Comparative Evaluation of VaR Models: Historical Simulation, GARCH-Based Monte Carlo, and Filtered Historical Simulation

TL;DR

This paper compares three VaR models, finding that GARCH-FHS outperforms others in accurately capturing tail risks and providing reliable risk forecasts, especially under fat-tailed distributions.

Contribution

It provides a comprehensive empirical evaluation of VaR models, highlighting the superior calibration and robustness of GARCH-FHS over traditional methods.

Findings

GARCH-FHS aligns closely with theoretical breach rates

HS and GARCH-N models show severe miscalibration

GARCH-FHS offers more conservative and robust tail risk estimates

Abstract

This report presents a comprehensive evaluation of three Value-at-Risk (VaR) modeling approaches: Historical Simulation (HS), GARCH with Normal approximation (GARCH-N), and GARCH with Filtered Historical Simulation (FHS), using both in-sample and multi-day forecasting frameworks. We compute daily 5 percent VaR estimates using each method and assess their accuracy via empirical breach frequencies and visual breach indicators. Our findings reveal severe miscalibration in the HS and GARCH-N models, with empirical breach rates far exceeding theoretical levels. In contrast, the FHS method consistently aligns with theoretical expectations and exhibits desirable statistical and visual behavior. We further simulate 5-day cumulative returns under both GARCH-N and GARCH-FHS frameworks to compute multi-period VaR and Expected Shortfall. Results show that GARCH-N underestimates tail risk due to its reliance on the Gaussian assumption, whereas GARCH-FHS provides more robust and conservative tail estimates. Overall, the study demonstrates that the GARCH-FHS model offers superior performance in capturing fat-tailed risks and provides more reliable short-term risk forecasts.