A risk manager at a large investment bank is assessing the predictive capabilities of two different GARCH(1,1) specifications for a high-volatility proprietary equity trading book. The goal is to understand the long-term implications for risk capital allocation, specifically regarding the unconditional variance and the persistence of volatility shocks.
Model A parameters:
$\omega_A = 0.000002$
$\alpha_A = 0.08$
$\beta_A = 0.89$
Current daily variance $\sigma_{t-1,A}^2 = 0.00005$
Recent daily return $r_{t-1,A} = 0.005$
Model B parameters:
$\omega_B = 0.000001$
$\alpha_B = 0.05$
$\beta_B = 0.93$
Current daily variance $\sigma_{t-1,B}^2 = 0.000045$
Recent daily return $r_{t-1,B} = 0.006$
Which of the following statements most accurately describes the long-term volatility characteristics implied by Model A compared to Model B?
A.Model A implies a higher long-term unconditional variance, and Model B exhibits greater persistence of volatility shocks.
B.Model B implies a higher long-term unconditional variance, and Model B exhibits greater persistence of volatility shocks.
C.Model A implies a higher long-term unconditional variance, and Model A exhibits greater persistence of volatility shocks.
D.Model B implies a lower long-term unconditional variance, but its higher $\beta$ parameter suggests slower mean reversion of current volatility compared to Model A.
Rationale:
The unconditional (long-run) variance for a GARCH(1,1) model is calculated as $\sigma^2 = \frac{\omega}{1 - \alpha - \beta}$, provided that $\alpha + \beta < 1$. The persistence of volatility shocks is measured by the sum $\alpha + \beta$. A higher sum indicates greater persistence, meaning shocks take longer to decay and volatility reverts to its long-run mean more slowly.
Comparing the results:
Model A's long-term unconditional variance (0.00006667) is higher than Model B's (0.00005).
Model B's persistence (0.98) is greater than Model A's (0.97).
Therefore, the statement that Model A implies a higher long-term unconditional variance and Model B exhibits greater persistence of volatility shocks correctly reflects these findings. The 'current daily variance' and 'recent daily return' values provided are irrelevant for calculating long-term unconditional variance or persistence and serve as distractors.
The option suggesting Model B implies a higher long-term unconditional variance incorrectly compares the unconditional variances. While Model B has a smaller $\omega$, its denominator $(1 - \alpha - \beta)$ is also smaller, resulting in a lower long-term variance overall compared to Model A.
The option stating that Model A implies a higher long-term unconditional variance but also greater persistence for Model A incorrectly compares the persistence values. Model A's persistence (0.97) is lower than Model B's (0.98).
The option indicating that Model B implies a lower long-term unconditional variance but its higher $\beta$ parameter suggests slower mean reversion of current volatility compared to Model A is partially correct but contains a common misinterpretation. While Model B does have a lower long-term unconditional variance and higher persistence, attributing slower mean reversion solely to a higher $\beta$ is inaccurate. Persistence, which dictates the speed of mean reversion, is determined by the sum $\alpha + \beta$, not just $\beta$ in isolation.
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