Portfolio Optimization & Risk Analysis

Build a complete portfolio optimization workflow, from parameter estimation to portfolio selection, with explicit risk criteria.

A 25-asset universe, with historical prices sourced through the yfinance library.

• Estimated expected returns and covariances with the Single-Index Model.
• Applied Vasicek beta shrinkage to reduce estimation error.
• Constructed the efficient frontier and derived the global minimum variance (GMV) and tangency portfolios.
• Implemented the Elton–Gruber optimal portfolio algorithm.
• Applied VaR-based risk criteria.

The workflow produces the efficient frontier, the weights of the GMV and tangency portfolios, the Elton–Gruber optimal portfolio and the related VaR measures.

• Estimates rely on historical data: past returns and covariances do not guarantee future ones.
• The Single-Index Model assumes securities are correlated only through the market. Sector and style correlations fall outside the model.
• Mean-variance optimization is highly sensitive to errors in expected returns. Beta shrinkage reduces them but does not remove them.
• VaR sets a loss threshold; it says nothing about the size of losses beyond it.

How much input quality drives the output: small changes in expected returns shift optimal weights materially, which is why estimation techniques matter as much as the optimization itself.