Physical layer network coding (PNC) was first introduced by Zhang et al. for two-way relay channels (TWRCs). By utilizing the PNC, we can complete two-way communications within two time slots, instead of three time slots required in non- PNC systems. Recently, the upper and lower bounds for a bit error rate (BER) of PNC have been analyzed for fading channels. In this paper, we derive an exact BER of the PNC for the TWRC over fading channels. We determine decision regions based on the nearest neighbor rule and partition them into several wedge areas to apply the Craig's polar coordinate form for computing the BER. We confirm that our derived analysis accurately matches with the simulation results.