The Markowitz Category

TL;DR

This paper provides an algebraic classification of Markowitz markets, revealing that simplified low-dimensional models for large markets are inherently limited to at most two dimensions when using mean-variance analysis.

Contribution

It introduces an algebraic framework for classifying Markowitz markets and shows the triviality of portfolio optimization without short selling constraints under this classification.

Findings

Classification of Markowitz markets up to isomorphism

Portfolio optimization becomes trivial in classified markets

Low-dimensional models are limited to two dimensions

Abstract

We give an algebraic definition of a Markowitz market and classify markets up to isomorphism. Given this classification, the theory of portfolio optimization in Markowitz markets without short selling constraints becomes trivial. Conversely, this classification shows that, up to isomorphism, there is little that can be said about a Markowitz market that is not already detected by the theory of portfolio optimization. In particular, if one seeks to develop a simplified low-dimensional model of a large financial market using mean--variance analysis alone, the resulting model can be at most two-dimensional.