observed (post-dividend) level, then a fraction of the excess is paid as a lump sum dividend. Ruin is declared when the observed surplus is negative. In this proposed periodic threshold-type dividend strategy, the insurer can have a ruin probability of less than one (as opposed to the periodic barrier strategy). The expected discounted dividends before ruin (denoted by V) will be analyzed. For arbitrary claim distribution, the general solution of V is derived. More explicit result for V is presented when claims have rational Laplace transform. Numerical examples are provided to illustrate the effect of randomized observations on V and the optimization of V with respect to b. When claims are exponential, convergence to the traditional threshold strategy is shown as the inter-observation times tend to zero.     

Optimal dividend policies with transaction costs for a class of jump-diffusion processes

This paper addresses the problem of finding an optimal dividend policy for a class of jump-diffusion processes. The jump component is a compound Poisson process with negative jumps, and the drift and diffusion components are assumed to satisfy some regularity and growth restrictions. Each dividend payment is changed by a fixed and a proportional cost, meaning that if ?? is paid out by the company, the shareholders receive k???K, where k and K are positive. The aim is to maximize expected discounted dividends until ruin. It is proved that when the jumps belong to a certain class of light-tailed distributions, the optimal policy is a simple lump sum policy, that is, when assets are equal to or larger than an upper barrier $\bar{u}^{*}$ , they are immediately reduced to a lower barrier $\underline{u}^{*}$ through a dividend payment. The case with K=0 is also investigated briefly, and the optimal policy is shown to be a reflecting barrier policy for the same light-tailed class. Methods to numerically verify whether a simple lump sum barrier strategy is optimal for any jump distribution are provided at the end of the paper, and some numerical examples are given.     

The impact of government ownership on dividend policy in China

We investigate the impact of State ownership on Chinese corporate dividend policy. We find that Chinese firms' dividend payout rates respond fairly quickly to earnings changes, and the average actual payout ratio for Chinese firms falls between the payout ratios for emerging-market and developed firms. These results are consistent with the dividend policies of developing economies in general. We also find that dividend payouts among dividend-paying firms, and the likelihood that a firm will pay a dividend, are increasing in State ownership. Our findings are consistent with the State's need for cash flow as a partial motivation for continued State ownership of a significant portion of the corporate economy, and support the agency and tax clientele explanations for dividend policy.     

The relative pricing of European dividend futures and their predictive abilities for index returns

We use dividend futures prices to derive a dividend future discount model. Arbitrage arguments postulate that the sum of discounted dividend futures prices should equal the index price, i.e. the sum of discounted dividends. We analyze whether this relation holds and find that the two valuation approaches lead to a different valuation of expected dividends. These observations indicate that dividend futures and index prices seem to provide the investor with different information on future dividends. We further show that the difference in valuation can be used to forecast index returns and show how an investment strategy can exploit this predictability.     

Inflation illusion and the dividend yield: evidence from the UK

We report evidence that the UK dividend yield and expected inflation are positively correlated from 1962 to 1997, but negatively correlated subsequently. Using a commonly used VAR (vector auto-regression)-based procedure we find strong evidence that the positive correlation is caused both by inflation illusion and the effect of inflation on required rates of return. We also find some evidence that it is caused by inflation rationally reducing expected real dividend growth. We find that Chen and Zhao's (2009. “Return Decomposition.” Review of Financial Studies 22 (12): 5213–5249) criticism of the VAR-based procedure has little empirical relevance but that the procedure can be highly sensitive to the choice of data period.     

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 dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints

In this paper, we consider a company whose surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments. With each dividend payment, there are transaction costs and taxes, and it is shown in Paulsen (Adv. Appl. Probab. 39:669?C689, 2007) that under some reasonable assumptions, optimality is achieved by using a lump sum dividend barrier strategy, i.e., there is an upper barrier $\bar{u}^{*}$ and a lower barrier $\underline{u}^{*}$ so that whenever the surplus reaches $\bar{u}^{*}$ , it is reduced to $\underline{u}^{*}$ through a dividend payment. However, these optimal barriers may be unacceptably low from a solvency point of view. It is argued that, in that case, one should still look for a barrier strategy, but with barriers that satisfy a given constraint. We propose a solvency constraint similar to that in Paulsen (Finance Stoch. 4:457?C474, 2003); whenever dividends are paid out, the probability of ruin within a fixed time T and with the same strategy in the future should not exceed a predetermined level ??. It is shown how optimality can be achieved under this constraint, and numerical examples are given.     

Expected returns and expected dividend growth

We investigate a consumption-based present-value relation that is a function of future dividend growth and find that changing forecasts of dividend growth are an important feature of the post-war U.S. stock market, despite the failure of the dividend–price ratio to uncover such variation. In addition, dividend forecasts are found to covary with changing forecasts of excess stock returns over business cycle frequencies. This covariation is important because positively correlated fluctuations in expected dividend growth and expected returns have offsetting effects on the log dividend–price ratio. The market risk premium and expected dividend growth thus vary considerably more than is apparent using the log divided–price ratio alone as a predictive variable.     

On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier

In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.