Robust estimation of value-at-risk through distribution-free and parametric approaches using the joint severity and frequency model: Applications in financial, actuarial, and natural calamities domains

Value-at-Risk (VaR) is a well-accepted risk metric in modern quantitative risk management (QRM). The classical Monte Carlo simulation (MCS) approach, denoted henceforth as the classical approach, assumes the independence of loss severity and loss frequency. In practice, this assumption does not always hold true. Through mathematical analyses, we show that the classical approach is prone to significant biases when the independence assumption is violated. This is also corroborated by studying both simulated and real-world datasets. To overcome the limitations and to more accurately estimate VaR, we develop and implement the following two approaches for VaR estimation: the data-driven partitioning of frequency and severity (DPFS) using clustering analysis, and copula-based parametric modeling of frequency and severity (CPFS). These two approaches are verified using simulation experiments on synthetic data and validated on five publicly available datasets from diverse domains; namely, the financial indices data of Standard & Poor's 500 and the Dow Jones industrial average, chemical loss spills as tracked by the US Coast Guard, Australian automobile accidents, and US hurricane losses. The classical approach estimates VaR inaccurately for 80% of the simulated data sets and for 60% of the real-world data sets studied in this work. Both the DPFS and the CPFS methodologies attain VaR estimates within 99% bootstrap confidence interval bounds for both simulated and real-world data. We provide a process flowchart for risk practitioners describing the steps for using the DPFS versus the CPFS methodology for VaR estimation in real-world loss datasets.