beta ^ rho i.e. correlation coeff #clarified

Update: I don’t have a intuitive feel for the definition of rho. In contrast, beta is intuitive, as the slope of the OLS fit

Defining formulas are similar for  beta and rho:

rho   = cov(A,B)/  (sigma_A . sigma_B)
beta = cov(A,B)/  (sigma_B . sigma_B) ,  when regressing A on B
= cov(A,B)/  variance_B

Suppose a high tech stock TT has high beta like 2.1 but low correlation with SPX (representing market return). If we regress TT monthly returns vs the SPX monthly returns, we see a cloud — poor fit i.e. low correlation coefficient. However, the slope of the fitted line through the cloud is steep i.e. high beta !

Another stock ( perhaps a boring utility stock ) has low beta i.e. almost horizontal (gentle slope) but well-fitted line, as it moves with SPX synchronously i.e. high correlation ! explains beta vs correlation. Both rho and beta measure the strength of relationship.

Rho is bounded between -1 and +1 so from the value you can get a feel. But rho doesn’t indicate how much (magnitude) the dependent variable moves in response to an one-unit change in the independent variable.

Beta of 2 means a one-unit change in the SPX would “cause” 2 units of change in the stock. However, rho value could be high (close to 1) or low (close to 0).


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