Optimisation of Consumption, Capital Injections and Dividends in Actuarial Mathematics

Optimal control models with different risk measures, infinite or finite time horizon play an important role in actuarial mathematics. By such problems, one considers a target functional that quantifies risks connected to an insurance portfolio augmented by additional components, such as dividend payments, investments, reinsurance etc. The main goal is to find a strategy, that minimizes/maximizes the underlying target functional, i.e. minimizes the risks or maximizes the benefits of the insurer. Thus, the choice of a risk measure is of crucial importance for solving control problems. The next step, when formulating a control problem, is the modeling of the surplus process of the considered insurance entity. Here, one has the choice between deterministic and stochastic modeling. Most economic problems are of a stochastic nature, due to the uncertainty about future system development. On the one hand, stochastic models approximate the real processes much better than deterministic ones, on the other - deterministic modeling enjoys a much greater ease of computability. However, it should be noticed that even in a deterministic setting it is only rarely possible to find a closed expression for the optimal strategy. The present project aims to consider three different risk measures: expected discounted consumption, capital injections and dividends under deterministic and stochastic surplus modeling. The first model describes the situation when the surplus of an economic agent is of deterministic nature. As an example one may think of households living from tourism (every summer regular income, every winter ``employment gap``). The discounting factor is stochastic (depends on the global macroeconomic situation, which cannot be assumed to be deterministic). We target to maximize the expected discounted consumption of the agent up to a finite deterministic time horizon. The second problem models the surplus of the considered insurance company by a Brownian motion with drift. In addition, the insurer is allowed to buy reinsurance and pay out dividends. It is also assumed, that in the case the surplus process becomes negative, the shareholders have to inject capital in order to facilitate a smooth business progress, which allows to consider the problem in the infinite time horizon. The model parameters, such as the drift, the volatility, the retention level of the reinsurance, the safety loadings of the insurer and reinsurer, evolve due to a Markov process. The main goal is to find a reinsurance strategy, depending on the initial capital and initial state of the Markov process, that would maximize the value of expected difference of discounted dividends and capital injections. Finally, we want also to contribute in the direction of maximizing dividend payments. Here, we again model the insurer`s surplus by a Brownian motion with drift and want to maximize the expected discounted dividend payments under a finite time horizon. If the surplus process remains non-negative until the final time, the remaining surplus becomes the final payment, i.e. we pay out whatever is left. On the other hand, too much consumption is penalized only through the fact that the insurer will be ruined earlier. In our model, the insurer will be additionally rewarded if he "stays alive" until the final time.

Imagine, an insurance company needs to be rated by an internal service. Does a company need more reinsurance or more reserves? When is it optimal to pay dividends? These and other questions must be answered by the responsible actuaries to the companys management.In order to quantify a risk connected to a company, you need a risk measure, which can say something about the economic well-being of the company. As a such one can use for instance the expected discounted dividend payments, capital injections or the expected discounted consumption. Once a risk measure has been chosen, one has to model the surplus process of the considered economic agent.In the project FWF-P26487-N25, we consider the problem of maximizing the expected consumption, dividend payments or minimizing capital injections of an insurance company or a household.The main goal is to find a strategy that minimizes / maximizes the underlying functional, i.e. the risks are minimized or the benefits are maximized. The logic is the following: If the company is able to pay high dividends, then it is a signal that the company is doing well. On the other hand, if the shareholders have to inject capital in order to avoid red numbers, the company is struggling. The randomness of the considered processes reflects the unpredictable economic fluctuations such as macroeconomic market changes, microeconomic inflation or tax risks.The performance of an arbitrary strategy cannot say much about the well-being of a company, but rather about the abilities of the companys management. Using the methods from the optimal control theory, we are looking for the strategies that maximize or minimize the chosen risks measures. In other words, we measure the risk assuming the optimal possible behaviour of the management, which yields a more objective picture.