Access, Success, and Persistence in Reshaping Education Act of 2019 or the ASPIRE Act This bill revises the Pell Grant Program to provide incentives for institutions of higher education (IHEs) to increase (1) enrollment of students who receive Pell Grants, and (2) graduation rates of their students. Specifically, the Department of Education (ED) must rank IHEs participating in student financial aid programs. First, ED must rank IHEs based on the percentage of first-time, full-time students who receive Pell Grants and are enrolled at the IHEs. The IHEs in the bottom 5% must improve the enrollment of those students by specific deadlines or pay a fee-per-student penalty. Second, ED must rank IHEs based on the percentage of first-time, full-time students who enroll at the school and graduate within six years. The bill gives IHEs in the bottom 5% the option of receiving funding to improve student graduation rates. Those IHEs must improve those rates by specific deadlines or pay a penalty. ED must use the fees to fund (1) grant programs for improving graduation rates; and (2) a bonus program for providing nonfinancial rewards to IHEs with proven records of affordability, access, and success for low- income, moderate-income, or working class students.