  We calculate the $D^0$-$\overline{D}^0$ mixing parameter $y$ in the
factorization-assisted topological-amplitude (FAT) approach, considering
contributions from $D^{0}\to PP$, $PV$, and $VV$ modes, where $P$ ($V$) stands
for a pseudoscalar (vector) meson. The $D^{0}\to PP$ and $PV$ decay amplitudes
are extracted in the FAT approach, and the $D^{0}\to VV$ decay amplitudes with
final states in the longitudinal polarization are estimated via the parameter
set for $D^{0}\to PV$. It is found that the $VV$ contribution to $y$, being of
order of $10^{-4}$, is negligible, and that the $PP$ and $PV$ contributions
amount only up to $y_{PP+PV}=(0.21\pm0.07)\%$, a prediction more precise than
those previously obtained in the literature, and much lower than the
experimental data $y_{\rm exp}=(0.61\pm0.08)\%$. We conclude that $D^{0}$ meson
decays into other two-body and multi-particle final states are relevant to the
evaluation of $y$, so it is difficult to understand it fully in an exclusive
approach.
