# How can I calculate time-varying Value at Risk (VaR) and Conditional VaR for return series?

I am working on ABT index and I calculated the return series. Also, I intend to fit a GARCH(1,1) model to the return series and then calculate the VaR and CVaR as the following image:

I used many approaches and as far as I know, the VaR and CVaR are one value per series. I do not understand the time-varying VaR/CVaR. Any help would be appreciated.

A GARCH model specifies the entire conditional distribution of a time series $$\{x_t\}$$ at each time point. E.g. an ARMA(p,q)-GARCH(r,s) model looks like this: \begin{aligned} x_t &= \mu_t + u_t, \\ \mu_t &= c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \theta_1 u_{t-1} + \dots + \theta_q u_{t-q}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $$D$$ is some probability distribution with zero mean and unit variance. Thus the distribution of $$x_t$$ is $$D(0,1)$$ scaled by $$\sigma_t$$ and shifted by $$\mu_t$$.
• Thank you so much. I still do not understand what this means ...from the distribution of the time point...? Can you give me some idea about how to perform it by simulation please? Commented May 6, 2023 at 8:57
• @Afshin, take the distribution $D(0,1)$, simulate $n$ realizations, multiply them by $\sigma_t$ and add $\mu_t$. That will correspond to a simulation of $n$ realizations of $x_t$. Commented May 6, 2023 at 9:56