can someone explain this to me? Performance evaluation.

CFA Level III
#1

Taken from CFAI. the highlighted statements i don’t understand. seem contradictory to me.

Potential systematic bias can also be identified through a set of correlation statistics. Consider the correlation between A = (P – B) and S = (B – M). The contention is that a manager’s ability to identify attractive and unattractive investment opportunities should be uncorrelated with whether the manager’s style is in or out of favor relative to the overall market. Accordingly, a good benchmark will display a correlation between A and S that is not statistically different from zero

#2

removed … since I think what I write below makes more sense…

#3

that part i understand… but then last statement confused me. it seems that last statement contradicts what you just said.

#4

Statistical remark: Correlation between A=(P-B) and S=(B-M) should be 0 on average.

Let us say this were not the case - and the correlation was 0.5 e.g. and you are a small cap manager. (Style).

It means your active returns are doing well when your style (Small cap) is in favor with the market. The active returns should do well - because you are making an active bet on them - whether or not your style is in favor with the market or not. (Only then are you doing well as a manager - and if that were the case - you would have a correlation of 0 between A and S).

#5

Statistical remark: Correlation between A=(P-B) and S=(B-M) should be 0 on average.

Let us say this were not the case - and the correlation was 0.5 e.g. and you are a small cap manager. (Style).

It means your active returns are doing well when your style (Small cap) is in favor with the market. The active returns should do well - because you are making an active bet on them - whether or not your style is in favor with the market or not. (Only then are you doing well as a manager - and if that were the case - you would have a correlation of 0 between A and S).