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An alternative definition of the information Ratio (sharpe ratio) is:

$IR = IC\sqrt{BR}$

I have been reading Grinold and Kahn. I have the following questions for calculating BR:

Q1. If 500 stocks are tracked and quarterly positions are taken in long only portfolio. (Would the BR = $500 \times 4$ ?)

Q2. If 500 stocks are tracked and quarterly positions are taken in long-short portfolio. (Would the BR = $500 \times 4 \times 2$ ?)

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The Sharpe Ratio and Information Ratio are not equivalent, be careful there.

I don't have Grinold & Kahn handy but I believe that it matches the contents of the CFA Curriculum which I have at hand where $\mathrm{IR}$ is defined as

$$\mathrm{IR} \approx \mathrm{IC} \sqrt{\mathrm{BR}}$$

The relation really is only approximate as in my opinion it is a rather fuzzy (but useful) concept dressed up in mathematical notation. So, I don't believe it to be wise to be too precise about the numbers.

Regarding your questions:

Q1: I agree with this formula but take into account the now deleted comment:

Because of correlations, 500 stocks cannot be truly considered 500 separate bets. The number of effective bets is much smaller than that.

If you have information that Telecom stocks go up, is that one bet on a sector or a bet on a number of stocks? How would you practically assign a piece of information to a number of bets?

Q2: In my opinion this is still $500 \times 4$. You make one choice on the portfolio weightings not two. Also, the caveat above still holds: if you say one of two related stocks will outperform the other and you go long and short to create a hedged position, is that one or two bets? However, it makes sense to not discuss investment constraints at all when calculating the $\mathrm{IR}$ at all as investment constraints are related to implementation not skill.

Final observation: It doesn't matter that much what the exact breadth is once it becomes large because of the square root.

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