  We study the non-Hermitian Anderson model on the ring. We provide the exact
rate of decay of the sensitivity of the eigenvalues to the non-Hermiticity
parameter $g$, on the logarithmic scale, as the Lyapunov exponent minus the
non-Hermiticity parameter. Namely, for $0 < g < \gamma(\lambda_{0})$ we show
that $-\frac{1}{n}\log|\lambda_{g}-\lambda_{0}|\sim \gamma(\lambda_{0})-g$ and
that the eigenvalue remains real for all such $g$. This provides an alternative
proof to that of Goldsheid and Sodin that the perturbed eigenvalue remains real
and specifies the exact rate at which the eigenvalue is exponentially close to
the unperturbed eigenvalue.
