Optimising dividends and consumption under an exponential CIR as a discount factor

Mathematical Methods of Operations Research https://doi.org/10.1007/s00186-020-00714-w ORIGINAL ARTICLE Optimising dividends and consumption under an exponential CIR as a discount factor Julia Eisenberg1 · Yuliya Mishura2 Received: 9 September 2019 / Revised: 18 April 2020 © The Author(s) 2020 Abstract We consider an economic agent (a household or an insurance company) modelling its surplus process by a deterministic process or by a Brownian motion with drift. The goal is to maximise the expected discounted spending/dividend payments under a discounting factor given by an exponential CIR process. In the deterministic case, we are able to find explicit expressions for the optimal strategy and the value function. For the Brownian motion case, we are able to show that for a special parameter choice the optimal strategy is a constant-barrier strategy. Keywords Hamilton–Jacobi–Bellman equation · Cox–Ingersoll–Ross process · Dividends · Brownian risk model · Consumption Mathematics Subject Classification Primary 93E20; Secondary 91B42 · 91B30 · 60H30 1 Introduction 1.1 General introduction An insurance company’s credit rating indicates its ability to pay customer’s claims. A bad credit rating can affect a company’s business plan, growth potential or even survival chances if new finance is needed to fulfil the capital requirements prescribed by Solvency II. The rating process run by a credit rating agency includes quantitative and qualitative analysis, where cash flow is one of the most important factors. Particular attention is paid to dividend payments, which are commonly believed to indicate a company’s financial health. B Julia Eisenberg 1 TU Wien, Vienna, Austria 2 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 123 J. Eisenberg, Y. Mishura Searching for the optimal strategy which maximises the value of expected discounted dividends under different constraints and in different setups has been a popular problem in actuarial mathematics for a long time. The papers by Shreve et al. (1984), Asmussen and Taksar (1997), Azcue and Muler (2005) are just some examples. For a detailed review we refer for instance to the survey by Albrecher and Thonhauser (2009). The papers mentioned above assume the discounting rate to remain constant up to the considered time horizon, often chosen to be infinite. Following the recent crisis with ultra low interest rates in Europe, the question arises whether the discounting of cash flows by a constant discounting rate could be considered as an admissible assumption. A stochastic discounting factor increases the dimension of the considered problem along with the complexity. Nevertheless, in recent years stochastic discounting has become a topical question in dividend maximisation problems. For instance Jiang and Pistorius (2012) model the interest rate by a positive deterministic function of the current state of a given Markov chain. If the drift of the underlying surplus process is positive in each state, they prove that it is optimal to adopt a regime-dependent barrier strategy; if the drift is small and negative in one state, the optimal strategy has a different form, which is explicitly identified for the two regimes case. Akyildirim et al. (2014) consider two macroeconomic factors: the interest rates and the issuance costs. Both factors are assumed to be governed by an exogenous Markov chain. The optimal dividend policy is characterised by dependence on these two factors: all things being equal, firms distribute more dividends when interest rates are high and less when issuing costs are high. Where Jiang and Pistorius (2012) use the fixed point theorem in order to obtain their results, Akyildirim et al. (2014) apply the direct approach by solving the corresponding ODEs, a method we will use in our paper. In the present paper, we take into account the time-varying interest by introducing a discounting factor given by an exponential Cox–Ingersoll–Ross (CIR) process. A CIR process is a squared diffusion process, which can attain non-negative values and hit zero for special parameters. Here, we would like to emphasise that in contrast to the usual financial setup, we are modelling the compound interest and not the short rate. Negative interest rates have governed the markets in the last several years, keeping insurance companies under pressure. Therefore, assuming a market interest rate to be given by a non-negative CIR process would not be realistic. The case of consumption maximisation with an Ornstein–Uhlenbeck process describing the interest rates has been considered in Eisenberg (2018). The logic behind the choice of a CIR process is that we would rather look at the preferences of the insurer than at the real market-given interest. The non-negativity of CIR processes describing the compound preference implies that for the insurer money today is preferable to money tomorrow. This idea conforms to the work of the famous Austrian economist Böhm-Bawerk claiming an always positive preference interest, see Von Böhm-Bawerk (1890). On the website of the Mises Institute (The Ludwig von Mises Institute for Austrian Economics 1999), named after the most famous student of Böhm-Bawerk economist Ludwig von Mises, one finds plenty of essays defending the thesis of Böhm-Bawerk and examining counterarguments. A detailed discussion of the topic certainly goes beyond the scope of the present paper. Hence, we refer for instance to an essay by Polleit (2015) and give below a digest of the theory that serves as an economic basis for the 123 Optimising dividends and consumption under an exponential… mathematical model presented in the following chapters. The theory of positive time preference distinguishes between the market and the originary interest rate. While the first is the interest rate observed on deposit or loan markets, the originary interest rate measures consumption today compared to consumption tomorrow, i.e. the time preference. According to Böhm-Bawerk and his student von Mises the originary interest rate and consequently the (compound) time preference are always positive. Some modern economists, for instance Alchian (2018), even call a positive time preference the “principle of rationality”. However, an insurance company cannot ignore negative market interest rates governing the European markets in the recent years. With this in mind, the oscillations of a CIR process can be interpreted as the dependence of the insurer’s preferences on the movements of the market interest rate: if the compound preference goes down the market rate is supposed to be negative and vice versa, whereas the compound preference always stays positive. Usually, when modelling interest rates in financially motivated problems one assumes CIR to be mean-reverting. However, under this assumption our problem would be ill-posed on the one hand, and would contradict the thesis in Von Böhm-Bawerk (1890), that the value of goods decreases by postponing (...truncated)