Portfolio Optimization Framework: Markowitz Efficient Frontier Implementation

🎯 Project Overview

Advanced portfolio optimization engine implementing Modern Portfolio Theory (Markowitz, 1952) for long-only equity portfolios. This project demonstrates quantitative finance principles, numerical optimization techniques, and data-driven asset allocation strategies applicable to institutional portfolio management.

Key Achievement: Constructed optimal portfolios achieving 1.313 Sharpe Ratio (33% improvement over minimum variance) through systematic risk-return optimization across 30 random US equities.

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📊 Performance Summary

Metric	Minimum Volatility	Maximum Sharpe	Improvement
Annual Return	9.77%	19.64%	+101%
Annual Volatility	11.13%	14.96%	+34%
Sharpe Ratio	0.878	1.313	+50%
Number of Assets	15	8	-47%
Largest Allocation	22.28%	42.14%	-
Top 5 Concentration	62.7%	91.2%	-

Data Period: 10 years of monthly returns (2015-2025)

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🚀 Key Features

• Dual Optimization Objectives:

  • Global Minimum Variance Portfolio (GMV)
  • Maximum Sharpe Ratio Portfolio (Tangency)

• Robust Data Processing:

  • Handles missing data and infinite returns (price anomalies)
  • Annualized covariance matrix estimation from monthly returns
  • Outlier detection and imputation

• Advanced Visualization:

  • Efficient Frontier with 10,000 Monte Carlo simulations
  • Dual-axis Risk-Return and Sharpe Ratio plots
  • Portfolio composition analysis (pie charts)
  • Comparative performance dashboards

• Numerical Optimization:

  • Sequential Least Squares Programming (SLSQP)
  • Long-only constraints (w ≥ 0, Σw = 1)
  • Iterative efficient frontier computation

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🛠️ Technical Stack

Core Technologies:

• Python 3.12: Primary language
• NumPy: Matrix operations and linear algebra
• Pandas: Time series data manipulation
• SciPy: Constrained optimization (scipy.optimize.minimize)
• Matplotlib/Plotly: Data visualization

Mathematical Framework:

• Quadratic Programming for portfolio variance minimization
• Mean-Variance Optimization (MVO)
• Covariance matrix estimation and annualization
• Sharpe Ratio maximization via negative objective function

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📈 Visualizations

Efficient Frontier

10,000 simulated portfolios with optimal portfolios highlighted

Portfolio Compositions

*Asset allocation breakdown for both optimal portfolios*

Performance Dashboard

Comprehensive comparative metrics and top holdings analysis

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🧮 Methodology

1. Data Preprocessing

Monthly returns with inf/NaN handling:

returns = prices.pct_change().dropna()
returns = returns.replace([np.inf, -np.inf], np.nan).fillna(returns.median())

2. Covariance Matrix Estimation

Annualized covariance from monthly data:

sigma_annual = returns.cov() * 12

3. Optimization Problem

Minimum Volatility:

Minimize: σ²ₚ = wᵀ Σ w
Subject to: Σwᵢ = 1, wᵢ ≥ 0

Maximum Sharpe:

Maximize: SR = (μₚ − r_f) / σₚ
Subject to: Σwᵢ = 1, wᵢ ≥ 0

Where:

• w = portfolio weights
• Σ = covariance matrix
• μₚ = portfolio expected return
• r_f = risk-free rate (3% assumed)

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🔬 Key Insights

Risk-Return Tradeoff

• +101% return for only +34% volatility increase (Min Vol → Max Sharpe)
• Efficient diversification: 15-asset Min Vol vs 8-asset Max Sharpe
• Concentration trade-off: Higher Sharpe achieved through focused bets (91% top 5)

Optimization Convergence

• SLSQP algorithm converged in <1s for 30-asset universe
• Monte Carlo validation: 10,000 portfolios confirm analytical frontier
• Numerical stability verified via eigenvalue decomposition (PSD matrix)

Practical Implications

• Min Vol suitable for conservative risk mandates (pension funds)
• Max Sharpe ideal for performance-focused strategies (hedge funds)
• Concentration risk management critical above 40% single-asset allocation

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📚 Theoretical Foundation

This implementation is based on:

1. Markowitz, H. (1952). "Portfolio Selection." Journal of Finance 7(1): 77-91.
2. Sharpe, W.F. (1966). "Mutual Fund Performance." Journal of Business 39(S1): 119-138.
3. Merton, R.C. (1972). "An Analytic Derivation of the Efficient Portfolio Frontier." Journal of Financial and Quantitative Analysis 7(4): 1851-1872.

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📧 Contact

Cauê Pileckas D'Agostinho
Computer Engineering Student @ Insper | Quant Trading Enthusiast 📍 São Paulo, Brazil
🔗 LinkedIn | GitHub

Interests: Algorithmic trading, machine learning in finance, derivatives pricing

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This project was developed as part of a comprehensive study in quantitative portfolio management and demonstrates production-ready code suitable for institutional asset management applications.