Beta and correlation have a tight linear relationship. You can solve the correlation with the Beta from the Capital Asset Pricing Model (CAPM) and the standard deviations of your two data sets. The CAPM measures the cost of equity of a company and the data sets include the overall index as well as the individual security that you are measuring.
Examine the CAPM for a specific stock. The formula is the cost of equity equals the risk-free rate of return plus the beta multiplied by the risk premium.
Ec = Rf + B(Rp)
Solve for the Beta based on the inputs or assumptions:
We can assume the following formula:
10% = 4% + B(12%) Beta = .5
Use one derivation for Beta which is equal to the correlation coefficient times the standard deviation of the stock divided by the standard deviation of the index.
B = CC X (S1/S2)
Solve for the Correlation Coefficient by inserting Beta and standard deviation figures.
.5 = CC X (.25/.2) CC = .4
