Optimization of dividend payments

Two main objectives of insurance companies are on the one hand side to maximize the dividends, which their shareholders receive, and on the other hand, which is the more conservative approach, to minimize the ruin probability of the company. Both approaches are classical by now. The latter one has its origins in the early 20th century, when F. Lundberg formulated his famous inequality, which says that, under certain model assumptions, one can bound the ruin probability by an exponentially decreasing function, depending on the initial wealth of the company. The former approach has its origin in the famous paper of B. de Finetti of 1957, where he criticized the ruin probability approach as too conservative and suggested rather to maximize expected discounted dividend payouts. He showed in the same paper that, if one uses a random walk model for the surplus of the company, it is optimal to use a so called barrier strategy, i.e. to pay out all the reserves, which are above a certain time-independent barrier. There are numerous generalisations and modifications to both approaches in the Actuarial literature. The aim of the project is to contribute in both directions. Our research will be focused on three main points: 1.) It seems to be an open problem, what the optimal strategy for an insurance company is, if one describes the endowment of the company by a so called diffusion approximation, if the objective is to maximize expected discounted dividend payments and if the time horizon of the problem is finite. For infinite horizon the solution of the problem is well known: it is again a barrier strategy, as in de Finettis model. We conjecture that in the finite time setting the optimal solution is also of barrier type, but this time with a time-dependent barrier. The solution of the problem will probably lead to a so called free boundary value problem. 2.) One possible modification of the original de Finetti approach is the use of utility functions. We want to investigate the following problem: Maximize the expected utility of discounted dividend payouts. Again we use a diffusion approximation process for the endowment , but this time we assume an infinite time horizon. There exists already a paper dealing with this problem by the project leader and co-authors. But this paper is in a certain sense incomplete, since we were not able to prove the existence of a solution of the integral equation, which describes the barrier function. The goal is now, to close this gap, by using new results on integral equations provided by the project leader in a recent paper. 3.) Finally, we want also to contribute in the direction of optimizing ruin probabilities. In recent years it was an active area of research, to investigate ruin probabilities of insurance companies, which invest in the stock market. This is also important for practitioners, since it is exactly what companies do in reality. Again there are numerous results in the literature so far. We want to concentrate on the so called "large claim case". In this model the distribution function of the individual claims has no exponential moment, which means that very large claims appear with a non-neglegible probability. This is of course a very relevant case for the financial industry. We want to investigate the asymptotics (for high enough endowment) of the optimal investment strategy for a certain subclass of the subexponential distributions, which are considered as the most important "heavy-tailed" distributions. Clearly, we expect that during our research we will find other topics, which are worthwhile to study, e.g. a sound numerical analysis in problem 1.) We apply for a funding of one post doc position over the period of three years.

In the project P 22449-N18 we considered the problem of maximizing the dividends or consumption in an insurance company or in a household.Imagine, you as a member of the board of directors of a big insurance company have to decide, how much of the profit should be paid to the shareholders.But why does it actually make sense to pay out dividends?Googling this question, one finds lots of information. One of the pro arguments is that the dividend payout can serve as an indicator of the company's financial well-being.The logical chain is as follows: if a company is able to pay out high dividends then its a signal for investors to intensify their activities.Assuming the target is to maximize the dividend payments, we model the surplus of a given insurance company with the help of different stochastic processes.Stochasticity reflects the unpredictable fluctuations in economic conditions. Examples for such fluctuations provide macroeconomic changes on the market, but also microeconomic risks connected to the inflation rates or corporate and personal taxation.Using optimal control methods, we search for the dividend strategy that will maximize the value of expected discounted dividends.Note that the dividend strategies depend on the time and on the initial capital.Often, an absolute amount of money is not very informative. That is why, we also consider the utility the insurance company can derive from the expected dividend payments. On the other hand, if we consider an individual or household, sometimes it does not make sense to model the surplus via a stochastic process. In most cases, an individual has a salary/wages which remains deterministic over a certain period of time. The fluctuations can be quite sophisticated, but still predictable.In this case, we cannot speak about dividends, we speak about consumption. Despite the complexity of the problem, the optimal consumption behaviour can be determined depending on the income function.