Equity portfolios generated by functions of ranked market weights

Abstract.

Dynamic equity portfolios can be generated by positive twice continuously differentiable functions of the ranked capitalization weights of an equity market. The return on such a portfolio relative to the market follows a stochastic differential equation that decomposes the relative return into two components: the logarithmic change in the value of the generating function, and a drift process that is of bounded variation. The method can be used to construct broad classes of stock portfolios, and has both theoretical and practical applications. Two applications of the method are presented: one offers an explanation for the size effect, the observed tendency of small stocks to have higher long-term returns than large stocks, and the other provides a rigorous analysis of the behavior of diversity-weighted indices, stock indices with weights that lie between capitalization weights and equal weights.