cvxcla 1.6.2

Critical line algorithm for the efficient frontier

📈 cvxcla - Critical Line Algorithm for Portfolio Optimization

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Quick Links: 📖 Documentation • 🐛 Report Bug • 💡 Request Feature • 💬 Discussions

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📋 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.

🧮 Why the Algorithm Works

The Markowitz problem is a quadratic program parametrized by a return target λ:

min  wᵀΣw - λ · μᵀw
s.t. Aw = b,  lb ≤ w ≤ ub

As λ sweeps from ∞ (maximize return) down to 0 (minimize variance), the solution traces the entire efficient frontier. The key insight is that between consecutive events, the optimal weights are a linear function of λ:

w(λ) = α + λ · β

This holds because the KKT optimality conditions are linear in λ whenever the active set — which assets sit at their bounds — is fixed. The algorithm exploits this in three steps:

1. Start at λ = ∞, where the portfolio concentrates on the highest-return asset within bounds (init_algo, called from _first_turning_point).

2. Solve the KKT system for the current active set to find α and β (_solve), then decrease λ until one of two events occurs (main loop):

   • a free asset hits its bound (leaves the free set), or
   • a blocked asset's KKT multiplier changes sign (enters the free set).

3. Update the active set (exactly one asset changes status) and repeat until λ ≤ 0.

Because only one asset changes per step and each step requires only a single linear solve, the algorithm traces the full frontier cheaply and exactly — no approximation needed.

✨ 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

pip install cvxcla

To include plotting support (Plotly and Kaleido):

pip install cvxcla[plot]

Development Setup

To set up a development environment:

1. Clone the repository:

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

2. Create a virtual environment and install dependencies:

   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:

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
lower_bounds = np.zeros(n)  # No short selling
upper_bounds = np.ones(n)  # No leverage

# Create a CLA instance
cla = CLA(
    mean = mean,
    covariance = covariance,
    lower_bounds = lower_bounds,
    upper_bounds = upper_bounds,
    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
max_sharpe_ratio, max_sharpe_weights = frontier.max_sharpe
print(f"Maximum Sharpe ratio: {max_sharpe_ratio:.6f}")
# Print first few weights to avoid long output
print(f"First 3 weights: {max_sharpe_weights[:3]}")

Maximum Sharpe ratio: 2.946979
First 3 weights: [0.         0.         0.08509841]

For visualization, you can plot the efficient frontier:

# Plot the efficient frontier
fig = frontier.plot(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:

make test

🧹 Code Quality

Format and lint the code with:

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.