Most previous researches on the rate analysis of the power allocation (PA) schemes in two-way decode-and- forward (DF) relay systems typically focus on maximizing the instantaneous objective rates. However, the ergodic rate analysis of the optimized rates is not well considered. In this paper, we investigate the ergodic rates of the PA schemes in a two-way DF relay system, where the physical-layer network coding (PNC) protocol is adopted. Specifically, under a total power constraint, we consider three heuristic PA schemes, in which the optimization objectives are: 1) maximizing the sum rate; 2) maximizing the sum rate with an additional rate fairness constraint; and 3) maximizing the minor rate of the two directions of the system. The cumulative distribution functions (CDFs) of the equivalent system SNR are approximately obtained and the analytical expressions for the calculations of the ergodic rates are theoretically derived. Numerical results show that the derived analytical rates are converged to the simulated results in high SNR regions.