  Luminous AGNs are usually selected by their non-stellar colours or their
X-ray emission. Colour selection cannot be used to select low-luminosity AGNs,
since their emission is dominated by the host galaxy. Objects with low X-ray to
optical ratio escape even the deepest X-ray surveys performed so far. In a
previous study we presented a sample of candidates selected through optical
variability in the Chandra Deep Field South, where repeated optical
observations were performed for the STRESS supernova survey. We obtained new
optical spectroscopy for a sample of variability selected candidates with the
ESO NTT telescope. We analysed the new spectra, together with those existing in
the literature and studied the distribution of the objects in U-B and B-V
colours, optical and X-ray luminosity, and variability amplitude. A large
fraction (17/27) of the observed candidates are broad-line luminous AGNs,
confirming the efficiency of variability in detecting quasars. We detect: i)
extended objects which would have escaped the colour selection and ii) objects
of very low X-ray to optical ratio. Several objects resulted to be
narrow-emission line galaxies where variability indicates nuclear activity,
while no emission lines were detected in others. Some of these galaxies have
variability and X-ray to optical ratio close to active galactic nuclei, while
others have much lower variability and X-ray to optical ratio. This result can
be explained by the dilution of the nuclear light due to the host galaxy. Our
results demonstrate the effectiveness of supernova search programmes to detect
large samples of low-luminosity AGNs. A sizable fraction of the AGN in our
variability sample had escaped X-ray detection (5/47) and/or colour selection
(9/48). Spectroscopic follow-up to fainter flux limits is strongly encouraged.
