Only very recently our community has acknowledged that the benefit of Structural Health Monitoring (SHM) can be properly quantified using the concept of Value of Information (VoI). The VoI is the difference between the utilities of operating the structure with and without the monitoring system, usually referred to as preposterior utility and prior utility. In calculating the VoI, a commonly understood assumption is that all the decisions to concerning system installation and operation are taken by the same rational agent. In the real world, the individual who decides on buying a monitoring system (the owner) is often not the same individual (the manager) who will actually use it. Even if both agents are rational and exposed to the same background information, they may behave differently because of their different risk aversion. We propose a formulation to evaluate the VoI from the owner's perspective, in the case where the manager differs from the owner with respect to their risk prioritisation. Moreover, we apply the results on a real-life case study concerning the Streicker Bridge, a pedestrian bridge on Princeton University campus, in USA. This framework aims to help the owner in quantifying the money saved by entrusting the evaluation of the state of the structure to the monitoring system, even if the manager's behaviour toward risk is different from the owner's own, and so are his or her management decisions. The results of the case study confirm the difference in the two ways to quantify the VoI of a monitoring system.