  We present a pipeline that allows recovering reliable information for all
four Stokes parameters with high accuracy. Its novelty relies on the treatment
of the instrumental effects already prior to the computation of the Stokes
parameters contrary to conventional methods, such as the M\"uller matrix one.
The instrumental linear polarization is corrected across the whole telescope
beam and significant Stokes $Q$ and $U$ can be recovered even when the recorded
signals are severely corrupted. The accuracy we reach in terms of polarization
degree is of the order of 0.1-0.2 %. The polarization angles are determined
with an accuracy of almost 1$^{\circ}$. The presented methodology was applied
to recover the linear and circular polarization of around 150 Active Galactic
Nuclei. The sources were monitored from July 2010 to April 2016 with the
Effelsberg 100-m telescope at 4.85 GHz and 8.35 GHz with a cadence of around
1.2 months. The polarized emission of the Moon was used to calibrate the
polarization angle. Our analysis showed a small system-induced rotation of
about 1$^{\circ}$ at both observing frequencies. Finally, we identify five
sources with significant and stable linear polarization; three sources remain
constantly linearly unpolarized over the period we examined; a total of 11
sources have stable circular polarization degree $m_\mathrm{c}$ and four of
them with non-zero $m_\mathrm{c}$. We also identify eight sources that maintain
a stable polarization angle over the examined period. All this is provided to
the community for polarization observations reference. We finally show that our
analysis method is conceptually different from the traditionally used ones and
performs better than the M\"uller matrix method. Although it was developed for
a system equipped with circularly polarized feeds it can easily be modified for
systems with linearly polarized feeds as well.
