Extreme Value Theory and 2D Non-Homogeneous Poisson Models for Estimating Value at Risk

Abstract

Extreme financial events have historically led to substantial market disruptions and losses for investors, institutions, and governments. Traditional risk assessment tools, such as Value at Risk (VaR), often fail to accurately capture these rare but severe losses due to their reliance on normal distribution assumptions. This limitation has driven the adoption of Extreme Value Theory (EVT), which offers a more robust framework for modeling tail risk using the Generalized Extreme Value (GEV) and Generalized Pareto Distributions (GPD). This study addresses a critical gap in the literature by integrating EVT with a Two-Dimensional Non-Homogeneous Poisson Process (2D-NHPP), allowing the distributional parameters—location, scale, and shape—to vary over time as linear functions of market volatility and interest rates. Unlike most existing models that assume the independence of extreme events and static risk levels, the proposed framework dynamically captures both the frequency and severity of extreme returns in response to changing economic conditions. Using daily data from the Nairobi Securities Exchange (NSE) 20 Share Index and Central Bank of Kenya interest rates from 2014 to 2023, the model parameters were estimated using the Maximum Likelihood Estimation (MLE) method. The result shows that volatility increases all the three measures, meaning that there will be higher variability and likelihood of extreme losses, while, interest rate increases are found to decrease the tail risk. As shown in the case of VaR estimates, the proposed approach is more responsive and accurate as compared to traditional methods. The study also establishes that 2D-NHPP model developed from EVT is a more accurate and flexible model for risk evaluation in emergent markets. Governments and regulatory bodies should embrace this model in order to enhance risk modeling, stress testing and policy making for their monetary institutions. Further studies should extend the scope of independent variables and compare the model in various markets to increase its scope and accuracy.