Portfolio Optimization (Markowitz Mean–Variance)

This project implements a classical Markowitz mean–variance portfolio optimization in Python.
It constructs the efficient frontier, identifies the minimum-variance and maximum-Sharpe (tangency) portfolios, and visualizes allocations and trade-offs.

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Project structure

portfolio-optimization/ ├─ main.ipynb # Jupyter notebook with full workflow ├─ requirements.txt # Dependencies ├─ README.md # Project overview ├─ LICENSE # MIT License ├─ .gitignore # Ignore build/checkpoint files └─ plots/ # Figures saved automatically

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Methodology

1. Data collection

   • Downloaded adjusted close prices from Yahoo Finance via yfinance.

2. Returns & covariance

   • Daily log returns → annualized expected returns (μ) and covariance matrix (Σ).

3. Optimization

   • Minimum-variance portfolio (lowest volatility).
   • Maximum-Sharpe portfolio (tangency portfolio, relative to risk-free rate).
   • Efficient frontier constructed by sweeping target returns.

4. Visualization

   • Efficient frontier with Capital Market Line.
   • Portfolio weights bar charts.

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Results & Interpretation

Data & setup

• Tickers: AAPL, MSFT, GOOGL, AMZN, META, JPM, XOM, SPY
• Sample: 2020-01-01 to 2025
• Risk-free rate: 3% annualized

Optimal portfolios

• Minimum-Variance Portfolio

  • Expected return: 14.02%
  • Volatility: 20.85 %
  • Weights: see plots/weights_minvar.png

• Max-Sharpe Portfolio

  • Expected return: 20.63 %
  • Volatility: 25.70 %
  • Sharpe ratio: 0.686
  • Weights: see plots/weights_maxsharpe.png

Figures

• Efficient Frontier:
• Portfolio Weights: ,

Interpretation

• The efficient frontier illustrates the classic trade-off between expected return and risk (σ).
• The minimum-variance portfolio concentrates on stable, low-volatility assets.
• The tangency portfolio maximizes the Sharpe ratio, balancing risk and return.
• Under no-short constraints, allocations are bounded between 0–1, producing a conservative frontier compared to unconstrained optimization.

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Extensions

• Use covariance shrinkage estimators (e.g., Ledoit–Wolf).
• Run rolling-window analysis for time-varying portfolios.
• Add sector/weight caps or transaction cost modeling.
• Compare with alternative risk measures (CVaR, MAD).

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Development log

• 2025-10-08: Initial project setup, data pipeline, returns/covariance.
• 2025-10-09: Implemented optimization (min-var, max-Sharpe) and efficient frontier.
• 2025-10-10: Added visualizations, results interpretation, and README polish.

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License

This project is licensed under the MIT License.