  Large-area sky surveys show that massive galaxies undergo at least one major
merger in a Hubble time. Ongoing pulsar timing array (PTA) experiments are
aimed at measuring the gravitational wave (GW) emission from binary
supermassive black holes (SMBHs) at the centres of galaxy merger remnants. In
this paper, using the latest observational estimates for a range of galaxy
properties and scaling relations, we predict the amplitude of the GW background
generated by the binary SMBH population. We also predict the numbers of
individual binary SMBH GW sources. We predict the characteristic strain
amplitude of the GW background to lie in the range $5.1\times10^{-16}<A_{\rm
yr}<2.4\times10^{-15}$ at a frequency of $(1\,{\rm yr})^{-1}$, with 95%
confidence. Higher values within this range, which correspond to the more
commonly preferred choice of galaxy merger timescale, will fall within the
expected sensitivity ranges of existing PTA projects in the next few years. In
contrast, we find that a PTA consisting of at least 100 pulsars observed with
next-generation radio telescopes will be required to detect continuous-wave GWs
from binary SMBHs. We further suggest that GW memory bursts from coalescing
SMBH pairs are not viable sources for PTAs. Both the GW background and
individual GW source counts are dominated by binaries formed in mergers between
early-type galaxies of masses $\gtrsim5\times10^{10}M_{\odot}$ at redshifts
$\lesssim 1.5$. Uncertainties in the galaxy merger timescale and the SMBH mass
- galaxy bulge mass relation dominate the uncertainty in our predictions.
