Asymmetries in the time-dependent rates of ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ and ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays are measured in a $pp$ collision data sample collected with the LHCb detector during LHC Run 1, corresponding to an integrated luminosity of $3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The asymmetries in effective decay widths between ${D}^{0}$ and ${\overline{D}}^{0}$ decays, sensitive to indirect $CP$ violation, are measured to be ${A}_{\mathrm{\ensuremath{\Gamma}}}({K}^{+}{K}^{\ensuremath{-}})=(\ensuremath{-}0.30\ifmmode\pm\else\textpm\fi{}0.32\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and ${A}_{\mathrm{\ensuremath{\Gamma}}}({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=\phantom{\rule{0ex}{0ex}}(0.46\ifmmode\pm\else\textpm\fi{}0.58\ifmmode\pm\else\textpm\fi{}0.12)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, where the first uncertainty is statistical and the second systematic. These measurements show no evidence for $CP$ violation and improve on the precision of the previous best measurements by nearly a factor of two.