Portfolio Optimization and Analysis

Modern Portfolio Theory (MPT), Monte Carlo simulation, and advanced risk analytics for quantitative portfolio management and risk measurement.

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Overview

#	Notebook	Details	Implementation
1	Portfolio Optimization using Monte Carlo Methods	Traditional portfolio optimization using SciPy SLSQP algorithm with Monte Carlo simulation to map the efficient frontier.	Docs
2	Portfolio Optimization - PyPortfolioOpt	Advanced portfolio optimization using PyPortfolioOpt library with dual optimization approach and discrete allocation functionality.	Docs
3	Portfolio Risk Analytics - VaR & ES	Calculate, compare, and backtest ~17 different VaR and Expected Shortfall models for market risk.	Docs

Details

1. Portfolio Optimization - Monte Carlo

Notebook: 01_portfolio_optimization_monte_carlo.ipynb

View - Github

Overview

Traditional portfolio optimization using SciPy SLSQP algorithm with Monte Carlo simulation to map the efficient frontier.

Key Features

• 50,000 Monte Carlo simulations for efficient frontier mapping
• SLSQP optimization maximizing Sharpe Ratio
• Logarithmic returns analysis
• Correlation analysis for diversification insights
• Comprehensive visualizations: efficient frontier, allocation charts, weight distributions

Methodology

• Objective: Maximize Sharpe Ratio
• Algorithm: Sequential Least Squares Programming (SLSQP)
• Constraints:
  • Weights sum to 100%
  • No short selling (0 ≤ w_i ≤ 1)
• Risk-Free Rate: 3.6% (1-year US Treasury yield)

Mathematical Framework

\(\mu_P = \sum_{i=1}^{n} w_i \mu_i \times 252\)

\[\sigma_P = \sqrt{w^T \Sigma w \times 252}\] \[SR = \frac{\mu_P - r_f}{\sigma_P}\]

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2. Portfolio Optimization - PyPortfolioOpt

Notebook: 02_portfolio_optimization_pyportfolioopt.ipynb

View - Github

Overview

Advanced portfolio optimization using PyPortfolioOpt library with dual optimization approach and discrete allocation functionality.

Key Features

• Dual optimization methods: SciPy SLSQP + PyPortfolioOpt
• Multiple optimization strategies: Max Sharpe, Min Volatility, Efficient Risk
• Discrete allocation: Convert continuous weights to integer shares
• Side-by-side comparison: Traditional vs. advanced methods
• Practical implementation: Real-world portfolio construction with $100,000 capital

Optimization Strategies

1. Maximum Sharpe Ratio
   • Optimal risk-adjusted returns
   • Best performance per unit of risk
2. Minimum Volatility
   • Conservative risk minimization
   • Lowest portfolio variance
3. Efficient Risk
   • Target volatility approach
   • Customizable risk tolerance

3. Portfolio Risk Analytics - VaR & ES

Notebook: 03_portfolio_risk_analytics_var_and_es.ipynb

View - Github

Overview

Calculate, compare, and backtest ~17 different VaR and Expected Shortfall models for market risk.

Key Features

• 17 VaR/ES methods: Parametric, non-parametric, and advanced historical
• Kupiec POF backtesting: Statistical validation of all methods
• Rolling window analysis: 250-day estimation windows
• Advanced methods: Bootstrapped, age-weighted, volatility-weighted, correlation-weighted
• Comprehensive visualizations: Heatmaps, comparison charts, temporal analysis

VaR/ES Methods Implemented

Parametric Methods (6)

1. Normal Distribution VaR/ES
2. Student-t Distribution VaR/ES
3. EWMA VaR (λ = 0.94)
4. Cornish-Fisher VaR

Non-Parametric Methods (4)

1. Historical Simulation VaR/ES
2. Kernel Density Estimation (KDE) VaR/ES

Advanced Historical Simulation (10)

1. Bootstrapped Historical Simulation (BHS) - 1,000 samples
2. Age-Weighted Historical Simulation (AWHS) - λ = 0.98
3. Volatility-Weighted Historical Simulation (VWHS)
4. Correlation-Weighted Historical Simulation (CWHS)
5. Filtered Historical Simulation (FHS) - Optional GARCH

Backtesting Framework

• Kupiec Proportion of Failures (POF) Test
• Likelihood Ratio Statistic: LR ~ χ²(1)
• Decision Rule: p-value < 0.05 → Reject model
• Metrics: Breach rate, LR statistic, p-value, adequacy assessment

Features

Portfolio Optimization

• Dual Optimization Approach: Compare traditional (SciPy) vs. modern (PyPortfolioOpt) methods
• Monte Carlo Simulation: 50,000 random portfolios mapping efficient frontier
• Discrete Allocation: Convert theoretical weights to actual share quantities
• Multiple Strategies: Max Sharpe, Min Volatility, Efficient Risk
• Comprehensive Analysis: Returns, volatility, correlations, Sharpe ratios

Risk Analytics

• 17 VaR/ES Methods: Complete methodology comparison
• Statistical Validation: Kupiec POF backtesting
• Advanced Techniques: Bootstrap, KDE, age-weighted, volatility-weighted
• Rich Visualizations: Heatmaps, comparison charts, temporal analysis
• Rolling Windows: Dynamic 250-day estimation

Repository

Source Code