A sample of 103 quasars from the Large Bright Quasar Survey (LBQS) has been observed with the VLA at 8.4 GHz to study the evolution of the radio luminosity distribution and its dependence on absolute magnitude. Radio data from pointed observations are now available for 359 of the 1055 LBQS quasars. The radio-loud fraction is constant at ~10% over the absolute magnitude range -28 <= MB <= -23, and it rises to ~20% (log R > 1) or ~35% (log L > 25) at the brightest absolute magnitudes in the sample. This nearly flat distribution differs markedly from those of the optically selected Palomar-Green (PG) Bright Quasar Survey and the X-ray selected Extended Medium Sensitivity Survey (EMSS), both of which have lower radio-loud fractions for absolute magnitudes fainter than MB = -24 and higher fractions at brighter magnitudes. The reason for the high radio-loud fraction at bright absolute magnitudes in the PG, compared to the LBQS and other optically selected quasar surveys, is unknown. The trend of increasing radio-loud fraction with absolute magnitude in the EMSS is due at least in part to a correlation between X-ray and radio luminosity. Combining the LBQS data with radio studies of high-redshift quasars leads to the conclusion that the radio-loud fraction in optically selected quasars does not appear to evolve significantly, aside from a modest increase at z ~1, from z = 0.2 to redshifts approaching 5, a result that is contrary to previous studies which found a decrease in radio-loud fraction with increasing redshift by comparing the low-z fraction in the PG to higher redshift samples.