Finite Unified Theories (FUTs) are N = 1 supersymmetric Grand Unified Theories (GUTs) which can be made finite to all‐loop orders, based on the principle of reduction of couplings, and therefore are provided with a large predictive power. Confronting the predictions of SU(5) FUTs with the top and bottom quark masses and other low‐energy experimental constraints a light Higgs‐boson mass in the range Mh ∼ 121–126 GeV was predicted, in striking agreement with the recent discovery of a Higgs‐like state around ∼ 125.5 GeV at ATLAS and CMS. Furthermore the favoured model, a finiteness constrained version of the MSSM, naturally predicts a relatively heavy spectrum with coloured supersymmetric particles above ∼ 1.5 TeV, consistent with the non‐observation of those particles at the LHC. Restricting further the best FUT's parameter space according to the discovery of a Higgs‐like state and B‐physics observables we find predictions for the rest of the Higgs masses and the supersymmetric particle spectrum.