  The accurate description of the optical spectra of insulators and
semiconductors remains an important challenge for time-dependent
density-functional theory (TDDFT). Evidence has been given in the literature
that TDDFT can produce bound as well as continuum excitons for specific
systems, but there are still many unresolved basic questions concerning the
role of dynamical exchange and correlation (xc). In particular, the role of the
long spatial range and the frequency dependence of the xc kernel $f_{\rm xc}$
for excitonic binding are still not very well explored. We present a minimal
model for excitons in TDDFT, consisting of two bands from a one-dimensional
Kronig-Penney model and simple approximate xc kernels, which allows us to
address these questions in a transparent manner. Depending on the system, it is
found that adiabatic xc kernels can produce a single bound exciton, and
sometimes two bound excitons, where the long spatial range of $f_{\rm xc}$ is
not a necessary condition. It is shown how the Wannier model, featuring an
effective electron-hole interaction, emerges from TDDFT. The collective,
many-body nature of excitons is explicitly demonstrated.
