Elsevier

Neurocomputing

Volume 98, 3 December 2012, Pages 90-100
Neurocomputing

Bacterial foraging based approaches to portfolio optimization with liquidity risk

https://doi.org/10.1016/j.neucom.2011.05.048Get rights and content

Abstract

This paper proposes a bacterial foraging based approach for portfolio optimization problem. We develop an improved portfolio optimization model by introducing the endogenous and exogenous liquidity risk and the corresponding indexes are designed to measure the endogenous/exogenous liquidity risk, respectively. Bacterial foraging optimization (BFO) is employed to find the optimal set of portfolio weights in the improved Mean-Variance model. BFO-LDC which is a modified BFO with linear deceasing chemotaxis step is proposed to further improve the performance of BFO. With four benchmark functions, BFO-LDC is proved to have better performance than the original BFO. And then comparisons of the results produced by BFO, BFO-LDC, particle swarm optimization (PSO), and genetic algorithms (GAs) for the proposed portfolio optimization model are presented. Simulation results show that BFOs can obtain both near optimal and the practically feasible solutions to the liquidity risk portfolio optimization problem. In addition, BFO-LDC outperforms BFO in most cases.

Introduction

Portfolio optimization (PO) is an investment problem which tries to maximize the expected return by selecting a proper combination of various securities among a large number of them in financial industry [1]. PO is a NP-hard and non-linear problem with many local optima. A lot of work has been done which attempted to solve this problem by a variety of techniques, such as cutting planes, interior point methods, decomposition etc, but exact solution methods failed to solve large-scale instances of the problem. The advent of evolutionary computation (EC) inspired as a new technique for optimal selection of portfolio assets. A number of different evolutionary computation approaches have been proposed to solve this problem, including genetic algorithms [2], simulated annealing [3], neural networks [4] and others [5], [6], [7].

Recently, bacterial foraging optimization (BFO) has emerged as a powerful technique for optimization problems [8], [9], [10]. It has been successfully applied to solve many real world problems like harmonic estimation [11], transmission loss reduction [12], active power filter for load compensation [13], power network [14], load forecasting [15], and stock market prediction [16]. In BFO, each bacterium updates its position using chemotaxis, swarming, reproduction, elimination, and dispersal. Among them the chemotaxis procedure is a key step in BFO.

In [17], we focused on investigating the ability of BFO to achieve high-quality solutions to general PO problem and also a new and more complicated PO problem, which was under liquidity risk environment. The liquidity risk is one of the most important adjustable parameters in PO problem. Bangia et al. [18] proposed a novel portfolio optimization model considering both the liquidity risk and the market risk by VAR, which named BDSS model and Heude–Wynendaele model. Consigi [19] studied the mean-VAR model in the case of fat tailed distribution. Berkowitz [20] applied the VAR to measure the bank liquidity risk. Anil et al. [21] constructed a new liquidity risk model with implications for market risk, but they ignored the endogenous liquidity risk.

In [17], we considered the difference between Chinese securities market (a centralized auction system) and foreign securities market (a market-maker system). An improved model using VAR to measure both market and liquidity risk was proposed, and three evolutionary computation techniques (GA, PSO and BFO) were applied to solve the new model with complex constrains. The comparative results illustrated that the improved PO model and the performance of BFO was relatively efficient. However, the improved PO model is a non-linear and complex optimization problem, the results produced by all the three approaches (GA, PSO and BFO) are not optimal solutions.

To solve this mixed-integer non-linear programming (NP-hard) more efficiently, we propose a modified BFO with the chemotaxis step varying dynamically as linear functions of iterations in this paper. In original BFO the chemotaxis step length is set as a constant value. There is not any mechanism to keep the balance of global search and local search. This restricts BFO to be applied to complex optimization problems. Our proposed method employed a linear decreasing chemotaxis step strategy, which allowed each bacterium kept a good balance between exploration and exploitation by decreasing its run-length unit linearly.

In order to demonstrate the performance of the proposed algorithm, BFO-LDC is applied into the improved portfolio optimization model and compared the test results with those of the original BFO, GA and PSO.

The rest of the paper is organized as follows. Section 2 gives a review of BFO and a description of the proposed algorithm BFO-LDC. In Section 3, it will be shown that BFO-LDC outperforms BFO on four benchmark functions. Section 4 describes the improved portfolio optimization model and a detailed design algorithm of BFO approaches for liquidity risk portfolio optimization. In Section 5, experimental settings and experimental results are given. Finally, Section 6 concludes the paper.

Section snippets

Bacterial foraging optimization (BFO)

Based on the biology and physics underlying the foraging behavior of Escherichia coli, Liu and Passino [9] exploited a variety of bacterial swarming and social foraging behaviors, discussed how the control system on the E. coli dictated and how foraging should proceed. In the bacterial foraging process, four motile behaviors (chemotaxis, swarming, reproduction, and elimination and dispersal) are mimicked.

  • (1)

    Chemotaxis: This process simulates the movement of an E. coli cell through swimming and

Benchmark functions

A set of well-known benchmark functions that are extensively used to compared both BFO-type and non-BFO-type bio-heuristic algorithm, were used to evaluate the performance, both in terms of solution quality and convergence rate, of the proposed algorithm. Our test suite includes 4 well-known benchmark functions, which present different difficulties to the algorithms to be evaluated. These benchmark functions can be grouped into unimodal functions (f1, f2) and multimodal functions (f3, f4). For

Liquidity risk portfolio optimization

The liquidity is the vitality of stock market. It is regarded as an important symbol of the maturity also a significant quality indicator of the stock market. The market provides sufficient liquidity to help investors to change their assets to cash. But it is not liquid all the time. It has friction and there must be liquidity risk. Modern portfolio analysis started from the work of Markowitz [23] who proposed the original Mean-Variance model, then Alexander and Baptista [24] established a new

Experimental settings

We choose four assets as the sample: PuFa Bank (600000), JiangXi Copper Industry (600362), ShangHai Automotive Industry (600124) and China Petrochemical Corporation (600028), which are from different industries, different places. The basic data about the assets were taken from January 1st in 2009 to December 30th in 2009, and we got the interrelated index value needed in the experiment based on them. We considered the different kinds of the investors, and four different risk factors λ are use

Conclusions

In this paper, we focused on solving the portfolio optimization problem with liquidity risk using BFO based methods. Instead of using standard Mean-Variance model, we proposed a new model using VAR measuring both market and liquidity risk. The improved portfolio model is a non-linear and complex optimization problem which is much harder to be solved by conventional techniques.

We employ a relatively new swarm intelligence based method, BFO to solve this model. In additional, a linear decreasing

Acknowledgment

This work is supported by National Natural Science Foundation of China (Grant no. 71001072), China Postdoctoral Science Foundation (Grant no. 20100480705), Science and Technology Project of Shenzhen (Grant no. JC201005280492), The Natural Science Foundation of Guangdong Province (Grant no. 9451806001002294), 863 Project (Grant no. 008AA04A105), and Project 801-000021 supported by SZU R/D Fund.

Ben Niu received the B.S. degree in mechanical engineering from Hefei Union University, Hefei, China, in 2001, the M.S. degree in enterprise information management from Anhui Agriculture University, Hefei, China, in 2004, and the Ph.D. degree in mechanical & electrical engineering from Shenyang Institute of Automation of the Chinese Academy of Sciences, Shenyang, China, in 2008. From 2008 to the present, he has been with the Department of Management Science of Shenzhen University. His main

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    Ben Niu received the B.S. degree in mechanical engineering from Hefei Union University, Hefei, China, in 2001, the M.S. degree in enterprise information management from Anhui Agriculture University, Hefei, China, in 2004, and the Ph.D. degree in mechanical & electrical engineering from Shenyang Institute of Automation of the Chinese Academy of Sciences, Shenyang, China, in 2008. From 2008 to the present, he has been with the Department of Management Science of Shenzhen University. His main fields of research are swarm intelligence, bio-inspired computing, multiobjective optimization and their applications on RFID system, business intelligence, and portfolio optimization.

    Yan Fan obtained her B.S. degree in information management and information system from the College of Management, Tianjin Polytechnic University, China, in July 2008. She is currently pursuing her M.S. degree in management science and engineering in College of Management, Shenzhen University. Her research interests include bacterial foraging optimization and its application on portfolio optimization and RFID network planning.

    Xiao Han obtained his combined bachelor's degree in management science and finance, from Finance and Economics University of Jiangxi, China, in July 2008. Currently he is a master student in College of Management, Shenzhen University. His research focus is computational intelligence, especially particle swarm optimization and bacterial foraging optimization.

    Bing Xue obtained the B.S. degree in information management and information system from Henan University of Finance and Economics, Zhengzhou, China, in 2007 and the M.S. degree in management science and engineering from Shenzhen University, Shenzhen, China, in 2010. She is currently working toward her Ph.D. degree in school of engineering and computer science in Victoria University of Wellington, Wellington, New Zealand. Her current research interest focus on swarm intelligence algorithms such as bacterial foraging optimization and particle swarm optimization, both for the portfolio selection problems.

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