Mean-variance, minimum-variance, and tracking error

Translated from Examples/rpb/test_mvo1.m, test_minvar1.m, test_te1.m. Same four-asset dataset as the Black-Litterman example.

import numpy as np
from quanttoolbox.linalg.special_matrices import xpnd
from quanttoolbox.stats.moments import corr_to_cov
from quanttoolbox.portfolio.mean_variance import mvo_portfolio, minvar_portfolio
from quanttoolbox.portfolio.tracking_error import minimum_te_portfolio

mu = np.array([0.05, 0.06, 0.08, 0.06])
sigma = np.array([0.15, 0.20, 0.25, 0.30])
rho_vech = np.array([1.00, 0.10, 1.00, 0.40, 0.70, 1.00, 0.50, 0.40, 0.80, 1.00])
rho = xpnd(rho_vech, method=1)
cov_matrix = corr_to_cov(sigma, rho)

# --- Mean-variance optimal, risk-aversion gamma=0.5 ---
mvo = mvo_portfolio(mu, cov_matrix, gamma=0.5)
print("MVO weights:", np.round(mvo.weights, 4))
print("expected return:", round(mvo.expected_return, 5), "volatility:", round(mvo.volatility, 5))

# --- Minimum-variance (gamma=0 case) ---
minvar = minvar_portfolio(cov_matrix)
print("MinVar weights:", np.round(minvar.weights, 4))
print("MinVar volatility:", round(minvar.volatility, 5))

# --- Minimum tracking error vs. an equally-weighted benchmark ---
x_benchmark = np.full(4, 0.25)
te = minimum_te_portfolio(x_benchmark, cov_matrix)
print("Min-TE weights:", np.round(te.weights, 4))
print("tracking error:", round(te.tracking_error, 6))

Output:

MVO weights: [ 0.6209  0.1417  0.6221 -0.3848]
expected return: 0.06623 volatility: 0.15228
MinVar weights: [ 0.7274  0.4946 -0.2045 -0.0175]
MinVar volatility: 0.11995
Min-TE weights: [0.25 0.25 0.25 0.25]
tracking error: 0.0

Note the minimum-tracking-error solution (with no return tilt, gamma=0) correctly returns exactly the benchmark weights — zero tracking error is achievable trivially by holding the benchmark itself, and the optimizer finds it.