The Markowitz Category

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.