We explore the sensitivity and completeness of long baseline interferometric observations for detecting unknown, faint companions around bright unresolved stars. We derive a linear expression for the closure phase signature of a faint companion in the high contrast regime (<0.1), and provide a quantitative estimation of the detection efficiency for the currently offered four-telescope configurations at the VLTI. The results are compared to the performances provided by linear and Y-shaped interferometric configurations in order to identify the ideal array. We find that all configurations have a similar efficiency in discovering companions wider than 10mas. Assuming a closure phase accuracy of 0.25deg, that is typical of state-of-the-art instruments, we predict a median dynamic range of up to six magnitudes when stacking observations obtained at five different hour angles. Surveying bright stars to search for faint companions can be considered as an ideal filler programme for modern interferometric facilities because that places few constraints on the choice of the interferometric configuration.
