Financial Risk Manager Part 1

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Explanation:

Explanation

Model Formulas:

EWMA: $\sigma_n^2 = \lambda \sigma_{n-1}^2 + (1-\lambda)r_{n-1}^2$
GARCH(1,1): $\sigma_n^2 = \omega + \alpha r_{n-1}^2 + \beta \sigma_{n-1}^2$

Given Conditions:

Both models have the same parameter on $\sigma_{n-1}^2$ (same $\beta$ )
$\sigma_{n-1}^2 = r_{n-1}^2$
Current variance is above long-run variance

Analysis:

In GARCH(1,1), the long-run variance is $\frac{\omega}{1-\alpha-\beta}$ . When current variance is above long-run variance, the GARCH model will forecast a lower variance than EWMA because:

GARCH has a mean-reversion component ( $\omega$ ) that pulls the forecast back toward the long-run average
EWMA has no mean-reversion - it's a weighted average that doesn't revert to any long-term level
With current variance above long-run level, GARCH's mean-reversion effect will reduce the forecast compared to EWMA

Therefore, GARCH(1,1) will forecast lower volatility than EWMA under these conditions.

Explanation:

Explanation

Model Formulas:

EWMA: $\sigma_n^2 = \lambda \sigma_{n-1}^2 + (1-\lambda)r_{n-1}^2$
GARCH(1,1): $\sigma_n^2 = \omega + \alpha r_{n-1}^2 + \beta \sigma_{n-1}^2$

Given Conditions:

Both models have the same parameter on $\sigma_{n-1}^2$ (same $\beta$ )
$\sigma_{n-1}^2 = r_{n-1}^2$
Current variance is above long-run variance

Analysis:

In GARCH(1,1), the long-run variance is $\frac{\omega}{1-\alpha-\beta}$ . When current variance is above long-run variance, the GARCH model will forecast a lower variance than EWMA because:

GARCH has a mean-reversion component ( $\omega$ ) that pulls the forecast back toward the long-run average
EWMA has no mean-reversion - it's a weighted average that doesn't revert to any long-term level
With current variance above long-run level, GARCH's mean-reversion effect will reduce the forecast compared to EWMA

Therefore, GARCH(1,1) will forecast lower volatility than EWMA under these conditions.

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The exponentially weighted moving average (EWMA) and the generalized autoregressive conditional heteroscedasticity (GARCH) are two well-recognized volatility models. Suppose we have a EWMA and a GARCH (1, 1). Both have the same parameter attached on the $\sigma_{n-1}^2$ , and $\sigma_{n-1}^2 = r_{n-1}^2$ . Further assume that $\sigma_{n-1}^2$ is currently above the long-run variance, which model will forecast a lower day $n$ volatility?

Exam-Like

The EWMA model.

33.3%

The GARCH (1, 1) model.

66.7%

The forecast is the same for both models.

Further information is required in order to make the comparison.