Paper: arXiv 2601.18124

Authors: Steven E. Pav

Abstract

We show that the Markowitz portfolio is a scalar multiple of another portfolio which replaces the covariance with the second moment matrix, via simple application of the Sherman-Morrison identity. Moreover it is shown that when using conditional estimates of the first two moments, this “Sherman-Morrison-Markowitz” portfolio solves the standard unconditional portfolio optimization problems. We argue that in multi-period portfolio optimization problems it is more natural to replace variance and covariance with their uncentered counterparts. We extend the theory to deal with constraints in expectation, where we find a decomposition of squared effects into spanned and orthogonal components. Compared to the Markowitz portfolio, the Sherman-Morrison-Markowitz portfolio downlevers by a small amount that depends on the conditional squared maximal Sharpe ratio; the practical impact will be fairly small, however. We present some example use cases for the theory.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a elegant mathematical re-framing of multi-period portfolio optimization using the Sherman-Morrison identity and second-moment matrices. However, it is purely theoretical in this excerpt, lacking empirical backtests, real-world data validation, or empirical rigor.

Research Flowchart

  flowchart TD
    A["Research Goal: Reformulate Markowitz Portfolio via Second Moment Matrix"] --> B["Inputs: First & Second Uncentered Moments, Asset Returns"]
    B --> C["Apply Sherman-Morrison Identity to Covariance Matrix"]
    C --> D["Derive Sherman-Morrison-Markowitz (SMM) Portfolio"]
    D --> E["Extend SMM to Multi-Period Settings & Expectation Constraints"]
    E --> F["Key Outcome: SMM solves unconditional optimization by downlevering Markowitz weight based on max Sharpe ratio"]

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