The sub-population of quasars reddened by intrinsic or intervening clouds of dust are known to be underrepresented in optical quasar surveys. By defining a complete parent sample of the brightest and spatially unresolved quasars in the COSMOS field, we quantify to which extent this sub-population is fundamental to our understanding of the true population of quasars. By using the available multiwavelength data of various surveys in the COSMOS field, we built a parent sample of 33 quasars brighter than J = 20 mag, identified by reliable X-ray to radio wavelength selection techniques. Spectroscopic follow-up with the NOT/ALFOSC was carried out for four candidate quasars that had not been targeted previously to obtain a 100% redshift completeness of the sample. The population of high AV quasars (HAQs), a specific sub-population of quasars selected from optical/near-infrared photometry, some of which were shown to be missed in large optical surveys such as SDSS, is found to contribute 21%+9-5 of the parent sample. The full population of bright spatially unresolved quasars represented by our parent sample consists of 39%+9-8 reddened quasars defined by having AV > 0.1, and 21%+9-5 of the sample having E(B-V) > 0.1 assuming the extinction curve of the Small Magellanic Cloud. We show that the HAQ selection works well for selecting reddened quasars, but some are missed because their optical spectra are too blue to pass the g-r color cut in the HAQ selection. This is either due to a low degree of dust reddening or anomalous spectra. We find that the fraction of quasars with contributing light from the host galaxy, causing observed extended spatial morphology, is most dominant at z ≲ 1. At higher redshifts the population of spatially unresolved quasars selected by our parent sample is found to be representative of the full population of bright active galactic nuclei at J< 20 mag. This work quantifies the bias against reddened quasars in studies that are based solely on optical surveys. Partly based on observations made with the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.