The standard general relativistic model of a razor-thin accretion disc around a black hole, developed by Novikov & Thorne (NT) in 1973, assumes the shear stress vanishes at the radius of the innermost stable circular orbit (ISCO) and that, outside the ISCO, the shear stress is produced by an effective turbulent viscosity. However, astrophysical accretion discs are not razor thin; it is uncertain whether the shear stress necessarily vanishes at the ISCO, and the magnetic field, which is thought to drive turbulence in discs, may contain large-scale structures that do not behave like a simple local scalar viscosity. We describe 3D general relativistic magnetohydrodynamic simulations of accretion discs around black holes with a range of spin parameters, and we use the simulations to assess the validity of the NT model. Our fiducial initial magnetic field consists of multiple (alternating polarity) poloidal field loops whose shape is roughly isotropic in the disc in order to match the isotropic turbulence expected in the poloidal plane. For a thin disc with an aspect ratio |h/r| ∼ 0.07 around a non-spinning black hole, we find a decrease in the accreted specific angular momentum of 2.9 per cent relative to the NT model and an excess luminosity from inside the ISCO of 3.5 per cent. The deviations in the case of spinning black holes are also of the same order. In addition, the deviations decrease with decreasing |h/r|. We therefore conclude that magnetized thin accretion discs in X-ray binaries in the thermal/high-soft spectral state ought to be well described by the NT model, especially at luminosities below 30 per cent of Eddington where we expect a very small disc thickness |h/r| ≲ 0.05. We use our results to determine the spin equilibrium of black hole accretion discs with a range of thicknesses and to determine how electromagnetic stresses within the ISCO depend upon black hole spin and disc thickness. We find that the electromagnetic stress and the luminosity inside the ISCO depend on the assumed initial magnetic field geometry. We consider a second geometry with field lines following density contours, which for thin discs leads to highly radially elongated magnetic field lines. This gives roughly twice larger deviations from NT for both the accreted specific angular momentum and the luminosity inside the ISCO. Lastly, we find that the disc's corona (including any wind or jet) introduces deviations from NT in the specific angular momentum that are comparable to those contributed by the disc component, while the excess luminosity of bound gas from within the ISCO is dominated by only the disc component. Based on these indications, we suggest that differences in results between our work and other similar work are due to differences in the assumed initial magnetic field geometry as well as the inclusion of disc gas versus all the gas when comparing the specific angular momentum from the simulations with the NT model.