The geometry of higher order modern portfolio theory

TL;DR

This paper explores the mathematical structure of advanced portfolio theory involving higher-order cumulants, analyzing critical points and the geometry of feasible portfolio sets to deepen understanding of complex investment models.

Contribution

It introduces a geometric and algebraic framework for generalized portfolio theory with higher-order utility functions, analyzing critical points and the portfolio set variety.

Findings

Critical points are finite under generic conditions

Discriminant locus characterizes multiplicity of critical points

Dimension and degree formulas for the portfolio set variety

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.