# 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.

# Covariance Matrix

Which of the following statements about covariance matrices is correct?

# Covariance Matrix

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 $$\mathbf{X}$$?

# Covariance Matrix

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?

# Covariance Matrix

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 $$B$$.

# Covariance Matrix

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!)

# Covariance Matrix

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

# Covariance Matrix

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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