Vast research on immigration lumps together native citizens' attitudes toward two different groups: the immigrant stock of non-naturalized resident aliens, and the immigrant flow, that is, the future arrival of foreigners seeking to enter and live in the country. Does popular opposition to immigration distinguish between the two, and if so, how? This article analyzes theoretically the reasons the stock and flow might induce different views among natives, and presents experimental evidence from the United States showing that natives are systematically more accepting of the former. The analysis indicates that this ‘stock premium’ partly stems from a sense of moral obligation toward people residing in the country. Replicating two widely cited experiments, the study shows that the stock–flow distinction has important implications for the interpretation of earlier findings on immigration attitudes, and for understanding voter preferences regarding policies designed to curtail immigration.