Analytic arguments have suggested that the existence of high-redshift quasars challenges hierarchical galaxy formation theories in which the majority of galaxies form late. We address this issue by using a series of numerical simulations of the biased cold dark matter model with and without gas dynamics. Potential quasar sites are identified with massive density peaks collapsing to form massive self-gravitating cold gas clouds. The Press–Schechter formalism, using a Gaussian filter with δc = 1.61, correctly predicts the number of high-mass dark haloes. Very efficient cooling and angular momentum transfer to the dark halo, however, allow the gas to become much denser than naive predictions would suggest. We find that there are > 103 times the required number of potential quasar sites at z = 4.5; at z = 8 there is already at least one site in our simulation volume of 106 Mpc3. Thus sites for quasars could easily form at significantly higher redshifts than those of observed quasars, even in models where most galaxy formation occurs late.