In the book titled "Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk" by Grinold & Kahn, the information ratio is defined as "the ratio of the expected annual residual return to the annual volatility of the residual return". The key concept is "residual return" in the definition, which is risk adjusted return, i.e. the so-called "alpha". However, in other books, articles and blogs on the Internet, the Information Ratio is always calculated as the ratio of expected annual active return to the annual volatility of the active return. I'm so confused about this.
So why are there two different definitions? Which one is more proper?
P.S., The active return is the portfolio return minus benchmark return.