Advanced Time Series: GARCH and Volatility Models
Model time-varying volatility using GARCH models for actuarial and financial applications on Exam MAS-II.
Volatility Clustering and Motivation
Financial and insurance time series often exhibit volatility clustering: periods of large price changes tend to be followed by more large changes, and periods of small changes tend to be followed by more small changes. Standard time series models like ARIMA assume constant conditional variance, which fails to capture this phenomenon. ARCH and GARCH models explicitly model the time-varying conditional variance, producing more accurate forecasts of future volatility and more realistic confidence intervals for return predictions.
Volatility clustering is visible in many actuarial datasets. Stock market returns, interest rate changes, and even insurance loss ratios show periods of elevated variability followed by calmer periods. Traditional actuarial models that assume constant variance throughout the observation period underestimate risk during volatile periods and overestimate it during calm periods. GARCH models provide a dynamic framework that adapts to changing volatility conditions.
ARCH Models
The Autoregressive Conditional Heteroscedasticity (ARCH) model, introduced by Engle, specifies the conditional variance as a function of past squared residuals. The ARCH(q) model defines sigma_t^2 = alpha_0 + alpha_1 * epsilon_{t-1}^2 + alpha_2 * epsilon_{t-2}^2 + ... + alpha_q * epsilon_{t-q}^2, where epsilon_t = sigma_t * z_t and z_t is a standard normal (or other standardized) innovation. The parameters must satisfy alpha_0 > 0 and alpha_i >= 0 for all i to ensure the conditional variance is positive.
The ARCH(1) model is the simplest case: sigma_t^2 = alpha_0 + alpha_1 * epsilon_{t-1}^2. If yesterday's residual was large (whether positive or negative), today's conditional variance is high, producing volatility clustering. The unconditional (long-run) variance is alpha_0 / (1 minus alpha_1), which exists and is finite when alpha_1 < 1. In practice, ARCH models often require many lags q to capture the persistence of volatility, leading to the GARCH generalization.
GARCH Models
The Generalized ARCH model (GARCH) adds lagged conditional variances to the ARCH specification. The GARCH(p,q) model is sigma_t^2 = alpha_0 + sum from i=1 to q of alpha_i * epsilon_{t-i}^2 + sum from j=1 to p of beta_j * sigma_{t-j}^2. The GARCH(1,1) model, sigma_t^2 = alpha_0 + alpha_1 * epsilon_{t-1}^2 + beta_1 * sigma_{t-1}^2, is by far the most widely used specification and captures most volatility dynamics in financial data.
The GARCH(1,1) parameters have clear interpretations. The constant alpha_0 sets the baseline variance level. The coefficient alpha_1 determines how strongly new information (yesterday's shock) affects today's volatility. The coefficient beta_1 measures the persistence of past volatility. The sum alpha_1 + beta_1 controls overall persistence: when this sum is close to 1, volatility shocks decay slowly and the model exhibits long memory in volatility. Stationarity of the variance process requires alpha_1 + beta_1 < 1, and the unconditional variance is alpha_0 / (1 minus alpha_1 minus beta_1).
Volatility Forecasting
One of the primary uses of GARCH models is forecasting future volatility. The one-step-ahead forecast from a GARCH(1,1) model is sigma_{t+1}^2 = alpha_0 + alpha_1 * epsilon_t^2 + beta_1 * sigma_t^2, which uses today's residual and today's conditional variance to predict tomorrow's variance. Multi-step forecasts can be computed recursively: sigma_{t+h}^2 converges to the unconditional variance alpha_0 / (1 minus alpha_1 minus beta_1) as the horizon h increases.
The speed of convergence to the long-run variance depends on alpha_1 + beta_1. When this sum is close to 1 (for example, 0.98), short-term volatility forecasts differ substantially from the long-run average, and the forecasts revert slowly. When the sum is smaller (for example, 0.85), forecasts revert to the long-run average more quickly. This behavior directly affects how actuaries project future risk: a high-persistence GARCH model implies that a recent period of high volatility will continue to influence risk estimates for many periods.
Asymmetric GARCH Extensions
Standard GARCH models treat positive and negative shocks symmetrically: a large positive return and a large negative return of equal magnitude produce the same increase in conditional variance. In practice, financial markets exhibit the "leverage effect," where negative returns increase volatility more than positive returns of the same magnitude. Several GARCH extensions capture this asymmetry.
The EGARCH (Exponential GARCH) model specifies ln(sigma_t^2) as a function of both the standardized residual epsilon_{t-1}/sigma_{t-1} and its absolute value. The log specification ensures positivity of the conditional variance without parameter constraints. The GJR-GARCH model adds an indicator term: sigma_t^2 = alpha_0 + (alpha_1 + gamma * I(epsilon_{t-1} < 0)) * epsilon_{t-1}^2 + beta_1 * sigma_{t-1}^2, where gamma > 0 captures the additional volatility from negative shocks. The IGARCH model (Integrated GARCH) sets alpha_1 + beta_1 = 1, implying that volatility shocks have a permanent effect. This is used when the data shows extremely persistent volatility.
Actuarial Applications
GARCH models have several important actuarial applications. In variable annuity pricing, the guarantee cost depends critically on the volatility of the underlying fund returns. Using a GARCH model instead of a constant-volatility model produces more accurate hedging strategies and reserve estimates, especially after periods of market turbulence. Interest rate volatility modeled with GARCH affects the valuation of interest-sensitive insurance products like universal life and fixed annuities.
In enterprise risk management, GARCH models capture the dynamic nature of market risk. Dynamic financial analysis (DFA) simulations that incorporate GARCH-type volatility produce more realistic projections of asset returns and surplus variability than those assuming constant volatility. Loss ratio volatility can also be modeled with GARCH if there is evidence of volatility clustering in underwriting results, though this is less common than financial market applications.
Common Pitfalls
Students often confuse the conditional variance (which varies over time in a GARCH model) with the unconditional variance (which is constant for a stationary GARCH process). The conditional variance at time t depends on the observed data up to time t, while the unconditional variance is a long-run average. Another common error is forgetting the stationarity condition alpha_1 + beta_1 < 1 when interpreting parameter estimates. If this condition is violated, the unconditional variance does not exist and the model is non-stationary.
When forecasting, remember that multi-step GARCH volatility forecasts converge to the unconditional variance. Do not extrapolate one-step-ahead forecasts linearly over multiple periods. Also note that GARCH models the conditional variance, not the conditional mean. If the mean process has ARMA dynamics, you need both an ARMA model for the conditional mean and a GARCH model for the conditional variance.