  Given a string $S$ of length $n$, its maximal unbordered factor is the
longest factor which does not have a border. In this work we investigate the
relationship between $n$ and the length of the maximal unbordered factor of
$S$. We prove that for the alphabet of size $\sigma \ge 5$ the expected length
of the maximal unbordered factor of a string of length~$n$ is at least $0.99 n$
(for sufficiently large values of $n$). As an application of this result, we
propose a new algorithm for computing the maximal unbordered factor of a
string.
