Why Equal Market Betas Do Not Imply Perfect Stock Correlation

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


# 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 by Serg (score 3, accepted)

https://quant.stackexchange.com/a/9231

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.

## Answer by htrahdis (score 1)

https://quant.stackexchange.com/a/9219

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.