Exact tests of equivalence and efficacy with a non-zero lower bound based on two independent binomial proportions for comparative trials are proposed. These exact tests are desirable for studies with small sample sizes. They generalize classical methods to include testing of null hypotheses of prespecified differences and can be used to demonstrate a new treatment's efficacy or its equivalence to a standard treatment. The proposed exact tests use unconditional distributions of the test statistics. Variances of test statistics are estimated via a constrained maximum likelihood method (Farrington and Manning). Data from oncology and vaccine clinical trials are used to illustrate the exact tests.