How do you forecast volatility multiple steps ahead using a GARCH(1,1) model?

Question

AcadiFi · Accepted Answer

Multi-step forecasting with GARCH(1,1) is a critical skill for FRM because risk managers need volatility estimates over holding periods longer than one day (e.g., 10-day VaR for regulatory capital).

## GARCH(1,1) Recap

The one-step-ahead variance forecast:

sigma^2_{t+1} = omega + alpha x epsilon^2_t + beta x sigma^2_t

Where alpha + beta < 1 for stationarity. The long-run (unconditional) variance is:

**V_L = omega / (1 - alpha - beta)**

## Multi-Step Forecast Formula

For h steps ahead:

**sigma^2_{t+h|t} = V_L + (alpha + beta)^{h-1} x (sigma^2_{t+1|t} - V_L)**

This is the mean-reverting forecast. As h grows, (alpha + beta)^{h-1} approaches zero, and the forecast converges to V_L.

## Worked Example

Briarwood Risk Advisors estimates a GARCH(1,1) for daily S&P 500 returns:
- omega = 0.000002
- alpha = 0.08
- beta = 0.90
- Today's conditional variance: sigma^2_{t+1|t} = 0.000320 (daily vol = 1.789%)

**Long-run variance:**
V_L = 0.000002 / (1 - 0.08 - 0.90) = 0.000002 / 0.02 = 0.000100
Long-run daily vol = sqrt(0.0001) = 1.00%

**Multi-step forecasts:**

| Horizon h | (alpha+beta)^{h-1} | sigma^2_{t+h} | Daily Vol |
|---|---|---|---|
| 1 | 1.000 | 0.000320 | 1.789% |
| 2 | 0.980 | 0.000316 | 1.777% |
| 5 | 0.922 | 0.000303 | 1.740% |
| 10 | 0.834 | 0.000283 | 1.683% |
| 20 | 0.665 | 0.000246 | 1.569% |
| 50 | 0.364 | 0.000180 | 1.342% |
| 100 | 0.133 | 0.000129 | 1.137% |

```mermaid
xychart-beta
    title "GARCH(1,1) Variance Forecast Convergence"
    x-axis ["1d", "5d", "10d", "20d", "50d", "100d", "200d"]
    y-axis "Daily Variance" 0.00008 --> 0.00035
    line [0.000320, 0.000303, 0.000283, 0.000246, 0.000180, 0.000129, 0.000104]
```

The forecast starts at 0.000320 (elevated) and gradually decays toward the long-run level of 0.000100.

## Aggregating to h-Day VaR

To compute the h-day variance (for VaR), sum the individual daily forecasts:

sigma^2_{h-day} = SUM_{i=1}^{h} sigma^2_{t+i|t}

This is more accurate than the naive sqrt(h) scaling rule, which assumes constant volatility.

**FRM exam tip:** Questions often test whether you recognize that GARCH multi-step forecasts converge to V_L, and that higher persistence (alpha + beta closer to 1) means slower convergence.

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How do you forecast volatility multiple steps ahead using a GARCH(1,1) model?

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