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

ArXiv ID: 2505.05646 “View on arXiv”

Authors: Xin Tian

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.

Keywords: Value-at-Risk, Filtered Historical Simulation, GARCH, risk modeling, tail risk estimation, Market Risk

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced statistical models like GARCH and filters like FHS, which involve complex time-series mathematics, but also implements empirical backtesting with breach frequency analysis and visual diagnostics, making it both mathematically dense and data/implementation-heavy.

  flowchart TD
    A["Research Goal: Comparative Evaluation of VaR Models<br>HS, GARCH-N, GARCH-FHS"] --> B["Data Input<br>Daily Returns for In-sample & Forecasting"]
    B --> C["Methodology: VaR Computation (5%)<br>1. Historical Simulation (HS)<br>2. GARCH-Normal (GARCH-N)<br>3. Filtered Hist. Sim. (GARCH-FHS)"]
    C --> D["Assessment: Accuracy & Breach Frequency<br>Compare Empirical Breaches vs 5%"]
    D --> E["Multi-Period Analysis (5-day)<br>Simulate Cumulative Returns<br>Compute VaR & Expected Shortfall"]
    E --> F["Key Findings & Outcomes"]
    F --> F1["HS & GARCH-N: Severe Miscalibration<br>High Breach Rates"]
    F --> F2["GARCH-FHS: Superior Performance<br>Aligns with Theoretical Levels"]
    F --> F3["GARCH-N: Underestimates Tail Risk<br>(Gaussian Assumption)"]
    F --> F4["GARCH-FHS: Robust Tail Estimates<br>Recommended for Risk Management"]