[Paper Review] Solving High-Order Portfolios via Successive Convex Approximation Algorithms

Rui Zhou, Daniel P. Palomar|arXiv (Cornell University)|Aug 3, 2020

Risk and Portfolio Optimization28 references4 citations

TL;DR

This paper proposes a successive convex approximation (SCA)-based algorithm framework to efficiently solve high-order portfolio optimization problems involving skewness and kurtosis, overcoming the non-convexity and high computational cost of higher moments. The method constructs convex subproblems iteratively, ensuring convergence to a stationary point with low computational complexity, enabling scalable solutions for high-dimensional portfolios.

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.

Motivation & Objective

• Address the limitations of the mean-variance framework in capturing real-world asset return distributions that are asymmetric and heavy-tailed.
• Overcome the computational intractability of high-order portfolio optimization due to non-convex third and fourth central moments (skewness and kurtosis).
• Develop a scalable, convergent, and efficient algorithm framework for high-dimensional portfolio optimization incorporating higher-order moments.
• Enable practical application of the mean-variance-skewness-kurtosis (MVSK) framework by replacing slow metaheuristic or DC-based methods with a fast, gradient-compatible SCA approach.

Proposed method

• Formulate high-order portfolio problems as non-convex optimization problems involving the third and fourth central moments of portfolio returns.
• Apply the successive convex approximation (SCA) framework to iteratively approximate the non-convex objective functions with convex subproblems.
• Construct convex approximations using first-order Taylor expansions of the non-convex terms, ensuring each subproblem is strongly convex and solvable via standard solvers.
• Derive closed-form expressions for the gradients and Hessian matrices of skewness and kurtosis functions to enable efficient computation.
• Ensure convergence to a stationary point by leveraging theoretical guarantees of the SCA algorithm under appropriate conditions.
• Integrate the SCA framework with existing efficient convex optimization solvers, enabling high-dimensional scalability.

Experimental results

Research questions

• RQ1 Can a convergent and efficient algorithm be designed to solve high-order portfolio optimization problems involving skewness and kurtosis?
• RQ2 How can the non-convexity of higher-order moments be effectively handled to enable scalable portfolio optimization?
• RQ3 What is the computational complexity and convergence behavior of the proposed SCA-based framework in high-dimensional settings?
• RQ4 How does the proposed algorithm compare in performance and convergence speed to existing metaheuristic or DC-based methods for MVSK portfolio optimization?
• RQ5 To what extent can the SCA framework be generalized to other formulations of high-order portfolio problems?

Key findings

• The proposed SCA-based algorithm framework converges to a stationary point of the non-convex high-order portfolio problem, ensuring theoretical reliability.
• The method achieves significantly faster convergence and lower computational cost compared to metaheuristic approaches like differential evolution or genetic algorithms.
• The algorithm is computationally efficient and scalable, making it suitable for high-dimensional portfolio problems where traditional gradient-based methods fail due to complexity.
• Theoretical analysis confirms boundedness of the Hessian’s spectral radius, supporting the convergence of the SCA iterations under the given constraints.
• Numerical experiments demonstrate the algorithm’s effectiveness in achieving desired trade-offs between mean, variance, skewness, and kurtosis in portfolio construction.
• The framework is applicable to various MVSK portfolio formulations, including tilting-based and constrained optimization problems, enhancing its practical versatility.