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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)?

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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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  • $\begingroup$ I have read Ding and Martin (2017), in which the authors provide their own version of FLAM. Though the concept 'breadth' in Grinold & Kahn (2000) is ambiguous, I still want to understand the proof in Grinold & Kahn (2000) from a mathematical point of view. Note my question is about the IC part, not the Breadth part. $\endgroup$
    – Gödel
    Nov 14, 2022 at 14:10
  • $\begingroup$ It will be a miracle if anybody can understand G&K's proof. The step in (6A.18) may not be defined at all, just like their many other formulas. Basically G&K's result is not valid and is based on some very confusing assumptions. At first you thought they are talking about a single factor model, but it looks like a k- (or BR-) factor model in their derivation. Their result does not hold even if it is based on multi-factor model just as you figured out. $\endgroup$
    – zack young
    Nov 15, 2022 at 2:19
  • $\begingroup$ Under their assumptions, for a single factor model, the result should be IR=ICsqrt(N) where N is the number of stocks in the universe. For a multi-factor model, the result should be IR=sqrt((IC(1)^2+IC(2)^2+...+IC(k)^2)*N)=ICsqrt(N*k) when all the ICs are the same. This is not an interesting case at all since the most important risk in factor investing is the uncertainty of factor returns. They assume that factor returns are constants which is not true in practice and if it is true then they have a money machine. $\endgroup$
    – zack young
    Nov 15, 2022 at 3:16
  • $\begingroup$ By the way, I have another question about Ding and Martin (2017). In this paper, $\Sigma_t$, defined by equation 8, is covariance matrix of residual returns, which is equivalent to $VR$ in G&K's notation. However, the rank of covariance matrix of residual returns is $N-1$, and it is a singular matrix. So I wonder why Ding and Martin (2017) uses the inverse of $\Sigma_t$ directly, not considering whether it is invertible or not. This usage of inverse of residual covariance matrix can also be found in Clarke, Silva and Thorley (2006), Buckle(2004) etc. $\endgroup$
    – Gödel
    Nov 17, 2022 at 5:13
  • $\begingroup$ You made a very good observation. Financial models are not like pure mathematical models. They are just approximation of the real world observations. Ding and Martin (2017) model is not perfect but it is a much closer approximation to the real world practice than those of G&K, Clarke, de Silva and Thorley (2002,2006). Your question can lead to the dispute if CAPM itself is a good description of the world. What is the market portfolio, is it S&P 500, or Russell 3000, or Whilshire 5000, or should it include all the risky assets in the world, such as gold, silver, crypto etc? $\endgroup$
    – zack young
    Nov 18, 2022 at 14:26

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