Conditional Value at Risk using GARCH models

Question

In this paper: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjSlIHYnMj1AhWqNOwKHZfHDhkQFnoECAkQAQ&url=https%3A%2F%2Fwww.mdpi.com%2F2076-3387%2F9%2F2%2F40%2Fpdf&usg=AOvVaw12ONJTGtEwL7xIYfPqbhBh they use the following equation for the Conditional Value at Risk using GARCH: $$ ES_q(X_{t})=\mu_{t}+\sigma_{t}ES_q(Z) $$ where $\mu$ is derived from the average prediction models and $\sigma$ is estimated from the volatility prediction models. $ES_q^t(Z)$ is the Conditional Value at Risk of the standardized residuals. I don't understand how to calculate that. Suppose that:

• $r_t$ are the logarithmic returns $r_t = \ln S_{t+1} - \ln S_t$;
• $\varepsilon_t = r_t - \bar{r}$ are the residuals, where $\bar{r}$ is the average return;
• $\sigma_0^2 = \frac{\omega}{1-\alpha -\beta}$ and $\sigma_t^2 = \omega+\alpha \varepsilon_{t-1}^2+\beta \sigma_{t-1}^2$ is the conditional variance at time zero and at time $t$ respectively;
• $\sqrt{\varepsilon_t^2}$ is the realized volatility and $\sqrt{\sigma_t^2}$ is the "fitted" volatility;

Then I don't understand if:

1. $ES_q(Z) = ES_q(\sqrt{\varepsilon^2}-\sqrt{\sigma^2})$ or $ES_q(Z) = ES_q(\varepsilon)$ or $ES_q(Z) = ES_q(r)$?
2. $ES_q(Z)$ is calculated for the entire time series or only for a sliding window?
3. $\mu_t=\bar{r}$ or $\mu_t=$ average loss of the time series?

How can I calculate Conditional Value at Risk using a GARCH model given my conditions?

Answer 1

Answer to question 1:

As mentioned in the linked paper, you estimate the Expected shortfall (ES) or the Value-at-risk (VaR) on the standardized residuals $z_t$, which could be calculated on the basis of a distributional assumption eg. a Gaussian distribution or Student's t-distribution. Remember that the return-process is on the form:

$$r_t \vert \mathcal{F}_{t-1} = \mu_t + \sigma_t \cdot z_t, \quad z_t \overset{iid}{\sim} D(0,1),$$ where $z_t$ are the standardized residuals. With regards to the paper, this implies that you estimate expected shortfall as:

$$ \text{ES}^{\alpha}_t\left(\frac{r_t - \mu}{\sigma_t} \bigg\vert \mathcal{F}_{t-1}\right) = \text{ES}_{t\vert t-1}^\alpha\left(z_t\right), $$ where you have an estimate for $\sigma_t$ via your GARCH model and $\mu = \bar{r}$ is your estimated average returns.

As an alternative source of information, you can follow Kevin Sheppards' Econometrics Notes where he has an entire chapter dedicated to Value-at-risk and Expected shortfall including parametric-methods involving GARCH-dynamics (see Chapter 8).

Answer to question 2:

You can estimate expected shortfall on the entire sample-set, as you can with a GARCH model. However, we are often interested in the out-of-sample performance for our specified risk-measures. In that case, you forecast Expected shortfall (or VaR) by doing a 1-step ahead forecast of your GARCH model and back out the estimate for the Expected shortfall. Here, you can rely on a sliding window on the GARCH model to get your next conditional volatility estimate.

In the end, it will also be a good idea to evaluate the performance of your estimates for ES and VaR. You can find a sub-chapter about in the Econometric notes linked above (see chapter 8.5). Another reasonable good paper on VaR forecasting and comparison can be found here if you need some more inspiration.

Answer to question 3:

The sentence, "$\mu$ is defined from the average prediction models" used by the authors in the aforementioned paper, is misleading and I believe their intention was to infer that $\mu_t$ can follow a user-specified mean-model. This could be something a lá the Autoregressive model or simply the average returns $\bar{r}$ as you have described above (be aware that the loss series is just the negation of $r_t$, ie. multiplied by -1).