Let $f$ be an orientation-preserving homeomorphism of the euclidean plane $\mathbb R^{2}$ that has a periodic point $z^{*}$ of period $q\geq 2$. We prove the existence of a fixed point $z$ such that the linking number between $z^{*}$ and $z$ is different from zero. That means that the rotation number of $z^{*}$ in the annulus $\mathbb R^{2}\setminus\{z\}$ is a non-zero element of $\mathbb R/\mathbb Z$. This gives a positive answer to a question asked by John Franks.