Are Three Matrices All You Need To Beat the Market? Observable Matrix Dynamics for Portfolio Optimization

2026-07-29

AI Summary

This study addresses the limitations of traditional mean-variance optimization, which relies on hard-to-estimate expected returns and assumes a static, single-period framework ill-suited for dynamic markets. The authors propose a dynamic portfolio construction method that dispenses entirely with expected returns and covariance matrices. Instead, it leverages only daily price, volume, and market capitalization data to construct three fixed-dimensional matrices: a return correlation distance matrix and two Markov transition matrices ranked by historical returns and volatility. Portfolio weights are driven by cross-sectional rankings, eliminating matrix inversion and enhancing robustness. The approach innovatively incorporates residual-distance diversification and a convex information-leader hedging mechanism. Out-of-sample tests over 2022–2024 and 2025–2026 yield Sharpe ratios of 1.08 and 1.44, with annualized returns of 20% and 56%, respectively—substantially outperforming both the market and classical benchmarks.

Abstract

We present a simple framework for dynamic portfolio management that uses nothing but daily prices, trading volumes, and market capitalizations. Its state is three fixed-size matrices built from the price history: the distance matrix of the return correlations and the transition matrices of two Markov chains that rank the S\&P 500 names monthly by trailing return and by trailing volatility. These three matrices rest on the price history alone, the same information Markowitz mean-variance optimization draws on, but they replace its expected-return vector and covariance matrix. Our method requires no matrix inversion, works on outlier-robust cross-sectional ranks, and is dynamic rather than single-period. Empirically the volatility rank is forecastable one step ahead while the return rank stays close to unforecastable. A portfolio built on the forecasts, a market-neutral momentum long-short blended with an opportunistic long-only sleeve, beats the market on two non-overlapping out-of-sample test sets, January 2022 to December 2024 and January 2025 to July 2026, at Sharpes of $1.06$ and $1.32$ against the market's $0.78$ and $1.14$, respectively, net of a five-basis-point trading cost and marked to market daily. It also outperforms the classical minimum-variance and maximum-diversification portfolios. Diversifying the long sleeve by residual distance adds a further edge on both periods, lifting the Sharpe to $1.08$ and $1.44$ and the annualized return from $18\%$ to $20\%$ and from $44\%$ to $56\%$, respectively. A convex information-leader overlay separately insures the market-neutral sleeve, buying convexity and a shallower drawdown at a small cost in return, the Sharpe unchanged.

Problem

• portfolio optimization
• market dynamics
• matrix representation
• out-of-sample performance
• Sharpe ratio

Innovation

• matrix dynamics
• portfolio optimization
• Markov chains
• market-neutral momentum
• outlier-robust ranking