  Pulsar timing arrays (PTAs) and the Laser Interferometer Space Antenna (LISA)
will open complementary observational windows on massive black-hole binaries
(MBHBs), i.e., with masses in the range $\sim 10^6 - 10^{10}\,$ M$_{\odot}$.
While PTAs may detect a stochastic gravitational-wave background from a
population of MBHBs, during operation LISA will detect individual merging
MBHBs. To demonstrate the profound interplay between LISA and PTAs, we estimate
the number of MBHB mergers that one can expect to observe with LISA by
extrapolating direct observational constraints on the MBHB merger rate inferred
from PTA data. For this, we postulate that the common signal observed by PTAs
(and consistent with the increased evidence recently reported) is an
astrophysical background sourced by a single MBHB population. We then constrain
the LISA detection rate, $\mathcal{R}$, in the mass-redshift space by combining
our Bayesian-inferred merger rate with LISA's sensitivity to spin-aligned,
inspiral-merger-ringdown waveforms. Using an astrophysically-informed formation
model, we predict a 95$\%$ upper limit on the detection rate of $\mathcal{R} <
134\,{\rm yr}^{-1}$ for binaries with total masses in the range $10^7 - 10^8\,$
M$_{\odot}$. For higher masses, i.e., $>10^8\,$ M$_{\odot}$, we find
$\mathcal{R} < 2\,(1)\,\mathrm{yr}^{-1}$ using an astrophysically-informed
(agnostic) formation model, rising to $11\,(6)\,\mathrm{yr}^{-1}$ if the LISA
sensitivity bandwidth extends down to $10^{-5}$ Hz. Forecasts of LISA science
potential with PTA background measurements should improve as PTAs continue
their search.
