Abstract

This paper deals with a behavioral portfolio decision problem with triangular fuzzy number return. A fuzzy sentimental mean model for behavioral portfolio decision is proposed by taking into account investor’s sentiment and multiple mental accounts. The presented behavioral portfolio decision model maximizes the fuzzy sentimental mean value of portfolio return and ensures the portfolio return of each mental account exceeding the given minimum triangular fuzzy number level with a given possibility degree. Then, multiple programming models are designed to solve the optimal behavioral portfolio strategy. Finally, a numerical example is given to illustrate the validity of the proposed approach.

1. Introduction

In 1952 Markowitz ¹ proposed the mean-variance portfolio decision model and provided a fundamental basis for modern portfolio selection theory by maximizing the expected return for a given level of risk. In addition, Shefrin and Statman ² proposed behavioral portfolio framework for asset choice under uncertainty based on prospect theory. In the behavioral portfolio process, each portfolio layer is associated with a particular aspiration level and resembles a separate mental account ³. After that, Ma ⁴ proposed a practical decision making method for behavioral portfolio choice, Yaz ⁵ studied a behavioral approach to efficient portfolio formation, Mukesh ⁶ and Amelia ⁷ developed multi-criteria behavioral portfolio decision models. Jin ⁸ developed multi-period and multi-objective behavioral portfolio approach. Also, Xie ⁹ studied the behavioral assets portfolio method based on sentiment recognition.

Recently, fuzzy set have been generally used in handling and describing imprecise and complex phenomena that often rise in business, financial and managerial systems. In uncertain portfolio decision scenario, the return of financial asset is conveniently evaluated by fuzzy number. Inspired by the idea of Markowitz’s M-V model, a lot of fuzzy portfolio model extensions have been proposed to deal with portfolio decision with fuzzy return and risk under fuzzy uncertain environment. For example, Wang ¹⁰, Fang ¹¹ studied the fuzzy portfolio selection problems. Bilbao Terol ¹² and Gupta ¹³ studied the portfolio models based on fuzzy decision theory and fuzzy programming technique. Tsaur ¹⁴ and Zhou ¹⁵ investigated fuzzy portfolio model with different investor’s attitudes. Zhang ^{16, 17} proposed some portfolio models based on possibilistic mean and variance. Rupak ¹⁸ presented the portfolio selection model based on fuzzy entropy and skewness. Mukesh ¹⁹, Yue ²⁰ and Zhang ²¹ proposed fuzzy higher order moment portfolio models. Liagkouras ²², Liu ^{23, 24}, Muresh ²⁵ and Zhang ²⁶ also discussed the fuzzy multi-period portfolio models.

However, in uncertain portfolio decision environment the portfolio return of each mental account are usually fuzzy, and the above-mentioned fuzzy portfolio decision models have not considered the investor’s sentiment and investor’s behavioral interacting factors. Therefore, we will provide a new methodology to build portfolios for behavioral investors that follow ethical, environmental and social considerations in their investment process. To do so, we construct fuzzy behavioral portfolio theory with mental accounts, multiple programming models and sentiment factors. In this work, we propose a fuzzy behavioral portfolio model to determine the asset allocation between the different mental accounts according to the investor’s sentiment and the historical fuzzy return data in financial market.

2. Preliminaries

Let us first review some basic concepts of triangular fuzzy numbers, which will be utilized in the following sections about fuzzy behavioral portfolio decision model with fuzzy number return.

Definition 1 ²⁷. A fuzzy set of real line with a normal, fuzzy convex and continuous membership function of bounded support is called a triangular fuzzy number if its membership function satisfies the following form:

and and are the left width and right width of fuzzy number , respectively.

Definition 2 ²⁷. is a triangular fuzzy number, the -level cut set of can be computed as

Definition 3 ¹⁴. Let ， be any two triangular fuzzy numbers, some basic operators are defined as

(1)

(2)

Theorem 1 ¹⁷. Let be any two triangular fuzzy numbers, then we can prove that

(1)

(2)

The proof is easy and omitted.

Definition 4 ²⁸. Let be a triangular fuzzy numbers with -level set

then the possibilistic mean value of fuzzy number is defined as

Definition 5 ²⁹. Let , be any two triangular fuzzy numbers, the degree of possibility of is defined as

Theorem 2 ²⁹. Let ,, be three triangular fuzzy numbers, then

(1) ;

(2) If If ;

(3) ;

(4) If then .

It can be proved easily by Definition 5 of possibility degree of triangular fuzzy numbers.

Definition 6 ⁹ Letbe the sentiment of investor on asset , be the fuzzy number return of asset, the sentiment influential function and the sentiment-adjusted fuzzy number return of asset, respectively, are defined as

Remark 1. The sentiment influential function is a increasing function. The higher is the investor’s sentiment on asset, the higher is the estimated fuzzy return of asset. And one can easily get the following results.

(i) If then , the fuzzy return of assetwill increase with the positive sentiment on asset.

(ii) If then , the fuzzy return of assetwill be unchanged with the rational sentiment on asset.

(iii) If then , the fuzzy return of assetwill decrease with the negative sentiment on asset.

3. The Formulation of Fuzzy Behavioral Portfolio Decision Model

In this section, we discuss the behavioral portfolio selection problem with fuzzy returns and investor’s sentiments. We first introduce the problem description and notations used in the following section. Then, we formulate the fuzzy behavioral portfolio model by maximizing the fuzzy sentimental mean of portfolio return.

Let us consider a behavioral portfolio selection problem with mental accounts. Each mental account consists risky assets. The return rates of risky assets are evaluated by triangular fuzzy numbers. Assume that the investor intends to allocate his/her wealth among the risky assets for making accounting investment plan in mental accounts. To make it easier to follow our exposition, we put together all the notations that will be used hereafter.

: the investment proportion of risky asset in mental account ;

: the lower boundary of investment proportion of risky asset in mental account ;

: the upper boundary of investment proportion of risky asset in mental account ;

: the importance degree of the holding mental account ;

: the unit transaction cost of risky assetin mental account ;

: the sentiment of investor on asset in mental account ;

: the sentiment influential function of investor on asset in mental account .

Assume that the whole investment process is self-financing, that is, the investor does not invest the additional capital during the portfolio selection process. Let be the triangular fuzzy number return of asset at mental account . According to the previous section, the sentimental mean value of triangular fuzzy number return for portfolio at mental account is determined by

Assume that the objective of the investor wants to maximize the expected sentimental return of portfolio over the whole mental accounts. At the same time, the fuzzy portfolio return at each mental account must achieve or exceed the given minimum fuzzy number return level with a certain possibility degree. Thus, the fuzzy behavioral portfolio decision selection problem with multi-accounts can be formulated as the following programming model denoted by (P1):

s.t.

where is the weight vector of all the mental accounts, is the importance degree of mental account And represents the given minimum aspiration fuzzy return level of the portfolio wealth regarding the -th mental account; is the given possibility degree level assuring that the fuzzy return of -th mental account greater than the given minimum aspiration fuzzy return level . In general, the lower level is the account mental, the greater is the parameter.

If we let be the sentiment function of investor on asset at mental account , then the above programming model (P1) can be transformed to the following optimization models (P11)-(P12) according to Definition 5 of possibility degree of fuzzy number returns and Theorem 1, 2.

(P11)

s.t.

The above model is equivalent to the following programming model.

(P12)

s.t.

4. Illustrative Example

Example 1. In order to express the idea of our model and the effectiveness of the proposed fuzzy behavioral portfolio method, we give an example for simulating the real transaction. For simplicity, in the example we consider two-mental accounting behavioral portfolio decision problem with fuzzy number returns. Assume that the financial market has two mental accounts . The lower-level mental account has three alternative financial assets . The high-level mental account has three alternative financial assets . All the financial assets in the above two mental accounts are selected from Shanghai Stock Exchange in China. To simulate the transaction, we collect the weekly closing pricing of assets from Jun 2018 to Jun 2019, with 1 yearly observations. By analyzing the stock historical data, the corresponding corporations’ financial reports and the future information, we can utilize the simple statistical frequency method ¹⁷ to assess the fuzzy return of assets in the above two mental accounts, . The evaluated fuzzy number return are listed in the following Table 1.

Suppose that the investor’s initial sentiment vector on the selected six financial assets is

If we choose as the sentiment influential function, we can compute the sentiment influential function value vector as

Also we can compute the possibilistic mean vector of the selected stock assets as

In this example we assume the lower boundary and upper boundary of investment proportion of risky asset at mental account are 0.1 and 0.3, respectively. Suppose , are the given minimum expected fuzzy return of the portfolio for mental account 1 and 2, respectively. And we let be the possibility degree that the fuzzy return of the first mental account exceeds the minimum aspiration and is the given possibility degree that the fuzzy return of the second mental account exceeds the minimum aspiration Suppose is the transaction costs of assets in and is the transaction costs of assets in , .

In order to obtain the corresponding portfolio strategy , we construct the following sentiment-adjusted fuzzy behavioral portfolio model.

(P2)

s.t.

Substituting data of fuzzy number return of assets in Table 1 into the model (P2) and according to Definition 4, 5 and Theorem 2 we can easily transform the above fuzzy behavioral portfolio model to the following four linear programming models (P21)-(P24).

(P21)

s.t.,

which is equivalent the following model ():

s.t.

(P22)

s.t., .

(P23)

s.t.,

(P24)

s.t.,

Since the different importance agree of each mental account affects the behavioral portfolio solution, in this paper we shall consider four types of investment importance vectors of the two mental accounts as W1=(0.1,0.9) , W2=(0.5,0.5), W3=(0.9,0.1), W4=(0.6,0.4).

Then, we apply the nonlinear optimization tools in Matlab software package to solve the above-mentioned programming models. Finally, we obtain the optimal behavioral portfolio strategy, which is the solver corresponding to maximum objective function value of portfolio. The optimal behavioral investment portfolio solution corresponding to the maximum sentimental mean regarding the different weight vector of mental accounts are easily computed as listed in the following Table 2.

5. Summary and Conclusion

In this paper, we consider the multi-account behavioral portfolio selection problem under fuzzy uncertain environment. We use the sentiment-adjusted mean value to measure triangular fuzzy number return of the behavioral portfolio. Furthermore, based on the possibility degree of triangular fuzzy number return of each mental account exceeding the given minimum fuzzy number level we present a sentiment-adjusted behavioral portfolio model with fuzzy return and investor’s sentiment. In order to solve the proposed model, we transform it into the equivalent programming models. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Acknowledgements

We thank the Editor and the anonymous referees for their good suggestions concerning this paper. This research is supported by the Natural Science Foundation of Guangdong Province, China (Nos. 2018A030313996 and 2017A030313435).

References