While Bacardi is still the top U.S. rum by volume market share, sales have been sagging. The brand's share fell to 34.3% last year from 35.4% in 2013, according to Euromonitor International. That is off the recent peak share of 39.7% in 2005, according to Euromonitor. The No. 2 U.S. rum brand, Diageo's Captain Morgan, has also lost share in recent years as smaller brands gain ground.