A crucial issue facing polymer-based solar cells is how to manage the energetics of the polymer/ fullerene blends to maximize short-circuit current density and open-circuit voltage at the same time and thus the power conversion efficiency. Here we demonstrate that the use of a naphthobisoxadiazole-based polymer with a narrow bandgap of 1.52 eV leads to high open-circuit voltages of approximately 1V and high-power conversion efficiencies of similar to 9% in solar cells, resulting in photon energy loss as small as similar to 0.5 eV, which is much smaller than that of typical polymer systems (0.7-1.0 eV). This is ascribed to the high external quantum efficiency for the systems with a very small energy offset for charge separation. These unconventional features of the present polymer system will inspire the field of polymer-based solar cells towards further improvement of power conversion efficiencies with both high short-circuit current density and open-circuit voltage.