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Covariance Matrix

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?

Assume the entries of the random vector \(\mathbf{X}\) are the prices of stocks, each of which has positive variance and is independent of the others. Then which of the following best describes the covariance matrix of \(\mathbf{X}\)?

Assume the entries of the random vector \(\mathbf{X}\) are the prices of stocks. What is the \(n^{\text{th}}\) entry along the diagonal of the covariance matrix equal to?

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!)

Consider the random vector \[\mathbf{X}=\left[\begin{array}{c} H\\ T\end{array}\right],\] where \(H\) and \(T\) are the number of heads and tails, respectively, that result from flipping a coin 5 times. Compute the covariance matrix for this vector.

Suppose we purchase two assets, \(A\) and \(B\), such that \(A\) is normally distributed with mean 5 and variance 2, and \(B\) is normally distributed with mean 3 and variance 1. If \(\text{cov}(A,B)=-0.5\), what is the variance of the return from investing in both of these assets?


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