Covariance matrices capture information about relationships between the elements of a vector. For example, in a financial portfolio, the covariance matrix gives information about the way the stocks interact with each other.

Which of the following statements about covariance matrices is correct?

The covariance matrix is extremely useful in financial mathematics for calculating the return from a portfolio of correlated assets. Suppose we have two assets \(A\) and \(B\), such that the return from \(A\) is normally distributed with mean \(\mu_A\) and variance \(\sigma_A^2\), and similarly for \(B\). These assets are correlated, so define \(\text{cov}(A,B)=\sigma_{AB}\). Let \(\Sigma\) be the covariance matrix for the random vector with first entry \(A\) and second entry \(B\).

If we purchase all assets, the total variance of the return on our investment will be \(\text{var}(A+B)=\mathbf{1}^T\Sigma\mathbf{1},\)where \(\mathbf{1}\) is the vector with every entry equal to 1. (Write out the formula for the variance of the sum of two correlated variables, and compare it to this result!)

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