The resolution of excess investment in Fuzzy portfolio selection using piecewise linear function

Abstract

Fuzzy portfolio selection on the guaranteed return rate has been proposed to study excess investment in different dimensions of risk preferences. However, the theory of behavioural finance proposes that the investment proportions of securities are not only determined by the intrinsic value of the security but also largely affected by the behaviour of investors. Therefore, the investment behaviour of an investor in the fuzzy portfolio selection should be considered in the excess investment for different securities at different return rates. In this study, we propose a piecewise linear function for the fuzzy expected return rates and use the new decision variables for investment proportion in each security. In addition, considering the investment behaviours of decision-makers, we use the concept of fuzzy contingent strategy to take into account the relationships of investment proportion among the securities in the fuzzy portfolio selection. Analysis results indicated that we can find the investment proportions in the portfolio with the possible results of shortage investment, full investment, or excess investments by the fuzzy constraint of total investment proportions. Finally, in the illustration with two experiments, investors could individually select the portfolio by the piecewise linear function in shortage investment, full investment, or excess investment.

KEYWORDS:

• Fuzzy portfolio selection
• excess investment
• guaranteed return rate
• piecewise linear function
• contingent strategy

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 08A72
• 90B50
• 90C70
• 90-10

1. Introduction

The portfolio model using mean-variance was proposed by optimizing the objective of the expected return rate with subject to the constrained risk [1]. Thereafter, many researchers in finance have focused on overcoming some criticisms of the traditional portfolio models by relaxing some of Markowitz’s rigid assumptions and dealing with different investment business environments [2–8]. In contrast with the Markowitz model, some behaviours or factors in the knowledge of experts, investors’ subjective opinions, used to evaluate the degree of uncertainty by investors cannot always be assumed to be random variables. Besides, using the previous return rates for portfolio selection cannot overall fit the new economy environment, where a lot of unknown factors are not considered in the portfolio selection. Therefore, in order to cope with this kind of uncertainty portfolio selection, the possibility theory is applied to overcome portfolio selection problems. Tanaka and Guo [9] extended traditional probability measures into fuzzy possibilities to the proposed fuzzy portfolio model. The fuzzy-mean and semi-variance method was applied to deal with the larger returns and avoid risk by eliminating the degree of asymmetry degree of return distributions [10]. Li et al. [11] proposed the fuzzy mean-variance model to measure the asymmetry of fuzzy portfolio return using the concept of skewness in the form of a third central moment, and then they designed a genetic algorithm integrating fuzzy simulation to derive the portfolio. Khan et al. [12] formulated quantum beetle antennae search and a variant of beetle antennae search which are used for solving portfolio selection. Then, they applied QBAS on real-world stock market data, the results showed that the proposed models outperformed swarm algorithms such as the particle swarm optimization and the genetic algorithm. Wang et al. [13] proposed a novel three-way decisions model by using a boundary region with three-way decisions theory to reduce decision risk. By constructing an outranked set for each alternative and a hybrid multi-criteria decision-making matrix, three strategies are proposed to design the three-way decisions model and an improved particle swarm optimization as the solution algorithm. García et al. [14] considered corporate social responsibility as one eligibility criterion and employed other return and risk measures for further constraints to model portfolio cardinality to analyze the trade-off between return, risk and corporate social responsibility in which the multicriteria portfolio selection problem is NP-hard. To solve this problem, the non-dominated sorting genetic algorithm II has been developed, and then applying the proposed portfolio selection methodology to measure socially responsible portfolios by ESG scores. Thereafter, numerous researchers have devoted themselves to the field of fuzzy portfolio selected with the objective of maximizing the fuzzy return rates and constraining the investment risk using possibility theory [15].

On the other, Mehlawat et al. [16] applied credibility theories to propose a multi-objective risk measure and evaluate the fuzzy portfolio selection. Garcia et al. [17] extended the stochastic mean-variance model to a credibilistic multi-objective model in which the uncertainty of the future returns and the liquidity of each risky asset are quantified to measure portfolio performance using trapezoidal fuzzy numbers. Zhou and Xu [18] proposed fuzzy portfolio selection for solving qualitative information represented as hesitant fuzzy elements where both of max-score rule and score-deviation trade-off rule are used to distinguish three types of risk behaviours for the investors. Yadav et al. [19] proposed a comprehensive three-stage sustainable financial portfolio selection approach under an intuitionistic fuzzy framework for ethical and sustainable portfolio selection in an intuitionistic fuzzy environment which can enable the investors to choose the portfolio best suited to their preferences. García et al. [20] extended the stochastic mean-semivariance model to a fuzzy multiobjective model, and they also considered liquidity to measure the performance of a portfolio in which the decision process of this novel approach takes into account not only the multidimensional nature of the portfolio selection problem but also realistic constraints by investors. Then, an empirical study using the data set of the Latin American Integrated Market to analyze the effectiveness and efficiency.

Thereafter, most researchers have focused on the multi-period fuzzy optimization problems for solving the multi-objective problems by genetic algorithm and neural networks [21–24].

From the above literature reviews, we can find that possibility or credibility theories are the major methods in modelling fuzzy portfolio selection based on vagueness information. Tsaur et al. [25–27] proposed the guaranteed return rates to be the thresholds of excess investment for each security in the fuzzy portfolio selection. Theoretically, an investor supposes that he can consistently supervise the market on the guaranteed return rates for excess investment in the portfolio selection, and thus the investor can maximize the expected return rates under the constrained risk. However, some criticisms from the theory of behavioural finance proposed that the investment proportions of securities are not only determined by the intrinsic value of the security but also largely affected by the behaviour of investors perceived from the economic environment. Therefore, the investment behaviour of a decision maker in the fuzzy portfolio affects portfolio selection and should be considered for controlling the investment proportions in the excess investment. That is, according to the economic environment, an investor's behaviour for portfolio selection might be in shortage investment, regular full investment, and excess investment. In order to cope with this issue, we propose a piecewise linear function into the fuzzy expected return rates, and then we use the separated decision variables in investment proportion for each security. In addition, considering the investment behaviours of an investor, we use the concept of fuzzy contingent strategy to take into account the relationships of investment proportion among the securities in the fuzzy portfolio selection. The objectives in the proposed fuzzy portfolio selection are summarized as follows:

1. We decompose the decision variables of security proportions on excess investments into two parts by a piecewise linear function, where the first part is for the decision variables of investment proportion in each security from the traditional fuzzy expected return rate, and the second part is for the decision variables of investment proportion in each security from the excess investment.

2. Next, for the investor's behaviour in the portfolio selection, we fuzzify the constraints to the sum of investment proportions for deriving the possible portfolio selection in shortage investment, full investment, and excess investment.

3. Finally, for controlling the priority among the securities, the fuzzy contingent strategy is proposed to evaluate the relationship among securities in investment proportions.

2. Preliminaries

In this section, we first introduce and define the fundamental algebraic operations of the fuzzy numbers and its defuzzification. In addition, fuzzy expected values, and fuzzy variances are introduced and defined by possibility theory.

2.1. Fuzzy numbers and their operations

A fuzzy number $\tilde{A}$ is described by a membership function as $u_{\tilde{A}}\left(x\right):X\to \left[0,1\right]$ which maps the elements of the universe of discourse X to the interval [0, 1]. Therefore, a fuzzy number can be defined as follows:

Definition 2.1

[28]

Let $\tilde{A}$ be a fuzzy number described as any fuzzy subset of the real line R with a membership function $u_{\tilde{A}}\left(x\right):R\to \left[0,1\right]$ satisfying the following conditions:

1. The fuzzy number $\tilde{A}$ is normal, if there exists an x ∈ R with $u_{\tilde{A}}\left(x\right)=1$;

2. $u_{\tilde{A}}\left(x\right)$ is convex, i.e. $u_{\tilde{A}}\left(\mathrm{\lambda x}+\left(1-\lambda \right)y\right)\ge min\left\{u_{\tilde{A}}\left(x\right),u_{\tilde{A}}\left(y\right)\right\}$, ∀ x, y∈ R and λ ∈ [0, 1];

3. $u_{\tilde{A}}\left(x\right)$ is upper semicontinuous, i.e. $\left\{x\in R:u_{\tilde{A}}\left(x\right)\ge \alpha \right\}={\tilde{A}}^{\alpha }$ is a closed subset of U for each α ∈ (0, 1];

4. The closure of the set $\left\{x\in R:u_{\tilde{A}}\left(x\right)\ge 0\right\}$ is a compact subset of R.

Definition 2.2

[28]

A fuzzy number $\tilde{A}$ is defined as LR-type fuzzy number as $\tilde{A}=(a,c_1,c_2{)}_{\mathrm{LR}}$, then the membership function of $\tilde{A}=(a,c_1,c_2{)}_{\mathrm{LR}}$ has the following form: $u_{\tilde{A}}\left(x\right)=\{\begin{array}{ll}L\left(\frac{a-x}{c_1}\right),& \text{if }x<a\\ 1,& \text{if }x=a\\ R\left(\frac{x-a}{c_2}\right),& \text{if }x>a\end{array}$where $a$ is the central value, and $c_1$ and $c_2$ are the left and right spread values.

Let $\tilde{A}$ and $\tilde{B}$ be fuzzy numbers of LR-type defined as $\tilde{A}=(a,c_1,c_2{)}_{\mathrm{LR}}$ and $\tilde{B}=(b,d_1,d_2{)}_{\mathrm{LR}}$, where $a\text{and}b$ are the central values, $c_1\text{and}{d}_1$ are the left spread values and $c_2\text{and}{d}_2$ are the right spread values of $\tilde{A}$ and $\tilde{B}$, respectively. Then, $\begin{array}{rl}\tilde{A}+\tilde{B}& ={\left(a,c_1,c_2\right)}_{\mathrm{LR}}+{\left(b,d_1,d_2\right)}_{\mathrm{LR}}={\left(a+b,c_1+d_1,c_2+d_2\right)}_{\mathrm{LR}};\\ \tilde{A}-\tilde{B}& ={\left(a,c_1,c_2\right)}_{\mathrm{LR}}-{\left(b,d_1,d_2\right)}_{\mathrm{LR}}={\left(a-b,c_1+d_2,c_2+d_1\right)}_{\mathrm{LR}}\end{array}$

Next, the multiplication to both positive fuzzy numbers $\tilde{A}$ and $\tilde{B}$ can be derived as. $\tilde{A}\otimes \tilde{B}=(a,c_1,c_2{)}_{\mathrm{LR}}\otimes (b,d_1,d_2{)}_{\mathrm{LR}}=(\mathrm{ab},a{d}_1+b{c}_1,a{d}_2+b{c}_2{)}_{\mathrm{LR}}$

2.2. The possibility theory used to derive Fuzzy mean andvariance values

Theorem 2.1

[29]

Let $\tilde{A}$ be a fuzzy number with differentiable membership function with $\alpha$-level set ${\tilde{A}}^{\alpha }=\left\{\mathrm{xR}:u_{\tilde{A}}\left(x\right)\ge \alpha \right\}=\left[a_1\left(\alpha \right),a_2\left(\alpha \right)\right],0\le \alpha \le 1$. The lower possibilistic mean values of fuzzy number is defined as $M_{\ast }\left(\tilde{A}\right)=2{\int }_0^1\alpha \cdot a_1\left(\alpha \right)\mathrm{d\alpha }$; and the upper possibilistic mean value of fuzzy number is defined as $M^{\ast }\left(\tilde{A}\right)=2{\int }_0^1\alpha \cdot a_2\left(\alpha \right)\mathrm{d\alpha }$. Then, the expected value of a fuzzy number $\tilde{A}$ is expressed as $M\left(\tilde{A}\right)={\int }_0^1\alpha \cdot \left[a_1\left(\alpha \right)+a_2\left(\alpha \right)\right]\mathrm{d\alpha }$.

By Theorem 2.1, the lower possibilistic mean and upper possibilistic mean values for $\tilde{A}+\tilde{B}$ can be obtained as Equations (1) and (2) as follows: (1) $\begin{array}{rl}{M}_{\ast }\left(\tilde{A}+\tilde{B}\right)& =M_{\ast }\left(\tilde{A}\right)+M_{\ast }\left(\tilde{B}\right)\end{array}$(1) (2) $\begin{array}{rl}{M}^{\ast }\left(\tilde{A}+\tilde{B}\right)& =M^{\ast }\left(\tilde{A}\right)+M^{\ast }\left(\tilde{B}\right)\end{array}$(2) Then, the sum of possibilistic mean value of $\tilde{A}\text{and}\tilde{B}$ are obtained as follows: (3) $\begin{array}{r}M\left(\tilde{A}+\tilde{B}\right)=\frac{M_{\ast }\left(\tilde{A}+\tilde{B}\right)+M^{\ast }\left(\tilde{A}+\tilde{B}\right)}{2}\end{array}$(3) Next, the lower and upper possibilistic variances of $\tilde{A}$ are defined as Equations (4) and (5), respectively [36]. (4) $\begin{array}{rl}\mathrm{Va}{r}_{\ast }\left(\tilde{A}\right)& =2\underset{0}{\overset{1}{\int }}\alpha {\left[M_{\ast }\left(\tilde{A}\right)-a_1\left(\alpha \right)\right]}^2\mathrm{d\alpha }\end{array}$(4) (5) $\begin{array}{rl}\mathrm{Va}{r}^{\ast }\left(\tilde{A}\right)& =2\underset{0}{\overset{1}{\int }}\alpha {\left[M^{\ast }\left(\tilde{A}\right)-a_2\left(\alpha \right)\right]}^2\mathrm{d\alpha }\end{array}$(5) In addition, for ranking the return rate of each security to the guarantee return rate, we use a popular ranking method for fuzzy numbers described as follows:

Theorem 2.2

[30]

Let $\tilde{A}=\left(a,c_1,c_2\right)$ and $\tilde{B}=\left(b,d_1,d_2\right)$ be triangular fuzzy numbers, the central values be a and b, and the left and right spread values be $c_1,c_2$, and $d_1,d_2$; and then we define the circumcenter of $\tilde{A}$ as $S_{\tilde{A}}=\left({\overline{x}}_0,{\overline{y}}_0\right)=\left(\frac{6a+\left(c_2-c_1\right)}{6},\frac{5-c_2{c}_1}{12}\right)$. The ranking function $R\left(\tilde{A}\right)$ which maps $\tilde{A}$ to a real number can be derived as $R\left(\tilde{A}\right)=\sqrt{{\left({\overline{x}}_0\right)}^2+{\left({\overline{y}}_0\right)}^2}$. If the ranking value $R\left(\tilde{A}\right)$ is bigger than $R\left(\tilde{B}\right)$, then the fuzzy number $\tilde{A}$ is bigger than fuzzy number $\tilde{B}$.

3. Piecewise linear function in Fuzzy portfolio selection

Fuzzy portfolio is proposed to optimize the investment proportion for each security under the objective of maximizing expected return with respect to the constrained risk in the vagueness environment. Fuzzy portfolio selection in the assumption of excess investment for the securities whose expected return rates are larger than the guarantee return rates has been proposed by Tsaur et al [25] as shown in Equation (6) (6) $\begin{array}{r}\tilde{R}=\sum _{j=1}^n{x}_j{\tilde{r}}_j+\sum _{j=1}^n\sum _{k=1}^m{x}_j\left(|{\tilde{r}}_j-{\tilde{p}}_k|+{\tilde{r}}_j-{\tilde{p}}_k\right)/\phantom{\left(|{\tilde{r}}_j-{\tilde{p}}_k|+{\tilde{r}}_j-{\tilde{p}}_k\right)2}2\end{array}$(6) where $x_j$ is the investment proportion for security j, the return rate ${\tilde{r}}_j=\left(r_j,c_j,d_j\right)$ is triangular fuzzy number with r_j being central value, and c_j, d_j being left and right spreads, j = 1, … , n, and the guaranteed return rate ${\tilde{p}}_k=\left(p_k,e_k,f_k\right)$ with central value $p_k$, and $e_k,f_k$ are left and right spread values, respectively. If the fuzzy return rates as ${\tilde{r}}_j$ for security j is larger than the selected guaranteed return rate ${\tilde{p}}_k$, then security j can be denoted as excess investment; otherwise, this security cannot be involved in excess investment.

3.1. A piecewise linear function in Fuzzy expected return rates

For an investor’s behaviour with maximization of the expected return, the decision in the excess investment always prefers to invest more proportions to the securities with higher return rates than those smaller securities. Therefore, the security proportions in an excess investment of Equation (6) should be constrained by the priority of the return rate for each security. In order to cope with this issue, we propose a piecewise linear function for the fuzzy expected return rates and use the separate decision variables in investment proportion for each security.

Definition 3.1:

The fuzzy expected return rate in excess investment on the guarantee return rates by a piecewise linear function is defined as follows: (7) $\begin{array}{r}\tilde{R}=\sum _{j=1}^n{x}_{j1}{\tilde{r}}_j+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left({\tilde{r}}_j-{\tilde{p}}_k\right),\mathrm{if}{\tilde{r}}_j\ge {\tilde{p}}_k\end{array}$(7)

Then, the new variables, $x_{j1}\text{and}{x}_{j2}$ over the proportion $x_j$ are set as follows: (8) $\begin{array}{r}{x}_j=x_{j1}+x_{j2}\end{array}$(8) where the range of $x_j$ is denoted as $l_j\le x_j\le u_j$, with the lower and upper bounds $l_j$ and $u_j$, then the ranges for $x_{j1}\text{and}{x}_{j2}$ can be denoted as $l_j\le x_{j1}\le u_{j1}$ and $0\le x_{j2}\le u_{j2}$ with the lower bounds $l_j$ and 0, and upper bounds $u_{j1}$ and $u_{j2}$, and $u_{j1}+u_{j2}=u_j$.

Then we can define the first part of Equation (7) as the traditional fuzzy expected return rate, and the second part as the excess investment of the fuzzy expected return rate. By Theorem 1, the lower and upper possibilistic mean values for ${\tilde{r}}_j$ and ${\tilde{r}}_j-{\tilde{p}}_k$ can be obtained as Equations (9)–(11) and Equations (12)–(14), respectively. (9) $\begin{array}{rl}{M}_{\ast }\left({\tilde{r}}_j\right)& =r_j-\frac{1}{3}{c}_j\end{array}$(9) (10) $\begin{array}{rl}{M}^{\ast }\left({\tilde{r}}_j\right)& =r_j+\frac{1}{3}{d}_j\end{array}$(10) (11) $\begin{array}{rl}M\left({\tilde{r}}_j\right)& =r_j+\frac{1}{6}\left(d_j-c_j\right)\end{array}$(11) (12) $\begin{array}{rl}{M}_{\ast }\left({\tilde{r}}_j-{\tilde{p}}_k\right)& =\left(r_j-p_k\right)-\frac{1}{3}\left(c_j+f_k\right)\end{array}$(12) (13) $\begin{array}{rl}{M}^{\ast }\left({\tilde{r}}_j-{\tilde{p}}_k\right)& =\left(r_j-p_k\right)+\frac{1}{3}\left(d_j+e_k\right)\end{array}$(13) (14) $\begin{array}{rl}M\left({\tilde{r}}_j-{\tilde{p}}_k\right)& =\left(r_j-p_k\right)+\frac{1}{6}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right]\end{array}$(14)

Finally, the expected possibilistic mean value for the proposed fuzzy return rate shown in Equations (7) can be obtained as Equations (15) (15) $\begin{array}{r}\begin{array}{rl}M\left(\tilde{R}\right)& =M\left(\sum _{j=1}^n{x}_{j1}{\tilde{r}}_j+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left({\tilde{r}}_j-{\tilde{p}}_k\right)\right)\\ & =\sum _{j=1}^n{x}_{j1}M\left({\tilde{r}}_j\right)+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}M\left({\tilde{r}}_j-{\tilde{p}}_k\right)=\sum _{j=1}^n{x}_{j1}\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right]\\ & +\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left\{\left(r_j-p_k\right)+\frac{1}{6}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right]\right\}\end{array}\end{array}$(15)

In addition, we can obtain the lower and upper possibilistic variances for the proposed fuzzy expected return rates shown as follows: (16) $\begin{array}{rl}\mathrm{Va}{r}_{\ast }\left(\tilde{R}\right)& =\frac{1}{18}{\left(\sum _{j=1}^n{c}_j{x}_{j1}+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left(c_j+f_k\right)\right)}^2\end{array}$(16) (17) $\begin{array}{rl}\mathrm{Va}{r}^{\ast }\left(\tilde{R}\right)& =\frac{1}{18}{\left(\sum _{j=1}^n{d}_j{x}_{j1}+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left(d_j+e_k\right)\right)}^2\end{array}$(17)

Then, we can obtain the standard deviation of the fuzzy return rates as follows: (18) $\begin{array}{r}\begin{array}{rl}\mathrm{SD}\left[\tilde{R}\right]& =\frac{1}{2}\left\{{\left[\mathrm{Va}{r}_{\ast }\left(\tilde{R}\right)\right]}^{1/2}+{\left[\mathrm{Va}{r}^{\ast }\left(\tilde{R}\right)\right]}^{\frac{1}{2}}\right\}\\ & =\frac{1}{6\sqrt{2}}\left[\sum _{j=1}^n\left(c_j+d_j\right)x_{j1}+\sum _{j=1}^n\sum _{k=1}^m{x}_{j2}\left(c_j+d_j+e_k+f_k\right)\right]\end{array}\end{array}$(18)

3.2. The contingent strategies for the investment proportions

As shown in Equation (7), the fuzzy expected return rate is proposed for considering the excess investment. If the investor focuses on the future economy for shortage investment, full investment, or excess investment, then the total investment proportions for the selected securities can be defined as ‘essentially equal to 1’ as Equation (19), which means that the total investment proportions can be possibly larger than 1 when excess investment, but in contrast to smaller than 1 when shortage investment. Therefore, the membership function of ‘essentially equal to 1’ implies the total investment proportions are absolutely less than or equal to 1 with the membership degree to be 1, and the investment proportions for the securities can be larger than 1 in excess investment with at least a membership degree under a given tolerance value q₁. Then, the membership ‘essentially equal to 1’ is defined as Equation (19), and its membership function is defined as Equation (20). (19) $\begin{array}{rl}& \sum _{j=1}^n{x}_{j1}+\sum _{j=1}^n{x}_{j2}\cong 1\end{array}$(19) (20) $\begin{array}{rl}& u_{\lesssim }\left(\sum _{j=1}^n{x}_{j1}+\sum _{j=1}^n{x}_{j2}\right)\\ & =\{\begin{array}{ll}1,& \left(\sum _{j=1}^n{x}_{j1}+\sum _{j=1}^n{x}_{j2}\right)\le 1\\ \frac{\left(1+q_2\right)-\left({\sum }_{j=1}^n{x}_{j1}+{\sum }_{j=1}^n{x}_{j2}\right)}{q_2}& 1\le \left(\sum _{j=1}^n{x}_{j1}+\sum _{j=1}^n{x}_{j2}\right)\le 1+q_2\\ 0& \text{otherwise}\end{array}\end{array}$(20) Therefore, the membership degree of 1 means that the investment proportions for the securities are absolutely less than or equal to 1, and that the investment proportions for the securities can be larger than 1 for excess investment under a tolerance value q₂ with at least a membership degree.

Next, in modelling the fuzzy portfolio selection for excess investment, an investor does not have sufficient information to judge his investment preference but just based on the fuzzy return rate of the security. Therefore, the higher return rate security should be invested to higher proportions in portfolio selection under excess investment. Suppose that we have n securities for portfolio selection whose fuzzy return rates are ranked as ${\tilde{r}}_1\le {\tilde{r}}_2\le \dots \le {\tilde{r}}_j\le {\tilde{p}}_k\le {\tilde{r}}_{j+1}\le \dots \le {\tilde{r}}_n$, then the securities from j + 1 to n are selected for excess investment. Therefore, in fuzzy portfolio selection with excess investment, we assume that an investor not only decides the portfolio selection on traditional fuzzy expected return rate, but also on hold pending the proportions by the return rates of securities in excess investments. Then, we can affect the portfolio selection by constraining the investment proportions of securities in excess investment which is defined as contingent strategies Therefore, the investor can diagnose portfolio selection problems in systematic ways from the priority of return rate for each security included in the excess investment. Based on this contingent strategy, we can denote the priority of the investment proportion for each security that is included in the excess investment as follows: (21) $\begin{array}{r}{x}_{j2}-x_{\left(j-1\right)2}\le p_j,\mathrm{\forall j},{\tilde{r}}_j>{\tilde{p}}_k\end{array}$(21)

In Equation (21), the investor expresses his decision for the priority between $x_{\left(j-1\right)2}$ and $x_{j2}$, and then his contingent strategy for each security can be decided. Therefore, based on the return rate, the investment proportion of security j in the excess investment is greater than the security j–1. However, based on the uncertainty of the future economy is possibly decided and affected by insufficient information or risk events, the investment proportions to the securities in the excess investment with contingent strategy can be denoted as a fuzzified constraint as essentially smaller than or equal to $p_j$ as follows: (22) $\begin{array}{r}{x}_{j2}-x_{\left(j-1\right)2}\stackrel{<}{\sim \phantom{{}_x}}{p}_j,\mathrm{\forall j},{\tilde{r}}_j>{\tilde{p}}_k\end{array}$(22)

Then the membership function of the fuzzy constraint can be defined as follows: (23) $\begin{array}{r}{u}_{\lesssim }\left(x_{j2}-x_{\left(j-1\right)2}\right)=\{\begin{array}{ll}1,& x_{j2}-x_{\left(j-1\right)2}\le p_j\\ \frac{\left[p_j+q_j\right]-\left[x_{j2}-x_{\left(j-1\right)2}\right]}{q_j}& p_j\le x_{j2}-x_{\left(j-1\right)2}\le p_j+q_j\\ 0& \text{otherwise}\end{array}\end{array}$(23)

Therefore, the membership degree of 1 means that the contingent decisions for $x_{j2}-x_{\left(j-1\right)2}$ are absolutely less than $p_j$, and that the difference of investment proportion between $x_{j2}$ and $x_{\left(j-1\right)2}$ can be larger than $p_j$ under a tolerance value $q_j$ with a membership degree. Then, based on the above discussion, the objective function, constrained risk and the fuzzy constrains for the total investment proportions, and the fuzzy contingent strategies, we can derive the fuzzy portfolio model (24) as follows: (24) $\begin{array}{rl}\text{Max}& \sum _{j=1}^n{x}_{j1}\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right]+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n\sum _{k=1}^m{x}_{j2}\left\{\left(r_j-p_k\right)+\frac{1}{6}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right]\right\}\\ s.t.& \frac{1}{6\sqrt{2}}\left[\sum _{j=1}^n\left(c_j+d_j\right)x_{j1}+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n\sum _{k=1}^m{x}_{j2}\left(c_j+d_j+e_k+f_k\right)\right]\le \sigma \\ & \sum _{j=1}^n{x}_{j1}+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n{x}_{j2}\cong 1\\ & x_{j2}-x_{\left(j-1\right)2}\stackrel{<}{\sim \phantom{{}_x}}{p}_j\mathrm{\forall j},{\tilde{r}}_j>{\tilde{p}}_k\\ & l_j\le x_{j1}\le u_{j1},j=1,2,\dots ,n\\ & 0\le x_{j2}\le u_{j2},\mathrm{\forall j}=1,2,\dots ,n;\end{array}$(24)

For solving model (24), an investor desires to set the maximum objective of the expected return rate to be possibly equal to at least $z_0$, where $z_0$ represents the selected lower bound of the objective function and thus its objective values to be larger than $z_0$ will be derived as membership degree 1; by contrast with possibly that the objective value can also be acceptable to be smaller than $z_0$ with a tolerance value $q_1$. For simplicity, we define the fuzzy objective ${\sum }_{j=1}^n{x}_j\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right]+{\sum }_{j=1}^n{\sum }_{k=1}^m{v}_j\left\{\left(r_j-p_k\right)+\frac{1}{6}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right]\right\}$ to be z, then the membership function of the fuzzy objective can be defined as follows: (25) $\begin{array}{r}{u}_{⪆}\left(z\right)=\{\begin{array}{ll}1,& z\ge z_0\\ 1-\frac{z_0-\text{z}}{q_1}& z_0-q_1\le z\le z_0\\ 0& \text{otherwise}\end{array}\end{array}$(25)

With the membership function of Equations (20), (23) and (25), we suggested that the investors can choose larger tolerance values $q_2$, $q_{2j}$, $q_1$ to obtain the optimal solution in a model (24). On the contrary, if choosing tolerance values still makes model (24) infeasible, then the larger tolerance values are required to be greater. In addition, Equations (20), (23) and (25) are required to be larger than the membership levels defined as follows: (26) $\begin{array}{rl}{u}_{⪆}\left(z\right)& =1-\frac{z_0-z}{q_1}\ge {\lambda }_1\end{array}$(26) (27) $\begin{array}{rl}{u}_{\lesssim }\left(\sum _{j=1}^n{x}_{j1}+\sum _{j=1}^n{x}_{j2}\right)& =\frac{\left(1+q_2\right)-\left({\sum }_{j=1}^n{x}_{j1}+{\sum }_{j=1}^n{x}_{j2}\right)}{q_2}\ge {\lambda }_2\end{array}$(27) (28) $\begin{array}{rl}{u}_{\lesssim }\left(x_{j2}-x_{\left(j-1\right)2}\right)& =\frac{\left[p_j+q_j\right]-\left[x_{j2}-x_{\left(j-1\right)2}\right]}{q_j}\ge {\lambda }_j,\mathrm{\forall j}{\tilde{r}}_j>{\tilde{p}}_k\end{array}$(28)

We can transform the Equations (26–28) to relax model (24), and thus the model (24) can be rewritten as model (29). In model (29), the objective value $w_1{\lambda }_1+w_2{\lambda }_2+\sum _{j=3}^n{w}_j{\lambda }_j$ is the satisfactory that the expected investment return can possibly be near to the least value of $z_0$ under the given tolerance value q₁, and the enlarged tolerance values of q₂ and $q_{2+\left(j-1\right)},\mathrm{\forall j}=2,3,\dots ,n$ are for the tolerance values for the total investment proportions, and contingent strategies. The larger objective value $w_1{\lambda }_1+w_2{\lambda }_2+\sum _{j=3}^n{w}_j{\lambda }_j$ does not imply a better solution but only shows the possibility for an optimal solution to be obtained under the given tolerance values. The weights w_j in the objective function are adopted as the arithmetic mean as the aggregation operator as 1/n. (29) $\begin{array}{rl}\text{Max}& w_1{\lambda }_1+w_2{\lambda }_2+\sum _{j=3}^n{w}_j{\lambda }_j\\ \mathit{\text{s.t.}}& \sum _{j=1}^n{x}_{j1}\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right]+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n\sum _{k=1}^m{x}_{j2}\left\{\left(r_j-p_k\right)+\frac{1}{6}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right]\right\}\\ & -{\lambda }_1{q}_1\ge z_0-q_1\\ & \frac{1}{6\sqrt{2}}\left[\sum _{j=1}^n\left(c_j+d_j\right)x_{j1}+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n\sum _{k=1}^m{x}_{j2}\left(c_j+d_j+e_k+f_k\right)\right]\le \sigma \\ & \sum _{j=1}^n{x}_{j1}+\sum _{\mathrm{\forall }j,{\tilde{r}}_j>{\tilde{p}}_k}^n{x}_{j2}+{\lambda }_2{q}_2\le 1+q_2\\ & \left[x_{j2}-x_{\left(j-1\right)2}\right]+{\lambda }_j\ast q_j\le [p_j+q_j],\mathrm{\forall j},{\tilde{r}}_j>{\tilde{p}}_k\\ & l_j\le x_{j1}\le u_{j1},j=1,2,\dots ,n;0\le x_{j2}\le u_{j2},\mathrm{\forall j}=1,2,\dots ,n\\ & 0\le {\lambda }_1\le 1,0\le {\lambda }_2\le 1,0\le {\lambda }_j\le 1,\mathrm{\forall j},{\tilde{r}}_j>{\tilde{p}}_k\end{array}$(29)

4. Two examples with application to the proposed model

4.1. Example 1

The secondary data is the closed prices for each week [31] collected from Shanghai Stock Exchange in the period between April 2002 and January 2004. According to the financial reports offered by companies’ information, we chose five securities to find the portfolio using the proposed model. Then, we can obtain the estimated fuzzy return rates for each security as ${\tilde{r}}_1=\left(0.073,0.054,0.087\right)$, ${\tilde{r}}_2=\left(0.105,0.075,0.102\right)$, ${\tilde{r}}_3=\left(0.138,0.096,0.123\right)$, ${\tilde{r}}_4=\left(0.168,0.126,0.162\right)$, ${\tilde{r}}_5=\left(0.208,0.168,0.213\right)$, where the first, second, and third values in the fuzzy return rates are central value, left and right spread values. In addition, the range of investment proportions for each security is derived as lower bound as (l₁, l₂, l₃, l₄, l₅) = (0.1, 0.1, 0.1, 0.1, 0.1), and upper bound as (u₁, u₂, u₃, u₄, u₅) = (0.4, 0.4, 0.4, 0.5, 0.6), respectively. In the illustrations, we select two guaranteed return rates from smaller to larger rates as ${\tilde{p}}_1=\left(0.1,0.05,0.05\right)\text{ and }{\tilde{p}}_2=\left(0.15,0.1,0.1\right)$ for solving the proposed model. Next, by Theorem 2, the ranking results between the fuzzy return rates and guaranteed return rates ${\tilde{r}}_1\le {\tilde{p}}_1\le {\tilde{r}}_2\le {\tilde{r}}_3\le {\tilde{p}}_2\le {\tilde{r}}_4\le {\tilde{r}}_5$.

4.1.1. Experiment 1

In this experiment, we select ${\tilde{p}}_1$ to be the guaranteed return rate whose defuzzy is larger than fuzzy return rate of security 1 and smaller than the other securities. Therefore, the securities 2–5 are selected to be included for excess investment, and the optimized fuzzy portfolio selection is proceeded by the following steps.

Step 1: Decide the range of the parameters.

In this step, we first decide the range of the investment proportions for the securities by their lower bound as (l₁, l₂, l₃, l₄, l₅) = (0.1, 0.1, 0.1, 0.1, 0.1), and upper bound as (u₁, u₂, u₃, u₄, u₅) = (0.4, 0.4, 0.4, 0.5, 0.6). For securities 1–3 with less fuzzy return rate in the past performance, the ranges for $x_{j1}\text{and}{x}_{j2}$ over $x_j$, j = 1, 2, 3; are divided into equal values as $0.1\le x_{11}\le 0.2,0.1\le x_{12}\le 0.2$, $0.1\le x_{13}\le 0.2$, and the higher return rate securities 4, 5 are ranged as $0.1\le x_{45}\le 0.3\text{and}0.1\le x_{25}\le 0.3$; and the other proportions are ranged to excess investment proportions as $0\le x_{22}\le 0.2,0\le x_{32}\le 0.2,0\le x_{42}\le 0.2$ and $0\le x_{52}\le 0.3$. Next, $z_0$ is the selected lower bound of the objective function, the given values $z_0$ are decided by the investor. Besides, in this experiment, the upper bound of constrained risk is selected as $\sigma =0.1$, and the parameters denoted as q₁ = 0.05 by considering the range that the objective value to be possibly smaller than the desired values $z_0$, q₂ = 0.4 is derived by considering the total security proportions to be possibly smaller than the desired values 1, and q₃ = q₄ = q₅ = 0.1 are the width for each contingent strategy to be possibly smaller than tolerance interval. Finally, the parameters of objective function of ${\lambda }_j$, j = 1, 2, … , 5, which shows the membership degree to which the solution for the fuzzy constraints to be possibly in the tolerance interval, and thus the weights w_j in the objective function are adopted as the arithmetic mean as the aggregation operator as 0.2.

Step 2: Formulate linear programming model for portfolio selection.

Input all the parameters to the model (29), and then we can derive the model (30) as follows: (30) $\begin{array}{rl}\text{Max}& 0.2{\lambda }_1+0.2{\lambda }_2+0.2{\lambda }_3+0.2{\lambda }_4+0.2{\lambda }_5\\ \mathit{\text{s.t.}}& 0.0785x_{11}+0.1095x_{21}+0.1425x_{31}+0.174x_{41}+0.2155x_{51}+0.0095x_{22}\\ & +0.0425x_{32}+0.074x_{42}+0.1155x_{52}-0.05{\lambda }_1\ge \left(z_0-0.05\right)\\ & 0.016617x_{11}+0.02086x_{21}+0.025809x_{33}+0.033941x_{41}+0.044901x_{51}\\ & +0.037595x_{22}+0.037595x_{32}+0.045726x_{42}+0.056686x_{52}\le 0.1\\ & x_{11}+x_{21}+x_{31}+x_{41}+x_{51}+x_{22}+x_{32}+x_{42}+x_{52}+0.4{\lambda }_2\le 1.4\\ & x_{32}-x_{22}+0.05{\lambda }_3\le 0.15\\ & x_{42}-x_{32}+0.05{\lambda }_4\le 0.15\\ & x_{52}-x_{42}+0.05{\lambda }_5\le 0.15\\ & 0.1\le x_{11}\le 0.2,0.1\le x_{21}\le 0.2,0.1\le x_{31}\le 0.2,0.1\le x_{41}\le 0.3,\\ & 0.1\le x_{51}\le 0.3,0\le x_{12}\le 0.2,0\le x_{22}\le 0.2,0\le x_{32}\le 0.2,\\ & 0\le x_{42}\le 0.2,0\le x_{52}\le 0.3,\\ & 0\le {\lambda }_1\le 1,0\le {\lambda }_2\le 1,0\le {\lambda }_j\le 1,j=3,4,5\end{array}$(30)

Step 3: Discussion and analysis.

Using the linear programming software, we can solve the investment proportions $x_j$ for security j from $x_{j1}\text{ and }{x}_{j2}$ in the Table 1. Clearly, the optimal portfolio in the first row of Table 1 is derived as (0.4000, 0.1000, 0.1000, 0.1000, 0.2599) which shows the total investment proportions are 0.8819 represented as shortage investment under the given constrained risk and the expected return rate is 13.00%. Next, as the lower bound $z_0$ increased as shown in the first column of Table 1, we can obtain different portfolios with shortage, full, or excess investments from the second to the last rows of Table 1. However, the portfolio is infeasible when the required lower bound $z_0$ is required to be larger than 0.25. By the proposed model, the contribution of the excess investment for each security can be obtained as the lower bound $z_0$ to be 0.23 and 0.25 shown in the last two row of Table 1. As shown in the 7-th row of Table 1, the total investment proportions for the securities is derived as (0.1000, 0.1393, 0.2000, 0.3000, 0.4000) which shows the excess investment where the total investment proportions are 1.1393, and the expected return rate is 18.00%. As shown in the last row of Table 1, the total investment proportions for the securities is derived as (0.1000, 0.2000, 0.2000, 0.3704, 0.4704) which shows the excess investment where the total investment proportions are 1.3408, and the expected return rate is 20.00%. In addition, from the second row to the last row of Table 1, we can find that the investment proportion $x_1$ is from 0.2 to 0.1 because return rate of security 1 is the smallest. By contrast, the investment security proportions x4 and x5 with higher expected return securities are increasing from 0.1 and 0.2819–0.3704 and 0.4704, respectively.

Table 1. The portfolio selection guaranteed return rate ${\tilde{p}}_1$.

Proportions Parameters	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{22}$	$x_{32}$	$x_{42}$	$x_{52}$	Total proportions	Expected Return Rates
z0 = 0.13	0.2000	0.2000	0.1000	0.1000	0.2819	0	0	0	0	0.8819	0.1300
z0 = 0.15	0.1806	0.2000	0.2000	0.1194	0.3000	0	0	0	0	1	0.1500
z0 = 0.17	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.19	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.21	0.1000	0.1000	0.2000	0.3000	0.3000	0	0	0	0	1	0.1642
z0 = 0.23	0.1000	0.1393	0.2000	0.3000	0.3000	0	0	0	0.1000	1.1393	0.1800
z0 = 0.25	0.1000	0.2000	0.2000	0.3000	0.3000	0	0	0.0704	0.1704	1.3408	0.2000

4.1.2. Experiment 2

In this experiment, we select ${\tilde{p}}_2$ to be the guaranteed return rate whose defuzzy is larger than fuzzy return rate of securities 1, 2, and 3, and smaller than the other securities. Therefore, the securities 4 and 5 are selected to be included for excess investment, and the optimized fuzzy portfolio selection is proceeded by the following steps.

Step 1: Decide the range of the parameters

In this step, we first decide range of the investment proportions for the securities by their lower bound as (l₁, l₂, l₃, l₄, l₅) = (0.1, 0.1, 0.1, 0.1, 0.1), and upper bound as (u₁, u₂, u₃, u₄, u₅) = (0.4, 0.4, 0.4, 0.5, 0.6). The ranges for $x_{j1}\text{and}{x}_{j2}$ over $x_j$, j = 1, 2, 3, 4, 5; are the almost the same as the experiment 1 except the proportion of security 4 changed as $0.1\le x_{41}\le 0.2$ and $0\le x_{42}\le 0.3$. This is because just only securities 4 and 5 are included in excess investment, we give more proportion of security 4 in the excess investment. Next, $z_0$ is the selected lower bound of the objective function, the given values $z_0$ are decided by the investor. Besides, in this experiment, the upper bound of constrained risk is selected as $\sigma =0.1$, and the parameters denoted as q₁ = 0.05 by considering the range that the objective value to be possibly smaller than the desired values $z_0$, q₂ = 0.4 is derived by considering the range that the objective value to be possibly smaller than the desired values $z_0$, and q₃ = 0.1 are the width for each contingent strategy to be possibly smaller than tolerance interval. Finally, the parameters of objective function of ${\lambda }_j$, j = 1, 2, 3 which shows the membership degree to which the solution for the fuzzy constraints to be possibly in the tolerance interval, and thus the weights w_j in the objective function are adopted as the arithmetic mean as the aggregation operator as 1/3.

Step 2: Formulate linear programming model for portfolio selection.

Input all the parameters to the model (29), and then we can derive the model (31) as follows: (31) $\begin{array}{rl}max& 0.3333{\lambda }_1+0.3333{\lambda }_2+0.3333{\lambda }_3\\ \mathit{\text{s.t.}}& 0.0785x_{11}+0.1095x_{21}+0.1425x_{31}+0.1740x_{41}+0.21550x_{51}+0.0240x_{42}\\ & +0.0655x_{52}-0.05{\lambda }_1\ge \left(z_0-0.05\right)\\ & 0.016617x_{11}+0.02086x_{21}+0.025809x_{31}+0.033941x_{41}+0.044901x_{51}\\ & +0.057511x_{42}+0.068472x_{52}\le 0.1\\ & x_{11}+x_{21}+x_{31}+x_{41}+x_{51}+x_{42}+x_{52}+0.4{\lambda }_2\le 1.4\\ & x_{52}-x_{42}+0.05{\lambda }_3\le 0.15\\ & 0.1\le x_{11}\le 0.2,0.1\le x_{21}\le 0.2,0.1\le x_{31}\le 0.2,0.1\le x_{41}\le 0.2,\\ & 0.1\le x_{51}\le 0.3,0\le x_{12}\le 0.2,0\le x_{22}\le 0.2,0\le x_{32}\le 0.2,\\ & 0\le x_{42}\le 0.3,0\le x_{52}\le 0.3,\\ & 0\le {\lambda }_1\le 1,0\le {\lambda }_2\le 1,0\le {\lambda }_3\le 1\end{array}$(31)

Step 3: Discussion and analysis.

Using the linear programming software, we can solve the investment proportions $x_j$ for each security j from $x_{j1}\text{ and }{x}_{j2}$ in the first row of Table 2. Clearly, the optimal portfolio in the first row of Table 1 is derived as (0.2000, 0.2000, 0.1000, 0.1000, 0.2819) which shows the shortage investment where the total investment proportions are 0.8819, and the expected return rate is 13.00%. Next, as the lower bound $z_0$ increased as shown in the second column of Table 2, we can obtain different portfolios with different total proportions or expected returns from the second to the last rows of Table 2. However, the portfolio is infeasible when the required lower bound $z_0$ is required to be larger than 0.23. By the proposed model, the contribution of the excess investment for each security can be obtained as the lower bound $z_0$ to be 0.22 and 0.23 shown in the last two row of Table 2. As shown in the 7-th row of Table 2, the total investment proportions for the securities is derived as (0.2000, 0.2000, 0.2000, 0.2000, 0.3679) which shows the excess investment where the total investment proportions are 1.1679, and the expected return rate is 17.00%. As shown in the last row of Table 2, the total investment proportions for the securities is derived as (0.2000, 0.2000, 0.2000, 0.2883, 0.4883) which shows the excess investment where the total investment proportions are 1.3766, and the expected return rate is 18.00%. In addition, the investment security proportions x4 and x5 with higher expected return securities are increasing from 0.1 and 0.2819 to 0.2883 and 0.4883, respectively.

Table 2. The portfolio selection guaranteed return rate ${\tilde{p}}_2$.

Proportions Parameters	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{42}$	$x_{52}$	Total Proportions	Expected Return Rates
z0 = 0.13	0.2000	0.2000	0.1000	0.1000	0.2819	0	0	0.8819	0.1300
z0 = 0.15	0.1806	0.2000	0.2000	0.1194	0.3000	0	0	1	0.1500
z0 = 0.17	0.1000	0.2000	0.2000	0.2000	0.3000	0	0	1	0.1577
z0 = 0.19	0.1000	0.2000	0.2000	0.2000	0.3000	0	0	1	0.1577
z0 = 0.21	0.1293	0.2000	0.2000	0.2000	0.3000	0	0	1.0293	0.1600
z0 = 0.22	0.2000	0.2000	0.2000	0.2000	0.3000	0	0.0679	1.1679	0.1700
z0 = 0.23	0.2000	0.2000	0.2000	0.2000	0.3000	0.0883	0.1883	1.3766	0.1800

4.1.3. The comparisons among Fuzzy portfolio models

In this subsection, we compare the proposed model to Zhang [31], Tsaur et al [25] and Chen & Tsaur [32] portfolio models whose results are shown in Table 3. For our proposed model, an investor needs to offer the desired objective for the expected return rate and the constrained risk as shown in the second column of Table 3 for deriving the portfolio. In contrast to Zhang’s model with only one chose portfolio as (0.1000, 0.1000, 0.1000, 0.1000, 0.6000), our proposed model can offer more diverse portfolio selections for an investor under smaller constrained risks. Second, Coefficient of variation (C.V.) is a possible indicator for the analysis performance of the invested portfolios, in which each alternative is provided by both or more of different models. It is noted that the smaller C.V. is, the better performance is. By comparing the results from the proposed model with the selected models in the fourth column of Table 3, an investor can apply our model for more diverse portfolios of investments with smaller values of C.V. Therefore, imbedded the piecewise linear function and incorporating fuzzy contingent strategies into portfolio selection can select a proper port-folio with lower investment risk and obtain higher expected return rate.

Table 3. The comparison among fuzzy portfolio models.

	σ (%)	M (%)	C.V.	x1	x2	x3	x4	x5	$\sum _{j=1}^{\text{5}}{x}_j$
Zhang [37]	4%	22.55%	0.1774	0.1	0.1	0.1611	0.1	0.5389	1
	5%	23.16%	0.2159	0.1	0.1	0.1	0.1	0.6	1
	6%	23.16%	0.2591	0.1	0.1	0.1	0.1	0.6	1
	7%	23.16%	0.3022	0.1	0.1	0.1	0.1	0.6	1
Tsaur et al.’s model [15] with a guaranteed return rate ${\tilde{p}}_1$	5%	10.50%	0.4762	0.2127	0.1	0.1	0.1	0.1	0.6127
	8%	17.46%	0.4582	0.4	0.1	0.1	0.1	0.2659	0.9659
	10%	21.59%	0.4632	0.2811	0.1	0.1	0.1	0.4189	1
	12%	25.63%	0.4682	0.121	0.1	0.1	0.1	0.5790	1
	15%	26.17%	0.5732	0.1	0.1	0.1	0.1	0.6	1
Chen & Tsaur’s model	6%	10.65%	0.5634	0.4000	0.1000	0.1142	0.1000	0.1	0.8142
	8%	14.48%	0.5525	0.3632	0.1000	0.3368	0.1000	0.1000	1
	10%	15.77%	0.6341	0.1560	0.1000	0.4000	0.2440	0.1000	1
	12%	16.87%	0.7113	0.1000	0.1000	0.3157	0.1000	0.3843	1
	15%	19.17%	0.7825	0.1000	0.1000	0.1000	0.1000	0.6679	1
Proposed model with a guaranteed return rate ${\tilde{p}}_2$	4% (z0 = 0.13)	13%	0.3077	0.2000	0.2000	0.1000	0.1000	0.2819	1
	4% (z0 = 0.15)	15%	0.2667	0.1806	0.2000	0.2000	0.1194	0.3000	1
	4% (z0 = 0.19)	15.77%	0.2536	0.1000	0.2000	0.2000	0.2000	0.3000	1
	4% (z0 = 0.21)	16%	0.2500	0.1293	0.2000	0.2000	0.2000	0.3000	1
	4% (z0 = 0.22)	17%	0.2353	0.2000	0.2000	0.2000	0.2000	0.3679	1.1393
	5% (z0 = 0.23)	18%	0.2778	0.2000	0.2000	0.2000	0.2610	0.4982	1.3408

4.2. Example 2

In the second example, we use the top 50 companies in Taiwan with the monthly data with the sampling period from January 2010 to September 2020. Gupta et al [33] proposed that a portfolio should not hold too many assets, and thus we group the fifty securities into four industries, and then using Earning per share to select the top three securities in each industry consisting a sample of twelve securities (n = 12) from Taiwan Stock Exchange. The possibility distribution in fuzzy return for each security can be defined as ${\tilde{r}}_j=\left(r_j,c_j,d_j\right)$ j = 1, … , 12, where $r_j$ is central value and $c_j$, $d_j$ are left and right spread values, respectively. As presented in Table 4, by Theorem 2, we can obtain the rank order of the collected fuzzy returns as ${\tilde{r}}_{12}<{\tilde{r}}_{11}<{\tilde{r}}_{10}<\dots <{\tilde{r}}_1$.

Table 4. The fuzzy return of each selected security

${\tilde{r}}_1=\left(0.1420,0.0713,0.1296\right)$	${\tilde{r}}_2=\left(0.1358,0.0490,0.0835\right)$	${\tilde{r}}_3=\left(0.0909,0.0499,0.0919\right)$
${\tilde{r}}_4=\left(0.0808,0.0203,0.0297\right)$	${\tilde{r}}_5=\left(0.0786,0.0162,0.0217\right)$	${\tilde{r}}_6=\left(0.0751,0.0226,0.0362\right)$
${\tilde{r}}_7=\left(0.0657,0.0205,0.0325\right)$	${\tilde{r}}_8=\left(0.0643,0.0244,0.0415\right)$	${\tilde{r}}_9=\left(0.0552,0.0133,0.0190\right)$
${\tilde{r}}_{10}=\left(0.0438,0.0079,0.0102\right)$	${\tilde{r}}_{11}=\left(0.0316,0.0110,0.0181\right)$	${\tilde{r}}_{12}=\left(0.0212,0.0051,0.0075\right)$

In this study, we set the estimated error of the 95% confidence interval of the average return from January 2010 to September 2020 to be the right spread of fuzzy return and the estimated error of the 90% confidence interval to be the left spread. In the real case data, we set the lower bounds of investment proportion x_j for security j are given by 0.05, and the upper bounds are given by 0.3. The threshold values for the guaranteed return rates are selected based on the ranking of the fuzzy return. We select the first guaranteed return rate as ${\tilde{p}}_3=(0.03,0.01,0.02)$ whose defuzzy is higher than ${\tilde{r}}_{12}$; the second guaranteed return rate ${\tilde{p}}_4=(0.07,0.03,0.04)$ is selected whose ranking is lower than ${\tilde{r}}_1,{\tilde{r}}_2,\dots ,{\tilde{r}}_6$. Therefore, for the guaranteed return rate ${\tilde{p}}_3$, the securities 1, 2, 3, … , 11 are excess investment. For the guaranteed return rate ${\tilde{p}}_4$, the securities 1–6 are excess investment.

4.2.1. Experiment 1

In this experiment, we select ${\tilde{p}}_3$ to be the guaranteed return rate, and thus the securities 1, 2, … , 11 are selected for excess investment. Then, the optimized fuzzy portfolio selection is proceeded by the following steps.

Step 1: Decide the range of the parameters

In this step, we first decide the range of the investment proportions for the securities by their lower bound as (l_j, l₂, l₃, l₄, l₅, l₆, l₇, l₈, l₉, l₁₀, l₁₁, l₁₂) = (0.1, 0.1, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05), and upper bound as (u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂) = (0.4, 0.4, 0.3, 0.3, 0.3, 0.3, 0.3, 0.2, 0.2, 0.2, 0.2, 0.2). Then, the new variable of investment proportions $x_{j1}$ for regular investment and $x_{j2}$ for excess investment over the investment proportion $x_j$ for security j is set as follows:

1. The ranges of investment proportions in the first part of expected return rate are denoted as 0.1 ≤ x₁₁ ≤ 0.2, 0.1 ≤ x₂₁ ≤ 0.2, 0.05 ≤ x₃₁ ≤ 0.15, 0.05 ≤ x₄₁ ≤ 0.15, 0.05 ≤ x₅₁ ≤ 0.15, 0.05 ≤ x₆₁ ≤ 0.15, 0.05 ≤ x₇₁ ≤ 0.15, 0.05 ≤ x₈₁ ≤ 0.1, 0.05 ≤ x₉₁ ≤ 0.1, 0.05 ≤ x_10,1 ≤ 0.1, 0.05 ≤ x_11,1 ≤ 0.1, 0.05 ≤ x_12,1 ≤ 0.2.

2. Then, the securities ranged to excess investment proportions are set as 0 ≤ x₁₂ ≤ 0.2, 0 ≤ x₂₂ ≤ 0.2, 0 ≤ x₃₂ ≤ 0.15, 0 ≤ x₄₂ ≤ 0.15, 0 ≤ x₅₂ ≤ 0.15, 0 ≤ x₆₂ ≤ 0.15, 0 ≤ x₇₂ ≤ 0.15, 0 ≤ x₈₂ ≤ 0.1, 0 ≤ x₉₂ ≤ 0.1, 0 ≤ x_10,2 ≤ 0.1, 0 ≤ x_11,2 ≤ 0.1

Step 2: Formulate linear programming model for portfolio selection

Next, $z_0$ is the selected lower bound of the objective function, the given values $z_0$ are decided by the investor. Besides, in this experiment, the upper bound of constrained risk is selected as $\sigma =0.08$, and the parameters denoted as q₁ = 0.05 by considering the range that the objective value to be possibly smaller than the desired values $z_0$, q₂ = 0.2 is derived by considering the total security proportions to be possibly smaller than the desired values 1, and q₃ = q₄ = … = q₁₂ = 0.05 are the width for each contingent strategy to be possibly smaller than tolerance interval. Finally, the parameters of the objective function of ${\lambda }_j$, j = 1, 2, … , 11, which shows the membership degree to which the solution for the fuzzy constraints to be possibly in the tolerance interval, and thus the weights w_j in the objective function are adopted as the arithmetic mean as the aggregation operator as 0.0909. Then, we can formulate the proposed fuzzy portfolio model.

Step 3: Discussion and analysis

Using the linear programming software, we can solve the investment proportions $x_j$ for security j from $x_{j1}\text{ and }{x}_{j2}$ in the Table 5. Clearly, the optimal portfolio in the last row of Table 5 which shows the total investment proportions are 1.2 represented as excess investment under the given constrained risk is 0.08 and the expected return rate is 11.9997%. Next, as the lower bound $z_0$ increased from 0.07 to 0.17 as shown in the first column of Table 5, we can obtain different portfolios with shortage (when z₀ = 0.07, Total investment proportion is 0.8962), full (when z₀ = 0.1 and 0.12, Total investment proportion are 1), or excess investments (z₀ = 0.16, 0.17, Total investment proportion are 1.125 and 1.2, respectively), and the portfolio is infeasible when the required lower bound $z_0$ is required to be larger than 0.17. By the proposed model, the higher expected return securities x₁ and x₂ shown in the last two row of Table 4 have the higher investment proportions than the other securities, where the investment proportion $x_1$ is increasing from 0.1 to 0.2599 and $x_2$ is increasing from 0.128 to 0.2599. By contrast, the investment security proportions from securities 5 to 12 do not make any change as 0.05, because their lower expected return than securities 1–4.

Table 5. The portfolio selection guaranteed return rate ${\tilde{p}}_3$.

z0 = 0.07 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0.028	0	0
z0 = 0.1 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.12 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.16 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.15	0.075	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0	0	0	0	0	0	0	0	0	0	0	0
z0 = 0.17 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.1202	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$	$x_{27}$	$x_{28}$	$x_{29}$	$x_{2,10}$	$x_{2,11}$	$x_{2,12}$
	0.0599	0.0599	0.01	0	0	0	0	0	0	0	0	0

4.2.2. Experiment 2

In this experiment, we select ${\tilde{p}}_4$ to be the guaranteed return rate whose defuzzy is larger than fuzzy return rate of securities from 7 to 12 and smaller than the securities from 1 to 6. Therefore, securities 1, 2, … , 6 are selected to be included for excess investment, and the optimized fuzzy portfolio selection is proceeded by the following steps.

Step 1: Decide the range of the parameters.

In this step, we first decide range of the investment proportions for the securities by their lower bound as (l_j, l₂, l₃, l₄, l₅, l₆, l₇, l₈, l₉, l₁₀, l₁₁, l₁₂) = (0.1, 0.1, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05), and upper bound as (u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂) = (0.4, 0.4, 0.3, 0.3, 0.3, 0.3, 0.3, 0.2, 0.2, 0.2, 0.2, 0.2). Then, the new variable of investment proportions $x_{j1}$ for regular investment and $x_{j2}$ for excess investment over the investment proportion $x_j$ for security j is set as follows:

1. The ranges of investment proportions in the first part of expected return rate are denoted as 0.1 ≤ x₁₁ ≤ 0.2, 0.1 ≤ x₂₁ ≤ 0.2, 0.05 ≤ x₃₁ ≤ 0.15, 0.05 ≤ x₄₁ ≤ 0.15, 0.05 ≤ x₅₁ ≤ 0.15, 0.05 ≤ x₆₁ ≤ 0.15, 0.05 ≤ x₇₁ ≤ 0.3, 0.05 ≤ x₈₁ ≤ 0.2, 0.05 ≤ x₉₁ ≤ 0.2, 0.05 ≤ x_10,1 ≤ 0.2, 0.05 ≤ x_11,1 ≤ 0.2, 0.05 ≤ x_12,1 ≤ 0.2.

2. Then, the securities ranged to excess investment proportions are set as 0 ≤ x₁₂ ≤ 0.2, 0 ≤ x₂₂ ≤ 0.2, 0 ≤ x₃₂ ≤ 0.15, 0 ≤ x₄₂ ≤ 0.15, 0 ≤ x₅₂ ≤ 0.15, 0 ≤ x₆₂ ≤ 0.15,

Step 2: Formulate linear programming model for portfolio selection.

In this step, we use the same parameters as experiment 1 where the upper bound of constrained risk is selected as $\sigma =0.08$, and the parameters denoted as q₁ = 0.05 by considering the range that the objective value to be possibly smaller than the desired values $z_0$, q₂ = 0.2 is derived by considering the total security proportions to be possibly smaller than the desired values 1, and q₃ = q₄ = … = q₁₂ = 0.05 are the width for each contingent strategy to be possibly smaller than tolerance interval. Finally, the parameters of objective function of ${\lambda }_j$, j = 1, 2, … , 7, which shows the membership degree to which the solution for the fuzzy constraints to be possibly in the tolerance interval, and thus the weights w_j in the objective function are adopted as the arithmetic mean as the aggregation operator as 0.14286. Besides, $z_0$ is the selected lower bound of the objective function, the given values $z_0$ are decided by the investor.

Step 3: Discussion and analysis.

Using the linear programming software, we can solve the investment proportions $x_j$ for security j from $x_{j1}\text{ and }{x}_{j2}$ in the Table 6. Clearly, the optimal portfolio in the last row of Table 6 which shows the total investment proportions are 1.1999 represented as excess investment under the given constrained risk is 0.08 and the expected return rate is 11.999%. Next, as the lower bound $z_0$ increased from 0.06 to 0.17 as shown in the first column of Table 6, we can obtain different portfolios with shortage (when z₀ = 0.06, and 0.07, Total investment proportion is 0.7 and 0.8455, respectively), full (when z₀ = 0.11, Total investment proportion are 1), or excess investments (z₀ = 0.16, 0.17, Total investment proportion are 1.125 and 1.1999, respectively), and the portfolio is infeasible when the required lower bound $z_0$ is required to be larger than 0.17. By the proposed model, the higher expected return securities x₁ and x₂ shown in the last two rows of Table 6 have higher investment proportions than the other securities, where the investment proportion $x_1$ is increasing from 0.1 to 0.2599 and $x_2$ is increasing from 0.1 to 0.2599. By contrast, the investment security proportions from securities 7 to 12 do not make any change as 0.05 because their expected return is smaller than the securities 1–6 under the guaranteed return rate ${\tilde{p}}_4$.

Table 6. The portfolio selection guaranteed return rate ${\tilde{p}}_4$.

z0 = 0.06 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.07 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.1	0.1	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0.0291	0.0291	0.0291	0.0291	0.0291
z0 = 0.11 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.16 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.15	0.075	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0	0	0	0	0	0
z0 = 0.17 σ = 0.08	$x_{11}$	$x_{21}$	$x_{31}$	$x_{41}$	$x_{51}$	$x_{61}$	$x_{71}$	$x_{81}$	$x_{91}$	$x_{10,1}$	$x_{11,1}$	$x_{12,1}$
	0.2	0.2	0.15	0.1202	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05
	$x_{21}$	$x_{22}$	$x_{23}$	$x_{24}$	$x_{25}$	$x_{26}$
	0.0599	0.0599	0.0099	0	0	0

4.3. Discussion

Based on the above two examples, we find that the resolution process for the excess investment in the fuzzy portfolio analysis has three advantages. First, we use the piecewise linear function to decompose the investment proportions from the ranges for $x_{j1}\text{and}{x}_{j2}$ over $x_j$, j = 1, 2, … , n for two parts of the traditional fuzzy portfolio and excess investment in the expected return rates. By this piecewise linear function, the contribution of each security in the excess investment can be derived. Second, we supposed that the investment behaviour of an investor is based on the future economy. Therefore, we used the fuzzy constraints in the constraints of expected return rates and the total investment proportions. When the future economy is worse, we give lower bound of the expected return rate to be smaller, the total investment proportions may be a shortage, and when the future economy is better, we give lower bound of the expected return rate to be larger, the portfolio can be changed from the shortage investment to the excess investment. Third, with the contingent strategy, the excess investment is found and reasonable when the investor’s attitude is positive for the future economy, and he will invest more assets in the higher return rate securities. In contrast with different guaranteed return rates at experiments 1 and 2 in examples 1 and 2, we also find that the trend of the three advantages of the proposed model is all the same. If the future economy is in positive growth, we suggest selecting a smaller guaranteed return rate for including more securities in excess investment.

5. Conclusions

Fuzzy portfolio selections for excess investment have been proposed by Tsaur et al. [25] on the guaranteed return rate. Their following researches focused on risk attitudes analysis by the dimensions of the excess investment. In this study, we suppose the excess investment for each security should be evaluated separately from the traditional fuzzy portfolio selection. In contrast with Tsaur et al. [25], our proposed model offers more precisely information, where new decision variables for investment proportions in those securities included in the excess investment are defined in our proposed model by piecewise linear function, and then the contribution of included securities in the excess investment can be evaluated. Next, once we focus on excess investment, our proposed model can be applied to reflect the investment behaviour of portfolio selection in different given lower bound of expected return rate. Therefore, three kind of portfolio with shortage investment, full investment, excess investment are derived from different perceived economy environment. In final, the fuzzy contingent strategy is used for solving the insufficient information in which the investor cannot easily judge the ranking proportions between securities selected in the excess investment, and thus we can reduce the effects of risks in consistently timing the market for deriving the expected return rates under the maximization to the objective function. For controlling the uncertainty risks, the contingent strategy is proposed to evaluate the investment attitude of an investor between securities in fuzzy constraints. Finally, investors could rank the investment proportions among the securities, and easily evaluate and analyze optimized portfolio by the proposed model.

The managerial implication of this study states that the portfolio selected from the traditional fuzzy portfolio model on the guaranteed return rate cannot fully reflect the market return rate of the securities. For sufficient asset investments, the security proportions in excess investment should be clearly defined and applied when the investor’s attitude toward the future can be perceived. Next, for future research, we expect a collaborative discussion between the analyzers and the investor involving several times of depth interviews are required to include in the evaluation and selection process for establishing contingent strategies. Therefore, future research should focus on the investment attitudes of the investors in different time period to our proposed model.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was funded with the financial support on outstanding talent project in Nanchang Vocational University.