  This paper aims to develop new mathematical and computational tools for
modeling the distribution of portfolio returns across portfolios. We establish
relevant mathematical formulas and propose efficient algorithms, drawing upon
powerful techniques in computational geometry and the literature on splines, to
compute the probability density function, the cumulative distribution function,
and the k-th moment of the probability function. Our algorithmic tools and
implementations efficiently handle portfolios with 10000 assets, and compute
moments of order k up to 40 in a few seconds, thus handling real-life
scenarios. We focus on the long-only strategy which is the most common type of
investment, i.e. on portfolios whose weights are non-negative and sum up to 1;
our approach is readily generalizable. Thus, we leverage a geometric
representation of the stock market, where the investment set defines a simplex
polytope. The cumulative distribution function corresponds to a portfolio score
capturing the percentage of portfolios yielding a return not exceeding a given
value. We introduce closed-form analytic formulas for the first 4 moments of
the cross-sectional returns distribution, as well as a novel algorithm to
compute all higher moments. We show that the first 4 moments are a direct
mapping of the asset returns' moments. All of our algorithms and solutions are
fully general and include the special case of equal asset returns, which was
sometimes excluded in previous works. Finally, we apply our portfolio score in
the design of new performance measures and asset management. We found our
score-based optimal portfolios less concentrated than the mean-variance
portfolio and much less risky in terms of ranking.
