What is filtered historical simulation and how does it fix the problems of standard HS?

Filtered Historical Simulation (FHS) is a hybrid method that combines the distributional flexibility of historical simulation with the dynamic volatility modeling of GARCH. It addresses the key weakness of standard HS — the inability to reflect current market conditions.

The Problem with Standard HS:

Standard HS treats all past returns equally. During a calm period, the VaR is low because the historical window is calm. When volatility suddenly spikes, HS is slow to react because the new volatile data is mixed with months of calm data.

FHS — The Two-Step Approach:

Step 1: Filter (GARCH)

Fit a GARCH(1,1) model to each risk factor's returns. Extract the standardized residuals:

z_t = r_t / sigma_t (where sigma_t is the GARCH conditional volatility)

These z_t are approximately i.i.d. (independent, identically distributed) — they've had the volatility clustering 'filtered out.'

Step 2: Simulate (Historical)

Resample the standardized residuals (with replacement), then scale them by the CURRENT conditional volatility:

r_simulated = sigma_{T+1} x z_i (where z_i is a randomly drawn past residual)

This produces scenarios that:

• Reflect today's volatility level (through sigma_{T+1})
• Preserve the empirical distribution shape (fat tails, skewness from historical z_t)
• Don't require a distributional assumption for the residuals

Example — Ashford Trading, equity portfolio:

Why FHS Is Superior:

1. Responsive: Uses GARCH to immediately adjust to current volatility
2. Non-parametric tails: Historical residuals capture fat tails without assuming a specific distribution
3. Correlation dynamics: If you use multivariate GARCH, correlations also update dynamically
4. No ghost effects: The GARCH filter removes the window-dependent jumps of standard HS
5. Better backtesting: Produces fewer clustered VaR exceptions

Limitations:

• Requires GARCH estimation for each risk factor (computational burden)
• GARCH model misspecification can introduce its own errors
• The standardized residuals may not be perfectly i.i.d.
• More complex to implement and explain

FRM Exam Focus:

• Know the two-step process: GARCH filter, then bootstrap residuals
• Understand why FHS adapts faster than standard HS
• Compare FHS to fully parametric and standard HS in terms of responsiveness, distributional assumptions, and backtesting performance

Master FHS and other VaR methods in our FRM Part II course.