Modelling of Risk Process With Expense‐Augmented Loss Under Economic Factors and Its Application to Aggregated General Insurance Data in Kenya

In this paper, we reformulate the classical risk model to consider economic factors such as taxation and real force of interest. In the model, the premiums are assumed to be compounded by increasing annuities over some time. The loss process is also presumed to be two mixed stochastic processes with weights that sum to 1. One of the process models claims and the other model’s expenses (expenses due to underwriting and also due to investments) incurred in an insurance company. These expenses are long‐tailed and positively skewed. The counts in the loss model are modeled as a dynamic Heston process which takes into consideration the economic factors, such as the real force of interest and taxation inherent in the insurance risk reserve process. Ruin probabilities are hence estimated from the Lundberg coefficient. The exact closed form of the coefficient is obtained to help in the graphical approximation of ruin probabilities. It is observed that ruin probabilities increase with the inclusion of economic factors in the model, as anticipated. The other risk models used for comparison produce significantly lower ruin probabilities, since the aspect of expenses is not included as part of the loss while applying the model, yet expenses also contribute negatively to the reserve of the portfolio. The result of the model can be adopted with the Insurance Regulatory Authority of Kenya and globally, since the data obtained is from all general insurance businesses for the stated period. The Kenyan government or any other country can adopt this risk process to help the insurance portfolio to have an accurate approximation of the financial positions, hence guarding against the possibility of ruin as a result of inaccurate estimation.