Portfolio Optimization: Markowitz to Machine Learning

Module: Fintech AI | Difficulty: Advanced

Mean-Variance Optimization

Risk Parity

Black-Litterman

Kelly Criterion

import numpy as np
from scipy.optimize import minimize

class PortfolioOptimizer:
    def __init__(self, returns, cov_matrix, risk_aversion=1.0):
        self.returns = returns; self.cov = cov_matrix; self.lam = risk_aversion
    def mean_variance(self):
        n = len(self.returns)
        def neg_utility(w):
            ret = w @ self.returns
            risk = w @ self.cov @ w
            return -(ret - self.lam/2 * risk)
        constraints = [{'type': 'eq', 'fun': lambda w: np.sum(w) - 1}]
        bounds = [(0, 1) for _ in range(n)]
        w0 = np.ones(n) / n
        result = minimize(neg_utility, w0, bounds=bounds, constraints=constraints)
        return result.x
    def risk_parity(self):
        n = len(self.returns)
        def risk_budget_objective(w):
            portfolio_risk = np.sqrt(w @ self.cov @ w)
            risk_contributions = w * (self.cov @ w) / portfolio_risk
            target = portfolio_risk / n
            return np.sum((risk_contributions - target)**2)
        constraints = [{'type': 'eq', 'fun': lambda w: np.sum(w) - 1}]
        w0 = np.ones(n) / n
        result = minimize(risk_budget_objective, w0, constraints=constraints)
        return result.x

Research Insight: Markowitz optimization is notoriously sensitive to input estimates. Black-Litterman addresses this by combining market equilibrium with investor views, producing more stable allocations. Machine learning approaches improve estimation of expected returns and covariances.