Estimating Effective Breadth in the Information Ratio
Summary
The document discusses the approximate relationship between information ratio, information coefficient, and breadth: IR is often expressed as IC multiplied by the square root of breadth. It cautions that this is a useful but imprecise framework and that information ratio and Sharpe ratio are not interchangeable.
Counting 500 stocks and four quarterly decisions as 2,000 independent bets can overstate breadth because correlated stocks may represent fewer effective decisions. A long-short portfolio does not automatically double the count: choosing portfolio weights is still one allocation decision, though whether a relative-value position constitutes one bet or two depends on the information being acted on. The note offers conceptual guidance rather than a precise counting rule, and emphasizes that the square-root relationship makes exact breadth estimates less consequential when breadth is already large.
Key ideas
- The information ratio is approximately the information coefficient times the square root of breadth.
- Correlations among stocks reduce the number of effective independent bets.
- A long-short portfolio does not necessarily double breadth because portfolio weights are chosen together.
- The framework is approximate, and exact breadth can be difficult to define consistently.
Tags
Full text
# How is breadth for Information Ratio Calculated
# 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 = $500 \times 4 \times 2$ ?)
## Answer by Bob Jansen (score 2, accepted)
https://quant.stackexchange.com/a/41282
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.