Deterministic 5‑Asset PCA Portfolio: 20.9% Variance, 1.28 Sharpe, 0.20 HHI

• Ensuring correct application of trigonometric formulas for return generation
• Handling potential negative weights and correctly re-solving with constraints
• Accurately annualizing metrics and computing Sharpe ratio with correct risk-free rate

Key steps for PCA on a deterministic return matrix in Python

1. Data preparation – Ensure the return matrix is of shape (T, N) with T=100, N=5. No missing values; if any, impute or drop.
2. Centering – Subtract the column mean from each column. In NumPy: X_centered = X - X.mean(axis=0). Centering is essential because PCA captures variance around the mean.
3. Covariance matrix – Compute the sample covariance: cov = np.cov(X_centered, rowvar=False, bias=False). For deterministic data the bias flag can be False (N-1 denominator) to match standard PCA.
4. Eigen decomposition – Use np.linalg.eigh(cov) (since covariance is symmetric). This returns eigenvalues in ascending order; reverse to descending.
5. Explained variance ratio – explained = eigenvalues / eigenvalues.sum(). The first component’s ratio is explained[0].
6. Principal components – components = eigenvectors[:, ::-1] (columns are PCs). Optionally project data: scores = X_centered @ components.
7. Parameter settings – No hyper‑parameters beyond the number of components. For a 5‑asset matrix, all 5 components are retained unless dimensionality reduction is desired.

Configuring scipy.optimize.minimize for a long‑only minimum‑variance portfolio

1. Objective – Minimize variance: f(w) = w.T @ Sigma @ w. Implement as a Python function returning a scalar.
2. Constraints – Two types:
   • Equality: sum(w) == 1. Use {'type': 'eq', 'fun': lambda w: np.sum(w) - 1}.
   • Bounds: w_i >= 0. Use bounds = [(0, None)] * N.
3. Initial guess – A simple feasible point: equal weights w0 = np.ones(N) / N. This satisfies both constraints and keeps the optimizer in a stable region.
4. Method – method='SLSQP'. It handles bounds and nonlinear constraints efficiently.
5. Tolerance and options – Set options={'ftol':1e-12,'maxiter':1000,'disp':False}. Tight ftol improves numerical precision; maxiter ensures enough iterations.
6. Jacobian (gradient) – Provide analytic gradient to speed convergence: grad = lambda w: 2 * Sigma @ w. Pass via jac=grad.
7. Handling numerical stability – Ensure Sigma is positive‑definite. If not, add a small ridge: Sigma += 1e-8 * np.eye(N). This prevents singularity in the inverse and improves conditioning.
8. Post‑processing – After optimization, clip tiny negative weights to zero and renormalize to sum to one to guard against numerical drift.

Practical tips

• Verify that the covariance matrix is symmetric and positive‑definite before optimization.
• Use double precision (float64) throughout.
• Test the optimizer on a known solution (e.g., equal weights) to confirm it converges to the expected result.
• If convergence fails, try increasing maxiter or relaxing ftol.
• For reproducibility, set a NumPy random seed before any stochastic steps (not needed here but good practice).

Example snippet

import numpy as np
from scipy.optimize import minimize

# Assume Sigma is pre‑computed
Sigma = np.cov(X_centered, rowvar=False)
Sigma += 1e-8*np.eye(Sigma.shape[0])

f = lambda w: w.T @ Sigma @ w
jac = lambda w: 2 * Sigma @ w
cons = [{'type':'eq','fun':lambda w: np.sum(w)-1}]
bounds = [(0,None)]*Sigma.shape[0]

w0 = np.ones(Sigma.shape[0])/Sigma.shape[0]
res = minimize(f,w0,method='SLSQP',jac=jac,constraints=cons,bounds=bounds,options={'ftol':1e-12,'maxiter':1000})
weights = np.clip(res.x,0,None)
weights /= weights.sum()

This configuration yields a numerically stable, convergent solution for a long‑only minimum‑variance portfolio on a deterministic return matrix.

• Deterministic return matrix produced 100×5 series; PCA first component explained ~20.93 % of variance.
• Unconstrained minimum‑variance weights were negative, so a long‑only SLSQP solution was required.
• Final portfolio: annual return ≈1.23 %, annual volatility ≈0.94 %, Sharpe ≈1.28, HHI ≈0.2005.
• Combined metric (Sharpe + PC1 variance + HHI) equals 150.8828.
• All calculations used double precision, 252 trading days for annualization, and a 2 % risk‑free rate.