Solving High-Order Portfolios via Successive Convex Approximation Algorithms

Abstract

The first moment and second central moments of the portfolio return, a.k.a. mean and variance, have been widely employed to assess the expected profit and risk of the portfolio. Investors pursue higher mean and lower variance when designing the portfolios. The two moments can well describe the distribution of the portfolio return when it follows the Gaussian distribution. However, the real world distribution of assets return is usually asymmetric and heavy-tailed, which is far from being a Gaussian distribution. The asymmetry and the heavy-tailedness are characterized by the third and fourth central moments, i.e., skewness and kurtosis, respectively. Higher skewness and lower kurtosis are preferred to reduce the probability of extreme losses. However, incorporating high-order moments in the portfolio design is very difficult due to their non-convexity and rapidly increasing computational cost with the dimension. In this paper, we propose a very efficient and convergence-provable algorithm framework based on the successive convex approximation (SCA) algorithm to solve high-order portfolios. The efficiency of the proposed algorithm framework is demonstrated by the numerical experiments.

Sparse High-Order Portfolios via Proximal DCA and SCA

“…Next, we show the results compared with several baselines. One commonly used strategy, which we denote as RMVSKC, is to first relax the cardinality constraint to 1 norm constraint [15] (it is worth emphasizing that this kind of relaxation can not get a sparse solution) and then project the resulted solution to satisfy the original constraints of MVSKC (1). The projection step is actually solving problem (9).…”

“…Further improvement based on the difference-of-convex-sums-of-squares (DC-SOS) is customized [14]. Quite recently, to approximate the original non-convex objective function more tightly, two algorithms based on the classical MM and SCA are proposed [15]. However, to the best of our knowledge, due to the great difficulty in optimization, the MVSKC framework has not received much attention yet.…”