VaR in case of ARMA-GARCH?

Question

How do I calculate the VaR in case of using an ARMA-GARCH approach? I am not good at time series, so I am more or less confused with the different possible notations of an ARMA-GARCH process. I hope the notation I use is correct.

Suppose I use the following AR(1)-GARCH(1,1):

\begin{align*} r_t=\delta + \phi_1 r_{t-1} + z_t\\ z_t = \sigma_t \epsilon_t\\ \epsilon_t \text{ is } N(0,1)\\ \sigma_t^2=\alpha_0+\alpha_1 z_{t-1}^2+\beta \sigma_{t-1}^2 \end{align*}

I mean, I would have to calculate a quantile at timepoint t. Normally in case of a more simpler approach: $r_t=\mu + \sigma_t*\epsilon_t$ and the $\epsilon_t$ is N(0,1) this would give $VaR=\mu + \hat{\sigma_t}*z_\alpha$ where $z_\alpha$ is the quantile of the standard normal. How to do this in case of the ARMA-GARCH approach? I now have no $\mu$, but an AR process?

I read some papers, they all give more or less the ARMA-GARCH formula and say that they generate the forecast, but they give no formula, how to do this.

Answer 1

To estimate $VaR$ of the one-step-ahead return (which is what you seem to need to do) you should do the following ($E_t$ denotes the conditional expectation given the values of the time series up to and including the moment $t$ have been observed):

1. Compute the forecast of the mean. In your case of $AR(p)$ process you should do the following: $$r_{t+1|t} = E_t(r_{t+1}) = E_t( \delta + \sum^p_{k=1} \phi_k r_{t+1-k} + \sigma_{t+1} \epsilon_{t+1})$$ which reduces to $r_{t+1|t} = \delta + \sum^p_{k=1} \phi_k r_{t+1-k}$, since $\epsilon_{t+1}$ has zero mean, and $\sigma^2_{t+1}$ is known given the observations up to $t$. In your case $$r_{t+1|t} = \delta + \phi_1 r_t$$

2. The volatility forecast is $\sigma^2_{t+1|t} = \alpha_0 + \alpha_1 \hat{z}^2_t + \beta \sigma^2_t$ where $\hat{z}_t = r_t - \delta - \phi_1 r_{t-1}$, and can be obtained similarly.

3. Now to get $VaR$ at the moment $t+1$, given the history up to $t$, you should just take the quantile of the standard normal, multiply it by the square root of $\sigma^2_{t+1|t}$ and then add $r_{t+1|t}$: $$VaR = r_{t+1|t} + \sqrt{\sigma^2_{t+1|t}} z_\alpha$$