  We characterize the fractional Dehn twist coefficient of a braid in terms of
a slope of the homogenization of the Upsilon function, where Upsilon is the
function-valued concordance homomorphism defined by Ozsv\'ath, Stipsicz, and
Szab\'o. We use this characterization to prove that $n$-braids with fractional
Dehn twist coefficient larger than $n-1$ realize the braid index of their
closure. As a consequence, we are able to prove a conjecture of Malyutin and
Netsvetaev stating that $n$-times twisted braids realize the braid index of
their closure. We provide examples that address the optimality of our results.
The paper ends with an appendix about the homogenization of knot concordance
homomorphisms.
