Portfolio Optimization using Monte Carlo Methods & PyPortfolioOpt

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Overview

This notebook demonstrates dual optimization approaches for portfolio construction, combining Modern Portfolio Theory (MPT), Monte Carlo simulation, and both traditional (SciPy) and advanced (PyPortfolioOpt) optimization techniques. The analysis spans 7 years from January 2019 to December 2025 across 9 diversified stocks.

Objective

Construct optimal investment portfolios using multiple optimization strategies:

  1. Traditional Optimization: SciPy SLSQP (Sequential Least Squares Programming)
  2. Advanced Optimization: PyPortfolioOpt library with multiple strategies
  3. Practical Implementation: Discrete allocation for real-world share purchases

Methodology

1. Data Collection & Processing

  • Data Source: Yahoo Finance API (yfinance)
  • Stock Universe: 9 stocks (AEP, TPL, BALL, A, SO, TJX, UNH, AXON, JPM)
  • Time Period: January 1, 2019 - January 1, 2026
  • Returns Calculation:
    • Simple daily returns: \((r_t - r_{t-1})/r_{t-1}\)
    • Logarithmic returns: \(R_i = \log(r_i / r_{i-1})\)

2. Statistical Analysis

  • Annualized Returns: \(\mu_p = \sum_{i=1}^{n} w_i \mu_i \times 252\)
  • Portfolio Volatility: \(\sigma_p = \sqrt{w^T \Sigma w \times 252}\)
  • Sharpe Ratio: \(SR = \frac{\mu_p - r_f}{\sigma_p}\)
  • Correlation Matrix: Computed for diversification analysis

3. Monte Carlo Simulation

  • Iterations: 50,000 random portfolio combinations
  • Purpose: Map efficient frontier across risk-return spectrum
  • Random Weight Generation: \(W_i\) where \(\sum W_i = 1\)
  • Visualization: Color-coded by Sharpe Ratio using viridis colormap

4. Dual Portfolio Optimization Approach

4.1 Traditional Optimization (SciPy SLSQP)

  • Algorithm: Sequential Least Squares Programming
  • Objective: Maximize Sharpe Ratio (minimize negative Sharpe)
  • Constraints:
    • Portfolio weights sum to 100%: \(\sum w_i = 1\)
    • No short selling: \(0 \leq w_i \leq 1\)
  • Method: Gradient-based optimization

4.2 Advanced Optimization (PyPortfolioOpt)

Three optimization strategies implemented:

  1. Maximum Sharpe Ratio
    • Objective: Optimal risk-adjusted returns
    • Uses mean historical returns and sample covariance
  2. Minimum Volatility
    • Objective: Conservative risk minimization
    • Focuses on lowest portfolio variance
  3. Efficient Risk (Target Volatility)
    • Objective: Target-volatility portfolio construction
    • Customizable risk tolerance (e.g., 20% volatility)

4.3 Discrete Allocation

  • Portfolio Value: $100,000
  • Algorithm: Greedy allocation maximizing portfolio value utilization
  • Output: Actual number of shares to purchase per stock
  • Practical Feature: Calculates leftover cash after allocation

Technical Implementation

Core Libraries

  • NumPy/Pandas: Numerical computation and data manipulation
  • Matplotlib/Seaborn: Data visualization
  • yfinance: Historical stock data retrieval
  • SciPy: Traditional optimization (SLSQP algorithm)
  • PyPortfolioOpt: Advanced portfolio optimization
    • EfficientFrontier: Core optimization class
    • risk_models: Covariance estimation
    • expected_returns: Return forecasting
    • DiscreteAllocation: Share quantity calculation

Key Functions

Traditional Approach (SciPy)

  1. initialize_weights() - Generate random portfolio weights
  2. calculate_portfolio_return() - Compute annualized returns
  3. calculate_portfolio_volatility() - Calculate portfolio risk
  4. calculate_sharpe_ratio() - Risk-adjusted performance metric
  5. negative_sharpe_ratio() - Objective function for minimization
  6. optimize_portfolio() - SLSQP wrapper function

Advanced Approach (PyPortfolioOpt)

  1. optimize_portfolio_pypfopt() - Multi-method optimization wrapper
  2. calculate_discrete_allocation() - Convert weights to share quantities
  3. plot_pypfopt_efficient_frontier() - Visualize efficient frontier curve

Configuration Parameters

  • Trading Days: 252 per year
  • Random Seed: 3407 (reproducibility)
  • Monte Carlo Portfolios: 50,000
  • Risk-Free Rate: 0.036 (3.6% - 1yr US Treasury yield, Dec 20, 2025)
  • Portfolio Value: $100,000 (for discrete allocation)
  • Covariance Estimation: Sample covariance with 252-day annualization

Key Results & Visualizations

1. Data Exploration

  • Stock price evolution time series
  • Daily logarithmic returns visualization
  • Return distribution histograms (150 bins)
  • Correlation heatmap with blue color scheme

2. Statistical Analysis

  • Annualized returns per stock
  • Annualized volatility metrics
  • Correlation matrix revealing diversification opportunities
  • Covariance matrix (annualized)

3. Monte Carlo Efficient Frontier

  • 50,000 simulated portfolios plotted
  • Risk-return trade-off visualization
  • Color gradient showing Sharpe Ratio spectrum
  • Optimal portfolios marked:
    • Red star: SciPy SLSQP optimal portfolio
    • Lime diamond: PyPortfolioOpt optimal portfolio

4. Optimization Results

SciPy SLSQP Results

  • Optimal portfolio weights (% allocation per stock)
  • Expected annual return
  • Expected annual volatility
  • Sharpe Ratio

PyPortfolioOpt Max Sharpe Results

  • Cleaned weights (removes insignificant positions)
  • Expected annual return
  • Expected annual volatility
  • Sharpe Ratio

PyPortfolioOpt Min Volatility Results

  • Conservative weight allocation
  • Minimized portfolio risk
  • Expected return and Sharpe Ratio

6. Comparative Analysis

Comparison table showing:

  • Method: SciPy SLSQP vs PyPortfolioOpt (Max Sharpe) vs PyPortfolioOpt (Min Volatility)
  • Expected Return: Annualized percentage
  • Volatility: Annualized risk measure
  • Sharpe Ratio: Risk-adjusted performance

7. Visualization Suite

Portfolio Weight Visualizations

  1. Pie Charts:
    • SciPy optimal allocation
    • PyPortfolioOpt optimal allocation (filtered for non-zero weights)
  2. Bar Charts:
    • SciPy weights with value labels
    • Side-by-side comparison (SciPy vs PyPortfolioOpt)
  3. Efficient Frontier Curves:
    • Monte Carlo scatter plot with both optimal portfolios
    • PyPortfolioOpt efficient frontier curve with:
      • Individual asset positions
      • Max Sharpe portfolio (red star)
      • Min Volatility portfolio (lime diamond)

8. Detailed Summary Statistics

Comprehensive table including:

  • Stock ticker
  • SciPy optimal weight (5 decimal precision)
  • PyPortfolioOpt optimal weight
  • Individual stock annual return
  • Individual stock annual volatility
  • Individual stock Sharpe Ratio

Key Insights

Dual Optimization Comparison

  • SciPy SLSQP:
    • Traditional gradient-based approach
    • Fast convergence
    • Suitable for standard constraints
  • PyPortfolioOpt:
    • Modern library with multiple strategies
    • Built-in weight cleaning
    • Discrete allocation functionality
    • Multiple optimization objectives available

Diversification Benefits

  • Correlation matrix reveals varying inter-stock relationships
  • Multi-sector exposure (utilities, retail, healthcare, finance, technology)
  • Portfolio volatility reduced through diversification
  • Optimal portfolios positioned on efficient frontier

Practical Implementation

  • Discrete allocation bridges theory and practice
  • Converts continuous weights to integer share quantities
  • Maximizes portfolio value utilization
  • Calculates remaining cash for liquidity

Risk-Return Trade-offs

  • Maximum Sharpe: Best risk-adjusted returns
  • Minimum Volatility: Conservative approach with lower risk
  • Clear visualization of efficient frontier
  • Multiple optimal portfolios for different investor preferences

Output Deliverables

  1. Visualizations (10+ charts):
    • Stock price evolution
    • Returns distribution histograms
    • Correlation heatmap
    • Monte Carlo efficient frontier (with dual optimal portfolios)
    • PyPortfolioOpt efficient frontier curve
    • Pie charts (2 methods)
    • Bar charts (individual and comparative)
    • Weight comparison charts
  2. Statistical Reports:
    • Annualized returns and volatilities
    • Covariance and correlation matrices
    • Optimal portfolio allocations (2 methods)
    • Performance metrics comparison
    • Detailed summary statistics
  3. Practical Outputs:
    • Discrete share allocation
    • Exact number of shares per stock
    • Investment amounts per position
    • Remaining cash calculation

Technical Advantages

PyPortfolioOpt Benefits

  • Flexibility: Multiple optimization methods
  • Robustness: Built-in weight cleaning and validation
  • Practicality: Discrete allocation for real trading
  • Extensibility: Easy to add custom objectives
  • Visualization: Built-in plotting capabilities

Comparison Framework

  • Side-by-side method comparison
  • Consistent performance metrics
  • Visual weight comparisons
  • Comprehensive summary tables

References

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