# Robustifying Markowitz

arXiv: https://arxiv.org/abs/2212.13996 · HTML: https://mlsanity.com/paper/2212.13996/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 68 references that are in the map.
- First posted on arXiv 28 Dec 2022.

## Builds on (most cited first, 17 of 17)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Being Robust (in High Dimensions) Can Be Practical](https://mlsanity.com/paper/1703.00893/) (2017)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Robust subgaussian estimation of a mean vector in nearly linear time](https://mlsanity.com/paper/1906.03058/) (2019)
- [Mean Estimation with Sub-Gaussian Rates in Polynomial Time](https://mlsanity.com/paper/1809.07425/) (2018)
- [Robust estimation via generalized quasi-gradients](https://mlsanity.com/paper/2005.14073/) (2020)
- [Fast Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1902.01998/) (2019)
- [User-Friendly Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1811.01520/) (2018)
- [Algorithms for Heavy-Tailed Statistics: Regression, Covariance Estimation, and Beyond](https://mlsanity.com/paper/1912.11071/) (2019)
- [Robust k-means Clustering for Distributions with Two Moments](https://mlsanity.com/paper/2002.02339/) (2020)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [Robust Covariance Estimation for Approximate Factor Models](https://mlsanity.com/paper/1602.00719/) (2016)
- [On Monte-Carlo methods in convex stochastic optimization](https://mlsanity.com/paper/2101.07794/) (2021)
- [Affine Invariant Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1902.03086/) (2019)

## Abstract (arXiv)

Markowitz mean-variance portfolios with sample mean and covariance as input parameters feature numerous issues in practice. They perform poorly out of sample due to estimation error, they experience extreme weights together with high sensitivity to change in input parameters. The heavy-tail characteristics of financial time series are in fact the cause for these erratic fluctuations of weights that consequently create substantial transaction costs. In robustifying the weights we present a toolbox for stabilizing costs and weights for global minimum Markowitz portfolios. Utilizing a projected gradient descent (PGD) technique, we avoid the estimation and inversion of the covariance operator as a whole and concentrate on robust estimation of the gradient descent increment. Using modern tools of robust statistics we construct a computationally efficient estimator with almost Gaussian properties based on median-of-means uniformly over weights. This robustified Markowitz approach is confirmed by empirical studies on equity markets. We demonstrate that robustified portfolios reach the lowest turnover compared to shrinkage-based and constrained portfolios while preserving or slightly improving out-of-sample performance.