# Square Information Ratio

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 !

• I’m with you. They’re the essentially same thing. Jun 2 '19 at 4:19

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:

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

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

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