ABSTRACT Cosmogenic neutrinos originate from interactions of cosmic rays propagating through the universe with cosmic background photons. Since both high-energy cosmic rays and cosmic background photons exist, the existence of high-energy cosmogenic neutrinos is certain. However, their flux has not been measured so far. Therefore, we calculated the flux of high-energy cosmogenic neutrinos arriving at the Earth on the basis of elaborate 4D simulations that take into account three spatial degrees of freedom and the cosmological time-evolution of the universe. Our predictions for this neutrino flux are consistent with the recent upper limits obtained from large-scale cosmic-ray experiments. We also show that the extragalactic magnetic field has a strong influence on the neutrino flux. The results of this work are important for the design of future neutrino observatories, since they allow to assess the detector volume and observation time that are necessary to detect high-energy cosmogenic neutrinos in the near future. An observation of such neutrinos would push multimessenger astronomy to hitherto unachieved energy scales.