Information Coefficient and Breadth in the Fundamental Law of Active Management

Summary

The document asks how a squared information coefficient can equal a sum of squared correlations when deriving the Fundamental Law of Active Management. The question distinguishes an aggregate measure from correlations associated with individual signals and notes that the book’s notation does not make the information coefficient’s definition clear.

The sole response recommends a later paper that reexamines the law and asserts that the traditional breadth term is flawed, proposing instead that information ratio depends on the mean information coefficient divided by its standard deviation. The response offers no derivation or supporting evidence in the document, so its strong conclusion should be treated as a claim to investigate rather than an established result. The exchange is useful for highlighting the importance of definitions, assumptions about signals, and the distinction between a theoretical relationship and its empirical interpretation. It does not resolve the original equation directly.

Key ideas

• The question concerns the relationship between a sum of squared signal correlations and a squared information coefficient.
• The source text does not clearly define the information coefficient used in the proof.
• The response challenges the traditional role of breadth in the Fundamental Law.
• The alternative information-ratio expression is asserted without a derivation in this exchange.
• The cited critique would need to be consulted to assess the claim and its assumptions.

Tags

# Confusion about the proof of Fundamental Law of Active Management in Grinold & Kahn (2000)


# Confusion about the proof of Fundamental Law of Active Management in Grinold & Kahn (2000)












I'm reading Grinold & Kahn (2000) for the proof of the Fundamental Law of Active Management.

I can't understand formula (6A.20) on page 168, which says:

> Finally, by assuming that all the signals have equal value, $$\zeta_b^2=\rho^2=IC^2 \qquad \text{(6A.20)}$$

According to (6A.18) at the end of page 167, $\zeta_b^2$ is sum of square of correlation coefficients: $$\zeta_b^2=\sum_{n=1}^{N}\rho_{n,b}^2 \qquad \text{(6A.18)}$$

where $\rho_{n,b}=corr(x_n,y_b)$. (I can't provide the whole proof here. You may read the book for more detail about $x_n$ and $y_n$.)

How can this sum of square of correlation coefficients be equal to $IC^2$(square of information coefficient)?

I think the $IC$ should be $corr(\theta_n,z_b)$ or $corr(x_n, y_b)$(I'm not sure about this, because the author doesn't give a clear formula for $IC$). But (6A.20) can't be derived from either of these two possibilities. Could anyone give some explanations about formula (6A.20)?

## Answer by zack young (score -1)

https://quant.stackexchange.com/a/73768

Please read the paper by Ding and Martin (2017), "The fundamental law of active management: Redux". The paper is available here: https://www.sciencedirect.com/science/article/pii/S0927539817300543

After you read the paper, you will realize that the Grinold and Kahn fundamental law is basically flawed. The real fundamental law is just

IR=IC_mean/IC_stdev

There is no such thing as breadth, period.

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