A software component should be trustworthy and behave in a secure manner as it will be reused many times. Despite extensive efforts, usually, it cannot be guaranteed that a developed software component is completely secure. Hence, its execution in the realworld needs to be monitored against its security specifications. Each time components are used to develop a component-based software (CBS), a new monitor has to be designed to observe the behavior of the CBS. This results in recurring costs as such monitors cannot be reused for other CBS. Moreover, development life cycle artifacts are usually not available when a pre-fabricated component is used to build a CBS. Given that, it is imperative that a specification-based security monitor is developed along with the monitored component (when all development artifacts are available) and is embedded in the component to increase the component's trustworthiness. In this paper, we identify the types of constraints that may be imposed by security specifications. These constraints should be taken into account while developing the software components and should also be monitored. Furthermore, we propose a design approach to develop components with built in monitors that are able to observe these security constraints. Components developed following this approach would be self-monitoring, promote greater reusability, and be more trustworthy. We evaluate our approach by analyzing the performance and design complexity of different versions of CBS. These versions are developed by following the traditional and proposed approaches for monitoring security aspects of CBS.