A new information-theoretic model is proposed for underlay-based cognitive radio (CR), which imposes rate limitation on the secondary user (SU), whereas the traditional systems impose either interference or transmit power limitations. The channel is modeled as a twin-user interference channel constituted by the primary user (PU) and the SU. The achievable rate of the SU is derived based on the inner bound formulated by Han and Kobayashi, where the PU achieves the maximum attainable rate of the single-user point-to-point link.We show that it is necessary for the SU to allocate its full power for the "public" message that can be decoded both by the SU and by the PU. We also demonstrate that it is optimal for the PU to allocate its full power for the "private" message that can only be decoded by the PU if the level of interference imposed by the PU on the SU is "ergodically strong." Similarly, it is optimal for the PU to allocate its full power for the public message that can be decoded both by the SU and PU if this interference is "ergodically weak." These findings suggest that this power allocation is independent of the level of interference imposed by the SU on the PU. Furthermore, the achievable rate is analyzed as a function of the average level of interference. An interesting observation is that if the level of interference imposed by the SU on the PU is "ergodically weak," the achievable rate becomes a monotonically increasing function of this interference, and it is independent of the level of interference imposed by the PU on the SU. Furthermore, we analyze the realistic imperfect channel estimation scenario and demonstrate that the channel estimation errors will not affect the optimal nature of the SU's power allocation. © 2015 IEEE.