Actuarial Notes

ARCH Model

An ARCH model (autoregressive conditionally heteroskedastic model) lets the variance of the error term in a time series model change over time: the variance of today's shock, given the past, is a linear function of the squares of recent shocks. It describes volatility clustering, the tendency of financial returns to have large moves, of either sign, followed by more large moves and quiet periods followed by quiet periods. The GARCH model generalizes it by adding past conditional variances to the equation; Frees (Regression Modeling with Actuarial and Financial Applications, Chapter 9) treats both.

\sigma_t^2 = \mathrm{Var}\left(\varepsilon_t \mid \varepsilon_{t-1}, \varepsilon_{t-2}, \ldots\right)

\sigma_t^2 = \omega + \gamma_1\varepsilon_{t-1}^2 + \cdots + \gamma_p\varepsilon_{t-p}^2

\sigma_t^2 = \omega + \sum_{j=1}^{p}\gamma_j\varepsilon_{t-j}^2 + \sum_{j=1}^{q}\delta_j\sigma_{t-j}^2

\mathrm{Var}(\varepsilon_t) = \frac{\omega}{1 - \sum_{j}\gamma_j - \sum_{j}\delta_j}

  • Symbols: \varepsilon_t is the shock in the mean model at time t, such as a return less its mean, or the error of an autoregressive model or a regression. \sigma_t^2 is its conditional variance. The second block is ARCH(p), the third is GARCH(p, q), and the fourth is the unconditional variance (for ARCH, every \delta_j = 0). The conditions \omega > 0, \gamma_j \geq 0 and \delta_j \geq 0 keep every conditional variance positive.
  • Volatility clustering: The shocks \varepsilon_t are uncorrelated, so their autocorrelation function looks like that of white noise, but their squares are positively autocorrelated, because a large \varepsilon_{t-1}^2 raises \sigma_t^2. Significant autocorrelation in the squared residuals of a fitted mean model is the sign that an ARCH term is needed. Even with normal shocks, the unconditional distribution has fatter tails than the normal.
  • Stationarity: The process is weakly stationary when \sum\gamma_j + \sum\delta_j < 1, and its unconditional variance is then the fourth block. The variance is therefore constant over the long run and changes only conditionally, from period to period. A sum close to 1 means that a burst of volatility dies away slowly.
  • Forecasting: ARCH changes the width of a forecast interval, not its centre: the point forecast comes from the mean model, and the interval uses the forecast of \sigma_{t+1} rather than a constant standard deviation. For GARCH(1, 1) the gap between the forecast variance and the long-run variance shrinks by the factor \gamma_1 + \delta_1 each period ahead. This is why a GARCH-type model of equity returns gives wider intervals after a turbulent period than a constant-volatility model does (see Time Series Forecast).

Forecast Interval for a Monthly Equity Return The monthly return on an insurer's equity portfolio follows y_t = 0.008 + \varepsilon_t, with ARCH(1) errors \sigma_t^2 = 0.001 + 0.30\,\varepsilon_{t-1}^2. Last month's return was -9.2\%. Find the forecast standard deviation of next month's return and a 95% forecast interval, and compare them with the interval that uses the unconditional variance.

Answer Last month's shock is the return less its mean, \varepsilon_{t-1} = -0.092 - 0.008 = -0.100.

\begin{align*}
\sigma_t^2 &= 0.001 + 0.30(-0.100)^2 \\
&= 0.001 + 0.003 \\
&= 0.004 \\
\sigma_t &= 0.0632
\end{align*}

The 95% interval is 0.008 \pm 1.96(0.0632) = 0.008 \pm 0.124, or -11.6\% to 13.2\%. The unconditional variance is 0.001/(1 - 0.30) = 0.001429, a standard deviation of 0.0378, which gives 0.008 \pm 0.074, or -6.6\% to 8.2\%. After a large loss the model widens next month's interval by about two-thirds, while the point forecast stays at 0.8\%.

Persistence of a GARCH(1, 1) Shock Monthly returns on a balanced fund have GARCH(1, 1) errors with \omega = 0.00001, \gamma_1 = 0.08 and \delta_1 = 0.90. Last month's conditional variance was 0.0009 and its shock was 0.05. Check that the process is stationary, find its long-run variance, and forecast next month's conditional variance.

Answer The process is stationary because \gamma_1 + \delta_1 = 0.98 < 1.

\begin{align*}
\mathrm{Var}(\varepsilon_t) &= \frac{0.00001}{1 - 0.98} \\
&= 0.0005 \\
\sigma_t^2 &= 0.00001 + 0.08(0.05)^2 + 0.90(0.0009) \\
&= 0.00001 + 0.0002 + 0.00081 \\
&= 0.00102
\end{align*}

The forecast standard deviation is \sqrt{0.00102} = 3.19\%, against a long-run \sqrt{0.0005} = 2.24\%. Most of the forecast comes from last month's high variance through \delta_1, not from the shock itself. With persistence of 0.98, the excess over the long-run variance shrinks by only 2% a month, so it takes about 34 months to halve.

How it appears on the exam

  • SRM asks candidates to describe autoregressive conditionally heteroskedastic models alongside exponential smoothing and autoregressive models. Multiple-choice statements test what volatility clustering is, which variance changes over time (the conditional one), and the stationarity condition.
  • Calculations: A question gives fitted ARCH or GARCH coefficients and recent shocks, then asks for the next conditional variance, the long-run variance, or a forecast interval, under SRM's objective on predicted values and confidence intervals.
  • Choosing the model: A question may show residuals whose ACF is flat while the ACF of their squares is not, and ask which model addresses it.

Exam traps

  • Shock, not return: The equation uses \varepsilon_{t-1}^2, the squared deviation from the mean model. Squaring the return itself, y_{t-1}^2, is wrong whenever the mean is not zero.
  • Conditional or unconditional: An ARCH process is heteroskedastic only conditionally. Its unconditional variance is constant when it is stationary, so "ARCH implies non-stationarity" is false.
  • The stationarity condition: It is the sum of all the \gamma_j and \delta_j that must be below 1, not each coefficient separately. The long-run variance has \omega in the numerator and 1 - \sum\gamma_j - \sum\delta_j in the denominator.
  • Two inputs in GARCH: \gamma_1 multiplies last period's squared shock and \delta_1 multiplies last period's conditional variance. Swapping them, or putting last period's squared shock in both places, gives the wrong forecast.
  • Variance, then standard deviation: A forecast interval uses \sigma_{t+1}, the square root of the forecast variance. Multiplying 1.96 by the variance understates the interval badly.