Minimum-Variance Portfolio Optimizer

This project implements a minimum-variance portfolio optimizer in Python using NumPy. It solves both:

• The standard minimum-variance portfolio problem (without return constraint)
• The minimum-variance portfolio with a specified expected return, optionally allowing short-selling

It uses the Lagrangian/KKT formulation and includes numerical stability handling (pseudoinverse fallback for ill-conditioned systems).

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🧠 What It Does

Given:

• A covariance matrix of asset returns $\Sigma$
• A vector of expected returns $\mu$
• An optional target return $\mu^*$

The optimizer solves:

$$ \min_w \ w^T \Sigma w \quad \text{subject to} \quad w^T \mu = \mu^*, \quad \sum w_i = 1 $$

If no target return is provided, it computes the global minimum-variance portfolio.

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🧪 Features

• ✅ Supports arbitrary number of assets
• ✅ Handles singular or ill-conditioned matrices using pseudoinverse
• ✅ Returns optimal weights and portfolio risk
• ✅ Easily extendable to include long-only constraints or regularization
• ✅ Plots efficient frontier for visualization

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📦 Requirements

• Python 3.x
• NumPy
• cvxpy
• matplotlib

Install dependencies:

pip install numpy