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We report linear response properties of the recently proposed boron fullerenes [N. Gonzalez Szwacki et al., Phys. Rev. Lett., 2007, 98, 166804]: magnetic susceptibilities, static dipole polarizabilities and dynamical polarizabilities (i.e. optical and near ultraviolet absorption spectra), calculated from first principles within the (time-dependent) density-functional theory framework. We find that all clusters except B80 are diamagnetic. The strong cancellation between diamagnetic and paramagnetic currents in B80 leads to a very small value for its susceptibility that turns out to be slightly paramagnetic. Static polarizabilities increase linearly with the number of B atoms. Furthermore, the absorption spectrum of B80 is very different from the one of its carbon counterpart C60, exhibiting a low absorption threshold of about 1.5 eV and many peaks in the visible and near ultraviolet. This can be understood by the analysis of the wavefunctions involved in the low energy transitions.