  Consider an equity market with $n$ stocks. The vector of proportions of the
total market capitalizations that belong to each stock is called the market
weight. The market weight defines the market portfolio which is a buy-and-hold
portfolio representing the performance of the entire stock market. Consider a
function that assigns a portfolio vector to each possible value of the market
weight, and we perform self-financing trading using this portfolio function. We
study the problem of characterizing functions such that the resulting portfolio
will outperform the market portfolio in the long run under the conditions of
diversity and sufficient volatility. No other assumption on the future behavior
of stock prices is made. We prove that the only solutions are functionally
generated portfolios in the sense of Fernholz. A second characterization is
given as the optimal maps of a remarkable optimal transport problem. Both
characterizations follow from a novel property of portfolios called
multiplicative cyclical monotonicity.
