# Do two stocks with the same beta have a correlation of 1?

If two stocks have the same beta over same time period, does it mean they are 100% correlated over that time period?

In a CAPM framework, a stock's beta is defined as

$$\beta_1={\rm Cov} (R_1, M) / {\rm Var} (M)$$

where

• $R_1$ is the return vector of security 1
• $M$ is the market return vector.

Equating two betas means ${\rm Corr}(M, R_1) \cdot {\rm Std} (R_1) = {\rm Corr} (M, R_2) \cdot {\rm Std} (R_2)$.

I'm not really sure where to go from here - the standard deviations of $R_1$ and $R_2$ might not be equal, and I'm not sure what the relation, if any is between the ${\rm Corr} (M, R2)$ and ${\rm Corr} (M, R1)$.

According to this paper, correlation is not transitive. If $R_1$ and $M$ are perfectly correlated, and $R_2$ and $M$ are perfectly correlated, it doesn't necessarily mean $R_1$ and $R_2$ are perfectly correlated.

• Commented Oct 20, 2013 at 3:17
• Corrected. Forgot to put the (-1) power to the market variance. Commented Oct 20, 2013 at 6:14
• In general, no. It does however put a lower bound on their correlation. Commented Oct 20, 2013 at 8:26
• Thanks, but the regression line you mention is the fit between the market returns and an asset's returns. Can we deduce anything about the correlation of two assets, both with the same beta? Commented Oct 20, 2013 at 16:07
• We can deduce something...see my answer here: quant.stackexchange.com/questions/529/… Commented Oct 21, 2013 at 1:19

Suppose, R1 = M. Then the claim Corr(M,R1) = Corr(M,R2) implies 1 = Corr(M,R2) for any R2, which is obviously wrong.