📊 Random Matrix Theory for Indian Stock Market Portfolio Optimization

Applying Marchenko-Pastur theory to denoise correlation matrices for improved portfolio construction on NIFTY 50 stocks.

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🎯 Overview

Problem: Standard Markowitz mean-variance optimization is notoriously sensitive to estimation errors in the covariance matrix. When the number of assets ($N$) is comparable to the number of observations ($T$), the empirical correlation matrix contains significant noise that degrades portfolio performance.

Solution: Random Matrix Theory (RMT) provides a rigorous mathematical framework to separate signal from noise in correlation matrices. By filtering out eigenvalues that are statistically indistinguishable from random noise (using the Marchenko-Pastur distribution as null hypothesis), we construct more robust portfolios.

Key Result: RMT-denoised portfolios demonstrate superior out-of-sample risk-adjusted returns compared to traditional Markowitz optimization on NIFTY 50 stocks (2019–2024).

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📐 Mathematical Framework

Marchenko-Pastur Distribution

For an $N \times T$ random matrix with i.i.d. entries of variance $\sigma^2/T$, the eigenvalue density converges to:

$$f_{MP}(\lambda) = \frac{Q}{2\pi\sigma^2} \cdot \frac{\sqrt{(\lambda_+ - \lambda)(\lambda - \lambda_-)}}{\lambda}$$

where $Q = T/N$ and the support bounds are:

$$\lambda_{\pm} = \sigma^2\left(1 \pm \sqrt{Q}\right)^2$$

Denoising Procedure

1. Eigendecompose: $C = V \Lambda V^\top$
2. Replace noise eigenvalues ($\leq \lambda_+$) with their mean
3. Reconstruct: $\tilde{C} = V \tilde{\Lambda} V^\top$
4. Rescale to unit diagonal: $\hat{C} = D^{-1/2} \tilde{C} D^{-1/2}$

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📊 Results

Method	Sharpe Ratio	Max Drawdown	Annual Return
Raw Markowitz	Baseline	Baseline	Baseline
RMT-Denoised	✅ Improved	✅ Reduced	✅ Competitive
Equal Weight (1/N)	Reference	Reference	Reference

Visualizations

Eigenvalue Spectrum	Correlation Heatmaps	Efficient Frontier
MP distribution overlay	Raw vs Denoised	3-method comparison

Cumulative Returns	Rolling Sharpe	Drawdown
Out-of-sample performance	Time-varying risk-adjusted returns	Risk analysis

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🚀 Quick Start

1. Clone the Repository

git clone https://github.com/adivx/rmt-portfolio.git
cd rmt-portfolio

2. Install Dependencies

pip install -r requirements.txt

3. Run the Notebook

jupyter notebook notebooks/rmt_portfolio_optimization.ipynb

4. Or Run as Python Script

python src/backtester.py

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📁 Project Structure

├── README.md                              # This file
├── requirements.txt                       # Python dependencies
├── config.py                              # Global configuration
├── create_notebook.py                     # Notebook generator script
│
├── src/
│   ├── __init__.py
│   ├── data_fetcher.py                    # NSE data download & preprocessing
│   ├── rmt_analysis.py                    # Core RMT: MP distribution, denoising
│   ├── portfolio_optimizer.py             # Markowitz optimization (3 methods)
│   ├── backtester.py                      # Out-of-sample backtesting engine
│   └── visualization.py                   # Dark mode visualization suite
│
├── notebooks/
│   └── rmt_portfolio_optimization.ipynb   # Main presentation notebook
│
└── data/                                  # Cached stock data (auto-generated)

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🔧 Configuration

All parameters are centralized in config.py:

Parameter	Default	Description
NIFTY50_TICKERS	50 stocks	NSE stock universe
START_DATE	2019-01-01	Data start date
END_DATE	2024-12-31	Data end date
TRAIN_YEARS	3	Training window
RISK_FREE_RATE	7%	India 10Y bond yield
REBALANCE_FREQ	Monthly	Portfolio rebalancing frequency

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📦 Dependencies

• yfinance — Yahoo Finance data API
• numpy — Numerical computing
• scipy — Scientific computing & optimization
• pandas — Data manipulation
• matplotlib — Visualization
• seaborn — Statistical visualization

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📚 References

1. Marchenko, V.A. & Pastur, L.A. (1967). Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sbornik, 1(4), 457–483.
2. Laloux, L., Cizeau, P., Bouchaud, J.-P., & Potters, M. (1999). Random matrix theory and correlated financial data. Phys. Rev. Lett., 82, 1909.
3. Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.
4. Bouchaud, J.-P. & Potters, M. (2003). Theory of Financial Risk and Derivative Pricing. Cambridge University Press.

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👤 Author

Aditya Kumar

• GitHub: @adivx
• LinkedIn: Aditya Kumar
• Email: aditya.kumar.x9182@gmail.com

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📄 License

This project is licensed under the MIT License — see LICENSE for details.

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