Let $X$ be a proper variety over a henselian discretely valued field. An
important obstruction to the existence of a rational point on $X$ is the index,
the minimal positive degree of a zero cycle on $X$. This paper introduces a new
invariant, the specialization index, which is a closer approximation of the
existence of a rational point. We provide an explicit formula for the
specialization index in terms of an $snc$-model, and we give examples of curves
where the index equals one but the specialization index is different from one,
and thus explains the absence of a rational point. Our main result states that
the specialization index of a smooth, proper, geometrically connected
$\mathbb{C}((t))$-variety with trivial coherent cohomology is equal to one.