To enable targeted therapies and enhance medical decision-making, biomarkers are increasingly used as screening and diagnostic tests. When using quantitative biomarkers for classification purposes, this often implies that an appropriate cutoff for the biomarker has to be determined and its clinical utility must be assessed. In the context of drug development, it is of interest how the probability of response changes with increasing values of the biomarker. Unlike sensitivity and specificity, predictive values are functions of the accuracy of the test, depend on the prevalence of the disease and therefore are a useful tool in this setting. In this paper, we propose a Bayesian method to not only estimate the cutoff value using the negative and positive predictive values, but also estimate the uncertainty around this estimate. Using Bayesian inference allows us to incorporate prior information, and obtain posterior estimates and credible intervals for the cut-off and associated predictive values. The performance of the Bayesian approach is compared with alternative methods via simulation studies of bias, interval coverage and width and illustrations on real data with binary and time-to-event outcomes are provided.