Beta and Correlation

[lRRAngle]

Hi David,

This kind of ties to another question I raised but I think it will be good to flush out the differences and similarities between these two metrics as I have been searching through the forum and haven't been able to find a thread that discusses this topic.

I did find online a resource for deriving the beta formula from the covaraince below:

Beta (a) = Cov(a,b) / Var (b)

Where a = dependent = stock, and b=independent

but what confused me is the following identity I found in same resource:
Beta (a) = Correlation (a,b) * [ Std.Dev. (b) / Std.Dev. (a) ]

I would have expected the second formula for Beta to be:
Beta (a) = Correlation (a,b) * [ Std.Dev. (a) / Std.Dev. (b) ] given that Correlation (a,b) = Cov (a,b) / [ Std.Dev (a) / Std.Dev (b) ] so that Std.Dev (a) cancels in the denominator and then you are left with Cov(a,b) / Var(b) , just as in the first formula I have above.

What am I missing?


Thanks

Edit:
Sorry this probably better fits the Quantitative forum.

 

[David Harper CFA FRM]

Hi @lRRAngle There are actually over a dozen forum posts about this (you can search "beta correlation" per your title); e g., https://forum.bionicturtle.com/threads/frm-exam-2007—q-28-expected-annual-return.3983
I just commented over here at https://forum.bionicturtle.com/thre...near-regression-stock-watson.5392/#post-59157

We don't want to write something like Beta(a) because beta is a relationship between two variables and, unlike correlation/covariance, the order matters. Just as we don't expect the slope of a regression line to remain the same if we switch the x-y axis variables, we want to specify β(x,y) or β(y,x). When we write β(y,x) it can be read as "beta of y with respect to x." When we see the beta of a stock is 1.2 (eg), it implicitly always means "beta of stock X's excess return with respect to market's excess return;" i.e., it is β(X, M) and never really β(M,X) because it is always really the regression of the stock's excess return (on the y axis, dependent) against the market's excess return (on the x-axis, independent).

Here is the key derivation, I hope this helps:

• Beta of y with respect to x = β(y,x) = covariance(y,x)/variance(x) = [ρ(y,x)*σ(x)*σ(y)]/σ^2(x) = [ρ(y,x)*σ(x)*σ(y)]/σ^2(x) = ρ(y,x)*σ(y)/σ(x). But imagine if you flipped the axes of scatterplot; you would not expect the slope of the regression line to remain unchanged:
  
• Beta of x with respect to y = β(x,y) = covariance(x,y)/variance(y) = [ρ(x,y)*σ(x)*σ(y)]/σ^2(y) = [ρ(x,y)*σ(x)*σ(y)]/σ^2(y) = ρ(x,y)*σ(x)/σ(y). Thanks,

 

[lRRAngle]

Very clear. Thanks David.

Hope you are having a nice relaxing weekend!