Optimal Dividend Payout with Path-dependent Drawdown Constraint

In this paper, we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e., the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We then derive sufficient conditions under which a two-curve strategy is optimal, and we show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for sufficiently large initial and maximum dividend rates. We also give a number of numerical illustrations in which the optimality of the two-curve strategy can be established for instances with smaller values of the maximum dividend rate and the concrete form of the curves can be determined. One observes that the resulting drawdown strategies nicely interpolate between the solution for the classical unconstrained dividend problem and that for a ratcheting constraint as recently studied in Albrecher et al. (SIAM J. Financial Math. 13:657–701, 2022). When the maximum allowed dividend rate tends to infinity, we show a surprisingly simple and somewhat intriguing limit result in terms of the parameter a for the surplus level above which, for a sufficiently large current dividend rate, a take-the-money-and-run strategy is optimal in the presence of the drawdown constraint.

Optimal consumption under a drawdown constraint over a finite horizon

Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

This paper proposes and studies an optimal dividend problem in which a two-state regime-switching environment affects the dynamics of the company's cash surplus and, as a novel feature, also the bankruptcy level. The aim is to maximize the total expected profits from dividends until bankruptcy. The company's optimal dividend payout is therefore influenced by four factors simultaneously: Brownian fluctuations in the cash surplus, as well as regime changes in drift, volatility and bankruptcy levels. In particular, the average profitability can assume different signs in the two regimes. We find a rich structure of the optimal strategy, which, depending on the interaction of the model's parameters, can be either of barrier-type or of liquidation-barrier type. Furthermore, we provide explicit expressions of the optimal policies and value functions. Finally, we complement our theoretical results by a detailed numerical study, where also a thorough analysis of the sensitivities of the optimal dividend policy with respect to the problem's parameters is performed.

This paper considers a discrete time optimal dividend payout problem with a risk probability criterion. Different from the expected discounted dividends that are widely studied in the existing literature, our emphasis is the risk probability that an insurance company's total discounted dividend fails to reach a given dividend goal by the time of ruin. We aim to find an optimal dividend policy to minimize such risk probability. More precisely, we establish a general model for optimal dividend problems based on a Markov decision process with varying discount factors and a probability criterion. The associated optimality equation and the existence of optimal policies under the probability criterion are investigated. For a special case with independently identically distributed incomes, we further characterize the properties of optimal value functions and the structures of optimal dividend policies. A value iteration‐type algorithm for computing value functions and optimal policies is developed. The convergence and error bound analyses of the algorithm are also derived. Finally, an optimal dividend policy in a concrete example is presented to demonstrate our main results.

Adopting a probabilistic approach we determine the optimal dividend payout policy of a firm whose surplus process follows a controlled arithmetic Brownian motion and whose cash‐flows are discounted at a stochastic dynamic rate. Dividends can be paid to shareholders at unrestricted rates so that the problem is cast as one of singular stochastic control. The stochastic interest rate is modeled by a Cox–Ingersoll–Ross (CIR) process and the firm's objective is to maximize the total expected flow of discounted dividends until a possible insolvency time. We find an optimal dividend payout policy which is such that the surplus process is kept below an endogenously determined stochastic threshold expressed as a decreasing continuous function of the current interest rate value. We also prove that the value function of the singular control problem solves a variational inequality associated to a second‐order, non‐degenerate elliptic operator, with a gradient constraint.

Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint $\min\Big\{v_t-\max\limits_{0\leq a\leq1}\Big(\frac{1}{2}\sigma^2a^2v_{xx}+\mu av_x\Big)+cv,\;v_x-1\Big\} = 0.$ Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.

In this paper, we devote our attention to the liquidity and risk management of a firm who faces two types of risks: a Poisson risk that can be insured for a fair premium and a Brownian risk that cannot be hedged nor insured. We apply a PDE method to solve this problem; the associated HJB equation is a Barenblatt equation with a gradient constraint in finite horizon. We not only show the existence of a classical solution to the problem, but also characterize the properties of the free boundaries arising from the HJB equation; especially, we are able to give a sufficient and necessary condition for the existence of the insurance free boundary.

Semismooth Newton methods with a shooting-like technique for solving a constrained free-boundary HJB equation

A model of optimal dividend payout is presented in which increased dividends lower agency costs but raise the transactions cost of external financing. The optimal dividend payout minimizes the sum of these two costs. A cross‐sectional test of the model relates dividend payout to the fraction of equity held by insiders, the past and expected future revenue growth of the firm, the firm's beta coefficient, and the number of common stockholders. The coefficients of all variables are significant in the predicted directions. The results indicate that investment policy influences dividend policy.

Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments

Numerical solution of an optimal investment problem with proportional transaction costs

How might model uncertainty and transaction costs impact retained earning & dividend strategies? An examination through a classical insurance risk model

The optimal dividend problem goes back to a paper that Bruno De Finetti presented to the International Congress of Actuaries in New York (1957). For a stock company that pays dividends to its shareholders, what is the strategy that maximizes the expectation of the discounted dividends (until possible ruin)? Jeanblanc-Picqué and Shiryaev (1995) and Asmussen and Taksar (1997) solved the problem in the Brownian motion model, when a ceiling is imposed for the dividend rate. Here we study the problem with the Brownian motion generalized to a compound Poisson process. In particular, we derive a rule for deciding between plowback and dividend payout, which is a key issue in corporate finance.

Dividend decision is a critical finance function since it involves determining the amount distributed to shareholders as earnings or the amount to reinvest internally. The determination of dividend pay-out is influenced by the liquidity position of the firm but the extent to which liquidity affects the dividend pay-out still remains a puzzle since most empirical studies conducted have reported inconsistent results and no universally accepted explanation for companies with adequate liquidity have observed uniform dividend payment behaviour. It is in this context that the study was set out to determine the effect of liquidity on dividend pay-out of a firm. The objectives of the study were; to determine the effect of profitability, cash flows and working capital on the firms’ dividend pay-out decisions. The study employed causal comparative research design on a target population of 61 firms listed at the NSE. Purposive sampling was used to select 30 firms which consistently paid dividends from the year 2008 to 2012. Data analysis was done using descriptive and inferential statistics. The study revealed that profitability plays a major role in dividend pay-out because of the higher coefficient as compared to cash flows and working capital and consequently the companies which posted higher profits translated this to higher dividends paid out to investors. The study recommends that firms should ensure that profits are stable, cash flows freely flow into the firm and working capital is efficiently managed so as to increase the firms’ dividend pay-out. The results would provide information to managers to determine an optimal dividend pay-out that would maximise the company’s stock price and thus lead to maximisation of shareholders wealth. The study also forms a basis for further research and adds knowledge to the existing body.