# Value-at-Risk formula with GARCH-model

I'm totally aware of that if we look at some loss process $$L_t$$, then $$\text{VaR}(\alpha)$$ is a quantile of the loss distribution. If we assume that $$L_t=-X_t$$ is the negative returns and they follow a GARCH-model, such that $$X_t=\sigma_t Z_t$$ where $$Z_t$$ is i.i.d. noise with mean 0 and variance 1, how do one prove the following formula: $$\text{VaR}_t(\alpha)=\sigma_t F_t^{-1}(\alpha)$$ where $$F_t^{-1}$$ is the inverse assumed distribution function for the noise process? I can't find the way through since I would expect $$F_t^{-1}(\alpha)$$ to be the inverse distribution function of $$\sigma_t Z_t$$. Which properties of the inverse distribution function are we using to get through?

$$Z_t$$ has some distribution with zero mean and unit variance. The $$\alpha$$-level quantile of $$Z_t$$ is $$F_{Z_t}^{-1}(\alpha)$$. If you scale the distribution of $$Z_t$$ by multiplying it by $$\sigma_t$$, you get that each quantile gets multiplied by $$\sigma_t$$ (recall that $$Z_t$$ has zero mean; otherwise an adjustment for a nonzero mean would be due). Your confusion likely stems from the following false intuition:

$$F_{Z_t}^{-1}$$ is the inverse assumed distribution function for the noise process <...> I would expect $$F_{Z_t}^{-1}(\alpha)$$ to be the inverse distribution function of $$\sigma_t Z_t$$.

(I have replaced $$F_t$$ in your notation by $$F_{Z_t}$$ when quoting.) The problem is that multiplication of $$Z_t$$ by $$\sigma_t$$ shifts the quantile; $$Z_t$$ and $$\sigma_t Z_t$$ do not have the same quantile unless $$\sigma_t=1$$.

In VaR it is the distribution function of your loss forecast. You make a forecast of volatility for time period $$t+h$$ as: $$\hat \sigma_{t+h}|I_t$$, which allows you to propose the distribution of losses $$\hat L_{t+h}|I_t$$. In particular for next period $$h=1$$ forecast $$\hat L_{t+1}|I_t$$ in GARCH it's very simple, since you assume the normal distribution of noise. Hence, $$F(L)$$ is the CDF of normal distribution $$\mathcal N(0,\hat \sigma^2_{t+1})$$. Notice that for $$t+1$$ in GARCH the volatility forecast is not stochastic, you know all the inputs at time $$t$$ to calculate $$\hat\sigma^2_{t+1}$$.

It gets a bit more complicated when you make $$h>1$$ steps ahead forecast. In this case the usual approach is to simulate volatility paths $$\hat\sigma_{t+i}$$ for $$i=1,\dots,h$$ recursively, you'll be sampling from $$r_i\sim\mathcal N(0,\hat\sigma^2_{t+i-1})$$ to get the next $$\hat\sigma_{t+i}$$ until reaching $$\hat\sigma_{t+h}$$. At which point you sample a loss $$L_{t+h}\sim\mathcal N(0,\hat\sigma^2_{t+h})$$. After repeating this in Monte Carlo setup, you get the set of $$\hat L_{t+h}$$ from which you can calculate VaR using a variety of techniques as simple as $$\alpha$$ quantiles.