A hybrid approach for portfolio selection with higher-order moments: Empirical evidence from Shanghai Stock Exchange

https://doi.org/10.1016/j.eswa.2019.113104Get rights and content

Highlights

  • A portfolio selection problem with higher-order moments is considered.

  • Machine learning algorithms are applied for data analysis and prediction in the stock market.

  • Genetic algorithm is used to solve the multi-objective optimization problem.

  • The out-of-sample performance of our model is significantly better than those of traditional ones.

  • Robustness is checked, compared with another two existing methods.

Abstract

Skewness and kurtosis, the third and fourth order moments, are statistics to summarize the shape of a distribution function. Recent studies show that investors would take these higher-order moments into consideration to make a profitable investment decision. Unfortunately, due to the difficulties in solving the multi-objective problem with higher-order moments, the literature on portfolio selection problem with higher-order moments is few. This paper proposes a new hybrid approach to solve the portfolio selection problem with skewness and kurtosis, which includes not only the multi-objective optimization but also the data-driven asset selection and return prediction, where the techniques of two-stage clustering, radial basis function neural network and genetic algorithm are employed. With the historical data from Shanghai stock exchange, we find that the out-of-sample performance of our model with higher-order moments is significantly better than that of traditional mean-variance model and verify the robustness of our hybrid algorithm.

Introduction

Portfolio selection problem has been one of the core issues of the modern investment theory. It originates from the mean-variance model by Markowitz (1952), which measured the expected return and risk of a portfolio by mean and variance, and thus first transformed the portfolio selection problem into a mathematical model. Based on this, researchers further extended the classical model by applying other risk measurements, such as semi-variance, absolute deviation, and semi-absolute deviation. See Steinbach (2001) for a review on these significant ones which considered only the first and second order moments of return distribution. However, these moments are inadequate in practice. Meanwhile, the performance of a portfolio selection model highly depends on the parameter estimation and the initial asset selection. We thus intend to build a hybrid algorithm, which includes not only data-driven asset selection and prediction, but also solving the portfolio selection problem with higher-order moments.

In the literature, the importance of higher order moments has attracted increasing attention. Liu (2004) and Maringer and Parpas (2009) proved that only when the investor’s utility function is quadratic or the asset’s yield obeys the normal distribution, the influence of higher-order moments can be ignored. Unfortunately, both of these two premises are hardly satisfied in the real world. Empirical studies found that the risk assets’ yields have fat tails and do not obey the normal distribution (Chunhachinda, Dandapani, Hamid, Prakash, 1997, Konno, Suzuki, 1992). There are also works (see e.g. Paravisini, Rappoport, & Ravina (2016)) observing the fact that an investor’s risk aversion degree and sensitivity will change with the wealth accumulation, whereas the quadratic utility function is unable to characterize this feature and thus is not practical in reality. Hence, it is necessary to consider higher-order moments when constructing the portfolio optimization model.

In the past decades, an increasing number of studies have involved skewness into the portfolio optimization model, and observed that the mean-variance-skewness (MVS) model performs better than the classical mean-variance one (see e.g. Adcock (2014); Briec, Kerstens, and Jokung (2007); Joro and Na (2006); Liu, Han, and Han (2016); Zhai, Bai, and Wu (2018)). Unfortunately, there are few studies on kurtosis, due to the difficulties in obtaining the optimal solution to the the higher-order-moment optimization problem, which is a multi-objective problem in essence. Maringer and Parpas (2009) applied stochastic optimization algorithms to extended mean-variance portfolio selection problems with either skewness or kurtosis being considered at a time. Saranya and Prasanna (2014) and Aksaraylı and Pala (2018) studied the mean-variance-skewness-kurtosis framework by constructing a polynomial goal programming model for the higher moments. Chen and Zhou (2018) applied multiobjective particle swarm optimization to deal with higher moments, but focused on the parameter uncertainty.

Inspired by the existing works on the mean-variance portfolio selection problem (see e.g., Li and Ng (2000) and Cui, Li, Wang, and Zhu (2012)), we, instead of construct a polynomial goal programming model, incorporate the trade-off parameters over the mean, variance, skewness and kurtosis to formulate a single target model. However, this constrained non-linear optimization problem is still difficult to solve. Traditional optimization algorithms for constrained non-linear optimization problem have quite low convergence rates, and may even not converge to a solution in extreme cases. Hence, we turn to machine learning algorithms to boast a high generalization and a fast convergence. In particular, we apply genetic algorithm, which can not only boast a fast searching speed, but also reduce the risk of falling into a local minimum trap, to solve our single target problem.

Moreover, we focus on how to improve the asset selection and return prediction with machine learning algorithm and historical data, instead of considering robust model with parameter uncertainty. In fact, a real portfolio selection problem starts with the selection of assets. Most present studies rely on investors’ experience to select risk assets as samples to build the prior portfolio, which appears subjective and lacks the support of scientific selection criteria. In addition, the performance of a portfolio selection model highly depends on the parameter estimations, such as the return vector and the risk measurements. The parameters estimated by the historical data may not be able to predict the future.

Our paper is thus also related to the works applying machine learning algorithms in various financial fields. Patel, Shah, Thakkar, and Kotecha (2015) and Chong, Han, and Park (2017) applied machine learning techniques for market analysis and prediction. See (Henrique, Sobreiro, & Kimura, 2019) for a review on the machine learning techniques applied to financial market prediction. Paiva, Cardoso, Hanaoka, and Duarte (2019) proposed a unique decision-marking model for day trading investments on stock market based on machine learning methodology with the classical mean-variance model. Varied clustering techniques are applied on financial data to identify the similarity inside the time series data (See e.g., Iorio, Frasso, D’Ambrosio, and Siciliano (2016); Zhang, Liu, Du, and Lv (2011) and the references therein). Iorio, Frasso, D’Ambrosio, and Siciliano (2018) furthermore applied the P-spline based clustering approach on financial data to build a portfolio. We also include the two-stage clustering method for asset selection in our hybrid algorithm. In fact, we apply machine learning algorithms for not only the market prediction and asset selection before the optimization problem, but also solving the multi-objective portfolio selection problem with higher order moments to identify the optimal portfolio.

To conclude, in this paper, we investigate the portfolio selection problem with skewness and kurtosis by applying a hybrid approach to select pre-diversified assets, predict the returns and optimize the portfolio.

First of all, to deal with this multi-objective problem, we introduce the risk preference to transfer it to a non-linear optimization problem and apply the genetic algorithm, which is a probability-based directional searching tool and is able to reduce the risk of falling into a local minimum trap, to numerically solve the considered portfolio optimization.

Second, instead of solving the portfolio selection problem with parameters directly estimated from the historical data, we propose a hybrid approach which includes the asset selection, return prediction and portfolio optimization. In particular, besides the genetic algorithm applied to solve the optimization problem, our hybrid approach also includes two-stage clustering and radial basis function neural network. The two-stage clustering, also known as Chameleon algorithm, is a statistical data analysis technique proposed by Karypis, Han, and Kumar (1999). The basic idea of this method is to classify a set of objects according to the inter-connectivity and closeness such that objects in a cluster are more similar to each other than those in the other cluster. We thus select assets with the two-stage clustering for a diversified portfolio. We then apply the radial basis function neural network on these selected assets to predict the future returns.

Finally, we apply the historical data from Shanghai Exchange to compare the performance of our model with those of mean-variance and mean-variance-skewness models and testify the robustness and efficiency of our hybrid algorithm.

The rest of this paper is organized as follows. Section 2 introduces the higher-order-moment portfolio optimization model and the genetic algorithm for solving it. Section 3 illustrates our hybrid approach, which includes not only the higher-order-moment portfolio optimization, but also the data-driven asset selection and return prediction for a pre-diversified portfolio selection. Section 4 performs the numerical experiments to test the model’s efficiency and reports the experimental results. Section 5 investigates the robustness of our algorithm. The last section concludes this paper.

Section snippets

Higher-order-moment portfolio optimization

In this section, we formulate our portfolio selection model with skewness and kurtosis, and introduce the genetic algorithm to solve this higher-order-moment portfolio optimization problem.

A hybrid approach for higher-order-moment portfolio optimization

In this section, we present our hybrid approach for the portfolio optimization problem, which includes not only the multi-objective optimization but also the data-driven asset selection and return prediction. We adopt the genetic algorithm mentioned in Section 2, and the techniques of two-stage clustering and radial basis function (RBF) neural network in the hybrid approach. We summarize the framework of the hybrid approach for solving higher-order-moment portfolio optimization problem in Fig. 2

Empirical analysis

In this section, we apply our hybrid algorithm, as stated in Section 3, to real transaction data from Shanghai Stock Exchange in China. We first briefly introduce the dataset we investigate, and then present our results of each step for our sample data in this section, according to the flowchart stated in Section 3.

Robustness of our hybrid method

This section further compares the solutions of the M-V-S-K model obtained by different algorithms, in order to test the robustness of our hybrid method. We also present some computational time analysis of our hybrid method and other alternatives.

For the purpose of a comparison, we choose another two algorithms, the mixed penalty function method (MPFM) and the simulated annealing algorithm (SA). The former one combines the advantages of exterior and interior penalty and is able to solve

Conclusion and future work

This paper studies the portfolio selection problem with higher-order moments. Aiming at extending the classical mean-variance model, this paper introduces skewness and kurtosis of the portfolio to construct the mean-variance-skewness-kurtosis (M-V-S-K) model. To solve our model, we transform the multi-objective optimization problem into a non-linear programming model with risk preferences. Furthermore, we propose a hybrid approach including three machine learning algorithms to select the

CRediT authorship contribution statement

Bilian Chen: Conceptualization, Methodology, Formal analysis, Validation, Supervision, Investigation, Writing - review & editing, Resources, Funding acquisition. Jingdong Zhong: Conceptualization, Methodology, Formal analysis, Software, Investigation, Data curation, Writing - original draft, Visualization. Yuanyuan Chen: Conceptualization, Methodology, Resources, Formal analysis, Investigation, Data curation, Visualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grants nos. 61772442, 11671335 and 61836005).

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