Ultimate ruin probabilities in the stochastic dual risk model compounded by investment income

Abstract

The ultimate ruin probability is a central problem in the operation of financial businesses since it can be used as a tool to manage and check the solvency levels of a company. This dissertation calculates the probability of a company that invests in both risk-free and risky assets. The company’s surplus is modelled by a stochastic dual risk process which takes into account a constant rate of expenses and random gains that occur occasionally. The surplus is invested in a risk-less asset such as a treasury bond and a risky asset such as a market index which is modelled by geometric Brownian motion. Anumerical solution for a second-order integro-differential equation (IDE) is obtained using a combination of the Finite Difference Method (FDM) and the Trapezoidal Rule for the derivative and integral terms, respectively. The accuracy of the results obtained is tested against a similar equation with a known exact solution. The ultimate ruin probabilities are obtained by solving a system of linear algebraic equations which result from the combination of the two methods for chosen parameters. The numerical examples provided for both light and heavy gain distributions further reinforce the accuracy and reliability of this approach. The study demonstrates that investments are crucial in maintaining a positive surplus for a long time as they reduce the probability of ruin. Investing in riskless assets such as bonds can guarantee a company’s survival indefinitely, as an increase in the force of interest lowers ruin probabilities. This probability indicates a higher risk, not bankruptcy or mismanagement, as volatility in the investment model increases the ruin probabilities and the overall risk. When the probability of ultimate ruin is high, it is imperative for company managers to take assertive measures to mitigate the risk. In this regard, investing has been proven to be a significant and effective strategy, as demonstrated by this study.