To maximize their efficacy, cognitive radios (CRs) need to be able to cope with the constantly changing spectrum environment, evolving spectrum access policies, and a diverse array of network application requirements. Policy-based cognitive radios address these challenges by decoupling the spectrum access policies from device-specific implementations and optimizations. These radios can invoke situation-appropriate, adaptive actions based on policy specifications and the current spectrum environment. A policy-based CR has a reasoning engine called a policy reasoner. The primary task carried out by the policy reasoner is evaluating the transmission requests with respect to the spectrum policies. In this paper, we describe the design of a policy reasoner that processes ontology-based spectrum policies. The main advantage of using ontology-based policies is that the policy reasoner can understand and process any spectrum policies authored by any organization by relying on the spectrum ontologies. In our implementation, the spectrum ontology defines the various dynamic spectrum access (DSA) concepts, models the domain of DSA networks in a machine-understandable manner, and uses SWRL (Semantic Web Rule Language) rules to represent spectrum policies. Unfortunately, ontological reasoning needed to process ontology-based spectrum policies incurs greater computation overhead compared to non-ontological reasoning. This drawback can be a critical one as it can impede a CR from meeting its real-time performance requirements. We have carried out a number of experiments, using our implementation, to evaluate whether a radio controlled by ontology-based policies can meet its real-time performance requirements. Based on our experimental results, we propose a set of guidelines for the design of ontology-based spectrum access policies.