Active share and active risk

CFA Level III Equities
#1

Hi, I have questions concerning active share and active risk, and this is the first example in page 327 Level 3 Book 3.

Manager A, Active Share 0.73, Active risk 2.8%, Number of positions 120
Manager B, Active Share 0.71, Active risk 6.0%, Number of positions 125
Solution: Managers A and B have a similar number of positions and similar Active Share. Manager B has much higher active risk. A high Active Share says only that a manager’s security-level weights are quite different from those of the index. A 0.5% underallocation to one security and a 0.5% over-allocation to another security will have the same impact on Active Share whether these two securities are in the same sector or in different sectors. Given similar levels of Active Share, it is likely that Manager B’s active risk is driven by active decisions at the sector level rather than at the security level. Clearly, they implement very different investment strategies. Although we cannot draw a direct conclusion about the ability of Manager B to outperform Manager A, we can assume that the realized outcomes of Manager B are likely to be much more dispersed about the benchmark (both in positive and negative directions) given the higher level of active risk.

  1. How could we infer “the realized outcomes of Manager B are likely to be much more dispersed about the benchmark" from the higher level of active risk.
    In my view, these 2 managers have similar active share and number of positions, so the higher active risk of Manager B comes from idiosyncratic risks, which means security selection. It’s possible for Manager B to choose any stock, and why “more dispersed about the benchmark"?

  2. Why “Manager B’s active risk is driven by active decisions at the sector level rather than at the security level”?
    I think it’s the opposite.

Thanks.

#2

What’s the definition of active risk?

Think of two scenarios:

  1. You underweight one healthcare stock and overweight another healthcare stock
  2. You underweight a healthcare stock and overweight a tech stock

Compared to the benchmark, which one do you believe will likely have a smaller difference in returns? Which will likely have a larger difference in returns?

#3

Hi magician, thanks for your reply.
About your hint, the 2 stocks in the scenario 1 have higher correlations than another 2 in the scenario 2, so the correlation in the first pair should be higher than that of the second pair, but I can’t see how security weights relate to correlations and active risk.

Besides, your hint is similar to another question - question 6 in page 387 Level 3 Book 3 as below.
Fund 3 held active positions in two automobile stocks—one was overweight by 1 percentage point(pp), and the other was underweight by 1pp. Fund 3 traded back to benchmark weights on those two stocks. In the second trade, Fund 3 selected two different stocks that were held at benchmark weights, one energy stock and one financial stock. Fund 3 overweighted the energy stock by 1pp and underweighted the financial stock by 1pp.
I think they’re the same type of question, but I just can’t figure out the logic.

#4

I totally understand your first reply, thank you : )