  It is proven that the recently found, renormalization-group invariant sum
rule for the soft scalar masses in softly-broken N=1 supersymmetric
gauge-Yukawa unified theories can be extended to all orders in perturbation
theory. In the case of finite unified theories, the sum rule ensures the
all-loop finiteness in the soft supersymmetry breaking sector. As a byproduct
the exact $\beta$ function for the soft scalar masses in the
Novikov-Shifman-Vainstein-Zakharov (NSVZ) scheme for softly-broken
supersymmetric QCD is obtained. It is also found that the singularity appearing
in the sum rule in the NSVZ scheme exactly coincides with that which has been
previously found in a certain class of superstring models in which the massive
string states are organized into $N = 4$ supermultiplets.
