An algebraic theory of portfolio allocation

Using group and majorization theory, we explore what can be established about allocation of funds among assets when asymmetries in the returns vector are carefully controlled. The key insight is that preferences over allocations can be partially ordered via majorized convex hulls that have been generated by a permutation group. Group transitivity suffices to ensure complete portfolio diversification. Point-wise stabilizer subgroups admit sectoral separability in fund allocations. We also bound the admissible allocation vector by a set of linear constraints the coefficients of which are determined by group operations on location and scale asymmetries in the rate of returns vector. For a distribution that is symmetric under a reflection group, the linear constraints may be further strengthened whenever there exists an hyperplane that separates convex sets.