Risk Analysis and Uncertainty Quantification in Insurance Ratemaking

Insurance ratemaking, which is the process of setting an adequate amount of premium for an insured entity, is an essential role of insurance actuaries. For the success of this process, they need to perform a delicate and sound statistical analysis of insurance data, considering all the information it contains. Recently, several works of literature that explore the Value-at-Risk (VaR) for premium calculation have been reported, such as Heras, Moreno, and Vilar-Zan´on (2018). Motivated by the importance of risk forecast in insurance ratemaking, this dissertation proposes diverse approaches to making inferences about risk measures and quantifying uncertainty. Specifically, I start by disputing the argument in Heras, Moreno, and Vilar-Zan´on (2018) that their two-step inference method with quantile regression at the second stage with categorical variables can make a better forecast of VaR of aggregate losses than usual simple nonparametric estimates. By constructing a confidence interval using a novel empirical likelihood method, I provide sound evidence of my disputing argument. I further expand the risk analysis in more general settings to make an inference about VaR using both categorical and continuous explanatory variables and to quantify uncertainty using a random weighted bootstrap method. Lastly, I propose a three-step inference method for forecasting quantile risk measures, such as VaR and Expected Shortfall (ES), at a high-risk level. I adopt a Generalized Pareto Distribution (GPD) with a dynamic threshold for modeling excess losses and prove that I have made an efficient and robust risk forecast. Empirically, I use a well-known Australian automobile insurance dataset to illustrate the developed methods