  Since Markowitz's mean-variance framework, optimizing a portfolio that
maximizes the profit and minimizes the risk has been ubiquitous in the
financial industry. Initially, profit and risk were measured by the first two
moments of the portfolio's return, a.k.a. the mean and variance, which are
sufficient to characterize a Gaussian distribution. However, it is broadly
believed that the first two moments are not enough to capture the
characteristics of the returns' behavior, which have been recognized to be
asymmetric and heavy-tailed. Although there is ample evidence that portfolio
designs involving the third and fourth moments, i.e., skewness and kurtosis,
will outperform the conventional mean-variance framework, they are non-trivial.
Specifically, in the classical framework, the memory and computational cost of
computing the skewness and kurtosis grow sharply with the number of assets. To
alleviate the difficulty in high-dimensional problems, we consider an
alternative expression for high-order moments based on parametric
representations via a generalized hyperbolic skew-t distribution. Then, we
reformulate the high-order portfolio optimization problem as a fixed-point
problem and propose a robust fixed-point acceleration algorithm that solves the
problem in an efficient and scalable manner. Empirical experiments also
demonstrate that our proposed high-order portfolio optimization framework is of
low complexity and significantly outperforms the state-of-the-art methods by 2
to 4 orders of magnitude.
