Numerical Optimization with Python - Portfolio Optimization Project

Authors

• Eldad Uzman
• Tom Ron
• Liron Zarhay

Overview

This project implements advanced portfolio optimization techniques using numerical optimization methods in Python. The focus is on Conditional Value-at-Risk (CVaR) optimization and Mean-Variance optimization for financial portfolio management, with additional features for risk attribution and walk-forward analysis.

Features

• CVaR Optimization: Implements Rockafellar & Uryasev's linear CVaR minimization approach
• Mean-Variance Optimization: Traditional Markowitz portfolio optimization
• Risk Attribution: Sharpe-style risk contribution analysis for portfolio components
• Walk-Forward Analysis: Time-series backtesting with rolling optimization windows
• Top-N Momentum Strategy: Simple momentum-based portfolio strategy for comparison
• Interactive Visualizations: Plotly-based charts for risk attribution and performance analysis
• Performance Profiling: Built-in profiling tools for optimization performance analysis

Core Algorithms

CVaR Optimization

The project implements Rockafellar & Uryasev's linear formulation for CVaR minimization:

def optimise_cvar(R, beta=0.95, target_return=None, short_cap=None):
    """
    Rockafellar-&-Uryasev linear CVaR minimisation.
    
    Parameters:
    - R: Returns DataFrame
    - beta: Confidence level (0, 1)
    - target_return: Optional return constraint
    - short_cap: Optional short selling constraint
    """

Mean-Variance Optimization

Traditional Markowitz portfolio optimization:

def optimize_mean_variance(R, target_return=None, short_cap=None):
    """
    Mean-variance optimisation with optional constraints.
    """

Risk Attribution

Sharpe-style risk contribution analysis:

def cvar_risk_contributions(R, weights, beta=0.95):
    """Sharpe-equivalent attribution for portfolio CVaR."""
    
def variance_risk_contributions(cov, weights):
    """Sharpe (Euler) attribution for portfolio variance."""

Data Format

The project expects stock price data in CSV format with:

• Date column (automatically detected)
• Price data in wide format (one column per stock)
• Or long format with columns: date, company, price

Example data structure:

date,MSFT,GOOGL,AAPL,...
2015-02-01,40.12,520.45,120.34,...
2015-02-02,40.89,525.12,121.67,...
...

Research Background

This project is based on academic research in portfolio optimization, particularly:

• Rockafellar & Uryasev (1999): CVaR optimization
• Markowitz (1952): Mean-variance optimization
• Sharpe value theory for risk attribution