📈 cvxcla - Critical Line Algorithm for Portfolio Optimization

📋 Overview

cvxcla is a Python package that implements the Critical Line Algorithm (CLA) for portfolio optimization. The CLA efficiently computes the entire efficient frontier for portfolio optimization problems with linear constraints and bounds on the weights.

The Critical Line Algorithm was introduced by Harry Markowitz in The Optimization of Quadratic Functions Subject to Linear Constraints and further described in his book Portfolio Selection.

The algorithm is based on the observation that the efficient frontier is a piecewise linear function when expected return is plotted against expected variance. The CLA computes the turning points (corners) of the efficient frontier, allowing for efficient representation of the entire frontier.

I gave the plenary talk at EQD's Singapore conference.

✨ Features

• Efficient computation of the entire efficient frontier
• Support for linear constraints and bounds on portfolio weights
• Multiple implementations based on different approaches from the literature
• Visualization of the efficient frontier using Plotly
• Computation of the maximum Sharpe ratio portfolio
• Fully tested and documented codebase

🚀 Installation

Using pip

bash pip install cvxcla

Development Setup

To set up a development environment:

1. Clone the repository:

   bash git clone https://github.com/cvxgrp/cvxcla.git cd cvxcla

2. Create a virtual environment and install dependencies:

   bash make install

This will:

• Install the uv package manager
• Create a Python 3.12 virtual environment
• Install all dependencies from pyproject.toml

🔧 Usage

Here's a simple example of how to use cvxcla to compute the efficient frontier:

```python

import numpy as np # Set a seed for reproducibility np.random.seed(42) from cvxcla import CLA

# Define your portfolio problem n = 10 # Number of assets mean = np.random.randn(n) # Expected returns cov = np.random.randn(n, n) covariance = cov @ cov.T # Covariance matrix lowerbounds = np.zeros(n) # No short selling upperbounds = np.ones(n) # No leverage

Create a CLA instance

cla = CLA( ... mean = mean, ... covariance = covariance, ... lowerbounds = lowerbounds, ... upperbounds = upperbounds, ... a = np.ones((1, n)), # Fully invested constraint ... b = np.ones(1) ... )

# Access the efficient frontier frontier = cla.frontier

# Get the maximum Sharpe ratio portfolio maxsharperatio, maxsharpeweights = frontier.maxsharpe print(f"Maximum Sharpe ratio: {maxsharperatio:.6f}") Maximum Sharpe ratio: 2.946979 # Print first few weights to avoid long output print(f"First 3 weights: {maxsharpe_weights[:3]}") First 3 weights: [0. 0. 0.08509841]

```

For visualization, you can plot the efficient frontier:

```python

Plot the efficient frontier (not run in doctests)

fig = plot_frontier(frontier, volatility=True) fig.show() ```

📚 Literature and Implementations

The package includes implementations based on several key papers:

📝 Niedermayer and Niedermayer

They suggested a method to avoid the expensive inversion of the covariance matrix in Applying Markowitz's critical line algorithm. Our testing shows that in Python, this approach is not significantly faster than explicit matrix inversion using LAPACK via numpy.linalg.inv.

📝 Bailey and Lopez de Prado

We initially started with their code published in An Open-Source Implementation of the Critical-Line Algorithm for Portfolio Optimization. We've made several improvements:

• Using boolean numpy arrays to indicate whether a weight is free or blocked
• Rewriting the computation of the first turning point
• Isolating the computation of λ and weight updates to make them testable
• Using modern and immutable dataclasses throughout

Our updated implementation is included in the tests but not part of cvxcla package. We use it to verify our results and include it for educational purposes.

📝 Markowitz et al

In Avoiding the Downside: A Practical Review of the Critical Line Algorithm for Mean-Semivariance Portfolio Optimization, Markowitz and researchers from Hudson Bay Capital Management and Constantia Capital present a step-by-step tutorial.

We address a problem they overlooked: after finding the first starting point, all variables might be blocked. We enforce that one variable labeled as free (even if it sits on a boundary) to avoid a singular matrix.

Rather than using their sparse matrix construction, we bisect the weights into free and blocked parts and use a linear solver for the free part only.

🧪 Testing

Run the test suite with:

bash make test

🧹 Code Quality

Format and lint the code with:

bash make fmt

📖 Documentation

• Online Documentation
• API Reference

👥 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

1. Fork the repository
2. Create your feature branch (git checkout -b feature/amazing-feature)
3. Run the tests to make sure everything works (make test)
4. Format your code (make fmt)
5. Commit your changes (git commit -m 'Add some amazing feature')
6. Push to the branch (git push origin feature/amazing-feature)
7. Open a Pull Request

📄 License

This project is licensed under the Apache License 2.0 - see the LICENSE file for details.

🔍 Related Projects

• PyCLA by Philipp Schiele - A previous implementation of the Critical Line Algorithm in Python.

• CLA by Martin Dengler - The original implementation by David Bailey and Marcos Lopez de Prado.