Portfolio Optimization with Robust Covariance and Conditional Value-at-Risk Constraints

ArXiv ID: 2406.00610 “View on arXiv”

Authors: Unknown

Abstract

The measure of portfolio risk is an important input of the Markowitz framework. In this study, we explored various methods to obtain a robust covariance estimators that are less susceptible to financial data noise. We evaluated the performance of large-cap portfolio using various forms of Ledoit Shrinkage Covariance and Robust Gerber Covariance matrix during the period of 2012 to 2022. Out-of-sample performance indicates that robust covariance estimators can outperform the market capitalization-weighted benchmark portfolio, particularly during bull markets. The Gerber covariance with Mean-Absolute-Deviation (MAD) emerged as the top performer. However, robust estimators do not manage tail risk well under extreme market conditions, for example, Covid-19 period. When we aim to control for tail risk, we should add constraint on Conditional Value-at-Risk (CVaR) to make more conservative decision on risk exposure. Additionally, we incorporated unsupervised clustering algorithm K-means to the optimization algorithm (i.e. Nested Clustering Optimization, NCO). It not only helps mitigate numerical instability of the optimization algorithm, but also contributes to lower drawdown as well.

Keywords: Portfolio Optimization, Covariance Estimation, Conditional Value-at-Risk (CVaR), Unsupervised Clustering, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced statistical methods like Ledoit-Wolf shrinkage, robust covariance estimators (Gerber), and CVaR constraints requiring semidefinite optimization, reflecting high mathematical density. It also demonstrates strong empirical rigor by backtesting on a decade of data, comparing benchmarks, and integrating unsupervised clustering (K-means) to improve implementation stability.

  flowchart TD
    A["Research Goal:<br>Robust Portfolio Optimization"] --> B["Data Input:<br>Large-cap Equities 2012-2022"]
    B --> C["Methodology I:<br>Robust Covariance Estimators<br>Ledoit-Shrinkage vs. Gerber-MAD"]
    B --> D["Methodology II:<br>CVaR Constraints & Nested Clustering Optimization NCO"]
    C --> E["Computational Process:<br>Out-of-sample Backtesting"]
    D --> E
    E --> F{"Key Outcomes"}
    F --> G["Gerber-MAD Covariance<br>Top Performer in Bull Markets"]
    F --> H["Robust Estimators alone<br>Fail during Extreme Events<br>e.g., Covid-19"]
    F --> I["CVaR + NCO Integration<br>Controls Tail Risk &<br>Reduces Drawdowns"]