The geometry of higher order modern portfolio theory

ArXiv ID: 2511.20674 “View on arXiv”

Authors: Emil Horobet

Abstract

In this article, we study the generalized modern portfolio theory, with utility functions admitting higher-order cumulants. We establish that under certain genericity conditions, the utility function has a constant number of complex critical points. We study the discriminant locus of complex critical points with multiplicity. Finally, we switch our attention to the generalization of the feasible portfolio set (variety), determine its dimension, and give a formula for its degree.

Keywords: Generalized Modern Portfolio Theory, Higher-order cumulants, Complex critical points, Discriminant locus, Algebraic geometry in finance, Portfolio Theory (General)

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with abstract algebraic geometry, using concepts like discriminant loci, degrees of varieties, and Bezout’s theorem to prove theorems about the number of complex critical points. It contains zero empirical data, backtests, or implementation details, relying purely on theoretical derivations under genericity conditions.

  flowchart TD
    A["Research Goal<br>Generalized Modern Portfolio Theory<br>with Higher-Order Cumulants"] --> B["Methodology<br>Algebraic Geometry & Critical Point Theory"]
    
    B --> C["Input Data<br>Utility Functions &<br>Portfolio Constraints"]
    C --> D["Computation<br>Find Constant Complex<br>Critical Points"]
    D --> E{"Analysis"}
    E --> F["Outcome 1<br>Discriminant Locus of<br>Multiplicity"]
    E --> G["Outcome 2<br>Feasible Portfolio Variety:<br>Dimension & Degree Formula"]
    
    D --> H["Validation<br>Genericity Conditions"]
    H --> B