# Correlation between two indexes

The Global Minimum Variance has an annual return standard deviation of 9.9%. Its correlation with the Standard & Poor's 500 Index is 0.45. What is the annual return standard deviation of the S&P 500?

I know that correlation is given by

Corr(A,B) = Cov(A, B)/ [StDev(A)*StDev(B)]

But in this problem, the covariance isn't given. Does that mean there's not enough information to answer the question?

Here is a clue: The GMVP has a fundamental property: it has the same covariance with every other efficient portfolio. You can assume that the S&P500 is an efficient portfolio. For details see here Covariance of a GMV portfolio with any asset This common covariance is usually denoted $\frac{1}{C}$ where $C = 1^T {\Sigma}^{-1} 1$
What is the covariance of the GMVP with itself: That's just the standard deviation squared, i.e. ${0.099}^2$
What is the covariance of the GMVP with the S&P500: By the above theorem it is the same number i.e. ${0.099}^2$
Now we apply the definition of correlation which you gave and we have $0.45 = \frac{{0.099}^2}{0.099 \sigma_{SP}}$. From this equation $\sigma_{SP}$ is found to be 0.22 or 22%