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Minimising a quadratic · Part 4 of 4

A mean–variance objective is $f(w) = \tfrac{1}{2} w^\top\Sigma w - \mu^\top w$ with $\Sigma = \begin{pmatrix}2 & 1\\ 1 & 2\end{pmatrix}$ and $\mu = (1, 2)^\top$ .

Suppose instead $\Sigma = \begin{pmatrix}2 & 2\\ 2 & 2\end{pmatrix}$ , the two assets being perfectly correlated, with the same $\mu$ . What happens to the minimisation?

AThere is a unique minimum, found with a pseudo-inverse
BThe minimum is the same as before, since the diagonal is unchanged
Cf is unbounded below: a riskless long–short pair earns return
DThere are infinitely many minimisers, all with the same value

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Definiteness, Cholesky and generating correlated normals

Suppose instead Σ=(2222)\Sigma = \begin{pmatrix}2 & 2\\ 2 & 2\end{pmatrix}Σ=(22​22​), the two assets being perfectly correlated, with the same μ\muμ. What happens to the minimisation?

More linear algebra questions

Suppose instead $\Sigma = \begin{pmatrix}2 & 2\\ 2 & 2\end{pmatrix}$ , the two assets being perfectly correlated, with the same $\mu$ . What happens to the minimisation?