Major precipitation accumulations associated with severe flooding (100s of mm) appear to be becoming more frequent in various locations around the world, and observations indicate that through the late 20th Century the magnitude of major accumulations increased. The Community Earth System Model (CESM) Large Ensemble is used to predict and understand future changes in frequency of these devastating events. The unique ability of a large ensemble is to aggregate the members to form a data period of several hundred years, so that exceptionally rare events, e.g., the 100-year accumulation, can be calculated, along with the future increase in their occurrence. Accumulation sizes are binned by the average recurrence interval (ARI) at each grid point, with ARI ranging up to 100 years, for the current and projected late 21st Century climates separately. For all ARIs, the frequency of exceedance of the given accumulation size increases in the future climate almost everywhere, in particular for the greatest accumulations, with the 100-year accumulations becoming about 3 times more frequent, averaged over the global land area, and about 10 times more frequent in parts of the tropics. Precipitation accumulation is approximately equal to the product of moisture, mass convergence, and duration, so that increases in frequency of a given accumulation size can be broken down into the increases due to each factor individually. For relatively small accumulation sizes (<1-year ARI), increases are almost precisely predicted by moisture increases, but for greater accumulation sizes (>10-year ARI), moisture is insufficient to explain the projected increase in frequency. In parts of the midlatitudes, duration increases are important for >10-year events, responsible for making events almost 1.5 times more frequent, but moisture is the most important factor. In the tropics and monsoon regions, increasing mass convergence, the dynamic influence, also impacts the >10-year events, responsible by itself for an approximate doubling of frequency, comparable to that of moisture increase. These projections represent a serious threat to flood risk around the world, so that the changes to duration and mass-convergence should be better understood in order to more accurately predict the future frequency of these major accumulations.