Value-at-Risk formula with GARCH-model

Question

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?

Answer 1

$Z_t$ 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 $\sigma_t=1$.

Answer 2

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.