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In the financial market, investors must deal with uncertain risk, and they also face background risk and many uncertain factors caused by their own characteristics. Considering the fuzzy nature of these factors as well as investors’ risk preferences, transaction costs, and so on, in order to reduce investment risk, an improved probability entropy measure is introduced, and a probability mean-lower semivariance-entropy model with different risk attitudes is established by using fuzzy sets and probability theory. To solve the portfolio model, an improved differential evolution algorithm is proposed and a numerical example is given. The numerical results show that the proposed algorithm is effective and that the model can disperse the financial risk to a certain extent and reasonably solve the portfolio problem under many different conditions.

The portfolio problem studies how to reasonably distribute the wealth in the hands of investors to different assets in order to realize the rapid growth of wealth and control investment risk. Markowitz [

Financial market risk is considered to be uncertain, and Shape [

In the investment process, people face not only financial risks but also background risks, including those related to labour income, proprietary income, real estate investment, unexpected expenses caused by health problems, health insurance, and so on. Different investors have different attitudes towards risk, and extremely rational investors are absolutely risk averse. How to diversify investment to reduce risk has become a hot issue that has been studied by researchers. As early as 1952, Markowitz put forward a view of the “Don’t put your eggs in the same basket,” which fully illustrates the importance of decentralized investment. At present, some scholars have used proportional entropy as a combined measure for decentralized portfolios. Huang [

The structure of the remainder of this paper is as follows. In Section

Li [

Li [

In view of the above formula of (

The upper and lower mean probabilities of trapezoidal fuzzy number

Via formulas (

The upper and lower semivariances of the probability of trapezoidal fuzzy number

From the above formulas (

Membership function of trapezoidal fuzzy number

The securities market is an extremely complex system. The returns and risk of securities are uncertain, and especially the influence of human factors on investment decisions cannot be ignored. In many cases, the returns and risks of securities can only be described in some vague languages, such as low risk and low return and high risk and high return. This makes investors make investment decisions in vague environments. For the convenience of the explanation, first of all, the relevant symbols involved in this paper are given as follows:

Suppose there are

The membership function of financial risk assets with risk attitude [

It is assumed that the return of the asset with background risk

Suppose the investment strategy is self-financing; that is, no new funds are injected into the portfolio adjustment process. For the transaction cost function, we use the commonly used

Thus, the equation containing background risks and transaction costs is expressed as follows:

Because the linear combination of the trapezoidal fuzzy numbers is a trapezoidal fuzzy number,

Via formulas (

From formula (

In the traditional mean-variance model, the variance is usually used to measure the risk in a portfolio. In fact, it is inappropriate to measure risk using the variance for the following reasons: (1) it only describes the degree of deviation of the return, and it does not describe the direction of the deviation; and (2) the variance does not reflect the losses of the portfolio. Therefore, this paper measures the risk using the lower semivariance of the probability of the rate of return on assets and measures the return using the mean probability of the rate of return on assets from the above formulas (

In recent years, according to information entropy theory, many scholars use the proportional entropy as an index to measure the degree of portfolio diversification. The concept of entropy was introduced by the German physicist Rudolph Clausius in 1850 and applied to thermodynamics to express the degree of confusion in the distribution of any kind of energy in space. The more chaotic an energy distribution is, the greater the entropy is. In 1948, Shannon [

It is assumed that a random test with

As a measure of the degree of chaos, entropy has the following properties:

Nonnegative:

Additivity: for independent events, the sum of entropies is equal to the entropy of sum

Extremum property: when the probability of the occurrence of all samples is equal, that is,

Asperity:

From the above properties, it is not difficult to find that the information entropy measures the uncertainty of the information. When the information entropy is larger, the uncertainty of the information is greater, and the utility value is smaller. In contrast, the smaller the information entropy is, the smaller the uncertainty of the information is, and the greater the utility value is.

In the investment portfolio research, many researchers use the information entropy to measure the degree of risk diversification in the portfolio, replace the probability of the sample appearance with the investment proportion of the assets in the portfolio, and obtain the measurement index of the degree of decentralization. The proportional entropy can be expressed as

According to the above analysis, this paper considers investors’ attitudes towards risk and their investment decisions on assets with background risk, takes the possible lower variance of the return on assets as the risk measure, measures the return on the basis of the mean probability of the return on assets, and uses the probability entropy as an effective tool for measuring the risk of the lower variance of the probability, which is all done to measure the degree of diversification of the asset portfolio. Therefore, the following probability mean-lower semivariance-entropy model with background risk and transaction costs considering the different risk attitudes of investors is established:

In the above two-objective programming problem (

Considering the conditional constraints of the model (

The differential evolution algorithm is similar to the genetic algorithm. Different from the genetic algorithm, it does not need to code and decode the feasible solutions. The initial population of the differential evolution algorithm is randomly generated, and the evolutionary population is formed by mutating, crossing, and selecting each individual in the population until the termination condition is satisfied.

The number of objective function variables in the differential evolution algorithm is the dimension

The evolution of the differential evolution algorithm [

Mutation operation. The mutation operation is carried out on the basis of the difference vector between the parent individuals. Let the currently evolved individual be

Cross-operation. The mutated individual

where

Selection operation [

The greedy selection strategy is used to select between the parent individual

To solve the above model, the differential evolution algorithm first generates the initial intermediate population. That is, it randomly initializes a group of intermediate particles in the feasible solution space and normalizes each intermediate particle to generate the initial population as follows:

The cross-probability is used to control individual

If the mutated individual is considered to be composed of only the random individuals, although it is advantageous to maintain the diversity of the population, the global search capability is strong, but the convergence speed is slow; and if the mutated individual only considers the

Step 1. Set the basic parameters including the population size

Step 2. Randomly generate the initial intermediate population for normalization operations and set the evolutionary algebra to

Step 3. Use formula (

Step 4. Judge whether the termination condition of the penalty function method is reached or the maximum number of iterations

Step 5. The mutation operation, cross-operation, and selection operation are performed according to equations (

Step 6. The evolutionary algebra to

The following examples will be used to illustrate the validity of the model. We assume that the return on assets of investors is a trapezoidal fuzzy number. Randomly select 5 stocks from the Shanghai Stock Exchange and estimate the probability distribution of trapezoidal fuzzy number of return on assets by analyzing the historical information of the relevant stocks [

The probability distribution of the rates of return of assets.

(0.0449, 0.0505, 0.0612, 0.0679) | |

(0.0447, 0.0502, 0.0608, 0.0675) | |

(0.1276, 0.1436, 0.1739, 0.1930) | |

(0.0466, 0.0524, 0.0635, 0.0705) | |

(0.0815, 0.0917, 0.1111, 0.1233) | |

(0.0400, 0.0450, 0.0545, 0.0605) |

We use the proposed differential evolution (DE) algorithm with random mutation and exponential increments to solve the model. The specific parameters of the algorithm are set as follows: a lot of experimental show that the population size

To solve this example, it is assumed that the return of the risk-free asset

From Tables

Portfolios, returns, and lower semivariances of risk-averse investors.

0.0405 | 0.0155 | 0.0025 | 0.0854 | 0.0969 | |

0.1287 | 0.0914 | 0.1334 | 0.0762 | 0.1187 | |

0.3133 | 0.3501 | 0.3662 | 0.3649 | 0.4062 | |

0.2048 | 0.2128 | 0.0767 | 0.1789 | 0.1618 | |

0.3126 | 0.3302 | 0.4212 | 0.2946 | 0.2163 | |

Return | 0.0953 | 0.1129 | 0.1198 | 0.1249 | 0.1260 |

Lower semivariance | 0.0040 | 0.0053 | 0.0059 | 0.0063 | 0.0067 |

Portfolios, returns, and lower semivariances of risk-neutral investors.

0.0159 | 0.0186 | 0.0893 | 0.0343 | 0.0577 | |

0.3018 | 0.1446 | 0.0560 | 0.1727 | 0.1087 | |

0.3689 | 0.3907 | 0.5883 | 0.3916 | 0.4267 | |

0.0372 | 0.1033 | 0.1900 | 0.1358 | 0.0333 | |

0.2762 | 0.3428 | 0.0764 | 0.2655 | 0.3735 | |

Return | 0.1011 | 0.1100 | 0.1149 | 0.1267 | 0.1327 |

Lower semivariance | 0.0144 | 0.0166 | 0.0182 | 0.0211 | 0.0233 |

Portfolios, returns, and lower semivariances of venture-seeking investors.

0.1610 | 0.2735 | 0.0042 | 0.0242 | 0.0254 | |

0.1025 | 0.0116 | 0.2412 | 0.2319 | 0.0096 | |

0.3431 | 0.3361 | 0.3619 | 0.2709 | 0.4505 | |

0.1114 | 0.0343 | 0.1438 | 0.1271 | 0.1810 | |

0.2819 | 0.3446 | 0.2490 | 0.3459 | 0.3335 | |

Return | 0.1219 | 0.1230 | 0.1261 | 0.1321 | 0.1470 |

Lower semivariance | 0.0628 | 0.0647 | 0.0673 | 0.0735 | 0.0864 |

According to Tables

The effective frontier of risk-averse investors with background risk.

The effective frontier of risk-neutral investors with background risk.

The effective frontier of risk-seeking investors with background risk.

Comparison of effective frontiers of investors under different risk attitudes with background risk.

As the risk attitude increases, so does the risk. Investors have different risk attitudes that affect their investment strategy choices. However, by comparing Tables

Table

Returns and lower semivariances of portfolios without background risk under different risk attitudes.

0.1021 | 0.1458 | 0.3093 | 0.0159 | 0.2609 | |

0.0208 | 0.0046 | 0.0587 | 0.3018 | 0.0162 | |

0.5447 | 0.4091 | 0.4348 | 0.3689 | 0.3130 | |

0.0349 | 0.2421 | 0.0172 | 0.0372 | 0.0347 | |

0.2974 | 0.1985 | 0.1800 | 0.2762 | 0.3752 | |

Return | 0.0661 | 0.0768 | 0.0825 | 0.0860 | 0.0976 |

Lower semivariance | 0.0007 | 0.0009 | 0.0010 | 0.0011 | 0.0014 |

0.0457 | 0.1792 | 0.0042 | 0.0343 | 0.0254 | |

0.1327 | 0.0276 | 0.2412 | 0.1727 | 0.0096 | |

0.3489 | 0.4417 | 0.3619 | 0.3916 | 0.4505 | |

0.2678 | 0.1045 | 0.1438 | 0.1358 | 0.1810 | |

0.2050 | 0.2469 | 0.2490 | 0.2655 | 0.3335 | |

Return | 0.0701 | 0.0723 | 0.0808 | 0.0832 | 0.1015 |

Lower semivariance | 0.0017 | 0.0018 | 0.0023 | 0.0024 | 0.0036 |

0.1717 | 0.0594 | 0.1610 | 0.0242 | 0.0178 | |

0.0862 | 0.0894 | 0.1025 | 0.2319 | 0.1987 | |

0.3849 | 0.4446 | 0.3431 | 0.2709 | 0.1669 | |

0.0204 | 0.0627 | 0.1114 | 0.1271 | 0.0398 | |

0.3368 | 0.3440 | 0.2819 | 0.3459 | 0.5768 | |

Return | 0.0726 | 0.0777 | 0.0845 | 0.0880 | 0.0954 |

Lower semivariance | 0.0035 | 0.0040 | 0.0049 | 0.0052 | 0.0060 |

Effective frontier for risk-averse investors without background risk.

Effective frontier for risk-neutral investors without background risk.

Effective frontier for risk-seeking investors without background risk.

Effective frontier comparison of investors under different risk attitudes without background risk.

Figure

Comparison of the effective frontiers with background risks under different risk attitudes.

Table

Comparison of the entropy factors in portfolios under different risk attitudes.

Entropy-containing | Return | 0.0953 | 0.1129 | 0.1198 | 0.1249 | 0.1260 | |

Lower semivariance | 0.0040 | 0.0053 | 0.0059 | 0.0063 | |||

No entropy | Return | 0.1148 | 0.1212 | 0.1231 | 0.1257 | 0.1276 | |

Lower semivariance | 0.0055 | 0.0060 | 0.0061 | 0.0065 | |||

Entropy-containing | Return | 0.1011 | 0.1100 | 0.1149 | 0.1267 | 0.1327 | |

Lower semivariance | 0.0144 | 0.0166 | 0.0182 | 0.0211 | |||

No entropy | Return | 0.1145 | 0.1178 | 0.1179 | 0.1279 | 0.1283 | |

Lower semivariance | 0.0178 | 0.0186 | 0.0189 | 0.0217 | |||

Entropy-containing | Return | 0.1219 | 0.1230 | 0.1261 | 0.1321 | 0.1470 | |

Lower semivariance | 0.0628 | 0.0647 | 0.0673 | 0.0735 | 0.0864 | ||

No entropy | Return | 0.1218 | 0.1245 | 0.1287 | 0.1485 | 0.1522 | |

Lower semivariance | 0.0642 | 0.0653 | 0.0698 | 0.0865 | 0.0894 |

For unconstrained optimization problem of the weight of

Table of relationship between

Risk attitude | Function value | |
---|---|---|

0.0868 | −0.5422 | |

0.2422 | −0.4125 | |

0.5345 | −0.1788 | |

0.7570 | −0.0959 | |

0.0136 | −0.5899 | |

0.3168 | −0.2547 | |

0.7060 | −0.1051 | |

0.8816 | −0.0348 | |

0.2088 | −0.3090 | |

0.4019 | −0.2091 | |

0.7975 | −0.0208 | |

0.9286 | 0.0300 |

Figure

In the financial market, investors have different perceptions of risk and different attitudes towards risk in the investment process. In this paper, the fuzzy portfolio problem under different risk attitudes is studied. We use the probability mean of the return on assets to measure the return and the lower semivariance to measure the risk. In addition, considering the different attitudes of investors to risk, background risk, and transaction costs, the probability entropy is used as an effective measure for the degree of diversification of an asset portfolio, and a probability mean-lower semivariance-entropy model is constructed. We use a differential evolution algorithm to solve the model and obtain five portfolio strategies under different risk attitudes. The effects of the risk attitude, background risk, and probability entropy on investors’ investment decisions are analyzed. Through the experimental results, it is found that the risk-averse investors avoid the risk, and the investors who like the risk seek the risk. Furthermore, the investment in assets with background risk will increase the total risk of the investors because the diversification effect of the entropy on the risk can make investors reduce the risks and improve the returns.

Five stocks are randomly selected from the Shanghai Stock Exchange, and the probability distribution of trapezoidal fuzzy number of return on assets is estimated by analyzing the historical information of the relevant stocks. The data in Table

The authors declare that they have no conflicts of interest.

This research was supported by the National Natural Science Foundation of China under Grant nos. 11961001 and 61561001, the Construction Project of First-Class Subjects in Ningxia Higher Education (NXYLXK2017B09), and the major proprietary funded project of North Minzu University (ZDZX201901).