How to calculate Information Ratio?

Question

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.

Answer 1

I know Information Ratio to be:

$IR = {E[R_p - R_b] \over \sqrt{var[R_p - R_b]}}$

Meaning the ratio ($IR$) is equal to the average excess return (Portfolio return - Benchmark return) divided by the standard deviation of excess returns relative to a benchmark. You could also say that this is the active return divided by the tracking error. Note that this formula can be misleading by way of producing negative numbers if/when a fund/manager uses leverage and creates excessive alpha compared to a benchmark. For this reason, you should be sure that the fund/manager is strongly correlated to the benchmark.

Variations of IR include using Jensens's Alpha as the numerator as well as the Geometric Information Ratio. Geometric IR is preferred for managers that use leverage.

References:

Informatio Ratio from Wikipedia

Jensen's Alpha

Geometric IR

Thoughts On Grinold & Kahn’s “Fundamental Law Of Active Management”