Managing the Risk via the Chi-Squared Distribution in VaR and CVaR with the Use in Generalized Autoregressive Conditional Heteroskedasticity Model

This paper develops a framework for quantifying risk by integrating analytical derivations of Value at Risk (VaR) and Conditional VaR (CVaR) under the chi-squared distribution with empirical modeling via Generalized Autoregressive Conditional Heteroskedasticity (GARCH) processes. We first establish closed-form expressions for VaR and CVaR under the chi-squared distribution, leveraging properties of the inverse regularized gamma function and its connection to the quantile of the distribution. We evaluate the proposed framework across multiple time windows to assess its stability and sensitivity to market regimes. Empirical results demonstrate the chi-squared-based VaR and CVaR, when coupled with GARCH volatility forecasts, particularly during periods of heightened market volatility.