Recently, Altug and Wagner  posed a question regarding the optimal behavior of the probability of error when channel coding rate converges to the capacity sufficiently slowly. They gave a sufficient condition for the discrete memoryless channel (DMC) to satisfy a moderate deviation property (MDP) with the constant equal to the channel dispersion. Their sufficient condition excludes some practically interesting channels, such as the binary erasure channel and the Z-channel. We extend their result in two directions. First, we show that a DMC satisfies MDP if and only if its channel dispersion is nonzero. Second, we prove that the AWGN channel also satisfies MDP with a constant equal to the channel dispersion. While the methods used by Altug and Wagner are based on the method of types and other DMC-specific ideas, our proofs (in both achievability and converse parts) rely on the tools from our recent work  on finite-blocklength regime that are equally applicable to non-discrete channels and channels with memory.