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Research Article

Research on Probability Mean-Lower Semivariance-Entropy

Portfolio Model with Background Risk

Qi Wu,

1

Yuelin Gao ,

1

,

2

and Ying Sun

2

1

Ningxia Province Key Laboratory of Intelligent Information and Data Processing, North Minzu University,

Yinchuan 750021, China

2

Ningxia Cooperative Innovation Center of Scientiﬁc Computing and Intelligent Information Processing,

North Minzu University, Yinchuan 750021, China

Correspondence should be addressed to Yuelin Gao; gaoyuelin@263.net

Received 29 March 2020; Accepted 5 June 2020; Published 17 July 2020

Guest Editor: Zaoli Yang

Copyright ©2020 Qi Wu et al. is is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. e

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.

1. Introduction

e 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 [1] put forward the classical

mean-variance (m-v) model in 1952, which provided the

theoretical basis for the portfolio problem and introduced

the era of quantitative analysis. e basic idea is to measure

returns via the expected return of an asset, to measure the

risk via the variance of the return of an asset, to minimize the

risk when the expected return of a portfolio is certain, or to

maximize the expected return of a portfolio when the risk

taken by the investor is certain. Many researchers, such as Yu

and Lee [2] and Shen et al. [3], have made many extensions

to this. e models mentioned above measure the risk using

the variance. Because the distribution of asset returns is

asymmetrical, using the variance to measure risk may sac-

riﬁce too much of the expected returns in the process of

eliminating the extreme low or high returns. To express or

measure the real investment risk in ﬁnancial market more

accurately, scholars have proposed new risk measurement

indicators that can be used instead of the variance. For

example, Petters and Dong [4] introduce the capital asset

pricing model (CAPM), the linear factor model and the

concepts of the value at risk, and the conditional value at risk

and related risk measurements. Kang and Li [5] propose a

uniﬁed framework to solve the distributed robust average

risk optimization problem, which uses the variance, VaR,

and CVaR as the three risk measures. Ma et al. [6] employ

the Lagrange dual method and the BSDE theory to tackle a

continuous-time m-v asset-liability management problem.

To explore the multiple heterogeneous relationships among

membership functions and criteria, a novel decision algo-

rithm is based on q-ROF set in literature [7] to deal with

these using interactive operators and Maclaurin symmetric

mean (MSM) operators.

Financial market risk is considered to be uncertain, and

Shape [8] is of the view that the uncertainty in ﬁnancial

markets cannot be predicted based on certainty. Risk, un-

certainty, and randomness are equal to each other. Proba-

bility theory is used to describe random uncertainty. Qin [9]

Hindawi

Mathematical Problems in Engineering

Volume 2020, Article ID 2769617, 13 pages

https://doi.org/10.1155/2020/2769617

ﬁrst gives a measure of the variance of portfolio returns,

veriﬁes it based on uncertainty theory, and then introduces

the corresponding mean-variance model. Zhang and Chen

[10] study the mean-variance portfolio selection problem

with system switching under the constraint of banning short

selling. However, a ﬁnancial market is an extremely complex

system, and the inﬂuence of human factors on investors’

decision-making in the investment process should not be

underestimated. Investors will make diﬀerent investment

decisions under the inﬂuence of social factors, psychological

factors, subjective will, and personal experience. In the light

of these factors, in 1970, Bellman and Zadeh [11] put for-

ward the theory of fuzzy decision-making. Tiryaki and Fang

et al. [12] describe the investor’s point of view with fuzzy

sets; construct two fuzzy Black–Litterman models with the

fuzzy view and the fuzzy random view, respectively; redeﬁne

the expected return and uncertainty matrix of view; and use

fuzzy methods to appropriately represent the view. It is

shown that the fuzzy method can better represent the in-

formation in the view and more accurately measure the

uncertainty. Tsaur [13] develops a fuzzy portfolio model that

focuses on diﬀerent investor risk attitudes and thus enables

fuzzy portfolio selection for investors with diﬀerent risk

attitudes. Yang et al. [14] proposed a deep-learning model,

and the sentiment dictionary is used to calculate sentiment

orientation, which is represented by the q-rung orthopair

fuzzy set. Due to the uncertainty eﬀects of COVID-19 and

limits of human cognition, Yang et al. [15] proposed a

decision support algorithm based on the novel concept of the

spherical normal fuzzy (SpNoF) set.

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 re-

searchers. 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 [16]

considers the credibility applying the mean-variance model

and mean-semivariance model to proportional entropy

fuzzy portfolios, as well as the clear form of the corre-

sponding model. Lassance and Vrins [17] accurately

quantify the uncertainty embedded in a distribution using a

target function that depends on the R´enyi entropy index and

considers the high-order moments. e portfolio generates a

number of minimum variance combinations that are su-

perior to the prior settings by minimizing the R´enyi entropy

in terms of the risk-working capital trade closure. Fang et al.

[18] consider the degree of diversiﬁcation of a portfolio. Lee

et al. [19] consider the limited control of total funds, such as

the total, risk, and liquidity, to achieve a distributed strategic

asset allocation with global constraints. Li et al. [20] discuss

fuzzy multiobjective dynamic portfolio optimization for

time-inconsistent investors, establish a model to

simultaneously maximize the cumulative combined objec-

tive function and minimize the cumulative portfolio vari-

ance, and design and propose a multiobjective dynamic

evolutionary algorithm as a possible solution to the pro-

posed model. A novel approach based on the genetic al-

gorithm (GA) for feature selection and parameter

optimization of support vector machine (SVM) is proposed

in literature [21]. Yang and Chang [22] proposed the in-

terval-valued Pythagorean normal fuzzy (IVPNF) sets by

introducing the NFN into IVPF environment.

e structure of the remainder of this paper is as follows.

In Section 2, the relevant concepts are given, including the

membership function of the trapezoid fuzzy number, the

mean probability value, the lower semivariance of the

probability, and so on. In Section 3, the establishment

process of the probability mean-lower semivariance-entropy

model is given. In Section 4, an improved diﬀerential

evolution algorithm is designed to solve the model. Section 5

uses Chinese stock market data to conduct the empirical

analysis, including the investors with diﬀerent risk attitudes.

It contains the two aspects of background risk and trans-

action costs and discusses the portfolio problem of how risk

attitudes and background risk aﬀect investors’ decision-

making. Finally, some conclusions are given in Section 6.

2. Preliminaries

Deﬁnition 1. Li [23] deﬁned a fuzzy set as

A∈F(U), for any

c∈[0,1]. en, fuzzy set

Awith a c-level set [

A]cis deﬁned

as

[

A]c�x|x∈U, μ

A(x)≥c

�a(c), a(c)

,(1)

where cis the conﬁdence level. a(c)and a(c)are denoted as

the left and right endpoints of the clevel set, respectively.

Deﬁnition 2. Li [23] deﬁned

A� (a, b, α,β)is a trapezoidal

fuzzy number. en, the membership function of the

trapezoidal fuzzy number with the risk attitude is deﬁned as

μ

A(x) �

1−a−x

α

k

, a −α≤x≤a,

1, a ≤x≤b,

1−x−b

β

k

, b ≤x≤b+β,

0,otherwise,

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

(2)

where kis a real number greater than 0, which is the ﬁtness

of the risk attitude. e [a,b] known as the trapezoidal fuzzy

number of peak area, αand β, respectively, is called

Awidth

of left and right, as shown in Figure 1. By ﬁnding the de-

rivative of the above membership function, we can know

that the smaller the kis, the more averse the investors are to

risk.

In view of the above formula of (2) of the trapezoidal

fuzzy number with the risk attitude and the deﬁnition of

2Mathematical Problems in Engineering

c-level set, the c-level set [23] of

Acan be obtained as

follows: [

A]c� [a, a] � [a−α(1−c)1/k, b +β(1−c)1/k].

e upper and lower mean probabilities of trapezoidal

fuzzy number

Awith the risk attitude [23] are, respectively,

denoted as

M+(

A) � 21

0

cb+β(1−c)1/k

dc�b+2k2β

(2k+1)(k+1),

(3)

M−(

A) � 21

0

ca−α(1−c)1/k

dc�a−2k2α

(2k+1)(k+1).

(4)

Via formulas (3) and (4), the mean value of the clear

probability of trapezoidal fuzzy number

Awith the risk

attitude [23] is denoted as

M(

A) � 1

2M−(

A) + M+(

A)

�a+b

2+k2(β−α)

(2k+1)(k+1).

(5)

e upper and lower semivariances of the probability of

trapezoidal fuzzy number

Awith the risk attitude are as

follows:

Var+(

A) � 21

0

c(M(

A) − a(c))2dc,(6)

Var−(

A) � 21

0

cM(

A) − a(c)

2dc.(7)

From the above formulas (6) and (7), the clear variance

of the probability of trapezoidal fuzzy number

Awith the

risk attitude [23] is denoted as

Var(

A) � 1

21

0

ca(c) − a(c)

2dc

�1

21

0

cb+β(1−c)1/k−a−α(1−c)1/k

2dc

�(b−a)2

4+k2(b−a)(α+β)

(2k+1)(k+1)+k2(α+β)2

4(k+1)(k+2).

(8)

3. Establishment of Model

e securities market is an extremely complex system. e

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. is

makes investors make investment decisions in vague envi-

ronments. For the convenience of the explanation, ﬁrst of all,

the relevant symbols involved in this paper are given as

follows:

ri� (ai, bi,αi,βi)represents the rate of return on ﬁ-

nancial risk asset i

rb� (ab, bb,αb,βb)represents the rate of return on

background risk asset b

cirepresents the unit transaction cost of asset i

X� (x1, x2,. . . , xn)represents the asset portfolio

xirepresents the investment proportion of risk asset i

krepresents the ﬁtness value of the risk attitude, where

kis a real number greater than 0

Cov(

ri,

rj)represents the possible covariance between

asset iand asset j

Cov(

ri,

rb)represents the possible covariance between

ﬁnancial risk asset iand background risk asset b

S(x)represents the likelihood entropy

Suppose there are nkinds of assets in the market, and the

rate of return

riof each asset is a trapezoidal fuzzy variable

ri� (ai, bi,αi,βi), i �1,2,. . . , n with the risk attitude. e

corresponding clevel set is denoted as

[

ri] � [ri1(c), ri2(c)] � [ai−αi(1−c)1/k, bi+βi(1−c)1/k].

e membership function of ﬁnancial risk assets with

risk attitude [23] is as follows:

μri(x) �

1−ai−x

αi

k

, ai−αi≤x≤ai,

1, ai≤x≤bi,

1−x−bi

βi

k

, bi≤x≤bi+βi,

0,otherwise.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

(9)

It is assumed that the return of the asset with background

risk

rbis also a trapezoidal fuzzy variable

rb� (ab, bb,αb,βb)

with the risk attitude and the corresponding clevel set is

denoted as [

rb] � [rb1(c), rb2(c)] � [ab−αb(1−c)1/k,

bb+βb(1−c)1/k]. e membership function [18] of the asset

with background risk is as follows:

µA

~(x)

1

0a − α a b b + β

x

Figure 1: Membership function of trapezoidal fuzzy number

A.

Mathematical Problems in Engineering 3

μrb(x) �

1−ab−x

αb

k

, ab−αb≤x≤ab,

1, ab≤x≤bb,

1−x−bb

βb

k

, bb≤x≤bb+βb,

0,otherwise.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

(10)

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 V-type function to represent it. ere-

fore, the total transaction cost of the portfolio is expressed

as

C�

n

i�1

cixi−x0

i

.(11)

us, the equation containing background risks and

transaction costs is expressed as follows:

R�

n

i�1

rixi+

rb−C. (12)

Because the linear combination of the trapezoidal fuzzy

numbers is a trapezoidal fuzzy number, n

i�1

rixi+

rb�

(n

i�1xiai+ab,n

i�1xibi+bb,n

i�1xiαi+αb,n

i�1xiβi+βb).

Via formulas (3)–(5), the mean value of the clear

probability of trapezoidal fuzzy number

Rwith the risk

attitude, background risk, and transaction cost is denoted

as

M(

R) � M

n

i�1

rixi+

rb−C

⎛

⎝⎞

⎠

�

n

i�1

xiM

ri

+M

rb

−cixi−x0

i

�

n

i�1

xi

ai+bi

2+k2βi−αi

(2k+1)(k+1)

+ab+bb

2

+k2βb−αb

(2k+1)(k+1)−cixi−x0

i

.

(13)

From formula (7), the lower semivariance of the clear

probability of the trapezoid fuzzy number

Ris expressed as

Var−(

R) � Var−

n

i�1

rixi+

rb−C

⎛

⎝⎞

⎠

�

n

i�1

x2

iVar−

ri

+2

n

i<j�1

xixjCov−

ri,

rj

+Var−

rb

+2

n

i�1

xiCov−

ri,

rb

�

n

i�1

x2

i

bi−ai

2

4+k2α2

i

(k+1)(k+2)+k2bi−ai

αi+βi

(2k+1)(k+1)+k4βi−αi

βi+3αi

[(2k+1)(k+1)]2

+

n

i<j�1

xixj

bi−ai

bj−aj

2+k2bi−ai

αj+βj

+bj−aj

αi+βi

(2k+1)(k+1)+2k2α2

i

(k+2)(k+1)

⎧

⎨

⎩

+2k4βi−αi

αj+βj

+2αiβj−αj

[(2k+1)(k+1)]2⎫

⎬

⎭

+bb−ab

2

4+k2α2

b

(k+1)(k+2)+k2bb−ab

βb+αb

(2k+1)(k+1)+k4βb−αb

βb+3αb

[(2k+1)(k+1)]2

+

n

i�1

xi

bi−ai

bb−ab

2+k2bi−ai

αb+βb

+bb−ab

αi+βi

(2k+1)(k+1)+2k2αiαb

(k+2)(k+1)

+2k4βi−αi

αb+βb

+2αiβb−αb

[(2k+1)(k+1)]2.

(14)

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

4Mathematical Problems in Engineering

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. erefore, 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 (11), (12). Assuming that the investor is

rational and short-selling is prohibited in the whole in-

vestment process, the following model is constructed:

min Var−(

R) � Var−

n

i�1

rixi+

rb−C

,

s.t. M

n

i�1

rixi+

rb

−

n

i�1

cixi−x0

i

≥r,

n

i�1

xi�1, xi≥0, i �1,2,..., n,

(15)

where rrepresents the minimum level of the investor’s net

return on the portfolio.

In recent years, according to information entropy the-

ory, many scholars use the proportional entropy as an index

to measure the degree of portfolio diversiﬁcation. e

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. e more chaotic an energy dis-

tribution is, the greater the entropy is. In 1948, Shannon [24]

ﬁrst introduced the concept of entropy into information

theory, which is deﬁned as follows.

It is assumed that a random test with nresults is per-

formed, and the discrete probability of each result is

pi(i�1,2,. . . , n). e information entropy is

S p1, p2,. . . , pn

� −

n

i�1

piln pi

,(16)

where pi(i�1,2,. . . , n)represents the probability of a

sample occurrence and it satisﬁes n

i�1pi�1.

As a measure of the degree of chaos, entropy has the

following properties:

(1) Nonnegative: S(p1, p2,. . . , pn) � − n

i�1piln(pi)≥0

(2) Additivity: for independent events, the sum of en-

tropies is equal to the entropy of sum

(3) Extremum property: when the probability of the

occurrence of all samples is equal, that is,

pi� (1/n)(i�1,2,..., n), its entropy reaches the

maximum, S(p1, p2,. . . , pn)≤S((1/n),(1/n),...,

(1/n)) � ln n

(4) Asperity: S(p1, p2,. . . , pn)is a symmetric concave

function for all variables

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. e proportional entropy can

be expressed as

En(x) � −

n

i�1

xiln xi,(17)

where xirepresents the investment proportion of asset

i(i�1,2,. . . n). From equation (16), when xi�1/n, the

value is the maximum; that is, the degree of decentralization

of the portfolio of assets is the largest. In real life, however,

investors will not allocate wealth to each asset in equal

proportions, which does not meet the investors’ investment

behaviour and when the rate of return

riof the assets is lower

than the rate of return rfof the risk-free assets, that is,

ri<rf(i�1,2,. . . , n); investors are not going to invest in

such assets, and their purpose is to reduce risk and increase

revenue through the moderate decentralization of their

assets. Based on the above analysis, in order to overcome the

deﬁciency of proportional entropy, Zhang et al. [25] con-

struct the probability entropy based on a decentralized

measurement index as follows:

S(x) � −

n

i�1

xiθxi

2ln ε+xiθxi

2

+1−xiθxi

2

ln 1 −xiθxi

2

,

(18)

where ε>0 is a suﬃciently small number, such as

ε�1.000e−7; max E(

ri−rf),0

/Var(

ri)represents the

proportion of remuneration ﬂuctuations in asset i;θ(xi) �

max E(

ri−rf),0

/ Var(

ri)/n

i�1max E(

ri−rf),0

/ Var

(

ri)} is the compensation factor of the investment pro-

portion of xi;E(

ri)and Var(

ri)represent the mean prob-

ability and variance of the probability of the i−th asset,

respectively; and rfrepresents the rate of return on risk-free

assets. Based on the probability entropy of the above de-

centralization measurement index, it can be seen that the

greater the proportion of the return ﬂuctuation of asset iis,

the greater the proportion of investment in asset iis. It is not

diﬃcult to ﬁnd that when E(

ri)<rf, the investment pro-

portion xi�0(i�1,2,. . . , n)of asset iis consistent with the

investment decisions of investors in practice.

According to the above analysis, this paper considers

investors’ attitudes towards risk and their investment de-

cisions 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. erefore, the following

probability mean-lower semivariance-entropy model with

background risk and transaction costs considering the dif-

ferent risk attitudes of investors is established:

Mathematical Problems in Engineering 5

min Var−(

R) � Var−

n

i�1

rixi+

rb−C

,

max S(x) � −

n

i�1

xiθxi

2ln ε+xiθxi

2

+1−xiθxi

2

ln 1 −xiθxi

2

s.t. M

n

i�1

rixi+

rb

−

n

i�1

cixi−x0

i

≥r

n

i�1

xi�1, xi≥0, i �1,2,..., n,

(19)

where rrepresents the minimum level of the investor’s net

return on the portfolio.

In the above two-objective programming problem (19),

there is no optimal solution in the strictest sense, and thus, it

is usually transformed into a single objective problem by

using the simple weighting method of the objective function.

e model is as follows:

min λVar−

n

i�1

rixi+

rb−C

+(1−λ)

n

i�1

xiθxi

2ln ε+xiθxi

2

+1−xiθxi

2

ln 1 −xiθxi

2

,

s.t. M

n

i�1

rixi+

rb

−

n

i�1

cixi−x0

i

≥r,

n

i�1

xi�1, xi≥0, i �1,2,..., n,

(20)

where λ∈[0,1]represents the preference coeﬃcient of the

investor. λ⟶0 indicates that investors pursue diverse

asset portfolios and tend to diversify their investment

strategies. λ⟶1 indicates that investors dislike diverse

asset portfolios.

Considering the conditional constraints of the model

(19), the problem (19) is transformed into the following

unconstrained optimization problem that is in the form of

penalty function:

min λVar−

n

i�1

rixi+

rb−C

⎛

⎝⎞

⎠+(1−λ)

n

i�1

xiθxi

2ln ε+xiθxi

2

+1−xiθxi

2

ln 1 −xiθxi

2

+Lp(x),(21)

where L>0 is called the penalty factor and it is a preset large

enough normal number. p(x) � (max 0, r −M(

rixi+

rb) +

n

i�1ci|xi−x0

i|})2+ (n

i�1xi−1)2is called the constraint

violation function, and it is easy to see that p(x)≥0.

4. Differential Evolution Algorithm

4.1. e Basic Principle of the Algorithm. e 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. e initial population

of the diﬀerential evolution algorithm is randomly gener-

ated, and the evolutionary population is formed by mu-

tating, crossing, and selecting each individual in the

population until the termination condition is satisﬁed.

e number of objective function variables in the dif-

ferential evolution algorithm is the dimension Dof the

algorithm’s search space, and NP is the size of the initial

intermediate population, which is generally set by scholars

according to the actual situation. e evolution operation of

the diﬀerential evolution algorithm is controlled by the

ﬁtness function. e ﬁtness function can be used to evaluate

6Mathematical Problems in Engineering

the relative value of the individual relative to the whole

population, and the ﬁtness function of this article is

expressed as

λVar−

n

i�1

rixi+

rb−C

⎛

⎝⎞

⎠+(1−λ)

n

i�1

xiθxi

2ln ε+xiθxi

2

+1−xiθxi

2

ln 1 −xiθxi

2

+Lp(x).(22)

e evolution of the diﬀerential evolution algorithm [26]

is as follows:

(1) Mutation operation. e mutation operation is

carried out on the basis of the diﬀerence vector

between the parent individuals. Let the currently

evolved individual be xt

i, where iis the serial number

of the current individual in the population and tis

the number of iterations. Randomly select three

individuals xt

r1,xt

r2, and xt

r3(r1≠r2≠r3≠i)from

the current population and add the vector diﬀerence

xt

r2−xt

r3between two individuals to xt

r1under the

action of scaling factor F. en, the individual vt+1

i

obtained from the mutation can be expressed as

vt+1

i�xt

r1+F xt

r2−xt

r3

.(23)

(2) Cross-operation. e mutated individual vt+1

iand the

current evolutionary individual xt

iof the population

intersect in a discrete crossing manner to generate

individual ut+1

iand the j−th component [26] of

individual ut+1

iis expressed as follows:

ut+1

i�vt+1

ij ,if rand ≤CR or j�randi(1, D),

xt

ij,otherwise,

⎧

⎨

⎩(24)

where rand is the uniformly distributed random

numbers over (0, 1) and randi(1, D)is a randomly

selected integer in 1,2,3,. . . D

{ }. To ensure that at

least one bit of ut+1

iis contributed by vt+1

iand

CR ∈[0,1]is the cross-probability, it is used to

control which variables in ut+1

iare contributed by

vt+1

iand xt

i. It can be seen that, with the increase in

the cross-probability CR, the contribution of vt+1

ito

ut+1

iis also increasing. When CR �1, ut+1

i�vt+1

i.

(3) Selection operation [27]:

xt+1

i�ut+1

i,if f ut+1

i

<f xt

i

,

xt

i,otherwise.

⎧

⎨

⎩(25)

e greedy selection strategy is used to select between

the parent individual xt

iand the experimental indi-

vidual ut+1

i, in which f(x)is the ﬁtness function, and

the individual with the best ﬁtness is selected. e

selection strategy of the diﬀerential evolution algorithm

is actually an elite reserve strategy.

4.2. Improvement of the Diﬀerential Evolutionary Algorithm

4.2.1. Normalization. To solve the above model, the dif-

ferential evolution algorithm ﬁrst generates the initial in-

termediate population. at 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:

xi�yi

n

i�1yi

,(26)

where n

i�1xi�1, i �1,2,..., n.

4.2.2. Exponential Increment Crossover Operator. e cross-

probability is used to control individual ut+1

i, which is

provided by the mutated individual vt+1

iand the current

individual xt

i. As the CR increases, the contribution of the

mutated individual vt+1

ito ut+1

iis greater; and when CR �1,

ut+1

i�vt+1

i. In this paper, the cross-probability factor with an

exponential increase in random iterations is used, and the

updated formula [27] is as follows:

CR �CRmin +CRmax −CRmin

∗exp −a∗1−t

Tmax

b

⎛

⎝⎞

⎠,

(27)

where a�40, b �4, Tmax is the maximum number of it-

erations, tis the current number of iterations, CRmin �0.1,

and CRmax �0.9. In this way, it can avoid the cross-factor is a

ﬁxed parameter that makes it fall into the local extreme

value, and it can well balance the global and local search

ability.

4.2.3. New Mutation Operation. 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 xt

best, although the local search capability

is strong and the accuracy is high, the algorithm will fall into

the local optimum. Combined with the characteristics of

these two methods, the eﬀects of a random individual xt

r1

and the optimal individual xt

best are simultaneously con-

sidered, the vector diﬀerence (xt

r2−xt

r3)of the two indi-

viduals is added to xt

r1and xt

best under the action of the

Mathematical Problems in Engineering 7

scaling factor F, and the mutated individual is obtained. e

mutation equation [28] selected in this paper is as follows:

vt+1

i�ηxt

r1+(1−η)xt

best +F xt

r2−xt

r3

,(28)

η�Tmax −t

Tmax

,(29)

F�η∗0.5+0.5,(30)

where η∈[0,1];Tmax is the maximum number of iterations;

tis the current number of iterations; and F∈[0,2]is a

scaling factor, which is used to control the scaling degree of

diﬀerence variables. It can be seen that Flinearly decreases as

tincreases. In the search process, ηgradually decreases from

1 to 0. at is, the weight of xt

r1to xt

best gradually increases,

which makes the population have better diversity in the early

stage and can ensure faster convergence speed and better

search accuracy in the later stage.

4.3. Speciﬁc Steps of the Algorithm

Step 1. Set the basic parameters including the pop-

ulation size NP, the contraction factor F, the maximum

number of iterations Tmax, the respective upper and

lower bounds CRmax and CRmin of the cross-probability

aand b, the penalty factor L, and the control error ε.

Step 2. Randomly generate the initial intermediate

population for normalization operations and set the

evolutionary algebra to t�1.

Step 3. Use formula (22) to calculate the ﬁtness value of

each individual, and the optimal ﬁtness value and the

optimal individual are obtained.

Step 4. Judge whether the termination condition of the

penalty function method is reached or the maximum

number of iterations Tmax is reached. If so, exit the

algorithm and output the optimal value; otherwise, the

diﬀerential evolution algorithm starts to iterate, and the

next step is carried out.

Step 5. e mutation operation, cross-operation, and

selection operation are performed according to equa-

tions (24), (25), and (27)–(30).

Step 6. e evolutionary algebra to t�t+1 and return

to step 3.

5. Empirical Analysis

e following examples will be used to illustrate the validity

of the model. We assume that the return on assets of in-

vestors 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 [23]. e related information is shown in

Table 1.

We use the proposed diﬀerential evolution (DE) algo-

rithm with random mutation and exponential increments to

solve the model. e speciﬁc parameters of the algorithm are

set as follows: a lot of experimental show that the population

size NP �30 and the maximum number of iterations Tmax �

200 can greatly reduce the running time of the algorithm. In

addition, the parameter CRmin �0.1, CRmax �0.9, a�40,

and b�4 in literature [27] enable the algorithm to better

balance the global and local search capability. Penalty factor

L�108and ε>0 is a suﬃciently small number, such as

ε�1.000e−7. All the tests are run in MATLAB 2015a, on an

Intel (R) Celeron (R) CPU G3900 @ 2.80 GHz, Windows 7.

To solve this example, it is assumed that the return of the

risk-free asset rfis 0.007, the transaction cost ratio of the

stock is 0.003, and the investors’ minimum net return re-

quirement for a portfolio is 0.085. en, the above intelligent

algorithm is used to solve the above model, and the in-

vestment strategy is obtained as follows.

From Tables 2–4, it can be seen that diﬀerent risk at-

titudes result in diﬀerent investment strategies. Investors

invest in the ﬁve kinds of assets and one kind of background

risk asset according to the risk attitude and the above ﬁve

investment strategies. e investment proportions in these

assets are also diﬀerent under the diﬀerent investment

strategies. When the risk attitude adaptation value k�0.5,

the proportion of stock 1 in the ﬁve investment strategies is

the smallest except for the fourth investment strategy, and

the proportion of stock 1 in the fourth investment strategy is

also very small. It can be seen that, regardless of the in-

vestment strategy, investors are very cautious about

investing in stock 1. However, the investment proportions in

the ﬁve investment strategies for stocks 1, 2, 4, and 5 are the

largest. When the risk attitude k�1.0, among the ﬁve in-

vestment strategies, stock 3 accounts for the largest in-

vestment proportion since investors prefer stock 3, and stock

1 accounts for the smallest proportion in investment

strategies 1, 2, and 4. When the risk attitude k�2.0, the

proportion of stock 5 in investment strategies 2 and 4 is the

largest, and compared with the other investment strategies,

stock 5 is also welcome by investors.

According to Tables 2–4, we can get that, with the in-

crease in the ﬁtness value of risk attitude, that is, investors’

attitude towards risk is changed from the aversion to

seeking, the lower semivariance is also increased, that is, the

risk is increasing, and the corresponding income is also

increasing. Figures 2–4 show the eﬀective frontiers of risk

averse, risk neutral, and risk-seeking investors when back-

ground risk is involved, and Figure 5 shows the comparison

diagram of the eﬀective frontiers of investors under the

diﬀerent risk attitudes with background risk. From Figure 5,

it can be intuitively seen that risk-averse investors avoid risk,

Table 1: e probability distribution of the rates of return of assets.

Assets Trapezoidal fuzzy number

Stock 1 (0.0449, 0.0505, 0.0612, 0.0679)

Stock 2 (0.0447, 0.0502, 0.0608, 0.0675)

Stock 3 (0.1276, 0.1436, 0.1739, 0.1930)

Stock 4 (0.0466, 0.0524, 0.0635, 0.0705)

Stock 5 (0.0815, 0.0917, 0.1111, 0.1233)

Asset with background risk (0.0400, 0.0450, 0.0545, 0.0605)

8Mathematical Problems in Engineering

and risk-loving investors seek risk because high risk is often

accompanied by high returns.

As the risk attitude increases, so does the risk. Investors

have diﬀerent risk attitudes that aﬀect their investment

strategy choices. However, by comparing Tables 2–4, we can

ﬁnd that, under the same risk attitude, the lower semi-

variance without background risk is much smaller than that

with background risk, and the corresponding return is much

smaller. Furthermore, the greater the background risk is, the

greater the risk that investors will bear, and so the impact of

background risk in the investment process cannot be

underestimated.

Table 5 shows the returns and lower semivariances of the

portfolios without background risk under diﬀerent risk

attitudes. Figures 6–8 show the eﬀective frontiers of risk

averse, risk neutral, and risk-seeking investors without

background risk under diﬀerent risk attitudes. Figure 9

compares the eﬀective frontiers of investors under diﬀer-

ent risk attitudes without background risk.

0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024

Lower semivariance

0.1

0.105

0.11

0.115

0.12

0.125

0.13

0.135

Expectation

Figure 3: e eﬀective frontier of risk-neutral investors with

background risk.

0.06 0.065 0.07 0.075 0.08 0.085 0.09

Lower semivariance

0.12

0.125

0.13

0.135

0.14

0.145

0.15

Expectation

Figure 4: e eﬀective frontier of risk-seeking investors with

background risk.

Table 4: Portfolios, returns, and lower semivariances of venture-

seeking investors.

Risk attitude Portfolio strategies

k�2.0

x10.1610 0.2735 0.0042 0.0242 0.0254

x20.1025 0.0116 0.2412 0.2319 0.0096

x30.3431 0.3361 0.3619 0.2709 0.4505

x40.1114 0.0343 0.1438 0.1271 0.1810

x50.2819 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

Table 3: Portfolios, returns, and lower semivariances of risk-

neutral investors.

Risk attitude Portfolio strategies

k�1.0

x10.0159 0.0186 0.0893 0.0343 0.0577

x20.3018 0.1446 0.0560 0.1727 0.1087

x30.3689 0.3907 0.5883 0.3916 0.4267

x40.0372 0.1033 0.1900 0.1358 0.0333

x50.2762 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

Table 2: Portfolios, returns, and lower semivariances of risk-averse

investors.

Risk attitude Portfolio strategies

k�0.5

x10.0405 0.0155 0.0025 0.0854 0.0969

x20.1287 0.0914 0.1334 0.0762 0.1187

x30.3133 0.3501 0.3662 0.3649 0.4062

x40.2048 0.2128 0.0767 0.1789 0.1618

x50.3126 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

4 4.5 5 5.5 6 6.5 7

Lower semivariance ×10–3

0.095

0.1

0.105

0.11

0.115

0.12

0.125

0.13

Expectation

Figure 2: e eﬀective frontier of risk-averse investors with

background risk.

Mathematical Problems in Engineering 9

Figure 10 shows the eﬀective frontier comparison dia-

gram with (without) background risk under diﬀerent risk

attitudes. It can be clearly and intuitively seen that the in-

vestment strategy with assets with background risk has high

risk, but it is also accompanied by high returns.

Table 5: Returns and lower semivariances of portfolios without

background risk under diﬀerent risk attitudes.

Risk attitude Portfolio strategies

k�0.5

x10.1021 0.1458 0.3093 0.0159 0.2609

x20.0208 0.0046 0.0587 0.3018 0.0162

x30.5447 0.4091 0.4348 0.3689 0.3130

x40.0349 0.2421 0.0172 0.0372 0.0347

x50.2974 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

k�1.0

x10.0457 0.1792 0.0042 0.0343 0.0254

x20.1327 0.0276 0.2412 0.1727 0.0096

x30.3489 0.4417 0.3619 0.3916 0.4505

x40.2678 0.1045 0.1438 0.1358 0.1810

x50.2050 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

k�2.0

x10.1717 0.0594 0.1610 0.0242 0.0178

x20.0862 0.0894 0.1025 0.2319 0.1987

x30.3849 0.4446 0.3431 0.2709 0.1669

x40.0204 0.0627 0.1114 0.1271 0.0398

x50.3368 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

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Lower semivariance

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

Expectation

×10–3

Figure 6: Eﬀective frontier for risk-averse investors without

background risk.

0 0.02 0.04 0.06 0.08 0.1

Lower semivariance

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Expectation

k = 0.5

k = 1.0

k = 2.0

Figure 5: Comparison of eﬀective frontiers of investors under

diﬀerent risk attitudes with background risk.

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

Lower semivariance

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

Expectation

×10–3

Figure 7: Eﬀective frontier for risk-neutral investors without

background risk.

Lower semivariance (%)

3.5 4 4.5 5 5.5 6

0.07

0.075

0.08

0.085

0.09

0.095

0.1

Expectation

×10–3

Figure 8: Eﬀective frontier for risk-seeking investors without

background risk.

10 Mathematical Problems in Engineering

Table 6 shows the comparison of the returns and the

lower semivariances under diﬀerent risk attitudes. It can be

seen that, under the same risk attitude, the lower semi-

variance of the portfolio without entropy is larger than the

lower semivariance of the portfolio with entropy, and the

corresponding return is higher than that with entropy.

However, when k�0.5, the last investment strategy has a

lower semivariance of entropy that is larger than the lower

0123456

Lower semivariance

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

Expectation

k = 0.5

k = 1.0

k = 2.0

×10–3

Figure 9: Eﬀective frontier comparison of investors under diﬀerent risk attitudes without background risk.

0 0.02 0.04 0.06 0.08 0.1

Lower semivariance

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Expectation

k = 0.5 including background risk

k = 1.0 including background risk

k = 2.0 including background risk

k = 0.5 not including background risk

k = 1.0 not including background risk

k = 2.0 not including background risk

Figure 10: Comparison of the eﬀective frontiers with background risks under diﬀerent risk attitudes.

Mathematical Problems in Engineering 11

semivariance without entropy; when k�2.0, the return of

the ﬁrst investment strategy with entropy is also larger than

that without entropy. Since the entropy is used to measure

the degree of decentralization of the portfolio and reduce the

risk, the entropy portfolio has a smaller risk.

For unconstrained optimization problem of the weight

of λvalues as shown in Table 7, we have done a lot of

experiments, respectively, to get λvalue and function value

under diﬀerent risk attitudes; it can be seen as the λvalue

increases, the function value will increase, under the con-

dition of risk attitude to adapt to the same value, when

λ⟶0, probability entropy contribution to the risk control;

when λ⟶1, the risk of the lower semivariance measure of

probability dominates.

Figure 11 more intuitively shows the changing trend of

weighting with diﬀerent values. When λ⟶0, investors

pursue diversiﬁed investment strategies and invest more

boldly. When λ⟶1, investors hate the diversity of

portfolios and invest more cautiously, which is also con-

sistent with the law of risk attitude.

6. Conclusions

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, con-

sidering the diﬀerent attitudes of investors to risk, back-

ground 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 semi-

variance-entropy model is constructed. We use a diﬀerential

evolution algorithm to solve the model and obtain ﬁve

portfolio strategies under diﬀerent risk attitudes. e eﬀects

of the risk attitude, background risk, and probability entropy

on investors’ investment decisions are analyzed. rough 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

Table 6: Comparison of the entropy factors in portfolios under diﬀerent risk attitudes.

Risk attitude Investment portfolios

k�0.5

Entropy-containing Return 0.0953 0.1129 0.1198 0.1249 0.1260

Lower semivariance 0.0040 0.0053 0.0059 0.0063 0.0067

No entropy Return 0.1148 0.1212 0.1231 0.1257 0.1276

Lower semivariance 0.0055 0.0060 0.0061 0.0065 0.0066

k�1.0

Entropy-containing Return 0.1011 0.1100 0.1149 0.1267 0.1327

Lower semivariance 0.0144 0.0166 0.0182 0.0211 0.0233

No entropy Return 0.1145 0.1178 0.1179 0.1279 0.1283

Lower semivariance 0.0178 0.0186 0.0189 0.0217 0.0224

k�2.0

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

Table 7: Table of relationship between lambda value and function

value under diﬀerent risk attitudes.

Risk attitude λFunction value

k�0.5

0.0868 −0.5422

0.2422 −0.4125

0.5345 −0.1788

0.7570 −0.0959

k�1.0

0.0136 −0.5899

0.3168 −0.2547

0.7060 −0.1051

0.8816 −0.0348

k�2.0

0.2088 −0.3090

0.4019 −0.2091

0.7975 −0.0208

0.9286 0.0300

0 0.2 0.4 0.6 0.8 1

Lamda value

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

Function value

k = 0.5

k = 1.0

k = 2.0

Figure 11: λcomparison chart under diﬀerent risk attitudes.

12 Mathematical Problems in Engineering

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.

Data Availability

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. e data in

Table 1 are selected from [23, 27]. All data and models

generated or used during the study are available within the

article.

Conflicts of Interest

e authors declare that they have no conﬂicts of interest.

Acknowledgments

is 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).

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