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Traditional portfolio theory uses probability theory to analyze the uncertainty of financial market. The assets’ return in a portfolio is regarded as a random variable which follows a certain probability distribution. However, it is difficult to estimate the assets return in the real financial market, so the interval distribution of asset return can be estimated according to the relevant suggestions of experts and decision makers, that is, the interval number is used to describe the distribution of asset return. Therefore, this paper establishes a portfolio selection model based on the interval number. In this model, the semiabsolute deviation risk function is used to measure the portfolio’s risk, and the solution of the model is obtained by using the order relation of the interval number. At the same time, a satisfactory solution of the model is obtained by using the concept of acceptability of the interval number. Finally, an example is given to illustrate the practicability of the model.

Portfolio selection refers to the way in which investors allocate a certain proportion of their wealth to a number of different assets so as to spread risks among multiple assets and obtain some stable returns. Markowitz [

Uncertainty exists everywhere, and scholars use various methods to study it [

Lai

The rest of this paper is organized as follows. In Section

In this paper, concepts and operations related to interval numbers will be used. This section will briefly review the relevant concepts.

(see [

The interval number is a special fuzzy number whose membership function takes value 1 over the interval and 0 anywhere else, as discussed in detail by Hansen [

(see [

Specifically, if

(see [

The notations used in this article are given below:

Let us consider a market consisting of a riskless assets and

Thus, the portfolio’s return

To set up a portfolio selection model, the following values need to be given.

First is the expected return of the portfolio’s return

The expected return on security

Secondly, the risk of the portfolio is as follows.

As mentioned in Section

The risk of the portfolio is given by

So,

Therefore, a portfolio selection model based on risk-return trade-off can be established:

As can be seen from equations (

The solution of model (

From Definition

This is an interval-valued linear programming problem. For the solution method of interval-valued linear programming, scholars have carried out a lot of research and put forward some solutions. For example, Yoon [

Chankong and Haimes [

The solution of model (

Because

According to Definition

In order to obtain the solution of (

Then, (

The optimal solution to (

The lower bounds of all the interval-valued efficient portfolios construct the interval-valued lower efficient frontier. The upper bounds of all the interval-valued efficient portfolios construct the interval-valued upper efficient frontier.

Meanwhile, based on the acceptability, (

According to Definition

Thus, a satisfactory solution of the model is obtained, and an acceptable efficient portfolio is obtained.

In order to illustrate the practicality of this model, we select five securities and one risk-free asset from the Chinese stock market for investment. Annual data from 2016 to 2020 were selected. Table

The expected return of the five securities.

Code | |
---|---|

[−0.557, 0.341] | |

[−0.170, 0.306] | |

[−0.381, 0.953] | |

[−0.316, 0.258] | |

[−0.503, 0.156] |

Let the risk-free asset be a treasury bond. We use the one-year treasury bond rate as the return rate of the riskless asset. So, we get the return on risk-free asset

Table

The investment proportion and risk for different

Risk | |||||||
---|---|---|---|---|---|---|---|

[0.04, 0.04] | 0.010 | 0.030 | 0.010 | 0.010 | 0.118 | 0.036 | 0.178 |

[0.04, 0.05] | 0.010 | 0.030 | 0.010 | 0.010 | 0.153 | 0.042 | 0.213 |

[0.04, 0.06] | 0.010 | 0.030 | 0.010 | 0.010 | 0.187 | 0.048 | 0.247 |

[0.04, 0.07] | 0.010 | 0.030 | 0.010 | 0.010 | 0.221 | 0.054 | 0.281 |

[0.04, 0.08] | 0.010 | 0.030 | 0.010 | 0.010 | 0.256 | 0.060 | 0.316 |

[0.04, 0.09] | 0.010 | 0.030 | 0.010 | 0.010 | 0.290 | 0.066 | 0.350 |

[0.04, 0.10] | 0.054 | 0.030 | 0.010 | 0.010 | 0.300 | 0.080 | 0.404 |

[0.04, 0.11] | 0.116 | 0.030 | 0.010 | 0.010 | 0.300 | 0.097 | 0.466 |

[0.04, 0.12] | 0.178 | 0.030 | 0.010 | 0.010 | 0.300 | 0.114 | 0.528 |

[0.04, 0.14] | 0.300 | 0.030 | 0.010 | 0.220 | 0.300 | 0.182 | 0.860 |

Some efficient portfolios for model (

As can be seen from Table

The lower limit of the minimum expected return rate remains unchanged. With the increase of the upper limit, the investment proportion of

As the minimum expected return rate increases, so does portfolio risk.

When the minimum expected return rate increases to a certain value, the model will have no feasible solution.

Parameter

The investment risk for different

Risk | Risk | ||
---|---|---|---|

0 | 0.190 | 0.55 | 0.107 |

0.1 | 0.182 | 0.6 | 0.097 |

0.2 | 0.134 | 0.7 | 0.086 |

0.3 | 0.128 | 0.8 | 0.078 |

0.4 | 0.115 | 0.9 | 0.056 |

0.5 | 0.109 | 1.0 | 0.045 |

The scatter diagram of portfolio risk and the risk preference coefficient

Model (

The investment proportion and risk for different

Risk | |||||||
---|---|---|---|---|---|---|---|

[0.03, 0.03] | 0.010 | 0.030 | 0.034 | 0.010 | 0.010 | 0.028 | 0.094 |

[0.03, 0.04] | 0.010 | 0.030 | 0.058 | 0.010 | 0.010 | 0.040 | 0.118 |

[0.03, 0.05] | 0.010 | 0.030 | 0.081 | 0.010 | 0.010 | 0.052 | 0.141 |

[0.03, 0.06] | 0.010 | 0.030 | 0.105 | 0.010 | 0.010 | 0.063 | 0.165 |

[0.03, 0.07] | 0.010 | 0.030 | 0.128 | 0.010 | 0.010 | 0.075 | 0.188 |

[0.03, 0.08] | 0.010 | 0.030 | 0.152 | 0.010 | 0.010 | 0.087 | 0.212 |

[0.03, 0.09] | 0.010 | 0.030 | 0.175 | 0.010 | 0.010 | 0.098 | 0.235 |

[0.03, 0.10] | 0.010 | 0.030 | 0.199 | 0.010 | 0.010 | 0.110 | 0.259 |

[0.03, 0.11] | 0.010 | 0.293 | 0.200 | 0.010 | 0.010 | 0.157 | 0.523 |

The investment proportion and risk for different

Risk | |||||||
---|---|---|---|---|---|---|---|

[0.03, 0.03] | 0.010 | 0.030 | 0.067 | 0.010 | 0.010 | 0.044 | 0.127 |

[0.03, 0.04] | 0.010 | 0.030 | 0.099 | 0.010 | 0.010 | 0.052 | 0.159 |

[0.03, 0.05] | 0.010 | 0.030 | 0.131 | 0.010 | 0.010 | 0.076 | 0.191 |

[0.03, 0.06] | 0.010 | 0.030 | 0.163 | 0.010 | 0.010 | 0.092 | 0.223 |

[0.03, 0.07] | 0.010 | 0.030 | 0.195 | 0.010 | 0.010 | 0.108 | 0.255 |

As can be seen from Table

With the increase of

When

It can be seen from Table

By comparing Tables

This paper takes the assets return as the interval number and uses the semiabsolute deviation function of the interval number to measure the portfolio’s risk. Therefore, a portfolio selection model with mean-semiabsolute deviation based on the interval number is constructed. In this model, firstly, the lower bound of the investors’ expected rate of return is also regarded as an interval number, which can better grasp investors’ psychology and measure investors’ expected return rate. Secondly, when solving the semiabsolute deviation portfolio selection model, a parameter which can reflect investors’ risk preference is introduced, and this parameter can reflect investors’ risk preference more intuitively. Finally, an application of the portfolio diversification problem is given by using a portfolio consisting of 5 risky assets and 1 risk-free asset. The results show that the introduced risk preference parameter can well reflect the investors’ attitude to risk, and the lower bound of the expected return rate of this method is more elastic. The model can be used more widely and can describe the expected return rate of investment portfolio and the investors’ attitude to risk more flexibly [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded by the National Natural Science Foundation of China (11401438), the National Social Science Foundation Project of China (13CRK027), and the social science planning project of Shandong Province (21CTJJ04).