Exercise: Computing $\Sigma$ for a Two-Factor Model and Checking PSD

Prerequisites: Covariance Matrices

Problem

Three assets follow a two-factor model:

X_i = B_{i,1}F_1 + B_{i,2}F_2 + \epsilon_i, \quad i = 1, 2, 3,

with loadings

B = \begin{pmatrix}1.0 & 0.3 \\ 0.8 & 0.6 \\ 0.5 & 1.0\end{pmatrix}, \quad \Omega = \text{Cov}(F) = \begin{pmatrix}0.04 & 0.005 \\ 0.005 & 0.01\end{pmatrix}, \quad D = \text{diag}(0.02, 0.03, 0.025).

(Think $F_1 =$ market factor, $F_2 =$ value factor, $\epsilon$ = stock-specific.)

Compute $\Sigma = B\Omega B^\top + D$ explicitly as a $3\times 3$ matrix.
Verify $\Sigma$ is symmetric.
Check that $\Sigma$ is PSD by computing its eigenvalues numerically. (Use numpy.linalg.eigvalsh.)
Compute the correlation matrix $\rho = D_\sigma^{-1}\Sigma D_\sigma^{-1}$ where $D_\sigma = \text{diag}(\sqrt{\Sigma_{ii}})$ . Identify the most- and least-correlated pair of assets.

Numerical precision: compute

B\Omega B^\top

first, then add

D

. Don't forget to add

D

to the diagonal only.

Jump to the solution when you're ready.