Date Published: March 13, 2019
Publisher: Public Library of Science
Author(s): Jianjian Wang, Feng He, Xin Shi, Baogui Xin.
Based on the Markowitz mean variance model, this paper discusses the portfolio selection problem in an uncertain environment. To construct a more realistic and optimized model, in this paper, a new general interval quadratic programming model for portfolio selection is established by introducing the linear transaction costs and liquidity of the securities market. Regarding the estimation for the new model, we propose an effective numerical solution method based on the Lagrange theorem and duality theory, which can obtain the effective upper and lower bounds of the objective function of the model. In addition, the proposed method is illustrated with two examples, and the results show that the proposed method is better and more feasible than the commonly used portfolio selection method.
With the mature development of the securities market, in the last decade, studies have paid increasing attention to the theory of portfolio selection. The first quantitative mean variance model for portfolio selection was developed by Markowitz , which considers the expected return and variance to be crisp numbers and seeks a balance between two objectives: maximizing the expected return and minimizing the risk in the portfolio selection. Since the 1950s, the quantitative methods for portfolio selection have been dramatically developed in both theories and applications. The deterministic portfolio model that Markowitz developed has been further extended by numerous scholars [2–8]. In these extended portfolio selection models, the coefficients in the objective function and constraint function are always determined as crisp values. However, because of the national economic situation, policy changes, investor psychology and many other factors, the securities market has a strong uncertainty, which causes the dynamic expected returns, risk loss rate and liquidity of the securities market . Moreover, the uncertainties increase the risk of decision-making on portfolio selection for investors. There are two popular approaches to address such uncertainties: (i) fuzzy programming and (ii) interval programming. Since the future returns of each securities cannot be correctly reflected by the historical data, particularly in an uncertain environment, investors can use the fuzzy set to estimate the vagueness of security returns and risk for the future [10–15], which is a good method to address the portfolio selection. The fuzzy programming treats the uncertain quantities as a fuzzy set with certain membership functions. Thus, the decision maker must have precise knowledge of the grade of membership function, which is not easy to obtain from the limited data that the decision maker often has in practice. In fact, another method to address the uncertainty in the portfolio selection problem assumes that the data are not well defined but can vary in given intervals . Hence, interval programming is appropriate to handle the imprecise input data. The existing literatures indicate that interval programming has become a popular topic in the research of portfolio selection because it can enrich the theory of optimization and provide the solution of the problem more practical significance.
(1) Definition of the interval number and interval matrix
Liu et al. (2015) showed that ignoring transaction costs often leads to invalid portfolio references, so this article introduces the concept of transaction costs . Suppose the investor purchases the risk securities xi(i = 1,2,…,n) to pay the transaction fee, the rate is ci, and the purchase amount does not exceed the given value ui, the transaction fee is calculated according to ui, then the transaction cost function is defined as follows
When considering the transaction cost, we may set the transaction cost function Ci(xi) as a linear function.
This section uses two numerical examples to illustrate the proposed method in this paper to solve a general interval quadratic programming model for portfolio selection. We solve the proposed model using the Lagrange dual method (method 1) in this paper and conventional method (method 2) in Section 3.3. To avoid the occasional results of an experiment and ensure the effectiveness of the results, this paper uses two examples to verify.
In the actual investment environment, considering the strong uncertainty in the securities market, the paper describes the uncertainties of the securities risk, return and corresponding liquidity with interval numbers and establishes a new general interval quadratic programming model for portfolio selection. Next, we propose a new efficient numerical method to solve the proposed model based on the Lagrange theorem and duality theory. To show the efficiency of the proposed Lagrange dual method, two numerical examples were illustrated. The numerical experiment results show that the proposed portfolio selection model is more feasible, and the Lagrange dual method is better than the traditional method in finding smaller solution intervals, which implies that smaller interval objective values correspond to smaller a risk of the portfolio. In addition, this provides a new investment idea for the securities investors. In the actual securities market, various forms of transaction costs likely affect the portfolio selection. However, this paper only considers the transaction cost as a linear function. There remains considerable research space to solve the quadratic programming model of portfolio selection for different forms of transaction costs.