Discrete Dynamics in Nature and Society

Volume 2016, Article ID 8417643, 11 pages

http://dx.doi.org/10.1155/2016/8417643

## Stochastic Portfolio Selection Problem with Reliability Criteria

^{1}Department of Economic Management, North China Electric Power University, Baoding 071003, China^{2}State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China

Received 8 October 2015; Accepted 7 February 2016

Academic Editor: Kamel Barkaoui

Copyright © 2016 Xiangsong Meng and Lixing Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Portfolio selection focuses on allocating the capital to a set of securities such that the profit or the risks can be optimized. Due to the uncertainty of the real-world life, the return parameters always take uncertain information in the realistic environments because of the scarcity of the a priori knowledge or uncertain disturbances. This paper particularly considers a portfolio selection process in the stochastic environment, where the return parameters are characterized by sample-based correlated random variables. To decrease the decision risks, three evaluation criteria are proposed to generate the reliable portfolio selection plans, including max-min reliability criterion, percentile reliability criterion, and expected disutility criterion. The equivalent linear (mixed integer) programming models are also deduced for different evaluation strategies. A genetic algorithm with a polishing strategy is designed to search for the approximate optimal solutions of the proposed models. Finally, a series of numerical experiments are implemented to demonstrate the effectiveness and performance of the proposed approaches.

#### 1. Introduction

Portfolio selection problems deal with how to allocate one capital to a given number of securities so that the involved return can be maximized in this process. When the return of each security is a constant, this problem can typically be formulated as a linear programming model and efficiently solved through the simplex method. Due to the uncertainty of the real-world applications, the actual return of each security usually cannot be prespecified in advance. In this case, how to effectively choose portfolio section strategy is a key problem for the investors in order to generate the least-risk plans and produce the expected return as much as possible. Along this line, Markowitz [1, 2] first proposed the mean-variance models in stochastic environments, in which the variance is used to quantify the existing risks in the uncertain return. In this method, a tolerance threshold is usually given for the portfolios. A portfolio selection plan is a safe (or a low-risk) strategy if the variance of its corresponding random return is not greater than this threshold. Based on this approach, a variety of researches about the portfolio selection with either random parameters or fuzzy parameters have been proposed in the literature, such as Markowitz et al. [3], Gao et al. [4], Yi et al. [5], Xing et al. [6], H. Levy and M. Levy [7], Chiu and Wong [8], A. Palczewski and J. Palczewski [9], Castellano and Cerqueti [10], Fu et al. [11], and Zhang et al. [12]. The second approach to represent the risk proposed by Markowitz [1, 2] is the semivariance. In comparison to the mean-variance method, the mean-semivariance approach can better handle the risks in case of asymmetrical security return distributions. Along this line, interested readers can refer to Zhang et al. [13], Huang [14], Najafi and Mushakhian [15], Yan et al. [16], Yang et al. [17], and so forth. Besides the aforementioned two approaches, some effective chance-constrained methods can also be adopted to characterize the risks such as Huang [14], Huang and Zhao [18], and Li et al. [19].

Different from the variance, semivariance and chance-constrained risk based methods, this paper aims to introduce the reliability term into the portfolio selection problem by using some reliability evaluation indexes, which can be used to optimize the low-risk portfolio strategies according to the real-world applications. We note that the current studies in the literature mainly focus on two specific uncertainties, that is, randomness and fuzziness, in which the returns of the involved securities are often assumed to be independent random variables or fuzzy variables (without correlations). Differently, this paper intends to propose a new representation method for the random return of each security on the basis of sample-based random data framework. This representation can allow for the correlations among returns of different securities in different periods. For instance, in each week (or month), we can collect realistic return data for individual securities, in which the week-specific or month-specific data can be regarded as having correlations among each other, and each week or month can be regarded as a considered stochastic sample with a specified probability. On the basis of these historical data, it is desirable for decision-makers to produce the reliable portfolio strategies to effectively reduce the investigation risk incurred by various uncertain factors. This research will particularly address this issue. To the best of our knowledge, no existing researches paid much attention to the data representation with inherent correlations.

As addressed above, the majority of existing studies always focus on the mean-variance (or semivariance) based approach to decrease the decision risks, in which the objective is often assumed to find the maximal expected return within the given risk threshold. However, in uncertain portfolio selection, the definition of an optimal portfolio may vary, since there are a large number of optimality criteria to measure the existing risks. Through adopting the sample-based random return to capture the randomness in the investigation process, this paper focuses on introducing three risk criteria into the portfolio selection problem to produce the risk-aversion planning, including the min-max criterion, percentile criterion, and expected disutility criterion, following the classical von Neumann and Morgenstern paradigm of decision under risk in economics [20]. In particular, these risk-aversion criteria have been successfully applied to some real-world applications, for instance, management, economics, and transportation (see [21–24]). In particular, with the sample-base random data representation and these risk measures, it is possible for us to transform the formulated models into linear (or mix-integer) programming, which can be expected to solve through either commercial optimization software or the variants of current exiting algorithms. In addition, to effectively solve the proposed models, we in particular design a genetic algorithm with the polishing strategy to search for the near-optimal solutions. Numerical experiments show that the solution quality can be improved greatly in comparison to the traditional genetic algorithm without the polishing strategy, demonstrating the effectiveness of the proposed approaches.

The remainder of this paper is organized as follows. In Section 2, we formulate the problem of interest with three risk evaluation criteria, and the equivalent linear or mixed-integer programming models are also deduced with specific proofs. In Section 3, a genetic algorithm is designed based on the polishing strategy to solve the proposed models when the model cannot be transformed into linear form or solved by commercial optimization software. Section 4 implements different experiments to test performance of the proposed models and algorithms. In Section 5, a conclusion is made finally.

#### 2. Formulation

Portfolio selection problem involves how to allocate the capital to a given number of securities so that the involved return can be maximized. Because of the uncertainty of the real-world investment environments, the returns of different securities are always set as independent uncertain variables over the entire decision process. This paper aims to handle the portfolio selection problem with a different input data representation, which is termed as sample-based random returns for different securities. Each sample corresponds to a distinct value (vector of values) that random return vector can take. By this method, it can be regarded that the elements in each sample vector are correlated with each other. Particularly, the historical return data over different periods (e.g., month, week) can be used as the input data in producing the least risk investment strategy. In the following, some relevant parameters and notations will be firstly given below.

*(i) Notations and Decision Variables. *Consider the following: : the index of securities. : the index of samples. : the probability of sample . : the return of the th security over the sample . : the return of the th security, which is denoted by a random variable. : the proportion invested for security .

*(ii) Constraints.* Since denotes the proportion invested for security , it is required that the sum of all the proportion should be a unity. Thus we need to consider the following constraints; that is,

*(iii) Objective Functions*. In the following discussion, we will specify different types of objective functions in formulating the problem of interest. We have a total of sets of historical data about the returns of different securities due to the portfolio activities in the past. In each set of historical data, we assume that all the returns of individual securities are correlated with each other. With the current set of historical data which may have a lot of uncertainties among different data set, we aim to find the most reliable decision-making plans to decrease the decision-making risks.

Typically, there are a lot of methods to handle the risks of the securities. Currently, we propose three types of methods to clearly handle the inherent uncertainties in the portfolio process. The first model is referred to as the max-min reliability model, detailed below.

##### 2.1. Max-Min Reliable Model of the Portfolio Selection Problem

According to the a priori information, we have a total of sample data about the returns of different securities. With this concern, we can actually produce types of total returns over different samples for each given portfolio selection strategy. Max-min model aims to find the most reliable portfolio plan across different samples such that the risk of decision-making can be decreased as much as possible. In detail, we denote the each sample-based total return by , , given below:To integrate these returns, we first calculate the minimal value over different returns as follows:Then the max-min reliable model can be formulated by In this model, it is easy to see that the objective function aims to optimize the lower bound of the returns over different samples. Typically, this is also a conservative decision-making model for the real-world applications.

Figure 1 gives an illustration for the random return for different solutions. As shown, different solutions might correspond to individual return values for different samples. According to the max-min criterion, the least realization value will be used as the evaluation of this solution. In Figure 1, we consider two solutions and . Typically, since , solution is better than in this criterion.