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

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

Parts 1–4 via Python

import numpy as np

B = np.array([[1.0, 0.3],
              [0.8, 0.6],
              [0.5, 1.0]])
Omega = np.array([[0.04, 0.005],
                  [0.005, 0.01]])
D = np.diag([0.02, 0.03, 0.025])

Sigma = B @ Omega @ B.T + D
print("Sigma =\n", Sigma.round(5))
# Sigma =
#  [[0.0609  0.0383  0.0335 ]
#   [0.0383  0.0644  0.0385 ]
#   [0.0335  0.0385  0.0460 ]]

# symmetric?
print("symmetric:", np.allclose(Sigma, Sigma.T))
# symmetric: True

# eigenvalues
eigs = np.linalg.eigvalsh(Sigma)
print("eigenvalues:", eigs.round(5))
# eigenvalues: [0.02 0.02474 0.12656]
# all positive → PSD (positive definite)

# correlation matrix
sigmas = np.sqrt(np.diag(Sigma))
rho = Sigma / np.outer(sigmas, sigmas)
print("rho =\n", rho.round(4))
# rho =
#  [[1.     0.611  0.631 ]
#   [0.611  1.     0.707 ]
#   [0.631  0.707  1.    ]]

Interpretation

$\Sigma$ is symmetric and positive definite — all eigenvalues are strictly positive. The smallest eigenvalue is $0.02$ , corresponding to the largest idiosyncratic component.
Largest eigenvalue $\approx 0.127$ captures the dominant "market direction" — roughly the direction of the first column of $B$ (all positive loadings).
Correlations range from $0.61$ to $0.71$ . Most correlated pair: assets 2 and 3 ( $\rho_{23} = 0.707$ ) — both have substantial loadings on the second factor ( $F_2$ ), which has lower variance but they share similar exposure. Least correlated pair: assets 1 and 2 ( $\rho_{12} = 0.611$ ).
$D$ adds variance, not correlation. If we had set $D = 0$ , all three assets would be perfectly correlated in pairs (rank-2 $\Sigma$ ).

Takeaways

Factor models produce PSD covariance matrices by construction — $B\Omega B^\top$ is PSD because $\Omega$ is PSD, and adding diagonal $D \succeq 0$ preserves PSD.
Low rank plus diagonal structure is the computational heart of real-world risk models. Storing $B \in \mathbb{R}^{n\times k}$ , $\Omega \in \mathbb{R}^{k\times k}$ , $D \in \mathbb{R}^n$ is $O(nk + k^2)$ instead of $O(n^2)$ — huge savings when $n = 10{,}000$ stocks and $k = 50$ factors.
Correlation matrix strips volatility information. $\rho$ captures the "shape" of dependence; $\Sigma$ captures shape × scale.
Eigenvalues give you the principal-component variances. The largest eigenvalue corresponds to the portfolio direction with the most variance — typically the "market" or a dominant risk factor.

Solution: Computing Σ\SigmaΣ for a Two-Factor Model and Checking PSD

Parts 1–4 via Python

Interpretation

Takeaways

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