Covariance-Aware Simplex Projection for Cardinality-Constrained Portfolio Optimization

TL;DR

CASP replaces Euclidean repair in cardinality-constrained portfolio optimization with a covariance-aware projection that minimizes tracking-error variance. The method uses a two-stage design (volatility-normalized asset selection plus Omega-metric projection) and an optional return-aware RA-CASP variant, validated on S&P 500 data (2020–2024). Results show substantial variance reduction for CASP-Basic and enhanced Sharpe for RA-CASP, with out-of-sample gains that transfer to realized performance and regime-dependent robustness. CASP thus provides a principled, drop-in replacement for Euclidean projection in metaheuristics, separating risk-based selection from covariance-aware projection to broaden the feasible risk-return-ESG trade-offs.

Abstract

Metaheuristic algorithms for cardinality-constrained portfolio optimization require repair operators to map infeasible candidates onto the feasible region. Standard Euclidean projection treats assets as independent and can ignore the covariance structure that governs portfolio risk, potentially producing less diversified portfolios. This paper introduces Covariance-Aware Simplex Projection (CASP), a two-stage repair operator that (i) selects a target number of assets using volatility-normalized scores and (ii) projects the candidate weights using a covariance-aware geometry aligned with tracking-error risk. This provides a portfolio-theoretic foundation for using a covariance-induced distance in repair operators. On S&P 500 data (2020-2024), CASP-Basic delivers materially lower portfolio variance than standard Euclidean repair without relying on return estimates, with improvements that are robust across assets and statistically significant. Ablation results indicate that volatility-normalized selection drives most of the variance reduction, while the covariance-aware projection provides an additional, consistent improvement. We further show that optional return-aware extensions can improve Sharpe ratios, and out-of-sample tests confirm that gains transfer to realized performance. CASP integrates as a drop-in replacement for Euclidean projection in metaheuristic portfolio optimizers.

Figures (3)

• Figure 1: Geometric intuition for covariance-aware projection. The gray triangle represents the feasible region (simplex) where portfolio weights are non-negative and sum to one. Given an infeasible candidate $z$, (a) Euclidean projection finds the nearest feasible point using circular iso-distance contours, treating assets as independent. (b) $\Omega$-metric projection (CASP) uses elliptical contours aligned with the covariance structure, finding the feasible point with minimum tracking error variance.
• Figure 2: Ablation study results across 500 random projections. (a) Portfolio variance: VolNorm+Euc shows that volatility-normalized selection explains most of the variance reduction, while CASP-Basic adds an incremental gain via $\Omega$-metric projection (15.7% total reduction vs Euclidean). (b) Sharpe ratio: return-aware methods achieve higher values but with reduced variance benefits. Dashed line indicates Euclidean baseline.
• Figure 3: Out-of-sample validation: in-sample vs. realized Sharpe ratios for 200 portfolio instances (train: 2020--2023, test: 2024). (a) Euclidean shows moderate correlation ($\rho \approx 0.29$). (b) CASP-Basic shows lower rank correlation ($\rho \approx 0.07$), consistent with its return-agnostic design. (c) RA-CASP achieves higher absolute performance with moderate correlation ($\rho \approx 0.25$).

Theorems & Definitions (5)

• Definition 1: Tracking-error metric (covariance-induced distance)
• Remark 1: Terminology
• Proposition 1: Tracking Error Interpretation
• Corollary 1: Diversification Benefit
• Remark 2: Two-stage heuristic vs global projection