It is proven that the recently found, renormalization-group invariant
sum rule for the soft scalar masses in softly-broken N=l supersymmetri
c gauge-Yukawa unified theories can be extended to all orders in pertu
rbation theory. Ln the case of finite unified theories, the sum rule e
nsures the all-loop finiteness in the soft supersymmetry breaking sect
or. As a byproduct the exact beta function for the soft scalar masses
in the Novikov-Shifman-Vainstein-Zakharov (NSVZ) scheme for softly-bro
ken supersymmetric QCD is obtained. It is also found that the singular
ity appearing in the sum rule in the NSVZ scheme exactly coincides wit
h that which has been previously found in a certain class of superstri
ng models in which the massive string states are organized into N=4 su
permultiplets. (C) 1998 Elsevier Science B.V. All rights reserved.