The Linear Fractional Portfolio Selection Problem

A simplified portfolio selection criterion suggested by Sharpe and Mao involves choosing at most n securities from a universe of m securities in order to maximize the portfolio's excess-return-to-beta ratio. This paper examines alternative solution procedures to achieve this objective, including a gradient procedure whose continuous Knapsack subproblems in m bounded variables are solved in O(m) time. The effect on the optimal portfolio of increasing n is discussed, as well as the relationship between the excess-return-to-beta ratio of an individual security and that of the optimal portfolio. The paper concludes with computational experience on problems with n ranging from 10 to 200 and m from 500 to 1,245.