The Two-Dimensional Non-Homogeneous Poisson Process Based On Extreme Value Theory for Value at Risk

Abstract

Estimating Value at Risk (VaR) is an important aspect in management and mitigation of risk for institutions, individual investors, and markets. Extreme events have in the past caused significant financial losses, sometimes leading to market crashes that have had a profoundly negative impact on many investors, companies, institutions, and governments. Hence, there is a need to accurately model and predict extreme events. The Extreme Value Theory which applies the Generalized Extreme Value Distribution and the Generalized Pareto Distribution offers a robust framework for modeling extreme events and tail risks. This study focused on the peak over the threshold method, where a two-dimensional nonhomogeneous Poisson process model based on extreme value theory was employed, and the model's parameters are linear functions of the interest rates and volatility. Volatility and interest rates have had a significant impact on investment returns and financial losses. The Nairobi Securities Exchange 20-share index and Central Bank of Kenya interest rates datasets were used, with daily observations spanning 10 years from January 2, 2014, to December 31, 2023. The maximum likelihood and optimization methods were employed to estimate the coefficients of the linear relationships between interest rates and volatility, as well as the location, scale, and shape parameters. The volatility variable was found to be positively related to all three model parameters, shape, scale, and location while interest rates were negatively related. Unlike the traditional model that assume risks are static, this approach assumes varying market risks and conditions with time. It becomes a reliable tool for markets experiencing frequent changes, which allows them to collect accurate data which would be challenging if the traditional method was used. Future studies should explore the non-linear associations and also other explanatory variables.