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%