# Are 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.

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– chrisaycock Oct 20 '13 at 3:17
Corrected. Forgot to put the (-1) power to the market variance. – nonbaryonic13 Oct 20 '13 at 6:14
In general, no. It does however put a lower bound on their correlation. – U2EF1 Oct 20 '13 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? – nonbaryonic13 Oct 20 '13 at 16:07
We can deduce something...see my answer here: quant.stackexchange.com/questions/529/… – Brian B Oct 21 '13 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.