A new spin on optimal portfolios and ecological equilibria

Garnier-Brun, Jerome; Benzaquen, Michael; Ciliberti, Stefano; Bouchaud, Jean-Philippe

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.