How the Information Ratio Squared Relates to Optimized Value Added

Summary

The document explains why the square of an information ratio can relate to value added in an active portfolio. It defines the ex-ante information ratio as expected alpha divided by active risk, then models value added as alpha minus a penalty proportional to the square of that risk. The penalty parameter represents risk aversion.

Substituting the information ratio into the value-added expression and maximizing with respect to active risk yields an optimal risk level proportional to the information ratio. Substituting that choice back into the objective makes maximum value added proportional to the squared information ratio, scaled by risk aversion. This gives the square a distinct interpretation within this optimization setup; it is not presented as equivalent to the regression coefficient of determination. The result depends on the stated quadratic risk penalty and ex-ante assumptions, so it is not a universal identity for every use of the information ratio. The answer points readers to a longer treatment but supplies no empirical test.

Key ideas

• The ex-ante information ratio is expected alpha divided by active risk.
• The value-added model subtracts a risk-aversion-weighted quadratic penalty from alpha.
• Maximizing the model sets optimal active risk proportional to the information ratio.
• Under these assumptions, maximum value added scales with the squared information ratio and inversely with risk aversion.
• This interpretation does not make the squared ratio equivalent to regression R-squared.

Tags

# Square Information Ratio


# 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 !

## Answer by Magic is in the chain (score 2, accepted)

https://quant.stackexchange.com/a/45880

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.