Portfolio Optimization using Monte Carlo Methods

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Overview

This notebook implements Modern Portfolio Theory (MPT) combined with Monte Carlo simulation to construct an optimal investment portfolio. The analysis covers a 7-year period from January 2019 to December 2025, utilizing 9 stocks across various sectors.

Objective

Maximize risk-adjusted returns (Sharpe Ratio) through optimal portfolio allocation while respecting diversification constraints.

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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
• Data Type: Daily adjusted close prices
• Returns Calculation: Logarithmic returns using \(R_i = \log(r_i / r_{i-1})\)

2. Statistical Analysis

• Annualized Returns: \(\mu^{Annual} = 252 \times \mu\)
• Annualized Covariance: \(\sigma_{ij}^{Annual} = 252 \times \sigma_{ij}\)
• Annualized Volatility: \(\sigma^{Annual} = \sqrt{252} \times \sigma\)
• Correlation Matrix: Computed to assess diversification benefits

3. Monte Carlo Simulation

• Iterations: 50,000 random portfolio combinations
• Weight Generation: Random weights \(W_i\) where \(\sum W_i = 1\)
• Portfolio Return: \(\mu_P^{Annual} = \sum W_i \mu_i^{Annual}\)
• Portfolio Volatility: \(\sigma_P^{Annual} = \sqrt{\sum_{ij} W_i W_j \sigma_{ij}^{Annual}}\)
• Purpose: Map the efficient frontier across risk-return spectrum

4. Portfolio Optimization

• Objective Function: Maximize Sharpe Ratio
• Sharpe Ratio: \((R_p - R_f) / \sigma_p\)
• Risk-Free Rate: 3.6% (1-year US Treasury yield as of Dec 20, 2025)
• Algorithm: SLSQP (Sequential Least Squares Programming)
• Constraints:
  • Portfolio weights sum to 100%
  • No short selling: \(0 \leq W_i \leq 1\)

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Technical Implementation

Libraries Used

• NumPy: Numerical computations
• Pandas: Data manipulation
• Matplotlib/Seaborn: Visualization
• yfinance: Stock data retrieval
• SciPy: Optimization algorithms

Key Functions

1. download_stock_data() - Historical price data retrieval
2. calculate_returns() - Compute daily and logarithmic returns
3. initialize_weights() - Generate random portfolio weights
4. portfolio_statistics() - Calculate return, volatility, and Sharpe ratio
5. generate_portfolios() - Monte Carlo simulation engine
6. optimize_portfolio() - SLSQP optimization wrapper

Configuration Parameters

• Trading Days: 252 per year
• Random Seed: 3407 (for reproducibility)
• Monte Carlo Portfolios: 50,000
• Risk-Free Rate: 0.036 (3.6%)

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Key Results & Visualizations

1. Data Visualization

• Stock price evolution over time
• Daily logarithmic returns time series
• Distribution histograms of returns

2. Statistical Analysis

• Annualized returns for each stock
• Annualized volatility metrics
• Correlation heatmap showing inter-stock relationships

3. Efficient Frontier

• 50,000 simulated portfolios plotted on risk-return space
• Color-coded by Sharpe Ratio (viridis colormap)
• Optimal portfolio marked with red star
• Clear visualization of risk-return trade-off

4. Optimal Portfolio

• Allocation: Pie chart and bar chart showing optimal weights
• Performance Metrics:
  • Expected Annual Return
  • Expected Annual Volatility
  • Sharpe Ratio

5. Summary Statistics

Comprehensive table including:

• Individual stock optimal weights
• Annual returns per stock
• Annual volatility per stock
• Individual Sharpe ratios

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Key Insights

Diversification Benefits

• Correlation matrix reveals varying relationships between stocks
• Portfolio volatility reduced through diversification
• Multi-sector exposure (utilities, retail, healthcare, finance, technology)

Risk-Return Trade-off

• Monte Carlo simulation clearly maps efficient frontier
• Optimal portfolio positioned on efficient frontier
• Maximum risk-adjusted return achieved for given volatility level

Optimization Success

• SLSQP algorithm successfully converges
• Constraints satisfied (weights sum to 1, no short positions)
• Optimal allocation maximizes Sharpe Ratio

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Output Deliverables

1. Visualizations:
   • Stock price evolution chart
   • Returns distribution histograms
   • Correlation heatmap
   • Efficient frontier scatter plot
   • Portfolio allocation pie chart
   • Portfolio weights bar chart
2. Statistical Reports:
   • Annualized returns table
   • Volatility metrics
   • Covariance matrix
   • Optimal portfolio allocation
   • Performance metrics summary
3. Optimal Portfolio:
   • Individual stock weights
   • Expected return and volatility
   • Sharpe Ratio
   • Complete performance summary

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Reference

GitHub Repository: Monte-Carlo-Portfolio-Allocation

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