Elsevier

Applied Soft Computing

Volume 12, Issue 2, February 2012, Pages 754-758
Applied Soft Computing

A portfolio selection model with borrowing constraint based on possibility theory

https://doi.org/10.1016/j.asoc.2011.10.017Get rights and content

Abstract

Compared with the conventional probabilistic mean–variance methodology, fuzzy number can better describe an uncertain environment with vagueness and ambiguity. In this paper, the portfolio selection model with borrowing constraint is proposed by means of possibilistic mean, possibilistic variance, and possibilistic covariance under the assumption that the returns of assets are fuzzy numbers. And a quadratic programming model with inequality constraints is presented when the returns of assets are trapezoid fuzzy numbers. Furthermore, Lemke algorithm is utilized to solve the model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the numerical example also show that the investor can make different decisions according to different requirements for the values of expected returns. And the efficient portfolio frontier of the model with borrowing constraints can be easily obtained.

Introduction

Portfolio selection is concerned with the problem of allocating one's wealth among alternative securities such that the investment goal can be achieved. The mean–variance model originally introduced by Markowitz [1], [2] plays an important role in the development of modern portfolio selection theory. The key principle of the mean–variance model is to use the expected return of a portfolio as the investment return and to use the variance of returns of the portfolio as the investment risk. Markowitz's portfolio model is a bi-criteria optimization problem where a reasonable trade-off between return and risk is considered-minimizing risk for a given level of return, or equivalently, maximizing expected return for a given level of risk, and the concerned model of this paper is the former. Typical researchers about the mean–variance portfolio selection include Sharpe [3], Merton [4], Perold [5], Pang [6], Elton and Gruber [7], Fang et al. [8] and Giove et al. [9].

There are many non-probabilistic factors that affect the financial market such that the return of risky asset is fuzzy uncertainty. Recently, a number of researchers investigated fuzzy portfolio selection problem. Bellman and Zadeh [10] proposed the basic fuzzy decision theory. Carlsson and Fullér [11] discussed some basic knowledge about possibilistic mean and variance of fuzzy numbers. Arenas et al. [12] proposed a fuzzy goal programming approach to solve the portfolio selection problem with three criteria: return, risk and liquidity. Huang et al. [13] discussed a new algorithm for the revised mean–variance model which was proposed to deal with the problem of uncertain portfolio selection. Wang and Zhu [14] gave an overview on the development of fuzzy portfolio selection using fuzzy approaches, quantitative analysis, qualitative analysis, the experts’ knowledge and manager's subjective opinions. Ida [15] considered portfolio selection problem with interval and fuzzy objective function coefficients as a kind of multiple objective problems including uncertainties. Abiyev and Menekay [16] presented that fuzzy logic was utilized in the estimation of expected return and risk. Managers could extract useful information and estimate expected return by using not only statistical data, but also economical and financial behaviors of the companies and their business strategies. Lacagnina and Pecorella [17] discussed that the financial market behavior was affected by several non-probabilistic factors such as vagueness and ambiguity and developed a multistage stochastic soft constraints fuzzy program with recourse in order to capture both uncertainty and imprecision as well as to solve a portfolio management problem. Huang [18], gave a new definition of risk for random fuzzy portfolio selection utilizing a different perspective that returns contained both randomness and fuzziness. Huang [19] also proposed two new models for portfolio selection in which the security returns were stochastic variables with fuzzy information. Vercher [20] provided new models for portfolio selection in which the returns on securities were considered fuzzy numbers rather than random variables. The corresponding optimal portfolio was derived using semi-infinite programming in a soft framework. The investment risk was approximated by mean intervals which evaluate the downside risk for a given fuzzy portfolio.

The rest of this paper is organized as follows. In Section 2, the possibilistic mean, the possibilistic variance, and the possibilistic covariance are discussed. In Section 3, the possibilistic efficient portfolio selection model with borrowing constraint is proposed as a quadratic programming. In Section 4, Lemke algorithm is introduced and the corresponding concrete computation steps are given. In Section 5, a numerical example is used to verify the effectiveness of the proposed model and algorithm used in this paper, and the corresponding efficient frontier of the portfolio model is obtained.

Section snippets

Possibilistic mean, variance and covariance

Let us introduce some definitions, which we shall need in the following section. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers is denoted by F. A γ-level set of a fuzzy number A is defined by [A]γ = {t  R|A(t)  γ} if γ > 0 and [A]γ = cl {t  R|A(t) > 0} (the closure of the support of A) if γ = 0. It is well known that if A is a fuzzy number, then [A]γ is a compact subset of R for all γ  [0, 1].

Possibilistic efficient portfolio selection model with borrowing constraint

In the conventional Markowitz's mean–variance model, the return rate of risky asset is considered as a random variable. It is well known that the returns of risky assets are in a fuzzy uncertain economic environment and vary from time to time, the future states of returns and risk of risky assets cannot be predicted accurately. Fuzzy number is a more powerful tool used to describe an uncertain environment with vagueness and ambiguity. Based on these factors, we consider the portfolio selection

Basic ideas of Lemke algorithm

Lemke algorithm is an effective algorithm of solving quadratic programming. Its basic ideas are follows: first amending the simplex method properly, and then solving the Kuhn–Tucker (K–T) point of the quadratic programming.

Now, we consider the following quadratic programming problem:minf(x)=12xHx+cx,s.t.Axb,x0.where H is an n × n symmetric matrix, c is an n-dimensional column vector, A is an m × n matrix, the rank of A is m, and b is an m-dimensional column vector.

By introducing the

Numerical example

In order to illustrate our proposed effective means and variances of the efficient portfolio in this paper, we considered a real portfolio selection example. In this example, we selected six stocks from Shanghai Stock Exchange, their returns ri(i = 1, 2, …, 6) are regarded as trapezoid fuzzy numbers. Based on the historical data and the future information and the advices of experts and the corporations’ financial reports, we could estimate their returns with the following possibility

Conclusions

Fuzzy number is a powerful tool used to describe an uncertain environment with vagueness and ambiguity to compare with the conventional probabilistic mean–variance methodology. In this paper, we have used the possibilistic means, variances and covariances to replace the probabilistic means, variances and covariances in Markowitz's mean–variance model, respectively. We propose a quadratic programming with inequality borrowing constraints based on the possibilistic theory when the returns of

Acknowledgements

This research was supported by the National Natural Science Funds of China under Grant No. X2gsB5101840 and supported by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011ZM0082. The authors are also grateful to a referee for his/her very helpful comments and suggestions.

References (20)

There are more references available in the full text version of this article.

Cited by (28)

  • A new mean-variance-entropy model for uncertain portfolio optimization with liquidity and diversification

    2021, Chaos, Solitons and Fractals
    Citation Excerpt :

    However, it is sometimes hard to achieve for complex financial markets. Based on fuzzy set theory [14], many researches have employed fuzzy variables to describe experts’ subjective estimates, such as [15–20]. Among these, entropy and liquidity are widely used in fuzzy portfolio optimization models [21–25].

  • Uncertain portfolio optimization problem under a minimax risk measure

    2019, Applied Mathematical Modelling
    Citation Excerpt :

    Mehlawat [17] considered two credibilistic mean-entropy models for portfolio optimization problem with multi-choice aspiration levels. Deng and Li [19] proposed a portfolio optimization model with borrowing constraint in the form of fuzzy mean-variance formulation. Li et al. [20] formulated a fuzzy mean-variance-skewness model.

  • Credibilistic multi-period portfolio optimization model with bankruptcy control and affine recourse

    2016, Applied Soft Computing Journal
    Citation Excerpt :

    Actually, fuzzy portfolio selection problem was researched from 1990s. Based on possibility theory, numerous portfolio selection models had been proposed, e.g., Carlsson et al. [4], Deng and Li [12], Tanaka and Guo [47] and Zhang et al. [58]. Due to the self-dual of possibility measure, Liu and Liu [34] defined a self-dual measure (i.e., credibility measure) to quantify the chance of occurrence of fuzzy events.

  • Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints

    2014, Fuzzy Sets and Systems
    Citation Excerpt :

    Most of the brokerage houses provide the opportunity to make an acquisition on different assets by borrowing the money from the brokerage. Some researchers studied the borrowing constraints, for example, Deng and Li [13] proposed a mean-variance fuzzy portfolio with borrowing constraint. Sadjadi et al. [43] proposed the fuzzy multiperiod portfolio selection with different rates for borrowing and lending.

  • Gradually tolerant constraint method for fuzzy portfolio based on possibility theory

    2014, Information Sciences
    Citation Excerpt :

    In the portfolio model, Arenas et al. [5] took three criteria into account: return, risk and liquidity. Considering the uncertain returns of risky assets as fuzzy numbers, Deng [13] obtained some new results on value ranges of risks for mean–variance portfolio models; Deng [14] also proposed a portfolio selection model with borrowing constraint based on possibility theory; Ida [19] considered the portfolio selection problem with interval and fuzzy objective function coefficients as a kind of multi-objective problem; Li et al. [21] presented a mean–variance-skewness model in 2010, which extended the fuzzy mean–variance model; Huang [18] presented a new perspective for the portfolio selection model; Zhang et al. [28] obtained the effective frontiers for portfolio possibilistic mean–variance models. Furthermore, some researchers also paid more attention to the multi-objective portfolio selection models.

View all citing articles on Scopus
View full text