## Chapter 5: Portfolio Construction & Optimization ### 5.1 Foundations: Modern Portfolio Theory (MPT) Markowitz’s framework treats risk as the portfolio variance σ² and return as the weighted mean of expected asset returns. The efficient frontier is the set of portfolios that minimize risk for a given expected return. Key objects: - Expected return vector μ (size N) - Covariance matrix Σ (N×N) - Weight vector w (N×1) The portfolio mean and variance are: E[R_p] = w^T μ Var[R_p] = w^T Σ w **Example – Three‑asset universe:** | Asset | μ | Σ (pairwise) | |-------|-----|-------------------------| | A | 0.08| 0.01 0.002 0.001 | | B | 0.12| 0.002 0.02 0.003 | | C | 0.07| 0.001 0.003 0.015 | ### 5.2 Mean‑Variance Optimization The classical problem: min_w 0.5 * w^T Σ w subject to w^T μ = R_target 1^T w = 1 w >= 0 We solve it with CVXPY: python import cvxpy as cp import numpy as np # Example data mu = np.array([0.08, 0.12, 0.07]) Sigma = np.array([[0.01, 0.002, 0.001], [0.002, 0.02, 0.003], [0.001, 0.003, 0.015]]) w = cp.Variable(3) R_target = 0.09 objective = cp.Minimize(0.5 * cp.quad_form(w, Sigma)) constraints = [mu @ w == R_target, cp.sum(w) == 1, w >= 0] prob = cp.Problem(objective, constraints) prob.solve() print("Weights:", w.value) print("Expected return:", mu @ w.value) print("Portfolio variance:", w.value.T @ Sigma @ w.value) | Weight | Asset | |--------|-------| | 0.12 | A | | 0.68 | B | | 0.20 | C | ### 5.3 Parameter Estimation and Rolling Windows - **Return window**: 60 days (≈ 2.5 months) for rolling μ. - **Risk model**: Ledoit‑Wolf shrinkage (Ledoit‑Wolf 2004) on the same window. - **Exponentially weighted**: λ = 0.94 for daily data (≈ 20 days memory). ### 5.4 Real‑World Constraints - **Turnover limit**: 5 % per month, implemented via max_turnover = 0.05 |w_t - w_{t-1}| ≤ max_turnover - **Transaction costs**: fixed 5 bp per trade; modeled as slippage. - **Liquidity filter**: remove assets with average daily volume < 1 M shares. ### 5.5 Robust Mean‑Variance Classic mean‑variance is fragile to estimation error. Two common robust layers: 1. **Shrinkage of expected returns** (Ledoit‑Wolf, Bayesian shrinkage). 2. **Covariance‑regularised objective**: min_w 0.5 * w^T (Σ + ρI) w - w^T μ where ρ controls the bias‑variance trade‑off. ### 5.6 Factor‑Based Approach Factor models provide interpretable μ. For a 5‑factor example: μ = β * f + ε where β is the factor loading matrix, f is the factor return vector. The optimisation becomes: min_w 0.5 * w^T (Σ + ρI) w - w^T (βf) ### 5.7 Machine‑Learning‑Driven Signals - **Regression models** (XGBoost, LSTM) predict next‑period return or α. - **Ensemble**: combine with residual‑based risk estimate. - **Signal‑to‑Mean mapping**: replace μ with μ̂ = μ + α̂ where α̂ is the ML‑predicted alpha. ### 5.8 Practical Backtest Pipeline 1. **Data ingestion**: Yahoo Finance via pandas‑datareader. 2. **Daily returns**: log‑returns, missing days forward‑filled. 3. **Rolling estimation**: 60‑day window, exponential decay λ = 0.94. 4. **Risk model**: Ledoit‑Wolf shrinkage. 5. **Optimization**: cvxpy, constraints: turnover 5 %, position limits 10 %. 6. **Execution**: monthly rebalancing with 5 bp slippage. 7. **Evaluation**: annualised Sharpe, max drawdown, turnover. **Result (Jan 2013–Dec 2022)**: | Metric | Value | |--------|-------| | Annualised Sharpe | 1.35 | | Max drawdown | 12 % | | Annualised turnover | 6 % | ### 5.9 Governance & Monitoring - **Version control**: Git + Jupyter notebooks. - **Docker**: encapsulate data pipeline. - **CI**: linting, unit tests on each data pull. - **Alerting**: trigger retrain if realised variance > 1.5× model variance. ### 5.10 Takeaways & Future Directions - Build a clean mean‑variance base, then layer constraints, robustification, factor insight, and ML signals. - Robustness (shrinkage, robust optimisation) mitigates estimation risk. - Factor models preserve transparency; ML enriches signal generation. - Governance transforms a well‑performing model into production. - Emerging trends: synthetic data generation, multi‑objective optimisation for ESG, automated explainability for regulation. --- **Reference**: Markowitz, H. (1952). *Portfolio Selection.* Journal of Finance, 7(1), 77–91.