Asymptotic Estimates of the Ruin Probality for an Insurer

The ruin probability of a company is the probability that the wealth of the company falls below zero at some future time. The wealth process of a typical insurance company is described by the claims it incurs and by its premium income. Since 1903, when F. Lundberg introduced a collective risk model based on a homogeneous Poisson claim process, the calculation of ruin probabilities in this setting has been very well understood . It is known that, if the claim sizes have exponential moments, the ruin probability decreases exponentially with the initial wealth. Also if the claim sizes have heavier tails, there exist numerous results in the literature. The overall aim of this project is to generalize these results to a setting where besides the risk process there exists some investment possibility, e.g. described by geometric Brownian motion. Then the insurer should have the possibility to reduce his ruin probability by choosing good investment strategies. In particular we want to investigate the asymptotic behaviour of the minimal ruin probability for large claims, when the distribution function of the claim sizes is of regular variation. Another goal of the project will be to look at the minimal ruin probability for small claims with exponential moments. We will use martingale methods to deduce Cramér-Lundberg bounds and asymptotics and to derive some information about the optimal investment strategy leading to the minimal ruin probability. Furthermore economically reasonable generalizations will be looked at, like the introduction of interest rates and dividend barriers. The results of the project shall provide valuable information for insurers or other companies with stochastic risks about opportunities and risks arising from investment possibilities. This is in line with the present emphasis - in practice as well as in theory - on asset-liability management, i.e., the consideration of risks on the liability side as well as on the asset side of the balance sheet. The wealth process of a typical insurance company is described by the claims it incurs and its premium income. Since 1903, when F. Lundberg introduced a collective risk model based on a homogeneous Poisson claim process, the calculation of ruin probabilities in this setting has been very well understood . It is known that, if the claim sizes have exponential moments, the ruin probability decreases exponentially with the initial wealth. Also if the claim sizes have heavier tails, there exist numerous results in the literature. The overall aim of this project is to generalize these results to a setting where besides the risk process there exists some investment possibility described by geometric Brownian motion. In particular we want to investigate the asymptotic behaviour of the ruin probability for large claims, when the distribution function of the claim sizes is of regular variation. It will be interesting to see whether the ruin probability decreases like the integrated tail distribution function (as it is the case without investment) or rather like the tail distribution itself (as it is the case for constant return on investment). Another goal of the project will be to look at the minimal ruin probability for claims with exponential moments. We will use martingale methods to deduce Cramér-Lundberg bounds and asymptotics and to derive some information about the optimal investment strategy leading to the minimal ruin probability. The same questions will also be looked at with the methods of asymptotic expansions and using the integro- differential equation that is obtained for this problem by using the Hamilton-Jacobi-Bellman approach. Furthermore economically reasonable generalizations will be looked at, like stochastic interest rates and dividend barriers. The results of the project shall give valuable information for insurers or other companies with stochastic risks, about opportunities and risks arising from modern investment possibilities.

Insurance comapnies have to estimate their risk in the one way or the other. A classical measure for the risk is the ruin probability of such a company. Let us briefly explain this concept by means of the classical model. The company starts with a certain endowment. As time goes by, an the one hand side the company earns premiums and an the other hand it has to pay for incoming claims. The premiums are offen modelled in a deterministic way, e.g. a constant amount of premiums per time unit. On the other hand side, the claims are random and one has to choose a certain stochastic model for them. To be a little bit more precise, one has to choose a model for the time inbetween two claims as well as a model for the sevirity of the claims. This is done by assigning certain probabilities for the possible time between the claims, and probabilities for the height of the claim. Finally one asks, what is the probability of ruin for such a company. A first estimate was given by Frederic Lundberg around 1900. He showed that, under certain assumptions an the model, the probability of ruin for the company decreases exponentially with the initial endowment of the company. Nowadays insurance companies are allowed to invest parts of their money in the stock market. The main topic of the present project was to give estimates for the ruin probability of a comany investing in the stock market. Therefore one has to add to the model described above a model for the behaviour of the prices an the stock market. We took the famous geometric Brownian motion. which is also used for the derivation of the well-known Black- Scholes option pricing formula. In our project we distinguished between so called light-tailed (small) claims and heavy-tailed (large) claims. Ort the other hand we disinguished between a company, which invests always a certain fraction of its wealth in the market and one, which tries to invest optimally in the market. Here "optimally" means that one tries to minimize the corresponding ruin probability. In all of these cases we were able to give either asymptotic values for the probability of ruin, when the initial endowment of the company is large, or to give at least estimates of this probability. It turned out that it is possible to reduce the ruin probability considerably by investing in the market in a clever way ( compared with a company, which does not invest at all).