We present a theoretical model that describes electrical spin-detection at a ferromagnet/semiconductor interface. We show that the sensitivity of the spin detector has strong bias dependence which, in the general case, is dramatically different from that of the tunneling current spin polarization. We show that this bias dependence originates from two distinct physical mechanisms: 1) the bias dependence of tunneling current spin polarization, which is of microscopic origin and depends on the specific properties of the interface, and 2) the macroscopic electron spin transport properties in the semiconductor. Numerical results show that the magnitude of the voltage signal can be tuned over a wide range from the second effect which suggests a universal method for enhancing electrical spin-detection sensitivity in ferromagnet/semiconductor tunnel contacts. Using first-principles calculations we examine the particular case of a Fe/GaAs Schottky tunnel barrier and find very good agreement with experiment. We also predict the bias dependence of the voltage signal for a Fe/MgO/GaAs tunnel structure spin detector.