Question on Active Share

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Hi, in the curriculum, Book 4, Reading 29, Section 3.1.4 Active Share and Active Risk, there was a closing paragraph for Active Share (just before the reading starts for Active Risk):
“If two portfolios are managed against the same benchmark (and if they invest only in securities that are part of the benchmark), the portfolio with fewer securities will have a higher level of Active Share than the highly diversified portfolio. A portfolio manager has complete control over his Active Share because he determines the weights of the securities in his portfolio.”
Assuming an example: Benchmark holds 6 stocks (A,B,C,D,E,F), Port X holds 4, Port Y holds 5. Based on the above paragraph, in this example, Port X should have a higher Active Share than Port Y.
But if one were to construct the ports in this way:
Benchmark stock weights:
– A: 2%
– B: 3%
– C: 10%
– D: 15%
– E: 30%
– F: 40%
Port X weights
– A: 0%
– B: 0%
– C: 15%
– D: 15%
– E: 30%
– F: 40%
Hence active share of X = (2+3+5)/2 = 5%
Port Y weights:
– A: 2%
– B: 3%
– C: 10%
– D: 15%
– E: 70%
– F: 0%
Hence active share of Y = 80/2 = 40%
Which means active share of the supposedly more concentrated port X is lesser than the less concentrated port Y, contradicting the above para.
Or are my calculations for Active Share wrong in the example?
Even if I introduce a Port Z holding only just 2 stocks with the following weights:
– E: 50%
– F: 50%

Active share of Z will still be lower! >> (2+3+10+15+20+10) / 2 = 30%
 
It would be nice if someone could pitch in, I was reading the same thing today and it got me confused as well.
Why are you dividing under/over weights by 2 though? Not that it will make any difference for your explanation but active share is calculated by a sum of overweight stocks (or under weight) compared to the benchmark. Overweight stocks are always the same as underweight stocks (Portfolio vs. Benchmark).
 
Maybe they are talking about a TOTAL number of stocks in the portfolio A, not just number of stocks X owned.
If portfolio A has 100 stocks in total and 10 stocks X while portfolio B has 70 stocks and 10 stocks X, then portfolio B will have a greater active share.
LOL
 
This is really a great question and it points to the wording in the text. The active share for port Y is 40%: ((0+0+0+0+.4+.4)/2), but your point remains the same.
Active share = 1/2 X SUM|Wfund,i - Windex,i|. It is a function of both the number and magnitude (absolute changes) of deviations from the benchmark.
Fewer securities would naturally result in increasing active share as you deviate from the benchmark, but if another manager (manager Y in this case) is going all in on stock E they will skew the active share in their favor, despite having a more “diversified” portfolio in terms of total securities held.
Just me thinking out loud here, but a great question.
 
Well the formula provided by the curriculum is to add all the overweights and underweights (ignoring the minus sign) and divide them by 2, but you’re not wrong to say that all overweights should equal to all underweights as well.
 
Hey, thanks for pointing it out! Have amended my original post and even added another Port Z holding just 2 stocks!
I’m really stuck on this as the text is really clear on the language – no ambiguity at all!
Moreover, my example of the benchmark weightings above is not that far fetched/extreme/rare of a scenario – just imagine the stocks represent a cluster of companies grouped by size and the weightings are market-cap weighted!
 
I am still scratching my head on this particular section of equity after trying both Schweser and CFAI.
Hoping magician will come to the rescue!
 
I think we must consider not only the number of securities but weighting scheme too! And in your example the way how you change weights is more important than number of holdings.
This is how looks “Active Share” WITHOUT changing weights (holding Equal weighting) but only excluding securities. As you see this is consistent with Curriculum.
 
No where does it says in the text that you have to apply equal weighting… in fact, I thought the text was quite explicit in that the manager has complete control over active share because he determines the weights in his portfolio.
 
The benchmark you made up is not what the authors were thinking about. Hence there are exceptions to the rule.
 
The benchmark I made up is suppose to be a reflection of most indexes using market cap weighting. Think about it, large caps dominate most indexes (80-90%). how is my made up benchmark in any way unrealistic compared to the real world? But that’s besides the point. shouldn’t the author already thought through these things before they commit into the text?
What I really wanna know is that I am not missing anything or misunderstanding any points on this topic.
 
Pretty sure this is just assuming equal weight. It may not explicitly state this, but it should be assumed unless they state otherwise. They are saying that, generally, if you have a portfolio with fewer positions than another, it will have a higher active share (assuming equal weighting) since each weight will differ more substantially relative to the weight in the highly diversified benchmark. Don’t overthink think it.
 
I am also stuck on this point…
well why should they assume equal weight without saying it explicitly? Is it that difficult to use plain clear and unambiguous sentences?
Anyway, what they are saying is wrong apart eith equal weight, but as I said… they are assuming it (implicitly) and as a consequence we are supposing…
 
Hey also make SURE YOU UNDERSTAND that concentrated means the opposite of diversified!!!!!!!!!! OP do not get that confused.
 
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