We apply GARCH $(1,1)$ model

$y_{t+1}=c+{}_{t+1}lo{n}_{t+1}$

${}_{t+1}^2={}_0+{}_1{}_t^{2}lo{n}_t^2+{}_t^2$

${}_{t}N(0,1)$

for the S&P $500$ index and we estimate parameters using $1$ year historical daily returns by

today. The estimated parameters are as follows $c=0,{}_0=0\mathrm{.}01,{}_1=0\mathrm{.}1,=0\mathrm{.}8.$

a$.$ If ${}_t=1$ and ${}_t=0\mathrm{.}4({}_t^2=0\mathrm{.}16),$ then what is the tomorrow conditional variance

${}_{t+1}^2?$

b$.$ The same condition as # a$,$ what is the tomorrow $99\%-$VaR$(Va{R}_{1\%})$ under the todays

information.