A new spin on optimal portfolios and ecological equilibria

TL;DR

This paper explores the mathematical structure of optimal portfolios and ecological equilibria under specific constraints, revealing complex solution landscapes and implications for decision-making in high-dimensional systems.

Contribution

It introduces a novel analysis of the solution landscape for constrained portfolio and ecological models with rank-one interactions, connecting to spin-glass physics.

Findings

Number of solutions grows as N^α with α ≤ 2/3

Solutions are sparsely distributed and quasi-degenerate

Presence of disorder chaos impacts decision strategies

Abstract

We consider the classical problem of optimal portfolio construction with the constraint that no short position is allowed, or equivalently the valid equilibria of multispecies Lotka-Volterra equations with self-regulation in the special case where the interaction matrix is of unit rank, corresponding to species competing for a common resource. We compute the average number of solutions and show that its logarithm grows as $N^\alpha$, where $N$ is the number of assets or species and $\alpha \leq 2/3$ depends on the interaction matrix distribution. We conjecture that the most likely number of solutions is much smaller and related to the typical sparsity $m(N)$ of the solutions, which we compute explicitly. We also find that the solution landscape is similar to that of spin-glasses, i.e. very different configurations are quasi-degenerate. Correspondingly, "disorder chaos" is also present in our problem. We discuss the consequence of such a property for portfolio construction and ecologies, and question the meaning of rational decisions when there is a very large number "satisficing" solutions.