Why this statement is correct?
“The active risk attributed to Active Share will be smaller in more diversified portfolios.”
Diversified portfolios can also have high active shares (portfolios can have large difference of holdings and weights with benchmark). As active share is large, active risk can also be large, even in diversified portfolio.
If you have more diversification you have less “concentration” and we would therefore expect a smaller difference in the standard deviation of of active return measured over a length of time.
thats the key here, is that they’re talking about this in the context of an index for example (think large cap core, so like a portfolio that seeks to replicate returns of the Russell 1000 and nothing else).
And since we now have more diversification, our beta of the portfolio will be closer to the beta of the benchmark (almost got you there didn’t I? The beta of the benchmark = 1) we will EXPECT there to be lower active risk (but, like obviously there’s still some idiosyncratic risks that can throw returns of the portfolio off from the benchmark. . . Duh).
edit: in short, remember that you can control active share, but you can’t control active risk. I think you’re also just getting caught up in the word “diversification”. In this context, the word diversification simply just means less concentration