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.

Answer

The answer is NO. It's mathematically incorrect. Simply look the correlation and covariance formulas. But here is a gedankenexperiment (thought experiment) that demonstrates that it's incorrect.

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