Estimating Covariance for Global Minimum Variance Portfolio: A Decision-Focused Learning Approach

ArXiv ID: 2508.10776 “View on arXiv”

Authors: Juchan Kim, Inwoo Tae, Yongjae Lee

Abstract

Portfolio optimization constitutes a cornerstone of risk management by quantifying the risk-return trade-off. Since it inherently depends on accurate parameter estimation under conditions of future uncertainty, the selection of appropriate input parameters is critical for effective portfolio construction. However, most conventional statistical estimators and machine learning algorithms determine these parameters by minimizing mean-squared error (MSE), a criterion that can yield suboptimal investment decisions. In this paper, we adopt decision-focused learning (DFL) - an approach that directly optimizes decision quality rather than prediction error such as MSE - to derive the global minimum-variance portfolio (GMVP). Specifically, we theoretically derive the gradient of decision loss using the analytic solution of GMVP and its properties regarding the principal components of itself. Through extensive empirical evaluation, we show that prediction-focused estimation methods may fail to produce optimal allocations in practice, whereas DFL-based methods consistently deliver superior decision performance. Furthermore, we provide a comprehensive analysis of DFL’s mechanism in GMVP construction, focusing on its volatility reduction capability, decision-driving features, and estimation characteristics.

Keywords: Decision-focused learning, Global minimum-variance portfolio, Portfolio optimization, Gradient-based optimization, Mean-squared error, Equity (Portfolio Management)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper is highly mathematically dense, featuring complex theoretical derivations and proofs, but also demonstrates strong empirical rigor with extensive experiments and a detailed, end-to-end implementation framework for decision-focused learning in portfolio optimization.

  flowchart TD
    A["Research Goal:<br>Optimize GMVP using<br>Decision-Focused Learning"] --> B["Data & Inputs:<br>Historical Equity Returns<br>Asset Covariance Matrix"]
    B --> C["Methodology:<br>Apply DFL to directly<br>optimize portfolio decisions<br>vs. traditional MSE methods"]
    C --> D["Computational Process:<br>Derive analytic gradient<br>of GMVP loss function<br>via Principal Components"]
    D --> E["Key Outcomes:<br>DFL reduces volatility &<br>outperforms MSE-based<br>estimators in decision quality"]