GARCH(1,1) Conditional and Unconditional Variance

Time Series · Easy · Free problem

The GARCH(1,1) model specifies daily returns as $r_t = \sigma_t \varepsilon_t$ with $\varepsilon_t \sim N(0,1)$ i.i.d., and conditional variance:

$\sigma_t^2 = \omega + \alpha\, r_{t-1}^2 + \beta\, \sigma_{t-1}^2$

You are given parameters $\omega = 0.000002$, $\alpha = 0.08$, $\beta = 0.91$, yesterday's return $r_{t-1} = -0.03$, and yesterday's conditional variance $\sigma_{t-1}^2 = 0.0004$.

  1. Compute today's conditional variance $\sigma_t^2$ and daily volatility $\sigma_t$.
  1. Compute the unconditional (long-run) variance and annualized volatility.
  1. What is the persistence $\alpha + \beta$, and what does it imply about the half-life of variance shocks?

Hints

  1. Just plug the given values directly into the GARCH(1,1) recursion $\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2$.
  2. For the unconditional variance, set $\sigma^2 = \omega + (\alpha + \beta)\sigma^2$ and solve. The stationarity condition $\alpha + \beta < 1$ must hold.
  3. The half-life of a variance shock under GARCH is $\ln 2 / (-\ln(\alpha + \beta))$. With $\alpha + \beta = 0.99$, shocks are extremely persistent.

Worked Solution

How to Think About It: GARCH(1,1) is the workhorse volatility model in finance. It captures two stylized facts: volatility clusters (big moves follow big moves) and mean-reverts (vol eventually returns to a long-run level). The conditional variance equation is just an exponentially weighted moving average of past squared returns, with a pull toward the long-run variance. When you see a large return (like $-3\%$ here), the model ratchets up tomorrow's vol forecast. How much it ratchets up depends on $\alpha$ (reaction to new shocks) and how quickly it fades depends on $\beta$ (persistence of old variance).

Quick Estimate: Yesterday's vol was $\sqrt{0.0004} = 2\%$ and we saw a $-3\%$ move -- that's a

.5\sigma$ event. With $\alpha = 0.08$, the shock contribution is $0.08 \times 0.0009 = 0.000072$, which adds roughly $0.000072$ to the variance baseline of $0.000364$. So today's variance should be around $0.000438$, giving vol of about
.09\%$ -- a modest uptick.

Formal Solution:

Part 1: Today's conditional variance

$\sigma_t^2 = \omega + \alpha\, r_{t-1}^2 + \beta\, \sigma_{t-1}^2$

$= 0.000002 + 0.08 \times (-0.03)^2 + 0.91 \times 0.0004$

$= 0.000002 + 0.08 \times 0.0009 + 0.91 \times 0.0004$

$= 0.000002 + 0.000072 + 0.000364 = 0.000438$

Daily volatility: $\sigma_t = \sqrt{0.000438} \approx 0.02093 = 2.09\%$

Yesterday's vol was $\sqrt{0.0004} = 2.00\%$, so today's forecast is higher due to the large negative return.

Part 2: Unconditional (long-run) variance

For stationarity, require $\alpha + \beta < 1$. Here $0.08 + 0.91 = 0.99 < 1$. The unconditional variance satisfies $\bar{\sigma}^2 = \omega + \alpha \bar{\sigma}^2 + \beta \bar{\sigma}^2$, giving:

$\bar{\sigma}^2 = \frac{\omega}{1 - \alpha - \beta} = \frac{0.000002}{1 - 0.99} = \frac{0.000002}{0.01} = 0.0002$

Long-run daily vol: $\bar{\sigma} = \sqrt{0.0002} \approx 1.414\%$

Annualized vol: $\bar{\sigma}_{\text{ann}} = 1.414\% \times \sqrt{252} \approx 22.4\%$

Note: today's conditional variance $0.000438$ is well above the long-run level $0.0002$. The model predicts vol will gradually decay back toward

.41\%$ daily.

Part 3: Persistence and half-life

Persistence: $\alpha + \beta = 0.99$

This is very high persistence -- shocks to variance decay extremely slowly. The half-life of a variance shock is:

$t_{1/2} = \frac{\ln 2}{-\ln(\alpha + \beta)} = \frac{\ln 2}{-\ln(0.99)} \approx \frac{0.693}{0.01005} \approx 69 \text{ days}$

This means if today's variance spikes, it takes about 69 trading days (roughly 3 months) for the excess variance to decay by half.

Practical Interpretation: - The decomposition of variance into components: $\omega = 0.000002$ (constant floor, tiny), $\alpha r_{t-1}^2 = 0.000072$ (reaction to yesterday's shock, 16% of today's variance), $\beta \sigma_{t-1}^2 = 0.000364$ (inherited from past, 83% of today's variance). - The model is heavily persistence-driven ($\beta = 0.91$), with moderate shock reactivity ($\alpha = 0.08$). This is typical for equity indices. - With $\alpha + \beta = 0.99$ close to 1, this model is nearly IGARCH -- variance shocks are almost permanent.

Answer:

  1. $\sigma_t^2 = 0.000438$, daily vol $\approx 2.09\%$
  2. $\bar{\sigma}^2 = 0.0002$, long-run daily vol $\approx 1.41\%$, annualized $\approx 22.4\%$
  3. Persistence $= 0.99$, half-life $\approx 69$ trading days

Intuition

GARCH(1,1) is the minimum viable volatility model for most financial applications. The conditional variance equation is really just a weighted average: $\omega/(1-\alpha-\beta)$ pulls you toward the long-run variance, $\alpha$ controls how much yesterday's surprise matters, and $\beta$ controls inertia. A high $\alpha + \beta$ (close to 1) means the model has a long memory -- volatility regimes persist for months. This particular parameterization ($\alpha + \beta = 0.99$) is very typical for equity index returns, where vol clusters are well-documented.

The key practical takeaway: the unconditional variance gives you the long-run risk level, but the conditional variance is what matters for short-term risk management. Today's conditional vol is

.09\%$ (above the long-run
.41\%$) because of the recent $3\%$ move. If you're sizing positions or computing VaR, you should use the conditional variance, not the unconditional one. The unconditional variance is useful for calibration and for understanding where vol is heading in the long run.

Open the full interactive solver →