The ratio of branching fractions ${\cal{R}}(D^{*-})\equiv {\cal{B}}(B^0 \to D^{*-} τ^+ ν_τ)/{\cal{B}}(B^0 \to D^{*-} μ^+ν_μ)$ is measured using a data sample of proton-proton collisions collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3$~$fb$^{-1}$. The $τ$ lepton is reconstructed with three charged pions in the final state. A novel method is used that exploits the different vertex topologies of signal and backgrounds to isolate samples of semitauonic decays of $b$ hadrons with high purity. Using the $B^0 \to D^{*-}π^+π^-π^+$ decay as the normalization channel, the ratio ${\cal{B}}(B^0 \to D^{*-} τ^+ ν_τ)/{\cal{B}}(B^0 \to D^{*-}π^+π^-π^+)$ is measured to be $1.97 \pm 0.13 \pm 0.18$, where the first uncertainty is statistical and the second systematic. An average of branching fraction measurements for the normalization channel is used to derive ${\cal{B}}(B^0 \to D^{*-} τ^+ ν_τ) = (1.42 \pm 0.094 \pm 0.129 \pm 0.054) \%$, where the third uncertainty is due to the limited knowledge of ${\cal{B}}(B^0\to D^{*-}π^+π^-π^+)$. A test of lepton flavor universality is performed using the well-measured branching fraction ${\cal{B}}(B^0 \to D^{*-} μ^+ν_μ)$ to compute ${\cal{R}}(D^{*-}) = 0.291 \pm 0.019 \pm 0.026 \pm 0.013$, where the third uncertainty originates from the uncertainties on ${\cal{B}}(B^0 \to D^{*-}π^+π^-π^+)$ and ${\cal{B}}(B^0 \to D^{*-} μ^+ν_μ)$. This measurement is in agreement with the Standard Model prediction and with previous measurements.