[Paper Review] Zero Variance Portfolio

Jinyuan Chang, Yi Ding+2 more|arXiv (Cornell University)|Feb 23, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper introduces Ridgelet and refined Ridgelet estimators to construct zero variance portfolios in high-dimensional MVP problems (N>>T), showing Ridgelet can generalize out-of-sample while Ridgeless fails, with theoretical and empirical support.

ABSTRACT

When the number of assets is larger than the sample size, the minimum variance portfolio interpolates the training data, delivering pathological zero in-sample variance. We show that if the weights of the zero variance portfolio are learned by a novel ``Ridgelet'' estimator, in a new test data this portfolio enjoys out-of-sample generalizability. It exhibits the double descent phenomenon and can achieve optimal risk in the overparametrized regime when the number of assets dominates the sample size. In contrast, a ``Ridgeless'' estimator which invokes the pseudoinverse fails in-sample interpolation and diverges away from out-of-sample optimality. Extensive simulations and empirical studies demonstrate that the Ridgelet method performs competitively in high-dimensional portfolio optimization.

Motivation & Objective

• Motivate and analyze minimum variance portfolio (MVP) in a high-dimensional, sample-poor regime (N>>T).
• Propose Ridgelet estimators that add a tiny ridge to the sample covariance to obtain a stable, generalizable ZVP.
• Show theoretical properties via random matrix theory and factor-model assumptions.
• Compare Ridgelet with Ridgeless and standard estimators through simulations and empirical data.

Proposed method

• Define ZVP and its exact solvability via linear constraints and the sample covariance.
• Introduce Ridgelet1: tau-regularized MVP solution  ω_hat_tau = (1^T S_tau^{-1} 1)^{-1} S_tau^{-1} 1 with S_tau = S_0 + tau I_N.
• Prove Ridgelet1 approximates the minimum L2-norm ZVP; contrast with Ridgeless which uses S_0^+ and is not the ZVP solution.
• Extend to Ridgelet2 by replacing I_N with a consistent idiosyncratic covariance estimator under a factor model.
• Provide asymptotic results for out-of-sample variance under different regimes (N/T), via Random Matrix Theory.
• Discuss double descent phenomena and optimality in the overparameterized regime (N>>T).

Figure 1: Risk curve of the MVP estimated with Ridgelet (left) and Ridgeless (right). The oracle minimum risk is a benchmark. The returns are generated from a factor model described in Section 3.1 .

Experimental results

Research questions

• RQ1 Can a simple ridge-augmented covariance estimator (Ridgelet) yield a zero in-sample variance portfolio that generalizes well out-of-sample in high-dimensional MVP settings?
• RQ2 How do Ridgelet and Ridgeless estimators compare in OOS performance as N grows relative to T under factor-model covariance structures?
• RQ3 Does incorporating a consistent idiosyncratic covariance estimator (Ridgelet2) achieve population-like optimality when N>>T?
• RQ4 What theoretical mechanisms explain double descent in MVP risk for Ridgelet and Ridgeless?
• RQ5 How do these methods perform empirically on real market data (e.g., S&P 500, Nikkei 225) relative to standard shrinkage methods?

Key findings

• Ridgelet1 approximates the exact zero-variance portfolio solution (ZVP) up to numerical error, while Ridgeless is not the ZVP solution.
• In the N/T regime, Ridgelet shows a double-descent risk pattern and can achieve OOS performance close to the oracle under certain conditions.
• In the N>>T regime, Ridgelet2 with a consistent idiosyncratic covariance estimator attains OOS risk approaching the population oracle (optimality) under a factor model.
• Ridgeless variance diverges in the N>>T regime, performing worse than an equal-weight or Ridgelet-based approaches.
• Empirical studies on S&P 500 and Nikkei 225 indicate Ridgelet competes favorably with LS and FNLS in high-dimensional settings.
• Theoretical results rely on Random Matrix Theory to characterize eigenvalue behavior and Stieltjes transforms in the asymptotic analysis.

Figure 2: Scree plot of daily returns of S&P 500 Index Constituent stocks. We compute the sample covariance matrix of daily returns of S&P 500 Index stocks between 2020 and 2023. The Y-axis shows the ratios of its principal eigenvalues over the sum of all eigenvalues.