Active Return and Active Risk

This is straight from the curriculum

  1. “Active risk increases when a portfolio becomes more uncorrelated with its benchmark.” - I don’t get it. Why is the active risk supposed to increase because our portfolio is not correlated to the benchmark? Is it because we are trying to imply that we are taking more active positions?

  2. “Overweighting or underweighting GM relative to Ford will generate some Active Share, it will typically not generate much active risk” - if two stocks are in the same industry, doesn’t it mean that the active risk will be higher?

  3. “overweighting or underweighting energy firms versus financial firms, small-cap firms versus large-cap firms, or growth firms versus value firms will certainly contribute more to active risk” - why? isn’t it supposed to lower the active risk because we have a diversified portfolio?

I’ll try:
First of all active risk = tracking error vs benchmark returns → It’s the difference in the standard deviation of returns on the portfolio vs. that of the benchmark. Ok, here we go:

  1. What would happen with tracking error (active risk) if you fill your portfolio with securities that don’t bring similar returns as the benchmark? (e.g. you fill your portfolio with large cap value stocks, but the benchmark is Russell2000 - I oversimplified it just for the sake of understanding)

  2. I presume that GM and Ford would have similar correlations to benchmark, so underweighting one and overweighting the other (of course in this case within the same sector) would have “offsetting effect” on active risk (tracking error).

  3. How do you call correlation between two stocks in two different sectors? Cross-correlation. Stock in energy sector paired with stock in financial sector will likely have lower correlation (of returns) than two stocks in financial or two stocks in energy sector. Lower the cross correlation, higher the active risk (it doesn’t have anything to do with diversification in this case, but tracking error).

I think you concentrated on diversification, and not tracking error.

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Why active risk attributable to active share is inversely proportional to the number of securities in the portfolio?

Thanks in advance for your answer.

More securities, more likely similar to the benchmark so lower active share and could be lower active risk/tracking error