📊 Factor-Based Portfolio Optimization

Demonstrating covariance estimation for minimum variance portfolios using Fama-French 5-factor model

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🎯 Overview

Implementation of factor-based covariance estimation for constructing long-only portfolios with constraints, comparing against traditional empirical approach. Out-of-sample backtest demonstrates statistical superiority in risk-adjusted returns (Sharpe +11.4%) with lower volatility and drawdown.

Problem: Sample covariance matrices are noisy with short windows (curse of dimensionality).
Solution: Structural decomposition via Fama-French factors reduces parameters from 95k → 2.6k.

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📈 Key Results (24-month rolling, 2015-2025)

Metric	Sample Cov	Factor-Based	Improvement
Sharpe Ratio	0.892	0.994	+11.4% 🏆
Volatility	15.29%	13.70%	-10.4% ✓
Max Drawdown	-23.02%	-16.86%	-26.8% ✓
ENB (Diversification)	8.41	10.09	+20.0% ✓
Monthly Turnover	15.09%	11.23%	-25.6% ✓

• ✅ Factor model wins 5 out of 6 metrics
• ✅ Superior CAGR despite similar mean return (volatility drag effect)
• ✅ Lower transaction costs (TC impact: 0.18% → 0.13%)

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🔬 Methodology

1. Covariance Estimation

Empirical (Baseline)

Σ_emp = (1/T) X'X  # Sample covariance
# Parameters: N(N-1)/2 = 95,046 (436 stocks)

Factor-Based (Proposed)

r_i = α_i + β_i'F + ε_i  # Fama-French 6-factor model
Σ_FF = B Σ_F B' + Δ      # Structured covariance
# Parameters: 436×6 + 21 + 436 = 2,673 (97% reduction)

Factors: Market, Size (SMB), Value (HML), Profitability (RMW), Investment (CMA), Momentum

2. Portfolio Optimization

Objective:

min w'Σw  s.t.  Σw_i = 1, 0 ≤ w_i ≤ 15%

Algorithm: SLSQP (Sequential Least Squares Programming) Constraints: Long-only + position caps

3. Backtest Framework

• Method: Rolling window out-of-sample
• Lookback: 24 months (sweet spot for factor model)
• Rebalancing: Monthly (101 periods)
• Universe: 30 random stocks (S&P 500 components)
• Risk-free adjustment: US 3-month T-Bill

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💻 Technical Stack

# Core
pandas, numpy, scipy.optimize

# Factor model regression
sklearn.linear_model (OLS with regularization)

# Visualization
matplotlib, seaborn

# Data sources
- Kenneth French Data Library (factors)
- Bloomberg (returns, market caps)

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🏗️ Project Structure

Project3.ipynb
├── Data Loading & Preprocessing
│   ├── Monthly returns (30 stocks)
│   ├── Fama-French 6 factors + Momentum
│   └── Risk-free rate adjustment
│
├── Covariance Estimation
│   ├── Sample covariance (empirical)
│   ├── Factor model regression (β estimation)
│   └── Structured covariance (B Σ_F B' + Δ)
│
├── Portfolio Optimization
│   ├── Closed-form MVP (unconstrained)
│   └── Constrained optimization (long-only + caps)
│
├── Out-of-Sample Backtest
│   ├── Rolling window (24 months)
│   ├── Performance metrics (Sharpe, vol, DD, ENB)
│   └── Transaction cost analysis
│
└── Visualization
    └── Comprehensive dashboard (9 panels)

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📊 Visualizations

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🚀 How to Run

# 1. Clone & install dependencies
pip install pandas numpy scipy matplotlib seaborn

# 2. Run notebook
jupyter notebook Project3.ipynb

# 3. Key functions
backtest_results, w_emp, w_ff = backtest_mvp_out_of_sample(
    R_m, df_factors, 
    lookback_months=24,
    long_only=True, 
    cap=0.15
)

metrics = analyze_oos_complete(backtest_results, w_emp, w_ff, 
                                rf_series=df_factors['RF'])
print_oos_report(metrics, lookback_months=24)

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🎓 Key Insights

1. Why Does Factor Model Win?

Curse of Dimensionality:

• Sample cov: 95k parameters / 24 observations = 0.00025 obs/parameter
• Factor model: 2.6k parameters / 24 observations = 0.009 obs/parameter (36× more robust)

Result: Factor model is more stable with short windows (18-30 months).

2. Trade-off: Lookback Window

Window	Sample Cov	Factor Model	Winner
12m	❌ Breaks	❌ Unstable	None
24m	Noisy	✅ Robust	Factor 🏆
36m	Stable	Slower	Sample slightly

Rule: Factor models excel with short-to-medium windows (18-30m).

3. Diversification (ENB)

ENB = 1/HHI measures "effective number of bets":

• Factor: 10.09 = equivalent to 10 independent bets
• Sample: 8.41 = equivalent to 8.4 independent bets
• 20% more diversification → explains lower volatility

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📚 Academic References

1. Fama-French (2015): "A five-factor asset pricing model" - Journal of Financial Economics
2. Ledoit-Wolf (2004): "Honey, I Shrunk the Sample Covariance Matrix" - Portfolio Management
3. Connor-Korajczyk (1988): "Risk and return in an equilibrium APT" - JFE
4. Markowitz (1952): "Portfolio Selection" - Journal of Finance

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👨‍💻 Author

Cauê Pileckas D'Agostinho - Quantitative Finance & Machine Learning
LinkedIn | GitHub

Computer Engineering Student @ Insper | Quant Trading Enthusiast

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