A lattice approach to the Beta distribution induced by stochastic dominance: Theory and applications

TL;DR

This paper develops a lattice framework for comparing Beta distributions under second-order stochastic dominance, simplifying risk assessment and optimizing portfolio choices based on distribution parameters.

Contribution

It introduces a novel lattice structure for Beta distributions under stochastic dominance and characterizes optimal investment strategies in a portfolio context.

Findings

Mean-preserving spread corresponds to increased variance.

Higher moments are irrelevant for risk comparison of Beta distributions.

Explicit characterization of portfolio optimality conditions.

Abstract

We provide a comprehensive analysis of the two-parameter Beta distributions seen from the perspective of second-order stochastic dominance. By changing its parameters through a bijective mapping, we work with a bounded subset D instead of an unbounded plane. We show that a mean-preserving spread is equivalent to an increase of the variance, which means that higher moments are irrelevant to compare the riskiness of Beta distributions. We then derive the lattice structure induced by second-order stochastic dominance, which is feasible thanks to the topological closure of D. Finally, we consider a standard (expected-utility based) portfolio optimization problem in which its inputs are the parameters of the Beta distribution. We explicitly characterize the subset of D for which the optimal solution consists of investing 100% of the wealth in the risky asset and we provide an exhaustive numerical analysis of this optimal solution through (color-coded) graphs.