On optimal periodic dividend strategies in the dual model with diffusion
Introduction
In actuarial risk theory, the introduction of a stochastic process formulation for the surplus of a company goes back to the early 20th century. The initial criterion for assessing the stability of a company was the probability of ruin, that is, the probability that the surplus ever becomes negative; see Bühlmann (1970). However, for the specifications of the surplus models to make economic sense, their drift (expected profit) is usually assumed to be positive. In the absence of surplus leakages, surpluses will hence grow (in average) to infinity, which does not make sense. Because of this, de Finetti (1957) first introduced an alternative formulation allowing explicitly for surplus leakages, called ‘dividends’. Note that by ‘dividends’, we consider any diminution of surplus that is made to the profit of the company’s owners according to the definition of ‘aggregate payout’ in Allen and Michaely (2003, p. 356). Many different models for the surplus of a company with dividends have been explored—see Avanzi (2009) and Albrecher and Thonhauser (2009) for comprehensive reviews.
When and how much dividends should be paid (a dividend strategy) is usually determined such that the expected present value of dividends until ruin is maximized. This pure maximization of dividends paid until ruin presents some issues. One of the two mentioned by Gerber (1974) is that the resulting optimal strategies are usually not realistic. In this paper, we restrict the form of the dividend strategy in order to partially address this issue, as explained below.
In most surplus models, unrestricted optimal dividend strategies lead to very irregular dividend payments, which is arguably an unrealistic feature. In reality, companies distribute dividends at regular time intervals (for instance, quarterly or annually) on the basis of balance sheets established at similar time intervals. Albrecher et al. (2011b) were the first to study random inter-dividend-decision times (in the Cramér–Lundberg model). However, their model does not allow ruin to happen in-between dividend payment times, de facto reducing to a modified discrete time model (see also Albrecher et al., 2011a, for related optimality results). Continuous monitoring of solvency with periodic dividends were first introduced by Albrecher et al. (2011c, with constant ‘intensity’ to ruin when the surplus is negative) in the Brownian risk model with exponential inter-dividend-decision times and Avanzi et al. (2013, with ruin defined as the first time the surplus hits 0) in the dual model with Erlang inter-dividend-decision times.
In the dual model without diffusion, the unrestricted dividend strategy that maximizes the expected present value of dividends is a continuous barrier strategy; see Bayraktar et al. (2013), whose results are extended in Bayraktar et al. (2014), with fixed transaction costs. In the dual model with diffusion and hyperexponential gains, a similar result has been established in Avanzi et al. (2011), in presence of capital injections as well. In a regime switching Brownian risk model, the optimality of the periodic barrier strategy is studied by Wei et al. (2012) when a liquidation-at-first-opportunity is not optimal. In this paper, we generalize the results of the last two references by showing that the periodic barrier strategy is still optimal in presence of hyperexponentially distributed gains and when inter-dividend-decision times are exponential. Its associated value function has a closed form representation. We also determine when a liquidation-at-first-opportunity strategy is optimal.
Section 2 defines formally the surplus model of the dual model with diffusion, and introduces the concept of periodic dividend strategies. The optimization problem considered in this paper is set up in Section 3, where admissible and optimal periodic strategies are defined, and an appropriate verification lemma is developed and proved in conjunction with its associated Hamilton–Jacobi–Bellman equation. We construct a candidate solution to the optimization problem in Section 4. We start by determining the expected present value of dividends under an arbitrary periodic dividend barrier and under a liquidation-at-first-opportunity strategy in Sections 4.1 Expected present value of dividends under an arbitrary periodic barrier, 4.2 Expected present value of dividends under a liquidation-at-first opportunity strategy (, respectively. In Section 4.3 we show that the former takes a particular form when the optimal level , whose existence and uniqueness is discussed in Section 4.4. These candidates are proven to be indeed optimal in Section 5.
Results are illustrated in Section 6. In Section 6.1, we investigate in detail the impact of parameters on the optimal strategy. In particular, we illustrate which parameter combinations lead to a liquidation-at-first-opportunity strategy to be optimal. Next, we show the impact of dividend-decision frequencies on the periodic strategy in Section 6.2. Lastly, we compare the continuous barrier strategy with the periodic barrier strategy in Section 6.3.
Section snippets
The dual model with diffusion
In the dual model with diffusion, the company surplus at time is described as where is the initial surplus, is the expense rate per unit of time and where is a compound Poisson process with intensity and jumps with distribution function . The process is a standard Brownian motion which is independent of , and which has a volatility of per unit of time. Throughout, we will assume that the distribution of the jumps in is
Definitions of admissible and optimal dividend strategies
Let denote the set of admissible periodic strategies. To be admissible, a strategy needs to have an associated aggregate dividend process that is a non-decreasing and -adapted with càdlàg sample paths and initial value . Dividend decision times in are stopping times with respect to with The dividend payout at decision time is for , which is measurable with respect to . While a dividend is allowed to bring the surplus back to a level
Expected present value of dividends under a periodic dividend barrier
Motivated by the structure of the HJB in (3.5), we postulated that the optimal periodic strategy is a periodic dividend barrier; see Section 3.3. In this section, we derive the expected present value of dividends paid until ruin and associated with a periodic barrier strategy and liquidation-at-first-opportunity strategy , respectively. Existence and uniqueness of an optimal level is also discussed.
The optimal periodic dividend strategy
In Section 3.2, we determined the conditions a periodic dividend strategy needs to satisfy to be optimal; see Lemma 3.4. In this section, we show that the periodic barrier strategy satisfies all of those conditions, thanks to the candidate value function obtained in Section 4.3. We first verify the optimality of the periodic barrier strategy with a strictly positive barrier when condition (4.47) holds, then show that a liquidation-at-first-opportunity is optimal when condition (4.47) does not
Numerical illustrations
In this section, we use some numerical examples to illustrate the results of this paper. Unless stated otherwise, we will use and such that the expected profit per time unit is . Remark 6.1 How time is defined impacts several parameters in our model: (number of gains per time unit), (number of dividend decision per time unit), and (volatility of the diffusion component per time unit). This has an impact on how results can be interpreted. In particular, choosing means that
Acknowledgments
Benjamin Avanzi acknowledges financial support from an Australian School of Business Special Research Grant. Vincent Tu acknowledges financial support from an Australian Postgraduate Award, as well as supplementary scholarships provided by the Australian School of Business, UNSW.
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