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We present the first systematic physical modelling of the time-lag spectrabetween the soft (0.3-1 keV) and the hard (1.5-4 keV) X-ray energy bands, as afunction of Fourier frequency, in a sample of 12 active galactic nuclei whichhave been observed by XMM-Newton. We concentrate particularly on the negativeX-ray time-lags (typically seen above $10^{-4}$ Hz) i.e. soft band variationslag the hard band variations, and we assume that they are produced byreprocessing and reflection by the accretion disc within a lamp-post X-raysource geometry. We also assume that the response of the accretion disc, in thesoft X-ray bands, is adequately described by the response in the neutral ironline (Fe k$\alpha$) at 6.4 keV for which we use fully general relativisticray-tracing simulations to determine its time evolution. These responsefunctions, and thus the corresponding time-lag spectra, yield much morerealistic results than the commonly-used, but erroneous, top-hat models.Additionally we parametrize the positive part of the time-lag spectra(typically seen below $10^{-4}$ Hz) by a power-law. We find that thebest-fitting BH masses, M, agree quite well with those derived by othermethods, thus providing us with a new tool for BH mass determination. We findno evidence for any correlation between M and the BH spin parameter, $\alpha$,the viewing angle, $\theta$, or the height of the X-ray source above the disc,$h$. Also on average, the X-ray source lies only around 3.7 gravitational radiiabove the accretion disc and the viewing angles are distributed uniformlybetween 20 and 60 degrees. Finally, there is a tentative indication that thedistribution of spin parameters may be bimodal above and below 0.62.

General relativistic modelling of the negative reverberation X-ray time delays in AGN