Exercise: Minimum-Variance Portfolio from a 3×3 $\Sigma$

Prerequisites: Covariance Matrices

Problem

Using the covariance matrix from the previous exercise:

1. Set up the optimisation: minimise $w^\top\Sigma w$ subject to $\mathbf{1}^\top w = 1$. Use a Lagrange multiplier to derive the closed-form solution $w^* = \frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^\top\Sigma^{-1}\mathbf{1}}.$

2. Compute $\Sigma^{-1}$ numerically and then $w^*$ for this $\Sigma$.

3. Compute the minimum-variance portfolio's variance $w^{*\top}\Sigma w^*$ and standard deviation (annualised volatility).

4. Interpret the weights. Compare the minimum-variance portfolio's weights to the equal-weighted portfolio $w = (1/3, 1/3, 1/3)$. Which asset gets the largest weight, and why? (Hint: look at the diagonals of $\Sigma$ and the off-diagonals.)

Hint

For part 1: the Lagrangian is $\mathcal{L}(w, \lambda) = w^\top\Sigma w - \lambda(\mathbf{1}^\top w - 1)$. First-order conditions give $2\Sigma w = \lambda\mathbf{1}$, so $w = (\lambda/2)\Sigma^{-1}\mathbf{1}$. The constraint $\mathbf{1}^\top w = 1$ determines $\lambda$.