  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.
