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 $A_1$, $A_2$ (dependent variables) and $B$ (independent variable). If $A_1$ and $B$ have $\beta=2.0$; correlation=0.8, and $A_2$ and $B$ have $\beta=0.5$; correlation=0.8, then both $A_1$ and $A_2$ are equally well explained by $B$.
Consider two series: A=$\{1,2,3,4,5,6,7,8,9,10\}$ and $B=2\times A$. 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.