We establish global well-posedness and asymptotic behavior of $BV$ solutions to a system of balance laws modeling traffic flow with nonconcave fundamental diagram. We prove the results by finding a convex entropy-entropy flux pair and verifying the Kawashima condition, the sub-characteristic condition, and the partial dissipative inequality in the framework of Dafermos. This problem is of specific interest since nonconcave fundamental diagrams arise naturally in traffic flow.