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I am learning about Fundamental Law of Active Management, and there seems to be two different Information Coefficient (IC) formulae presented.

Though I myself am not a CFA candidate, these appear to be presented especially in CFA curriculums.

I am very confused about the difference of the two, as to which one is the correct formula. If both are correct, then what would be the context in which either of them is preferred over the other?

1.
Calculated using number of correct bets (Nc) and number of total bets(N).
Most commonly presented like in here.
enter image description here

2.
Correlation coefficient of Realised Active Return and Forecast Active Return
Presented in this video.
enter image description here

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    $\begingroup$ Yes, there are two different uses of "information coefficient", very confusing. $\endgroup$
    – noob2
    Aug 15 at 15:10
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Paraphrasing some quote:

"they are different but same but still different"

In reality the number of correct bets $N_c$ is the number of times the analyst was correct predicting the direction of a stock (either up or down), therefore the first formula gives information only about how the analyst performed w.r.t the direction, while the second formula is susceptible to the "shape" i.e. non-linearity between prediction and realization. Most of the time the two formulas should be approximately equal, but can diverge.

In fact, the first formula $IC=2 \left( \frac{N_c}{N} \right)-1$ is just special case of the second formula when variables are transformed to binary.

To see this, we first transform realized and forecast return variables to binary variables giving 1 when return is positive and 0 when negative i.e: $$X=\mathbf{1}_{R_A>0}$$ $$Y=\mathbf{1}_{\mu>0}$$

Then $COR(X,Y)$ is just a Phi coefficient i.e. $$COR(X,Y)=\frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{(n_{10}+n_{11})(n_{00}+n_{01})(n_{00}+n_{10})(n_{01}+n_{11})}}$$ where:

  • $n_{11}$ - number of occurences where $X=1$ and $Y=1$ i.e. we forecasted up move and it was up move
  • $n_{10}$ - number of occurences where $X=1$ and $Y=0$ i.e. we forecasted up move but it was down move
  • $n_{01}$ - number of occurences where $X=0$ and $Y=1$
  • $n_{00}$ - number of occurences where $X=0$ and $Y=0$

It is easy to see that when the classes are balanced i.e. $n_{11}=n_{00}$ and $n_{10}=n_{01}$ (the analyst was correct the same number of times predicting the up move as predicting the down move) then we have equality i.e. $$COR(X,Y)=2 \left( \frac{N_c}{N} \right)-1$$ but in reality they are rarely balanced, in that case we just have approximation: $$COR(X,Y) \approx 2 \left( \frac{N_c}{N} \right)-1$$

Therefore I would see the first IC formulas as approximation to correlation between binary variables and the second formula as correlation between raw variables. It is interesting to calculate both and see if there is major difference as this would indicate non-linearity in the forecasts. For example, if the agent perfectly forecasts the direction of a stock move, but is too optimistic in that i.e. constitently predicts higher returns than true returns when stock goes up and less negative returns than true returns when stock goes down, then IC calculated with first formula would be 1 (100% accuracy in direction), but $CORR(R_{Ai},\mu)<1$ as there is some non-linearity in the forecast vs realized return.

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