Optimal dividends and capital injection under dividend restrictions

Mathematical Methods of Operations Research https://doi.org/10.1007/s00186-020-00720-y ORIGINAL ARTICLE Optimal dividends and capital injection under dividend restrictions Kristoffer Lindensjö1 · Filip Lindskog1 Received: 6 August 2019 / Revised: 5 June 2020 © The Author(s) 2020 Abstract We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital—corresponding to absorption at 0—implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. We also consider the possibility for the owner to choose a stopping time at which the insurance company is liquidated and the owner obtains a liquidation value. The uncontrolled surplus process is a Wiener process with drift. Keywords Bankruptcy · Capital injection · Dividend restrictions · Insolvency · Issuance of equity · Optimal dividends · Reflection and absorption · Singular stochastic control · Solvency constraints Mathematics Subject Classification 49J15 · 49N90 · 93E20 · 91B30 · 91G80 · 97M30 B Kristoffer Lindensjö Filip Lindskog 1 Department of Mathematics, Stockholm University, Stockholm, Sweden 123 K. Lindensjö, F. Lindskog 1 Introduction Insurance risk was originally studied in terms of ruin probability. This approach may, however, underestimate risk since insurance companies are realistically more interested in maximizing company value than minimizing risk and an alternative approach is therefore to study optimal dividend policies—in the sense of maximizing the expected value of the sum of discounted future dividend payments—as suggested by De Finetti in the 1950s; see De Finetti (1957). A vast literature on various versions of the optimal dividend problem has since emerged. Surveys of the topic include Albrecher and Thonhauser (2009), Avanzi (2009) and Taksar (2000); see also Alvarez (2018) and Schmidli (2008). A common type of formulation of the optimal dividend problem corresponds to assuming that the only cash flow that may occur between the insurance company and its owner is from the insurance company to its owner, and in this formulation the insurance company typically goes bankrupt when the surplus process becomes negative; see e.g. De Finetti (1957), Jeanblanc-Picqué and Shiryaev (1995) and Shreve et al. (1984). In other words, the owner of the insurance company is not allowed to inject capital into the insurance company in this formulation of the problem. Another common type of formulation corresponds to assuming that the owner of the insurance company is obliged to inject capital so as to keep the surplus process non-negative; see e.g. Avram et al. (2007), Kulenko and Schmidli (2008) and Shreve et al. (1984). Hence, bankruptcy can never occur in this formulation of the problem. Further reviews of these two formulations of the optimal dividend problem can be found in Albrecher and Thonhauser (2009, Section 4) and Avanzi (2009, Section 5). Corporations in financial and insurance markets in the real world, however, typically have the possibility of both going bankrupt and raising equity capital from its owner (capital injection). One of the first papers to take both of these market characteristics into account simultaneously is Løkka and Zervos (2008) which studies a singular stochastic control problem corresponding to the optimal dividend problem with the possibility of both capital injection and bankruptcy, under the assumption that the uncontrolled surplus process is a Wiener process with drift. The authors find that depending on the parameters of the model it is either optimal to pay dividends in order to reflect the surplus process at an upper barrier and never inject capital, or to pay dividends and inject capital in order to always reflect the surplus process at an upper barrier and at 0. The results of Løkka and Zervos (2008) are in Zhu and Yang (2016) extended to a general Itô diffusion model with a growth restriction for the drift function. In Gajek and Kuciński (2017) the optimal dividend problem with proportional costs for capital injection is studied under the assumption that the owner can at any time choose to liquidate the insurance company and thereby receive a liquidation value. The uncontrolled surplus process, which is defined as the surplus in excess of a parameter interpreted as a solvency capital requirement, follows a spectrally negative Lévy process. Corporations in financial and insurance markets are also regulated by supervisory authorities. In order to take this characteristic into account (Paulsen 2003) studies the optimal dividend problem in a model with solvency constraints, meaning that it is not allowed to pay dividends unless the surplus process exceeds a given constant—in 123 Optimal dividends and capital injection under dividend… the present paper called dividend payout barrier. Capital injection is not considered. The author finds that it is optimal to use a reflection strategy with the reflection barrier being the maximum of the dividend payout barrier and the reflection barrier that would have been optimal without regulation. The uncontrolled surplus process is a fairly general Itô diffusion. The results of Paulsen (2003) are in He et al. (2008) extended by the introduction of the possibility of reinsurance. The optimal dividend problem has also been studied in the finance literature, where both Décamps et al. (2011) and Hugonnier and Morellec (2017) study models allowing for both bankruptcy and capital injection. In these papers there are fixed costs for capital injection, implying that the solutions involve lump sum, i.e. impulse control type, capital injections; Hugonnier and Morellec (2017) moreover consider a liquidation value and study liquidity and leverage requirements. The uncontrolled state process in Décamps et al. (2011) is a Wiener process with drift. The uncontrolled state process in Hugonnier and Morellec (2017) is a Wiener process with drift and exponentially distributed jumps. In He and Liang (2009) both fixed and proportional transaction costs for capital injection, as well as rein (...truncated)