I came across this note from a book -
"..the correlation between the active and passive portfolios is greater when the $\beta$ of the active portfolio is higher.."
The author runs regression of active portfolio on passive portfolio.
This does not make sense to me. If I understand correctly, beta in a single variable linear regression analysis gives us the best relationship/fit between two variables (or the slope of independent vector on dependent vector), but it is not indicative of strength of that relationship. We need to look at correlation or $R^2$ value to determine how good is the relationship between the variables. As long as beta is statistically significant, we can look at $R^2$ to determine the strength of relationship. Here are couple of examples:
Consider three variables A1, A2 (dependent variables) and B (independent variable). If A1 and B have $\beta=2.0$; correlation=0.8, and A2 and B have $\beta=0.5$; correlation=0.8, then both A1 and A2 are equally well explained by B.
Consider two series: A=1,2,3,4,5,6,7,8,9,10 and B=2*A. In this case, correlation between A and B is 1. However, regression of A on B gives $\beta$ as 0.5 and B on A gives 2.0. However, each variable is completely explained by the other.
Is my understanding correct? Please highlight if I am missing something here.