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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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  • $\begingroup$ Your definition of beta is wrong. $\endgroup$
    – chrisaycock
    Oct 20, 2013 at 3:17
  • $\begingroup$ Corrected. Forgot to put the (-1) power to the market variance. $\endgroup$
    – nonbaryonic13
    Oct 20, 2013 at 6:14
  • $\begingroup$ In general, no. It does however put a lower bound on their correlation. $\endgroup$
    – U2EF1
    Oct 20, 2013 at 8:26
  • $\begingroup$ 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? $\endgroup$
    – nonbaryonic13
    Oct 20, 2013 at 16:07
  • $\begingroup$ We can deduce something...see my answer here: quant.stackexchange.com/questions/529/… $\endgroup$
    – Brian B
    Oct 21, 2013 at 1:19

2 Answers 2

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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.

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  • $\begingroup$ Thanks, that made it clear. It's funny how then in the industry people mix it up without giving much thought. $\endgroup$
    – nonbaryonic13
    Oct 22, 2013 at 22:39
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beta refers to the fact that on an average the stock has a degree of correlation with the movement of the index. the important thing is "on an average" because two different stocks may have the same beta but this average may have different weightages of different parts of that time period. so lets say that in the first part of the data, stock1 is not correlated much with the market, but stock2 is moderately correlated. now in the second part, stock1 is heavily correlated but stock2 is again moderately correlated. the values are such that the overall calculations put equal values for the beta for the two stocks for the entire time period. hence same beta does not imply mutual correlation.

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