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Markowitz Portfolio Construction at Seventy

Stephen Boyd, Kasper Johansson, Ronald Kahn, Philipp Schiele, Thomas Schmelzer

TL;DR

Markowitz++ extends the classical mean-variance framework by embedding robust optimization, regularization, and practical trading costs within a convex optimization setting. It combines probabilistic return models, factor structures, soft constraints, and cost forecasts to tame estimation risk while preserving tractable solutions. Through extensive back-testing on a 74-asset universe, Markowitz++ demonstrates improved out-of-sample risk-return profiles and controllable turnover and leverage, powered by efficient solvers and domain-specific languages. The work also provides two companion software packages, underscoring its practical relevance and potential for broad adoption in quantitative portfolio construction.

Abstract

More than seventy years ago Harry Markowitz formulated portfolio construction as an optimization problem that trades off expected return and risk, defined as the standard deviation of the portfolio returns. Since then the method has been extended to include many practical constraints and objective terms, such as transaction cost or leverage limits. Despite several criticisms of Markowitz's method, for example its sensitivity to poor forecasts of the return statistics, it has become the dominant quantitative method for portfolio construction in practice. In this article we describe an extension of Markowitz's method that addresses many practical effects and gracefully handles the uncertainty inherent in return statistics forecasting. Like Markowitz's original formulation, the extension is also a convex optimization problem, which can be solved with high reliability and speed.

Markowitz Portfolio Construction at Seventy

TL;DR

Markowitz++ extends the classical mean-variance framework by embedding robust optimization, regularization, and practical trading costs within a convex optimization setting. It combines probabilistic return models, factor structures, soft constraints, and cost forecasts to tame estimation risk while preserving tractable solutions. Through extensive back-testing on a 74-asset universe, Markowitz++ demonstrates improved out-of-sample risk-return profiles and controllable turnover and leverage, powered by efficient solvers and domain-specific languages. The work also provides two companion software packages, underscoring its practical relevance and potential for broad adoption in quantitative portfolio construction.

Abstract

More than seventy years ago Harry Markowitz formulated portfolio construction as an optimization problem that trades off expected return and risk, defined as the standard deviation of the portfolio returns. Since then the method has been extended to include many practical constraints and objective terms, such as transaction cost or leverage limits. Despite several criticisms of Markowitz's method, for example its sensitivity to poor forecasts of the return statistics, it has become the dominant quantitative method for portfolio construction in practice. In this article we describe an extension of Markowitz's method that addresses many practical effects and gracefully handles the uncertainty inherent in return statistics forecasting. Like Markowitz's original formulation, the extension is also a convex optimization problem, which can be solved with high reliability and speed.
Paper Structure (102 sections, 55 equations, 4 figures, 2 tables)

This paper contains 102 sections, 55 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The original Markowitz idea
  3. Alleged deficiencies
  4. It's sensitive to data errors and estimation uncertainty.
  5. It implicitly assumes risk symmetry.
  6. We should maximize expected utility.
  7. It considers only the first and second moments of the return.
  8. It's a greedy method.
  9. Robust optimization and regularization
  10. Robust optimization.
  11. Regularization.
  12. The high level story.
  13. Convex optimization
  14. Solvers.
  15. Domain-specific languages.
  16. ...and 87 more sections

Figures (4)

  • Figure 1: Tuned parameters over time.
  • Figure 2: Parameter tuning results.
  • Figure 3: Yearly annualized metrics for the equal weight portfolio, basic Markowitz, and tuned Markowitz++.
  • Figure 4: Timing results for the Markowitz problem with a single backtest setting.