Minimizing the probability of ultimate ruin by excess of loss reinsurance and Investments

Insurance companies provide protection to individuals against possible unexpected losses. While insuring individual risks, the insurance company is itself exposed to a risk of its surplus becoming negative. Therefore in this research, we consider a risk management strategy where the insurance company chooses to re-insure its surplus under excess of loss reinsurance arrangement and invests into both risky and risk free assets.We model the wealth dynamics of an insurance company by a risk process perturbed by diffusion. This process is then compounded by another return on investment process of Black-Scholes type. These two processes combined, form the risk process used in this dissertation. The Hamilton-Jacob-Bellman equation for this problem is then derived as well as its corresponding Volterra Integra Differential Equation of the second kind which is then transformed into a linear Volterra Integral Equation of second kind. We solve this integral equation numerically using the block-by-block method for different retention levels for chosen parameters. The results show that, the higher the rate of investment, the lower the ruin probability. Furthermore, the study reveals that, for a given initial capital, the ruin probability keeps on declining as the retention level for reinsurance increases. However, after a certain level, the probabilities begin rising again, giving an indication of the optimal retention level for Excess of Loss reinsurance.