  We construct a parametric SED model which is able to reproduce the average
observed SDSS-UKIDSS-WISE quasar colours to within one tenth of a magnitude
across a wide range of redshift $(0<z<5)$ and luminosity $(-22>M_i>-29)$. This
model is shown to provide accurate predictions for the colours of known quasars
which are less luminous than those used to calibrate the model parameters, and
also those at higher redshifts $z>5$. Using a single parameter, the model
encapsulates an up-to-date understanding of the intra-population variance in
the rest-frame ultraviolet and optical emission lines of luminous quasars. At
fixed redshift, there are systematic changes in the average quasar colours with
apparent i-band magnitude, which we find to be well explained by the
contribution from the host galaxy and our parametrization of the emission-line
properties. By including redshift as an additional free parameter, the model
could be used to provide photometric redshifts for individual objects. For the
population as a whole we find that the average emission line and host galaxy
contributions can be well described by simple functions of luminosity which
account for the observed changes in the average quasar colours across
$18.1<i_\textrm{AB}<21.5$. We use these trends to provide predictions for
quasar colours at the luminosities and redshifts which will be probed by the
Rubin Observatory LSST and ESA-Euclid wide survey. The model code is applicable
to a wide range of upcoming photometric and spectroscopic surveys, and is made
publicly available.
