  This paper derives the analytic form of the $h$-step ahead prediction density
of a GARCH(1,1) process under Gaussian innovations, with a possibly asymmetric
news impact curve. The contributions of the paper consists both in the
derivation of the analytic form of the density, and in its application to a
number of econometric problems. A first application of the explicit formulae is
to characterize the degree of non-Gaussianity of the prediction distribution;
for some values encountered in applications, deviations of the prediction
distribution from the Gaussian are found to be small, and sometimes not. the
Gaussian density as an approximation of the true prediction density. A second
application of the formulae is to compute exact tail probabilities and
functionals, such as the Value at Risk and the Expected Shortfall, that measure
risk when the underlying asset return is generated by a Gaussian GARCH(1,1).
This improves on existing methods based on Monte Carlo simulations and
(non-parametric) estimation techniques, because the present exact formulae are
free of Monte Carlo estimation uncertainty. A third application is the
definition of uncertainty regions for functionals of the prediction
distribution that reflect in-sample estimation uncertainty. These applications
are illustrated on selected empirical examples.
