Square Information Ratio

Question

I have read the following sentence : " The information ratio measures the active management opportunities, and the square of the information ratio indicates our ability to add value " ( In the Grinold's book about Active Portfolio Management).

I do not understand the second part. For me, the information ratio or its square measure the same thing, the possibility of extracting value from the market, on a different scale.

Is the square of the IR like the $R^2$ in statistics for linear regression ( with the fact that quadratic error = variance + square of the biais) ?

Thank you for your help !

Answer 1

You can define information ratio on ex-ante basis, so you will be using the expected values, and this definition is called alpha omega:

$IR=\frac{\alpha}{\omega}$

Let’s represent the risk reversion by $\lambda$ then the value add is:

$VA=\alpha-\lambda \omega^2$

Substituting for alpha:

$VA=IR \omega -\lambda \omega^2$

Now the value add is maximised at:

$\frac{d IR}{d\omega}=IR-2\lambda\omega=0$

$\omega=\frac{IR}{2\lambda}$

And if you substitute this into the value add equation, you get your result:

$VA=IR \omega -\lambda \omega^2$

$VA=IR \frac{IR}{2\lambda}-\lambda \frac{IR^2}{4\lambda^2}$

$VA=\frac{IR^2}{4\lambda}$

It is very well explained in section 4.2 of this article:

S. L. Blatt: An In-Depth Look at the Information Ratio (2004)

https://web.wpi.edu/Pubs/ETD/Available/etd-0824104-155216/unrestricted/Blatt.pdf