Based on Hill's condition and a field–fluctuation approach, a new derivation is given for the shift property of effective compliances for both planar and three–dimensional composites. The derived relations are exact, and hold for any kind of microstructures and anisotropy. To provide a link to the well–known shift property of Cherkaev, Lurie & Milton in plane elasticity, special reference is given to two–dimensional composites and voided materials with isotropic constituents, but covering both overall inplane isotropy and orthotropy. This method is substantially simpler than the stress–invariance approach commonly adopted in the literature and it provides a new means of addressing the shift characteristics of the effective compliances. By this approach, several universal relations governing the effective compliances of three–dimensional and two–dimensional composites are also found, to our knowledge, for the first time.