Proofs – Tidy Finance

Optimal Portfolio Choice Minimum variance portfolio The minimum variance portfolio weights are given by the solution to \[\omega_\text{mvp} = \arg\min \omega'\Sigma \omega \text{ s.t. } \iota'\omega= 1,\] where \(\iota\) is an \((N \times 1)\) vector of ones. The Lagrangian reads \[ \mathcal{L}(\omega) = \omega'\Sigma \omega - \lambda(\omega'\iota - 1).\] We can solve the first-order conditions of the Lagrangian equation: \[ \begin{aligned} & \frac{\partial\mathcal{L}(\omega)}{\partial\omega} =.