On The Continuous Statistical Maxwell Distribution for Obtaining the Risk in Economy

Abstract

This paper introduces an approach to financial risk quantification by utilizing the Maxwell distribution as a foundation for deriving closed-form expressions of two essential risk measures: expected shortfall and value at risk. In contrast to classic solvers that predominantly depend on normality assumptions, the framework integrates these Maxwell-based formulations within a GARCH model structure, providing a theoretically grounded and computationally efficient alternative for risk assessment. The proposed methodology is empirically evaluated through forecasting on actual stock market data, demonstrating its practical effectiveness and robustness in capturing tail risk dynamics. This novelty stems from the fact that, unlike the Gaussian distribution, which systematically underestimates tail events due to its thin-tailed nature, the Maxwell distribution naturally accommodates heavier right tails and yields closed-form expressions for both VaR and ES. This provides a new analytical alternative for risk modeling beyond classical Gaussian-based frameworks.