An SF340 Captain reported overtaking slower traffic 500 ft below. In response to a TCAS RA he climbed to avoid traffic and was questioned by ATC.

Narrative

[We were] level at 5;000 feet in cruise flight on vectors. The Controller issued a warning for traffic at approximately 15 miles and 500 feet below traveling in our direction. The First Officer acknowledged the call and we be begun to monitor the TCAS and look for the traffic visually. We noticed the traffic on the TCAS approximately 10 miles ahead at 500 feet below our altitude. We then noticed the traffic on the TCAS had climbed approximately 200 feet and still traveling in our direction. Soon after; the Controller queried the traffic regarding altitude and informed the traffic to maintain the assigned altitude (4;500 feet). We continued to monitor the traffic as it continued to approach. As the traffic approached the TCAS indication turned yellow and stated TRAFFIC. We still had not acquired the traffic visually. The traffic indication on the TCAS turned red and the CLIMB; CLIMB warning sounded. I immediately disengaged the autopilot and began a climb as the First Officer informed ATC that we were responding to a Resolution Alert. At approximately 6;300 feet the TCAS stated MONITOR VERTICAL SPEED at which time I leveled off. The Controller queried why we had climbed so high and we informed him that we responded to an RA and it was for safety. We descended back to our assigned altitude and the flight continued uneventfully to the destination.