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Research on Probability Mean-Lower Semivariance-Entropy Portfolio Model with Background Risk

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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.
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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 􏽥
AF(U), for any
c[0,1]. en, fuzzy set 􏽥
Awith a c-level set [􏽥
A]cis defined
as
[􏽥
A]cx|xU, 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) �
1ax
α
􏼒 􏼓k
, a αxa,
1, a xb,
1xb
β
􏼠 􏼡k
, b xb+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] � [aα(1c)1/k, b +β(1c)1/k].
e upper and lower mean probabilities of trapezoidal
fuzzy number 􏽥
Awith the risk attitude [23] are, respectively,
denoted as
M+(􏽥
A) � 2􏽚1
0
cb+β(1c)1/k
􏽨 􏽩dcb+2k2β
(2k+1)(k+1),
(3)
M(􏽥
A) � 2􏽚1
0
caα(1c)1/k
􏽨 􏽩dca2k2α
(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(βα)
(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) � 2􏽚1
0
c(M(􏽥
A) − a(c))2dc,(6)
Var(􏽥
A) � 2􏽚1
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
2􏽚1
0
ca(c) − a(c)
􏼂 􏼃2dc
1
2􏽚1
0
cb+β(1c)1/kaα(1c)1/k
􏼐 􏼑􏽨 􏽩2dc
(ba)2
4+k2(ba)(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)] � [aiαi(1c)1/k, bi+βi(1c)1/k].
e membership function of financial risk assets with
risk attitude [23] is as follows:
μ􏽥ri(x) �
1aix
αi
􏼠 􏼡k
, aiαixai,
1, aixbi,
1xbi
βi
􏼠 􏼡k
, bixbi+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)] � [abαb(1c)1/k,
bb+βb(1c)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
μ􏽥rb(x) �
1abx
αb
􏼒 􏼓k
, abαbxab,
1, abxbb,
1xbb
βb
􏼒 􏼓k
, bbxbb+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
i1
cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌.(11)
us, the equation containing background risks and
transaction costs is expressed as follows:
􏽥
R􏽘
n
i1􏽥
rixi+􏽥
rbC. (12)
Because the linear combination of the trapezoidal fuzzy
numbers is a trapezoidal fuzzy number, 􏽐n
i1􏽥
rixi+􏽥
rb
(􏽐n
i1xiai+ab,􏽐n
i1xibi+bb,􏽐n
i1xiαi+αb,􏽐n
i1xiβi+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
i1􏽥
rixi+􏽥
rbC
􏽘
n
i1
xiM􏽥
ri
 􏼁+M􏽥
rb
 􏼁cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌
􏽘
n
i1
xi
ai+bi
2+k2βiαi
 􏼁
(2k+1)(k+1)
􏼠 􏼡+ab+bb
2
+k2βbαb
 􏼁
(2k+1)(k+1)cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌.
(13)
From formula (7), the lower semivariance of the clear
probability of the trapezoid fuzzy number 􏽥
Ris expressed as
Var(􏽥
R) � Var􏽘
n
i1􏽥
rixi+􏽥
rbC
􏽘
n
i1
x2
iVar􏽥
ri
 􏼁+2􏽘
n
i<j1
xixjCov􏽥
ri,􏽥
rj
􏼐 􏼑+Var􏽥
rb
 􏼁+2􏽘
n
i1
xiCov􏽥
ri,􏽥
rb
 􏼁
􏽘
n
i1
x2
i
biai
 􏼁2
4+k2α2
i
(k+1)(k+2)+k2biai
􏼁 αi+βi
 􏼁
(2k+1)(k+1)+k4βiαi
􏼁 βi+3αi
 􏼁
[(2k+1)(k+1)]2
􏼨 􏼩
+􏽘
n
i<j1
xixj
biai
􏼁 bjaj
􏼐 􏼑
2+k2biai
􏼁 αj+βj
􏼐 􏼑+bjaj
􏼐 􏼑 αi+βi
 􏼁􏽨 􏽩
(2k+1)(k+1)+2k2α2
i
(k+2)(k+1)
+2k4βiαi
􏼁 αj+βj
􏼐 􏼑+2αiβjαj
􏼐 􏼑􏽨 􏽩
[(2k+1)(k+1)]2
+bbab
 􏼁2
4+k2α2
b
(k+1)(k+2)+k2bbab
􏼁 βb+αb
 􏼁
(2k+1)(k+1)+k4βbαb
􏼁 βb+3αb
 􏼁
[(2k+1)(k+1)]2
+􏽘
n
i1
xi
biai
􏼁 bbab
 􏼁
2+k2biai
􏼁 αb+βb
 􏼁+bbab
􏼁 αi+βi
 􏼁􏼂 􏼃
(2k+1)(k+1)+2k2αiαb
(k+2)(k+1)
􏼨
+2k4βiαi
􏼁 αb+βb
 􏼁+2αiβbαb
 􏼁􏼂 􏼃
[(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
i1􏽥
rixi+􏽥
rbC
􏼠 􏼡,
s.t. M 􏽐
n
i1􏽥
rixi+􏽥
rb
􏼠 􏼡􏽐
n
i1
cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌r,
􏽐
n
i1
xi1, xi0, 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(i1,2,. . . , n). e information entropy is
S p1, p2,. . . , pn
 􏼁 􏽘
n
i1
piln pi
 􏼁,(16)
where pi(i1,2,. . . , n)represents the probability of a
sample occurrence and it satisfies 􏽐n
i1pi1.
As a measure of the degree of chaos, entropy has the
following properties:
(1) Nonnegative: S(p1, p2,. . . , pn) � − 􏽐n
i1piln(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)(i1,2,..., 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
i1
xiln xi,(17)
where xirepresents the investment proportion of asset
i(i1,2,. . . n). From equation (16), when xi1/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(i1,2,. . . , 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
i1
xiθxi
 􏼁
2ln ε+xiθxi
 􏼁
2
􏼠 􏼡􏼢
+1xiθxi
 􏼁
2
􏼠 􏼡ln 1 xiθxi
 􏼁
2
􏼠 􏼡􏼣,
(18)
where ε>0 is a sufficiently small number, such as
ε1.000e7; max E(􏽥
rirf),0
􏽮 􏽯/Var(􏽥
ri)represents the
proportion of remuneration fluctuations in asset i;θ(xi) �
max E(􏽥
rirf),0
􏽮 􏽯/ Var(􏽥
ri)/􏽐n
i1max E(􏽥
rirf),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 ith 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 xi0(i1,2,. . . , 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
i1􏽥
rixi+􏽥
rbC
􏼠 􏼡,
max S(x) � − 􏽐
n
i1
xiθxi
 􏼁
2ln ε+xiθxi
 􏼁
2
􏼠 􏼡+1xiθxi
 􏼁
2
􏼠 􏼡ln 1 xiθxi
 􏼁
2
􏼠 􏼡􏼢 􏼣
s.t. M 􏽐
n
i1􏽥
rixi+􏽥
rb
􏼠 􏼡􏽐
n
i1
cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌r
􏽐
n
i1
xi1, xi0, 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
i1􏽥
rixi+􏽥
rbC
􏼠 􏼡+(1λ)􏽐
n
i1
xiθxi
 􏼁
2ln ε+xiθxi
 􏼁
2
􏼠 􏼡+1xiθxi
 􏼁
2
􏼠 􏼡ln 1 xiθxi
 􏼁
2
􏼠 􏼡􏼢 􏼣,
s.t. M 􏽐
n
i1􏽥
rixi+􏽥
rb
􏼠 􏼡􏽐
n
i1
cixix0
i
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌r,
􏽐
n
i1
xi1, xi0, i 1,2,..., n,
(20)
where λ[0,1]represents the preference coefficient of the
investor. λ0 indicates that investors pursue diverse
asset portfolios and tend to diversify their investment
strategies. λ1 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
i1􏽥
rixi+􏽥
rbC
+(1λ)􏽘
n
i1
xiθxi
 􏼁
2ln ε+xiθxi
 􏼁
2
􏼠 􏼡+1xiθxi
 􏼁
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
i1ci|xix0
i|})2+ (􏽐n
i1xi1)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
i1􏽥
rixi+􏽥
rbC
+(1λ)􏽘
n
i1
xiθxi
 􏼁
2ln ε+xiθxi
 􏼁
2
􏼠 􏼡+1xiθxi
 􏼁
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(r1r2r3i)from
the current population and add the vector difference
xt
r2xt
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
ixt
r1+F xt
r2xt
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 jth component [26] of
individual ut+1
iis expressed as follows:
ut+1
ivt+1
ij ,if rand CR or jrandi(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
ivt+1
i.
(3) Selection operation [27]:
xt+1
iut+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:
xiyi
􏽐n
i1yi
,(26)
where 􏽐n
i1xi1, 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
ivt+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 a1t
Tmax
􏼠 􏼡b
,
(27)
where a40, 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
r2xt
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
iηxt
r1+(1η)xt
best +F xt
r2xt
r3
􏼐 􏼑,(28)
ηTmax t
 􏼁
Tmax
,(29)
Fη0.5+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 t1.
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 tt+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, a40,
and b4 in literature [27] enable the algorithm to better
balance the global and local search capability. Penalty factor
L108and ε>0 is a sufficiently small number, such as
ε1.000e7. 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 k0.5,
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 k1.0, 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 k2.0, 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.
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
k2.0
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
k1.0
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
k0.5
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
k0.5
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
k1.0
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
k2.0
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 k0.5, 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 k2.0, 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
λ0, probability entropy contribution to the risk control;
when λ1, the risk of the lower semivariance measure of
probability dominates.
Figure 11 more intuitively shows the changing trend of
weighting with different values. When λ0, investors
pursue diversified investment strategies and invest more
boldly. When λ1, 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
k0.5
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
k1.0
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
k2.0
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
k0.5
0.0868 0.5422
0.2422 0.4125
0.5345 0.1788
0.7570 0.0959
k1.0
0.0136 0.5899
0.3168 0.2547
0.7060 0.1051
0.8816 0.0348
k2.0
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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Mathematical Problems in Engineering 13
... Additionally, this risk measurement only considers the extent to which actual returns deviate from expected returns, whereas true losses refer to fluctuations below the mean of returns [3][4][5][6][7][8]. In order to be more in line with social reality, mean-semivariance portfolio models have been proposed and are widely used [9][10][11][12]. ...
... is the mean return of asset in the targeted period; is the mean return of the portfolio in the targeted period. Equation (9) is the objective function of the model and represents minimizing the risk of the portfolio (the lower half of the variance); Equation (10) ensures that the return of the portfolio is greater than the investor's expected return ; and Equations (11) and (12) indicate that the variables take values in the range [0, 1], and the total investment ratio is 1. ...
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