  We present calculations of the absorption spectrum of semiconductors and
insulators comparing various approaches: (i) the two-particle Bethe-Salpeter
equation of Many-Body Perturbation Theory; (ii) time-dependent
density-functional theory using a recently developed kernel that was derived
from the Bethe-Salpeter equation; (iii) a scheme that we propose in the present
work and that allows one to derive different parameter-free approximations to
(ii). We show that all methods reproduce the series of bound excitons in the
gap of solid argon, as well as continuum excitons in semiconductors. This is
even true for the simplest static approximation, which allows us to reformulate
the equations in a way such that the scaling of the calculations with number of
atoms equals the one of the Random Phase Approximation.
