Dimensions Analysis to Excess Investment in Fuzzy Portfolio Model from the Threshold of Guaranteed Return Rates

Abstract

Portfolio selection is a major topic for investors to allocate their assets and maximize their profit under constrained risk. For uncertain investment behavior in a vagueness environment, some researchers have devoted themselves to this field of fuzzy portfolio models for portfolio selection. Especially, Tsaur, Chiu and Huang in 2021 defined guaranteed return rates to excess investment for securities whose return rates are bigger than the guaranteed return rates in the fuzzy portfolio selection. However, an independent investor has original ideas in investment, and thus we need to consider more types of risk attitudes for an investor’s portfolio selection when the guaranteed return rates are used to excess investment. To manage the excess investment by the risk preference, a new concept of s dimensions of excess investment is introduced to perceive the risk attitude of an investor for portfolio selection. Finally, we present a numerical example of a portfolio selection problem to illustrate the proposed model. This example shows that the higher dimensions of excess investment derive lower expected return rates with lower constrained risk than that of dimension s = 1; and we suggest lower risk preference should select a higher dimension of excess investment. Then, the dimension of excess investment s = 2 can be applied for portfolio selection when the risk preference is lower.

1. Introduction

Markowitz [1] proposed mean-variance portfolio selection method to maximize the expected value of a portfolio’s return under certain variability constraints, and then a lot of researchers devoted themselves to the field of portfolio models based on the probability theory [2,3,4,5,6,7,8]. The traditional portfolio models assumed that future returns can be precisely reflected by the historical data, whereas we cannot obtain precise probability distributions because there are some events changing the economic environment. In contrast, non-probabilistic factors or events that usually affect most of investor portfolio selection include linguistic descriptions of investment decision making, the new social economy, and international political conflicts such as Russian forces invading Ukraine, people’s cognitive and psychological reactions related to investment behaviors after the spread of COVID-19, etc. Therefore, fuzzy portfolio models are proposed for solving non-probabilistic portfolio selection where the defined returns of risky assets as fuzzy variables [9,10,11,12,13]. For instance, Zhang and Nie [14] presented the notions of lower and upper-possibilistic variances and covariances of fuzzy numbers in fuzzy portfolio analysis. Huang [15] proposed two fuzzy-mean semi-variance models to deal with situations wherein investors intend to obtain high returns while avoiding risk by eliminating asymmetry in the degree of return distributions. Liu and Zhang [16] formulated a mean-semivariance portfolio selection model to reflect investor’s aspiration levels for the objectives of maximizing the terminal wealth and minimizing the cumulative risk, and then they transformed the proposed model into a single objective mixed-integer nonlinear programming problem by a genetic algorithm for a solution. Guo et al. [17] considered a fuzzy multi-period portfolio selection problem with V-shaped transaction costs and then designed a fuzzy simulation-based genetic algorithm for illustration. Gupta et al. [18] proposed two intuitionistic fuzzy portfolio selection models for optimistic and pessimistic scenarios. Tsaur et al. [19] proposed a fuzzy portfolio model using a fuzzy goal programming model to manage investment during the COVID-19 pandemic period. Besides, some scholars applied the credibility theory to model fuzzy portfolio selection measures [20,21,22]. For example, Zhang et al. [23] proposed two credibilistic MV portfolio models and applied a quadratic programming approach to derive the optimal investment strategy. Mehlawat et al. [24] proposed a multi-objective function with variance and CVaR as risk measures for performance evaluation in the fuzzy portfolio selection models.

From the above literature reviews, the research on fuzzy portfolio models has used possibility or credibility theory to make decision results in portfolio selection, whereas the habitual behaviors on risk analysis in portfolio selection still lack sufficient research. For example, Li et al. [25] proposed an expected regret function to minimize the distance between maximum return and investment return. Yue and Wang [26] proposed a new entropy function as an objective function and formulated a novel fuzzy multi-objective weighted possibilistic higher order moment portfolio model, and then solved the proposed model by a multi-objective evolutionary algorithm, where the results indicated the efficiency and effectiveness of the proposed model and algorithm. Guo et al. [27] introduced a capital gain tax to fuzzy portfolio selection, where fuzzy random variables were employed to model uncertain returns of risky assets in a Markov-regime switching market, and a bi-objective mean-variance model was formulated and solved by a time-varying numerical integral-based particle swarm optimization algorithm in obtaining the efficient frontier of the portfolio in the sense of Pareto dominance. Liagkouras and Metaxiotis [28] examined a multi-period portfolio optimization problem with transaction costs and fuzzy variables to quantify the variance of fuzzy returns with respect to maximizing the terminal wealth and minimizing the cumulative risk of portfolios over the entire investment horizon. Zhou et al. [29] propose a stock portfolio selection problem based on varying conservative-neutral-aggressive attitudes by maximizing the return and minimizing the risk subject to constraints of transaction cost and value at risk. Therefore, the risk behavior of an investor is an important factor for portfolio selection [29,30,31]. With respect to the excess investment proposed by Tsaur et al. [32] and their following research [33,34], where excess investment is considered for some securities whose return rates are bigger than the threshold of guaranteed return rates. Without considering the risk behavior of an investor, we cannot suggest the portfolio selection based on her/his risk attitude. In order to cope with the overcome for completed investment, we consider the dimension of excess investment studied by Tsaur et al. [32], and thus risk attitude is considered in the fuzzy portfolio selection.

This paper is organized as follows. Section 2 provides a brief introduction of the definition fuzzy numbers. In Section 3, the dimension of excess investment a fuzzy portfolio model is proposed. In Section 4, an example using the proposed model is presented. Finally, conclusions are discussed in Section 5.

2. Preliminaries

In this section, fuzzy numbers are introduced, including some fundamental concepts of algebraic operation with fuzzy numbers in operation and defuzzy, fuzzy expected values, and fuzzy variances for modeling. Therefore, the following sections can be clearly understood.

Let $\tilde{A}$ be a fuzzy number, i.e., such fuzzy subset $\tilde{A}$ 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 [35]:

(i.)

$\tilde{A}$ is normal, i.e., there is an x ∈ R such that $u_{\tilde{A}}\left(x\right)=1$;

(ii.)

$u_{\tilde{A}}\left(x\right)$ is convex, i.e., $u_{\tilde{A}}$(λx + (1−λ)y) ≥ min{$u_{\tilde{A}}$(x), $u_{\tilde{A}}$(y)}, ∀ x, y∈ R and λ ∈ [0, 1];

(iii.)

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

(iv.)

the 0-level set ${\tilde{A}}_0$ is a compact subset of R.

Let $\tilde{A}$ and $\tilde{B}$ be fuzzy numbers of LR-type defined as $\tilde{A}={\left(a,c_1,c_2\right)}_{LR}$ and $\widetilde{B }={\left(b,d_1,d_2\right)}_{LR}$, where $a and b$ are the central values, $c_1 \mathrm{and} d_1$ are the left spread values and $c_2 \mathrm{and} d_2$ are the right spread values of $\tilde{A}$ and $\tilde{B}$, respectively. Then,

$$\tilde{A}+\tilde{B}={\left(a,c_1,c_2\right)}_{LR}+{\left(b,d_1,d_2\right)}_{LR}={\left(a+b,c_1+d_1,c_2+d_2\right)}_{LR};$$

$$\tilde{A}-\tilde{B}={\left(a,c_1,c_2\right)}_{LR}-{\left(b,d_1,d_2\right)}_{LR}={\left(a-b,c_1+d_2,c_2+d_1\right)}_{LR}$$

If $\tilde{A}$ and $\tilde{B}$ are positive fuzzy numbers, then

$$\tilde{A}\mathsf{\otimes }\tilde{B}={\left(a,c_1,c_2\right)}_{LR}\mathsf{\otimes }{\left(b,d_1,d_2\right)}_{LR}={\left(ab,a{d}_1+b{c}_1,a{d}_2+b{c}_2\right)}_{LR}$$

Suppose that $\tilde{A}$ is a fuzzy number with differentiable membership function, and $\alpha$-level set ${\tilde{A}}^{\alpha }=\left[a_1\left(\alpha \right),a_2\left(\alpha \right)\right],0\le \alpha \le 1$. Carlsson and Fullér [36] define the lower possibilistic mean value of fuzzy number $\tilde{A}$ as $M_{*}\left(\tilde{A}\right)=2{{\displaystyle \int }}_0^1\alpha ·a_1\left(\alpha \right)d\alpha$; and the upper possibilistic mean value of fuzzy number $\tilde{A}$ as $M^{*}\left(\tilde{A}\right)=2{{\displaystyle \int }}_0^1\alpha ·a_2\left(\alpha \right)d\alpha$. Then its expected value is expressed as $M\left(\tilde{A}\right)={{\displaystyle \int }}_0^1\alpha ·[a_1\left(\alpha \right)+a_2\left(\alpha \right)]d\alpha$. Next, Then, the sum of the lower and upper possibilistic mean values of the fuzzy numbers $\tilde{A} \mathrm{and} \tilde{B}$ can be derived as follows [36]:

$$M_{*}\left(\tilde{A}+\tilde{B}\right)=M_{*}\left(\tilde{A}\right)+M_{*}\left(\tilde{B}\right)$$

(1)

$$M^{*}\left(\tilde{A}+\tilde{B}\right)=M^{*}\left(\tilde{A}\right)+M^{*}\left(\tilde{B}\right)$$

(2)

Thus, the possibilistic mean value of $\tilde{A}+\tilde{B}$ can be obtained as

$$M\left(\tilde{A}+\tilde{B}\right)=\frac{M_{*}\left(\tilde{A}+\tilde{B}\right)+M^{*}\left(\tilde{A}+\tilde{B}\right)}{2}$$

(3)

Then, the lower possibilistic variance of $\tilde{A}$ can be defined as follows [36]:

$$Va{r}_{*}\left(\tilde{A}\right)=2{{\displaystyle \int }}_0^1\alpha {\left[M_{*}\left(\tilde{A}\right)-a_1\left(\alpha \right)\right]}^{2}d\alpha .$$

The upper possibilistic variance of fuzzy numbers $\tilde{A}$ can be defined as follows [36]:.

$$Va{r}^{*}\left(\tilde{A}\right)=2{{\displaystyle \int }}_0^1\alpha {\left[M^{*}\left(\tilde{A}\right)-a_2\left(\alpha \right)\right]}^{2}d\alpha .$$

Ranking fuzzy numbers plays a crucial role in decision making, however, granted most of the ranking methods in the previous research, the ranking procedures cannot discriminate fuzzy quantities, and some are counterintuitive [37]. In order to address the disadvantage of ranking fuzzy numbers, we rank the fuzzy numbers as follows:

Theorem 1

[37]. Suppose that $\tilde{A}=\left(a, c_1, c_2\right)$ and $\tilde{B}=\left(b, d_1, d_2\right)$ are triangular fuzzy numbers, where a and b are the central values and $c_1, c_2$, and $d_1, d_2$ are the left and right spread values, the circumcenter of a fuzzy number $\tilde{A}$ is defined as $S_{\tilde{A}}=\left({\overline{x}}_0, {\overline{y}}_0\right)=\left(\frac{6a+\left(c_2-c_1\right)}{6}, \frac{5-c_2{c}_1}{12}\right)$, and then we use a ranking function $R\left(\tilde{A}\right)$ to map the fuzzy number $\tilde{A}$ to a real number as $R\left(\tilde{A}\right)=\sqrt{{\left({\overline{x}}_0\right)}^2+{\left({\overline{y}}_0\right)}^2}$. If $R\left(\tilde{A}\right)>R\left(\tilde{B}\right)$ , then $\tilde{A}>\tilde{B}$.

3. Dimension Analysis to Guaranteed Rate of Return

In a fuzzy portfolio selection, we define investment proportion x_j for security j and return rate to be triangular fuzzy number ${\tilde{r}}_j=\left(r_j, c_j, d_j\right)$, where r_j is its central value, and c_j, d_j are left and right spreads, j = 1,…, n, respectively. Then, the expected fuzzy returns can be obtained as $\tilde{R}={{\displaystyle \sum }}_{j=1}^n{x}_j{\tilde{r}}_j$. Most researchers have devoted themselves to the field of portfolio selection when investors encounter vagueness in the investment environment. Tsaur et al. [32] proposed the concept of a guaranteed return rate to evaluate the excess investment for securities whose return rates are higher than the guaranteed return rate. Then, the dimension between the return rate for security j and the selected guaranteed return rate is measured by one dimension of excess investment. In reality, different investors have different investment behaviors, and thus the one-dimension excess investment cannot fit all of the possible behaviors in different investors constrained by bounded risks to solve the optimal expected return rates in fuzzy portfolio selection. Therefore, in the proposed fuzzy portfolio model, we assume that we have n securities, and the ordering of the defuzzy ranking for fuzzy return rates are $R\left({\tilde{r}}_1\right)<R\left({\tilde{r}}_2\right)<\cdots <R\left({\tilde{r}}_n\right)$, then the ordering of the fuzzy return rates are ${\tilde{r}}_1<{\tilde{r}}_2<\cdots <{\tilde{r}}_n$, in which excess investment in m securities (m ≤ n) are made based on the different guaranteed return rate defined as ${\tilde{p}}_k=\left(p_k, e_k, f_k\right)$, where $p_k$ is its central value, and $e_k, f_k$ are its left and right spread values, respectively. If the fuzzy return rate ${\tilde{r}}_j$ is larger than the guaranteed return rate ${\tilde{p}}_k$, then security j is allocated excess investment. By considering the s dimension of excess investment, we can formulate the expected fuzzy returns as follows:

$$\tilde{R}={\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left[\raisebox{1ex}{$\left(\left|{\tilde{r}}_j-{\tilde{p}}_k\right|+{\tilde{r}}_j-{\tilde{p}}_k\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right]}^s,s\ge 1$$

(4)

The guaranteed return rate ${\tilde{p}}_k$, k = 1, 2,…, m, is selected by the decision maker based on his investment behavior, and we assume that the defuzzy rankings for the guaranteed return rates are $R\left({\tilde{p}}_1\right)<R\left({\tilde{p}}_2\right)<\cdots <R\left({\tilde{p}}_m\right)$, then the ordering of the guaranteed return rates are ${\tilde{p}}_1<{\tilde{p}}_2<\cdots <{\tilde{p}}_m$. To solve Equation (4), if the fuzzy return rate ${\tilde{r}}_j=\left(r_j, c_j, d_j\right)$, j = 1,…, n, is larger than ${\tilde{p}}_k=\left(p_k, e_k, f_k\right)$ and $R\left({\tilde{r}}_j\right)>R\left({\tilde{p}}_k\right)$ [37], then excess investment will be made on security j; otherwise, no excess investment will be made, then the s dimension of excess investment can be formulated as follows:

$${\left[\raisebox{1ex}{$\left(\left|{\tilde{r}}_j-{\tilde{p}}_k\right|+{\tilde{r}}_j-{\tilde{p}}_k\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right]}^s=\left\{\begin{array}{cc}{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s& ifR\left({\tilde{r}}_j\right)>R\left({\tilde{p}}_k\right)\\ 0& o.w.\end{array}\right.$$

(5)

Next, the lower and upper possibilistic mean values for the s dimension excess investment ${\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$ are defined as $M_{*}{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$ and $M^{*}{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$, ∀ k = 1, 2,…, m, as follows:

$$\begin{array}{ll}{M}_{*}{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s=& 2{{\displaystyle \int }}_0^1\alpha \times {\left({\tilde{r}}_j-{\tilde{p}}_k\right)}_{j1}^s\left(\alpha \right)d\alpha \\ & ={\left(r_j-p_k\right)}^s-\frac{1}{3}s{\left(r_j-p_k\right)}^{s-1}\left(c_j+f_k\right)\end{array}$$

(6)

$$\begin{array}{ll}{M}^{*}{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s=& 2{{\displaystyle \int }}_0^1\alpha \times {\left({\tilde{r}}_j-{\tilde{p}}_k\right)}_{j2}^s\left(\alpha \right)d\alpha \\ & ={\left(r_j-p_k\right)}^s+\frac{1}{3}s{\left(r_j-p_k\right)}^{s-1}\left(d_j+e_k\right)\end{array}$$

(7)

where ${\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s=\left[{\left(r_j-p_k\right)}^s,s{\left(r_j-p_k\right)}^{s-1}\left(c_j+f_k\right),s{\left(r_j-p_k\right)}^{s-1}\left(d_j+e_k\right)\right]$ is a fuzzy number whose $\alpha$-level set is defined as ${\left[{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]}^{\alpha }=\left[{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}_{j1}^s\left(\alpha \right), {\left({\tilde{r}}_j-{\tilde{p}}_k\right)}_{j2}^s\left(\alpha \right)\right]$ for all $\alpha \in \left[0, 1\right]$. Then, we can obtain the crisp possibilistic mean value $M{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$ defined as follows:

$$M{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s={\left(r_j-p_k\right)}^s+\frac{1}{3}s{\left(r_j-p_k\right)}^{s-1}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right],\forall k>0$$

(8)

In the same way, the lower possibilistic and upper possibilistic mean values of the fuzzy returns ${\tilde{r}}_j$, j = 1,…, n, which are defined as $M_{*}\left({\tilde{r}}_j\right)$ and $M^{*}\left({\tilde{r}}_j\right)$, are derived as follows:

$$M_{*}\left({\tilde{r}}_j\right)=2{{\displaystyle \int }}_0^1\alpha \times r_{j1}\left(\alpha \right)d\alpha =r_j-\frac{1}{3}{c}_j,\forall k>0$$

(9)

$$M^{*}\left({\tilde{r}}_j\right)=2{{\displaystyle \int }}_0^1\alpha \times r_{j2}\left(\alpha \right)d\alpha =r_j+\frac{1}{3}{d}_j,\forall k>0$$

(10)

where ${\tilde{r}}_j$ is a fuzzy number whose $\mathsf{\alpha }$-level set is defined as ${\left[{\tilde{r}}_j\right]}^{\alpha }=\left[r_{j1}\left(\alpha \right), r_{j2}\left(\alpha \right)\right]$ for all $\alpha \in \left[0, 1\right]$. Then, we can obtain the crisp possibilistic mean value $M\left({\tilde{r}}_j\right)$ as follows:

$$M\left({\tilde{r}}_j\right)=\frac{\left[M_{*}\left({\tilde{r}}_j\right)+M^{*}\left({\tilde{r}}_j\right)\right]}{2}=r_j+\frac{1}{6}\left(d_j-c_j\right),\forall k>0,$$

(11)

Without the loss of generality, we assume that ${\tilde{p}}_1<{\tilde{p}}_2<\cdots <{\tilde{p}}_m<{\tilde{r}}_1<{\tilde{r}}_2<\cdots <{\tilde{r}}_n$, then the combination of the possibilistic mean value of Formula (11) with respect to all securities that must have return rates larger than those of guaranteed return rates in portfolio selection is obtained as follows:

$$\begin{gathered} M\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]={\displaystyle \sum }_{j=1}^n{x}_{j}M\left({\tilde{r}}_j\right)+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{j=1}^m{x}_{j}M{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s \\ ={\displaystyle \sum }_{j=1}^n{x}_j\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right] \\ +{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^s+\frac{1}{3}s{\left(r_j-p_k\right)}^{s-1}\left[\left(d_j+e_k\right)-\left(c_j+f_k\right)\right],\forall k>0 \end{gathered}$$

(12)

Then, the lower and upper possibilistic variances of the return rates ${{\displaystyle \sum }}_{j=1}^n{x}_j{\tilde{r}}_j+{{\displaystyle \sum }}_{j=1}^n{{\displaystyle \sum }}_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$ are derived as follows:

$$Va{r}_{*}\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]=\frac{1}{18}{\left[{\displaystyle \sum }_{j=1}^n{c}_j{x}_j+s{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^{s-1}\left(c_j+f_k\right)\right]}^2,\forall k>0$$

(13)

$$Va{r}^{*}\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]=\frac{1}{18}{\left[{\displaystyle \sum }_{j=1}^n{d}_j{x}_j+s{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^{s-1}\left(d_j+e_k\right)\right]}^2,\forall k>0$$

(14)

The standard deviation of ${{\displaystyle \sum }}_{j=1}^n{x}_j{\tilde{r}}_j+{{\displaystyle \sum }}_{j=1}^n{{\displaystyle \sum }}_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s$ can be solved as follows:

$$\begin{gathered} SD\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right] \\ =\frac{1}{2}\{{\left\{{\mathit{Var}}_{*}\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]\right\}}^{1/2} \\ +{\left\{{\mathit{Var}}^{*}\left[{\displaystyle \sum }_{j=1}^n{x}_j{\tilde{r}}_j+{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left({\tilde{r}}_j-{\tilde{p}}_k\right)}^s\right]\right\}}^{1/2}\} \\ =\frac{1}{6\sqrt{2}}\left[{\displaystyle \sum }_{j=1}^n\left(c_j+d_j\right)x_j+s{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^{s-1}\left(c_j+f_k+d_j+e_k\right)\right] \end{gathered}$$

(15)

Analogous to portfolio selection from Markowitz’s mean-variance model, we proposed the concept of s dimension excess investment into the fuzzy portfolio model. Therefore, the objective function shown in Equation (12) is maximized, while the possibilistic standard deviation shown in Equation (15) represents the risk of the portfolio constrained by the upper bound of an investor’s desired values. From this viewpoint, the proposed possibilistic mean-standard deviation model of portfolio selection in considering the concept of s-dimension excess investment can be obtained as follows:

$$\begin{gathered} Max {\displaystyle \sum }_{j=1}^n{x}_j\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right]+s{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^s+\frac{1}{3}s{\left(r_j-p_k\right)}^{s-1}[\left(d_j-e_k\right)-\left(c_j-f_k\right)] \\ s.t. \frac{1}{6\sqrt{2}}\left[{\displaystyle \sum }_{j=1}^n\left(c_j+d_j\right)x_j+s{\displaystyle \sum }_{j=1}^n{\displaystyle \sum }_{k=1}^m{x}_j{\left(r_j-p_k\right)}^{s-1}\left(c_j+f_k+d_j+e_k\right)\right]\le \sigma \\ {\displaystyle \sum }_{j=1}^n{x}_j=1 \\ {l}_j\le x_j\le u_j, j=1, 2,\dots ,n \end{gathered}$$

(16)

where l_j and u_j are the lower and upper bounds on proportion x_j, j = 1, 2, …, n, respectively.

Theoretically, the value of excess investment for jth security with respect to kth guaranteed return rate is ranged as $0<\left(r_j-p_k\right)<1$, and thus the bigger the value of s to the dimension of excess investment, the smaller the value of ${\left(r_j-p_k\right)}^s, s\ge 1$. Therefore, if the dimension value s is approaching to be infinity, then the s dimension of excess investment will be approaching 0 as ${\left(r_j-p_k\right)}^s\approx 0$ and then the proposed model can be approximately as follows:

$$\begin{gathered} Max {\displaystyle \sum }_{j=1}^n{x}_j\left[r_j+\frac{1}{6}\left(d_j-c_j\right)\right] \\ s.t. \frac{1}{6\sqrt{2}}\left[{\displaystyle \sum }_{j=1}^n\left(c_j+d_j\right)x_j\right]\le \sigma \\ {\displaystyle \sum }_{j=1}^n{x}_j=1 \\ {l}_j\le x_j\le u_j, j=1, 2,\dots ,n \end{gathered}$$

(17)

4. Illustration of the Proposed Method

4.1. Data Description and Model Explanation

The collected data are the closed prices for each week where the research period is from April 2002 to January 2004 in Shanghai Stock Exchange [38]. Based on the historical data, the corporations’ financial reports and future information, five securities were chosen for analyzing portfolio selection. The possibility distributions for fuzzy return rates were estimated as the following: ${\tilde{r}}_1=$(0.073, 0.054, 0.087), ${\tilde{r}}_2=$ (0. 105, 0.075, 0.102), ${\tilde{r}}_3=$ (0.138, 0.096, 0.123), ${\tilde{r}}_4=$ (0.168, 0.126, 0.162), ${\tilde{r}}_5=$ (0.208, 0.168, 0.213), where r_j is the center value defined the first value for the j-th fuzzy return rate, the second and third values are left and right spread values defined as c_j, and d_j, j = 1, 2, 3, 4, 5, respectively. The lower and upper bounds of investment proportion x_j for security j are as (l₁, l₂, l₃, l₄, l₅) = (0.1, 0.1, 0.1, 0.1, 0.1), and (u₁, u₂, u₃, u₄, u₅) = (0.4, 0.4, 0.4, 0.5, 0.6), respectively. Next, The guaranteed return rates are selected as ${\tilde{p}}_1=\left(0.1, 0.05, 0.05\right)$ as larger than ${\tilde{r}}_1$; ${\tilde{p}}_2=\left(0.15, 0.1, 0.1\right)$ is selected based on the rough averages of ${\tilde{r}}_1$, and ${\tilde{r}}_5$, and ${\tilde{p}}_3=\left(0.2, 0.1, 0.15\right)$ is selected to be smaller than ${\tilde{r}}_5$, where the first value in each guaranteed return rate is the center value defined as p_k, and the second and third values are the left and right spread values defined as e_k, and f_k, k = 1, 2, 3. Next, we distinguish the fuzzy returns into four groups; the first group consists of security 1 for regular investment because their fuzzy return rates are less than guaranteed return rate ${\tilde{p}}_1=\left(0.1, 0.05, 0.05\right)$, whereas the other securities 2, 3, 4, 5 are more than ${\tilde{p}}_1=\left(0.1, 0.05, 0.05\right)$ for excess investment. The second group consists of securities 1, 2, 3 and is a regular investment because their fuzzy return rates are less than the guaranteed rate of return as ${\tilde{p}}_2=\left(0.15, 0.1, 0.1\right)$; by contrast, the fuzzy return rates of securities 4, 5 are more ${\tilde{p}}_2$ for excess investment. The third group requires the securities 1, 2, 3, 4 to be regular investments, and then we set the third guaranteed return rate ${\tilde{p}}_3=\left(0.2, 0.1, 0.15\right)$ to relax the security 5 to excess investment [1]. In the final group, we propose a hybrid analysis to include more guaranteed return rates ${\tilde{p}}_1,{\tilde{p}}_2, \mathrm{and} {\tilde{p}}_3$ in the fuzzy portfolio model, in this analysis, only security 1 is for regular investment, on the other way, securities 2, 3, 4, and 5 are relaxed for different scenario of excess investment. The above-planned four groups and the related portfolio selection results are discussed in the next subsection.

4.2. Results and Discussions

To clearly describe the proposed model, we suggest two dimension (s = 2) of excess investment to derive the fuzzy portfolio selection. We first rank the fuzzy return rates ${\tilde{r}}_1, {\tilde{r}}_2, {\tilde{r}}_3, {\tilde{r}}_4,{\tilde{r}}_5$ to the guaranteed return rates ${\tilde{p}}_1,{\tilde{p}}_2, \mathrm{and} {\tilde{p}}_3$. According to the defuzzy method [38], we can map all the fuzzy numbers to real numbers and obtain the rank results as ${\tilde{r}}_1<{\tilde{p}}_1<{\tilde{r}}_2<{\tilde{r}}_3<{\tilde{p}}_2<{\tilde{r}}_4<{\tilde{p}}_3<{\tilde{r}}_5$.

Step 1: Use the guaranteed return rate to formulate the fuzzy portfolio model.

In this step, we use the guaranteed return rate ${\tilde{p}}_1=\left(0.1, 0.05, 0.05\right)$ to analyze the excess investment in the portfolio under two dimension (s = 2) of excess investment where the defuzzy value of ${\tilde{p}}_1$ is smaller than the defuzzy values of ${\tilde{r}}_2, {\tilde{r}}_3, {\tilde{r}}_4, \mathrm{and} {\tilde{r}}_5$. Therefore, the fuzzy portfolio model with s = 2 can be obtained as follows:

$$\begin{gathered} Max 0.0785x_1+0.109615x_2+0.144628x_3+0.180256x_4+0.230404x_5 \\ s.t. 0.141x_1+0.021186x_2+0.028667x_3+0.04016x_4+0.057146x_5\le \sigma \\ {x}_1+x_2+x_3+x_4+x_5=1 \\ 0.1\le x_1, x_2,x_3\le 0.4;0.1\le x_4\le 0.5;0.1\le x_5\le 0.6 \end{gathered}$$

(18)

Step 2: Portfolio analysis under different risk levels

In this step, for solving the proposed model with s = 2 with guaranteed return rate ${\tilde{p}}_1$, the risk of the investment is constrained by the upper bound of the investor’s desired values from 4.5% to 5.8%, and the portfolio results are shown in

Table 1. The efficient portfolio with s = 2 in a guaranteed return rate ${\tilde{p}}_1$.

	Risk	4.4%	4.5%	4.7%	5%	5.3%	5.5%	5.7%	5.8%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.1236	0.1	0.1	0.1	0.1	0.1	0.1
x3			0.4	0.265	0.12	0.1	0.1	0.1	0.1
x4			0.2764	0.435	0.5	0.3584	0.2406	0.1229	0.1
x5			0.1	0.1	0.1791	0.3416	0.4594	0.5771	0.6
Expected Return Rate			0.15211	0.15859	0.16769	0.17658	0.18249	0.18954	0.18954

. If the constrained risk is selected to be smaller than 4.5%, then the portfolio is infeasible. By contrast, if the constrained risk is selected to be larger than 5.8%, then its optimal portfolio will be the same as the constrained risk of 5.8%. Therefore, the optimal portfolio is obtained as x₁ = 0.1, x₂ = 0.1, x₃ = 0.1, x₄ = 0.1 and x₅ = 0.6, in which the expected return rate is 18.954% with the constrained risk 5.8%. By the constrained risk from 4.5% to 5.8%, we can find that the investment proportion of security 1 is at its lower bounds as 0.1 in different constrained risk because its return rate of security is less than the guaranteed return rate ${\tilde{p}}_1$; investment proportion for securities 2 and 3 are decreasing from the constrained risk between 4.5% and 5.8%, and the investment proportion for securities 4 is in the increasing process but changes to decreasing when security 5 of the constrained risk is near 5.8%. Next, we change the selected guaranteed return rates to ${\tilde{p}}_2 \mathrm{and} {\tilde{p}}_3$, respectively. In

Table 2. The efficient portfolio with s = 2 in a guaranteed return rate ${\tilde{p}}_2$.

	Risk	8.2%	8.3%	8.5%	8.7%	8.9%	9%	9.3%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1	0.1
x4			0.4569	0.3381	0.2193	0. 1005	0.1	0.1
x5			0.2431	0.3619	0.4807	0.5995	0.6	0.6
Expected Return Rate			0.16901	0.17495	0.18081	0.18667	0.18669	0.18669

, with s = 2 and the guaranteed return rate ${\tilde{p}}_2$, the risk of the investment is constrained by the upper bound of the investor’s desired values from 8.3% to 9.3%, and the portfolio results are shown in Table 2. The optimal portfolio is obtained as x₁ = 0.1, x₂ = 0.1, x₃ = 0.1, x₄ = 0.1 and x₅ = 0.6, in which the expected return rate is 18.669% under the constrained risk of 9%. By the constrained risk from 8.3% to 9%, we can find that the investment proportion of securities 1, 2, 3 are all at their lower bounds as 0.1 in different constrained risks because their return rates are less than the guaranteed return rate ${\tilde{p}}_2$. In addition, we can find that investment proportion for securities 4 is in the decreasing process but for securities 5 it is increasing when the constrained risk is between 8.3% and 9%, because the return rate of security 4 is less than that of security 5, the investment proportion of security 4 will be gradually decreased, and the security 4 will be gradually decreased when the constrained risk is increased. In

Table 3. The efficient portfolio with s = 2 in a guaranteed return rate ${\tilde{p}}_3$.

	Risk	11.0%	11.1%	11.5%	12%	12.5%	13%	13.5%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1	0.1
x4			0.1	0.1	0.1	0.1	0.1	0.1
x5			0.6	0.6	0.6	0.6	0.6	0.6
Expected Return Rate			0.17977	0.17977	0.17977	0.17977	0.17977	0.17977

, if the constrained risk is selected to be smaller than 11.1%, then the portfolio is infeasible. By contrast, if the constrained risk is selected to be larger than 11.1%, then its optimal portfolio will be the same as the constrained risk of 11.1% as x₁=0.1, x₂ = 0.1, x₃ = 0.1, x₄ = 0.1 and x₅ = 0.6, in which the expected return rate is 17.977%. Therefore, compare to Table 1, Table 2 and Table 3, we can find that the larger the guaranteed return rate we select, the smaller the expected return rate with larger constrained risk.

Step 3: Comparison to dimensions of excess investment with s = 1 and s = 2

The dimension of the excess investment in the model (16) is shown as ${\left(r_j-p_k\right)}^s, s\ge 1$, where the excess investment ranges on (0, 1) as $0<\left(r_j-p_k\right)<1$, and thus the bigger the value of s, the smaller the excess investment as ${\left(r_j-p_k\right)}^s, s \ge 1$. Since ${\left(r_j-p_k\right)}^2<\left(r_j-p_k\right)$, the corresponding constrained risk from a higher dimension of excess investment should be lower than that of dimension s = 1. Therefore, a different investor with a different risk attitude should adopt a different dimension of excess investment. As shown in

Table 4. The efficient portfolio with s = 1 in a guaranteed return rate ${\tilde{p}}_1$.

	Risk	7.2%	7.3%	7.7%	8.3%	8.7%	9.3%	9.5%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.3645	0.1715	0.1	0.1	0.1	0.1
x3			0.3355	0.4	0.177	0.1	0.1	0.1
x4			0.1	0.2285	0.5	0.4517	0.178	0.1
x5			0.1	0.1	0.123	0.2483	0.522	0.6
Expected Return Rate			0.17570	0.19672	0.22237	0.23793	0.26106	0.26765

,

Table 5. The efficient portfolio with s = 1 in a guaranteed return rate ${\tilde{p}}_2$.

	Risk	12.2%	12.3%	12.5%	12.7%	13%	13.3%	14%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1	0.1
x4			0.459	0.3677	0.2765	0.396	0.1	0.1
x5			0.241	0.3323	0.4235	0.5604	0.6	0.6
Expected Return Rate			0.19622	0.20393	0.21164	0.22320	0.22655	0.22655

and

Table 6. The efficient portfolio with s = 1 in a guaranteed return rate ${\tilde{p}}_3$.

	Risk	15.4%	15.5%	16%	17%	18%	19%	20%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1	0.1
x4			0.1	0.1	0.1	0.1	0.1	0.1
x5			0.6	0.6	0.6	0.6	0.6	0.6
Expected Return Rate			0.18355	0.18355	0.18355	0.18355	0.18355	0.18355

, the portfolio selections under different constrained risks are listed for s = 1. The major difference between Table 1, Table 2 and Table 3, and Table 4, Table 5 and Table 6 can be found that the excess investment with dimension s = 1 has a bigger expected return and higher constrained risk than that of dimension s = 2. Therefore, for the excess investment in the proposed fuzzy portfolio selection, we suggest to the investors whose degree of risk preference is higher should adopt the dimension of the excess investment to be 1; in contrast, the investors whose risk preference is lower should adopt dimension of the excess investment to be higher than 2.

Step 4: Sensitivity analysis to excess investment in different dimensions

In this step, in order to find the results of a different dimension, s affects the expected return rate and the constrained risk. We proceed with a sensitivity analysis to examine the excess investment in different dimensions s as 3 and 4, and then we solve the proposed portfolio selection model under the guaranteed return rates ${\tilde{p}}_1,{\tilde{p}}_2 \mathrm{and} {\tilde{p}}_3$, respectively. The portfolio selections under different constrained risks are shown in

Table 7. The efficient portfolio with s = 3 in a guaranteed return rate ${\tilde{p}}_1$.

	Risk	4.4%	4.5%	4.7%	5%	5.3%	5.4%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.1214	0.1	0.1	0.1	0.1
x4			0.5	0.3739	0.1302	0.1	0.1
x5			0.1786	0.3261	0.5698	0.6	0.6
Expected Return Rate			0.16216	0.16915	0.17959	0.18088	0.18088

,

Table 8. The efficient portfolio with s = 3 in a guaranteed return rate ${\tilde{p}}_2$.

	Risk	7.9%	8%	8.1%	8.3%	8.5%	8.6%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1
x4			0.4842	0.398	0.2255	0.1	0.1
x5			0.2158	0.302	0.4745	0.6	0.6
Expected Return Rate			0.16389	0.16750	0.17471	0.17996	0.17996

and

Table 9. The efficient portfolio with s = 3 in a guaranteed return rate ${\tilde{p}}_3$.

	Risk	10.9%	11%	11.5%	12%	12.5%	13%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1
x4			0.1	0.1	0.1	0.1	0.1
x5			0.6	0.6	0.6	0.6	0.6
Expected Return Rate			0.17975	0.17975	0.17975	0.17975	0.17975

for s = 3, and

Table 10. The efficient portfolio with s = 4 in a guaranteed return rate ${\tilde{p}}_1$.

	Risk	6.0%	6.1%	6.3%	6.5%	6.7%	6.8%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.2785	0.1299	0.1	0.1	0.1
x4			0.4215	0.5	0.373	0.1	0.1
x5			0.1	0.1701	0.327	0.6	0.6
Expected Return Rate			0.15342	0.16102	0.16851	0.17988	0.17988

,

Table 11. The efficient portfolio with s = 4 in a guaranteed return rate ${\tilde{p}}_2$.

	Risk	7.9%	8%	8.1%	8.3%	8.5%	8.6%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1
x4			0.4693	0.3785	0.1969	0.1	0.1
x5			0.2307	0.3215	0.5031	0.6	0.6
Expected Return Rate			0.16443	0.16820	0.17574	0.17976	0.17976

and

Table 12. The efficient portfolio with s = 4 in a guaranteed return rate ${\tilde{p}}_3$.

	Risk	10.9%	11%	11.5%	12%	12.5%	13%
Proportion
x1		Infeasible Solution	0.1	0.1	0.1	0.1	0.1
x2			0.1	0.1	0.1	0.1	0.1
x3			0.1	0.1	0.1	0.1	0.1
x4			0.1	0.1	0.1	0.1	0.1
x5			0.6	0.6	0.6	0.6	0.6
Expected Return Rate			0.17975	0.17975	0.17975	0.17975	0.17975

for s = 4. The trend of the portfolio selections in expected return rates and constrained risks are similar to the above step for different guaranteed return rates ${\tilde{p}}_1,{\tilde{p}}_2 and {\tilde{p}}_3$. The major character of results among s = 2, s =3, and s = 4 all solves optimal portfolio with security proportions as x₁ = 0.1, x₂ = 0.1, x₃ = 0.1, x₄ = 0.1, and x₅ = 0.6, but the expected return rates are shown that s = 2 > s = 3 > s = 4; whereas the constrained risks of different dimensions in solving the optimal portfolio are shown as the order of s = 2 < s = 3 < s = 4 under the guaranteed return rates ${\tilde{p}}_1$. This is because excess investment ranges on (0, 1) as $0<\left(r_j-p_k\right)<1$, and thus the bigger the value of s, the smaller the excess investment as ${\left(r_j-p_k\right)}^s, s\ge 1$. Therefore, the objective value of the optimal portfolio under bigger s should be less than that of smaller s. Besides, as the value of s approaches to be bigger, the lower constrained risk can be found to obtain the feasible portfolio selection. Therefore, for s = 2, s = 3, and s = 4, the expected return rates of the optimal portfolio are shown as the order of s = 2 > s = 3 > s = 4; whereas the constrained risks of different dimensions in solving the optimal portfolio are shown as the order of s = 2 < s =3 < s = 4. However, as the larger guaranteed return rates are selected for modeling the proposed model, the optimal portfolio in the expected return rates and constrained risks will be gradually equal because seldom securities are selected to be an excess investment.

5. Conclusions

The scientific novelty of this research focuses on the risk attitudes to investigate the risk attitude effects to portfolio selections by the dimensions of excess investment in a fuzzy portfolio model. Most importantly, if the fuzzy portfolio model for excess investment in different dimensions meets expectations, this approach will be useful to obtain the portfolio for the selected securities. The conclusion section can be extended as:

(1)

The numerical results. The dimensions of the excess investment are proposed in the fuzzy portfolio selection. The numerical results show that the selected dimension to the excess investment by an investor with a different degree of risk preference can be operated by one or two dimensions to the excess investment. For lower-risk preference investors, we suggest selecting two dimensions of excess investment to obtain the optimal portfolio with lower constrained risk; by contrast, one dimension of excess investment is adopted for obtaining a bigger expected return rate with higher constrained risk.

(2)

Limitations. Because different dimensions of excess investment affect the portfolio selection in our proposed model, it is important to formulate a process to evaluate the risk attitude of an investor. The limitation of this study should be considered for an excellent measure scale and being evaluated on the risk priority of past investment events, and then we can confirm the risk attitudes to make the portfolio selection.

(3)

Future research. In order to maximize the expected return rate under constrained risk, efforts to manage different types of risk attitudes play a major role in portfolio selection. This is because there are still numerous investors whose investment behaviors are not fully understood and cannot be applied to our proposed model.