We investigate theoretically the combined effect of fiber chromatic dispersion and self-phase modulation (SPM) on multi-channel subcarrier multiplexed (SCM) optical transmission systems in terms of the detected RF carrier power and SPM-induced power gain after transmission over single-mode fiber (SMF) links. According to the calculated power gain due to the SPM effect at the transmission distance of P3dB using the detected radio-frequency (RF) carrier power after photo-detection, the power gain is significantly degraded with large optical modulation index (OMI), small SCM channel spacing, and large fiber launching power because of the increased interaction between subcarrier channels. The nonlinear phase shift due to linear and nonlinear fiber characteristics is investigated to explain these results in detail. The numerical simulation results show that the OMI per SCM channel has to be smaller than 10 % for the fiber launching power of 10 dBm to guarantee prevention of SPM-induced power gain degradation below 0.5 dB for the SCM system with the channel spacing of 100 MHz. This result is expected to be utilized for the optical transmission systems using the SCM technology in future radio-over-fiber (RoF) networks.