# How is breadth for Information Ratio Calculated

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 = $500 \times 4 \times 2$ ?)

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.

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.