Markowitz Portfolio Optimization

A Monte Carlo approach to mean-variance portfolio optimization.

Objective

This project implements a simple Markowitz portfolio optimization using Monte Carlo simulation. It identifies optimal long-only portfolios (maximum Sharpe ratio and minimum volatility) from a universe of 5 stocks, then backtests the resulting allocations against an equal-weight portfolio and the S&P 500 benchmark.

Asset Universe

Ticker	Sector
AAPL	Technology
MSFT	Technology
JNJ	Healthcare
PG	Consumer Staples
JPM	Financials

Benchmark: SPY (S&P 500 ETF)

Data

• Source: Yahoo Finance (adjusted close prices)
• Period: January 2018 to present (~7 years)
• Frequency: Daily

Methodology

Returns Computation

• Daily simple returns: r_t = (P_t / P_{t-1}) - 1
• Annual return: Compounded (CAGR), not arithmetic mean
  • CAGR = (cumulative_return)^(1/years) - 1
• Annual volatility: std(daily_returns) * sqrt(252)
• Sharpe ratio: (CAGR - Rf) / volatility
  • Risk-free rate: 3% annual

Portfolio Optimization

• Monte Carlo simulation of 10,000 random portfolios
• Constraints: long-only (weights >= 0), fully invested (weights sum to 1)
• Identification of:
  • Max Sharpe portfolio: highest risk-adjusted return
  • Min Volatility portfolio: lowest annualized standard deviation

Backtesting

• Initial investment: $10,000
• Strategies compared:
  • Max Sharpe portfolio
  • Min Volatility portfolio
  • Equal-weight portfolio (20% each)
  • SPY benchmark
• Metrics: total return, CAGR, volatility, Sharpe ratio, maximum drawdown

Project Structure

portfolio-optimization/
├── notebooks/
│   └── portfolio_optimization.ipynb
├── requirements.txt
└── README.md

Requirements

numpy
pandas
yfinance
matplotlib

Usage

pip install -r requirements.txt
jupyter notebook notebooks/portfolio_optimization.ipynb

Data downloads automatically from Yahoo Finance.

Limitations

• Optimization is based on historical data; future performance may differ.
• Monte Carlo sampling approximates the efficient frontier but may not find the global optimum.
• Transaction costs and rebalancing frictions are not modeled.
• Results are sensitive to the sample period chosen.

License

MIT License

References

• Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.