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Active risk between high correlation pair trade and low correlation pair trade

In the Schweser note, underweighting a pharmaceutical stock in order to overweight another pharmaceutical stock will certainly increase active share because the weights of the portfolio will be different to the weights in the benchmark. However, if the two pharmaceutical stocks are highly correlated, the portfolio will not behave markedly different from the benchmark, hence active risk is not likely to substantially increase. On the other hand , underweighting a pharmaceutical company and overweighting a security with a low correlation to the pharmaceutical company, such as a consumer discretionary company, will likely increase both active share and active risk

The Example indicates that high correlation pair trade would have lower active risk than the low correlation pair trade.

Another question from the book,

Question: After further research, Nichols adds Portfolio D’s manager in the list of managers approved for use by JAG. One of the appeals of manager D was the manager’s use of paired trades. For example, in the most recent quarter the manager had in place a pairs trade between an industrial and a consumer durables stock. The industrial was overweighted by 1.5% and the consumer durable was underweighted by 1.5% versus the portfolio’s benchmark. The trade was not successful and the manager reversed the trade by restoring both positions to an equal weight versus the benchmark. In its place the manager initiated a new pairs trade between two different consumer staples companies. This time the over and underweights were only 1.0%.

Determine whether the active risk of Manager D most likely increased or decreased as a result of the two pairs trade actions taken.

Answer:

The effect on active risk is unclear:

  • Decreasing the over/underweights would decrease the active risk.
  • But active risk is also affected by covariance within the pair and the covariance likely went up, increasing the active risk. The covariance (and correlation) should increase because the manager went from a pair of stocks in different sectors to a pair in one sector.

The answer indicates that the higher correlation pair trade would lead to higher active risk. 

Is it a contradiction between two examples? 

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This was already asked here: https://www.analystforum.com/forums/cfa-forums/cfa-level-iii-forum/91370133

ANd I believe the question made a mistake. The CFAI text eoc #6 on Reading 29 has a similar question:

“Ap reviews quarterly holdings reports for Fund 3. In comparing the two most recent quarterly reports, he notices differences in holdings that indicate that Fund 3 executed two trades, with each trade involving pairs of stocks. Initially, 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.”

Answer: Active Risk Increased: Active risk is affected by the degree of cross-correlation. The correlation of two stocks in different sectors is most likely lower than the correlation of two stocks in the same sector. Therefore, the correlation of the energy/financial pair is most likely lower than that of the automobile/automobile pair. Because both positions were implemented as an overweight and underweight, the lower correlation of the two stocks in the new position should contribute more to active risk than the two-stock position that it replaced.

What about Active Share, 125mph? 

I know the curriculum says that, regardless of correlation, under- or overweighting by the same amount will have no impact on Active Share. But why is this, exactly? Over-/ underweighting two correlated stocks, versus over-/underweighting two uncorrelated stocks, how can it remain unchanged?

Many thanks!