Paper: arXiv 2611.09834

Authors: Ziyao Wang, Svetlozar T. Rachev, Frank J. Fabozzi

Abstract

Investors turn signals into holdings through decision rules: buy the top decile, hold stocks with high momentum and low volatility, exclude stocks that fail a screen. Many such rules are Boolean functions of threshold statements about signal ranks, and writing them this way shows both what a rule holds and how often it trades. A Walsh decomposition splits the return of any rule into exposures to individual signals and to interactions among signals; an intersection and a union of the same two screens, for example, have opposite interaction exposures. The same coefficients, together with the persistence of the underlying statements, determine turnover. Interactions are less persistent than their components, so rules that load on higher-order interactions trade more, and thresholds discard information, so at equal risk threshold implementations of interactions trade more than smooth ones. In U.S. stocks from 1973 to 2024, these predictions hold closely: predicted and realized persistence have correlations of at least 0.999, predicted turnover matches realized turnover with a slope of 1.02, and the same interactions are more persistent as products of ranks than as threshold rules. Trading costs rise with order, and because premia fall with order, 58% of single-signal portfolios have absolute premia above their quoted-spread costs over 1995-2024, against 24% of two-signal and 6% of three-signal interaction portfolios. Out of sample, interaction portfolios raise the Sharpe ratio of single-signal tangency portfolios, and over 1995-2024 they reject a six-factor model that conventional sorts on the same signals do not reject.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 9.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper introduces a highly novel framework using Walsh decomposition for Boolean functions in portfolio construction, demonstrating significant mathematical depth. It is backed by extensive empirical validation on U.S. stock data from 1973-2024, rigorously testing predictions on persistence, turnover, and trading costs. The findings offer practical implications for factor investing and asset pricing model testing.

Research Flowchart

  flowchart TD
    A[Research Goal: Understand Logic, Structure, and Performance of Factor Portfolios using Boolean Rules] --> B{Methodology: Boolean Rules, Walsh Decomposition, Persistence/Turnover Analysis};
    B --> C[Data Inputs: U.S. Stocks (1973-2024), Signal Ranks, Trading Costs (Quoted Spreads)];
    C --> D{Computational Processes: <br>1. Define Factor Portfolios as Boolean Functions <br>2. Walsh Decomposition of Returns <br>3. Predict & Realize Persistence/Turnover <br>4. Compare Threshold vs. Smooth Implementations <br>5. Factor Model Rejection Tests};
    D --> E[Key Findings: <br>1. Persistence/Turnover Predictions Highly Accurate (corr > 0.999, slope = 1.02) <br>2. Interactions Less Persistent, Higher Turnover <br>3. Thresholds Increase Turnover vs. Smooth <br>4. Trading Costs Rise with Interaction Order <br>5. Interaction Portfolios Improve Sharpe Ratio & Reject Factor Models];