The present research formulates, stochastically, the expense-augmented insurance portfolio loss process. This is achieved by developing a weighted mixed compound stochastic process which constitutes expenses driven by a Gamma process and claims driven by an exponential distribution. Their intensity process is driven by affine and known as the basic affine jump-diffusion process (BAJDP), which is tractable and integrable and aids in capturing the randomness inherent in insurance data. Monte Carlo simulations are conducted to estimate the intensity and the effectiveness of the simulated process by the moment matching method. The main aim of the present study is to formulate an expense-augmented insurance loss process that includes claims and other expenses as part of the loss to an insurance company as opposed to existing models such as the classical risk model, which only models the process with claims. The model is tested with data obtained from the IRA, Kenya website. The descriptive statistics generated show only Gross premium data are normal (p>0.05) using the Shapiro–Wilks test of normality; the same is confirmed with Q-Q plots of the data. The model parameters are estimated through maximum likelihood estimates. The model is compared with three other existing models including a classical (HPP) loss model, classical (NHPP) loss, and a Gaussian loss model using a measure of probability of loss called the loss ratio, in which it stands out as a better loss model. Finally, the regression model that relates expenses and claims shows statistical significance at 5% significance level, and it can be used for respective prediction. The present model gives desired results since it takes into consideration other expenses in an insurance portfolio, and also loss is reduced significantly as compared with the existing models.
Odiwuor et al. (Wed,) studied this question.