Estimating Conditional Value at Risk from GARCH Standardized Residuals

# Conditional Value at Risk using GARCH models


# Conditional Value at Risk using GARCH models












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:

- $ES_q(Z) = ES_q(\sqrt{\varepsilon^2}-\sqrt{\sigma^2})$ or $ES_q(Z) = ES_q(\varepsilon)$ or $ES_q(Z) = ES_q(r)$?

- $ES_q(Z)$ is calculated for the entire time series or only for a sliding window?

- $\mu_t=\bar{r}$ 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 by Pleb (score 1, accepted)

https://quant.stackexchange.com/a/69603

#### 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).