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entropy
Article
A Risk-Free Protection Index Model for Portfolio
Selection with Entropy Constraint under
an Uncertainty Framework
Jianwei Gao * and Huicheng Liu
School of Economics and Management, North China Electric Power University, Beijing 102206, China;
lhuicheng@ncepu.edu.cn
*Correspondence: gaojianwei111@sina.com; Tel.: +86-10-6177-3151
Academic Editor: Kevin H. Knuth
Received: 21 December 2016; Accepted: 15 February 2017; Published: 21 February 2017
Abstract:
This paper aims to develop a risk-free protection index model for portfolio selection based
on the uncertain theory. First, the returns of risk assets are assumed as uncertain variables and subject
to reputable experts’ evaluations. Second, under this assumption, combining with the risk-free interest
rate we define a risk-free protection index (RFPI), which can measure the protection degree when the
loss of risk assets happens. Third, note that the proportion entropy serves as a complementary means
to reduce the risk by the preset diversification requirement. We put forward a risk-free protection
index model with an entropy constraint under an uncertainty framework by applying the RFPI,
Huang’s risk index model (RIM), and mean-variance-entropy model (MVEM). Furthermore, to solve
our portfolio model, an algorithm is given to estimate the uncertain expected return and standard
deviation of different risk assets by applying the Delphi method. Finally, an example is provided to
show that the risk-free protection index model performs better than the traditional MVEM and RIM.
Keywords: portfolio selection; risk free protection index; entropy constrain; uncertain variable
1. Introduction
Portfolio selection focuses on the optimal allocation of one’s wealth to obtain maximum profitable
return under minimum risk control. Since Markowitz [
1
] first proposed the classic mean-variance
model (MVM), many researchers have suggested new methods or elements to get numerous variants
of the MVM for portfolio selection (e.g., minimum-variance model [
2
], mean-variance-skewness
model [
3
], mean-semivariance model [
4
]). Their research can be regarded as the extension to the
classic portfolio theory which is based on probability and statistics theory. The security returns are
all assumed to be random variables and their expected value and variance are obtained from the
sample of available historical data. Considering the complexity of the security market in the real world,
the non-uniqueness of randomness as a kind of uncertainty and the lack of enough historical data
to reflect the future performances of security returns in some real life cases, many scholars began to
regard security returns as fuzzy variables which rely on experienced experts’ evaluations instead of
historical data. Thus, fuzzy portfolio optimization theory is developed and has been mainly studied
based on following three methods: (i) Fuzzy set theory [
5
]; (ii) Possibility measure [
6
,
7
]; (iii) Credibility
measure [8–10].
However, paradoxes arise when fuzzy variables are utilized to describe the subjective estimations
of security returns in the above three methods [
11
]. For instance, if a security return is regarded as
a fuzzy variable, then it can be characterized by a membership function. We suppose that a security
return is the triangular fuzzy variable
ξ= (−
0.1, 0.5, 1.1
)
(see Figure 1). Based on the membership
function, it is easy to obtain that
Pos{ξ=0.5}=
1 (or
Cr{ξ=0.5}=
0.5), which means that the
Entropy 2017,19, 80; doi:10.3390/e19020080 www.mdpi.com/journal/entropy
Entropy 2017,19, 80 2 of 12
security return is exactly 0.5 with belief degree 1 in possibility measure (or 0.5 in credibility measure).
However, this is unreasonable because the degree belief of exactly 0.5 should be almost zero.
In addition, we also get from the possibility theory that
Pos{ξ6=0.5}=Pos{ξ=0.5}=
1
(or
Cr{ξ6=0.5}=Cr{ξ=0.5}=
0.5. It implies that the two events of the return being exactly
0.5 and not being exactly 0.5 have the same degree belief both in possibility measure and credibility
measure, and they are equally likely to happen. This conclusion is contradictory and unacceptable to
our judgment.
Entropy 2017, 19, 80 2 of 12
membership function, it is easy to obtain that P s = 0.5} = 1ο{
ξ
(or Cr = 0.5} = 0.5{
ξ
),which means that
the security return is exactly 0.5 with belief degree 1 in possibility measure (or 0.5 in credibility
measure). However, this is unreasonable because the degree belief of exactly 0.5 should be almost
zero. In addition, we also get from the possibility theory that P s 0.5} = P s = 0.5} = 1ο{ξ≠ ο{ξ (or
Cr 0.5} = Cr = 0.5} = 0.5{
ξ
≠{
ξ
. It implies that the two events of the return being exactly 0.5 and not
being exactly 0.5 have the same degree belief both in possibility measure and credibility measure,
and they are equally likely to happen. This conclusion is contradictory and unacceptable to our
judgment.
Figure 1. Membership function of a security return
()
=0.1, 0.5, 1.1ξ −.
To deal with the above situation, Liu [12–15] proposed an uncertain measure and further
developed the uncertainty theory, which has been used in various areas (e.g., insurance, medical
care, environment and education) especially in the study of portfolio optimization [16–19]. Qin, et al.
[20] first studied mean-variance model in the uncertain environment. Zhu [21] considered a
continuous-time uncertain portfolio optimization problem. Liu and Qin [22] proposed a mean
semi-absolute deviation model for uncertain portfolio selection. Different from the above studies on
risk measurement, some scholars took the risk-free interest rate into consideration in the uncertain
portfolio optimization. Huang [23] first put forward a risk index model, Huang and Qiao [24]
modeled the multi-period problem, Huang and Ying [11] further considered the portfolio adjusting
problem. These studies proved that the above-mentioned paradoxes can be solved when the
uncertain variable is used to describe human imprecise estimations of security returns [11,24].
However, we find that these researchers usually focused on the weight of risk assets for
uncertain portfolio selection problem and ignored the protective screening function of risk-free
asset. As a result, the capital allocation is usually too centralized or decentralized. In this paper, we
study the portfolio selection problem under the framework of the uncertainty theory. In particular,
we extend the work of Huang, et al. [11,23,24] by proposing a risk-free protection index model with
entropy constraint for portfolio selection problem. Firstly, to introduce the protective screening
function of risk-free asset in guaranteeing the expected return of portfolio selection as the loss of risk
assets happens at a certain confidence level, we put forward a risk-free protection index (RFPI).
Secondly, considering that the Mean-variance selection framework without entropy constraint may
result in concentrative allocation, we further add proportion entropy constraint to the RFPI model to
meet the preset diversification requirement, which can prevent the concentrative allocation. Finally,
we propose a risk-free protection index model with proportion entropy constraint for portfolio
selection problem under uncertainty framework. The RFPI model can evaluate the protection made
by risk-free asset when the risk assets happen to lose at a certain confidence level, i.e., it can measure
the protective effect of risk-free asset on risk assets.
The rest of the paper is organized as follows: Section 2 introduces the knowledge about
uncertain variables and entropy constraint in finance. In Section 3, we first present RIM for uncertain
portfolio and the MVEM for diversified fuzzy portfolio. Then we further propose a risk-free
protection index model with entropy constraint in uncertainty environment and give an algorithm
to solve the portfolio selection model. Illustrative example is given in Section 4. Section 5 draws the
conclusion.
Figure 1. Membership function of a security return ξ=(−0.1, 0.5, 1.1).
To deal with the above situation, Liu [
12
–
15
] proposed an uncertain measure and further
developed the uncertainty theory, which has been used in various areas (e.g., insurance, medical
care, environment and education) especially in the study of portfolio optimization [
16
–
19
].
Q
in, et al. [20]
first studied mean-variance model in the uncertain environment. Zhu [
21
] considered
a continuous-time uncertain portfolio optimization problem. Liu and Qin [
22
] proposed a mean
semi-absolute deviation model for uncertain portfolio selection. Different from the above studies on
risk measurement, some scholars took the risk-free interest rate into consideration in the uncertain
portfolio optimization. Huang [
23
] first put forward a risk index model, Huang and Qiao [
24
] modeled
the multi-period problem, Huang and Ying [
11
] further considered the portfolio adjusting problem.
These studies proved that the above-mentioned paradoxes can be solved when the uncertain variable
is used to describe human imprecise estimations of security returns [11,24].
However, we find that these researchers usually focused on the weight of risk assets for uncertain
portfolio selection problem and ignored the protective screening function of risk-free asset. As a result,
the capital allocation is usually too centralized or decentralized. In this paper, we study the portfolio
selection problem under the framework of the uncertainty theory. In particular, we extend the work of
Huang, et al. [
11
,
23
,
24
] by proposing a risk-free protection index model with entropy constraint for
portfolio selection problem. Firstly, to introduce the protective screening function of risk-free asset in
guaranteeing the expected return of portfolio selection as the loss of risk assets happens at a certain
confidence level, we put forward a risk-free protection index (RFPI). Secondly, considering that the
Mean-variance selection framework without entropy constraint may result in concentrative allocation,
we further add proportion entropy constraint to the RFPI model to meet the preset diversification
requirement, which can prevent the concentrative allocation. Finally, we propose a risk-free protection
index model with proportion entropy constraint for portfolio selection problem under uncertainty
framework. The RFPI model can evaluate the protection made by risk-free asset when the risk assets
happen to lose at a certain confidence level, i.e., it can measure the protective effect of risk-free asset on
risk assets.
The rest of the paper is organized as follows: Section 2introduces the knowledge about uncertain
variables and entropy constraint in finance. In Section 3, we first present RIM for uncertain portfolio
and the MVEM for diversified fuzzy portfolio. Then we further propose a risk-free protection index
model with entropy constraint in uncertainty environment and give an algorithm to solve the portfolio
selection model. Illustrative example is given in Section 4. Section 5draws the conclusion.
Entropy 2017,19, 80 3 of 12
2. Knowledge about Uncertain Variables and the Entropy Constraint
To use uncertain variables to describe the security returns and consider the portfolio selection
problem with an entropy constraint, this section first introduces some necessary knowledge about
uncertain variables, and then presents an entropy constraint to finance.
2.1. The Expected Value, Variance and Distribution of an Uncertain Variable
Liu [
12
,
13
] presented an uncertain variable and an uncertain measure. We suppose that
Γ
is
a nonempty set,
ζ
is a
σ
-algebra over
Γ
, each element
Λ∈ζ
is an event, and
M{Λ}
is the occurrence
possibility measure of Λ. The function Mis called an uncertain measure if it satisfies four axioms:
(i)
Axiom 1: (Normality) M{Γ}=1;
(ii)
Axiom 2: (Self-duality) M{Λ}+M{Λc}=1;
(iii)
Axiom 3: (Countable subadditivity) M∞
∪
i=1
Λi≤
∞
∑
i=1
M{Λi}for every event {Λi};
(iv)
Axiom 4: (Product measure) Let
(Γk
,
ζk
,
Mk)
be uncertain spaces for
k=
1, 2,
···
,
n
, then product
uncertain measure is M=M1∧M2∧··· ∧ Mn.
Let
M
be an uncertain measure. The triplet
(Γ
,
ζ
,
M)
is called an uncertainty space. An uncertain
variable is a measurable function
ξ
:
(Γ,ζ,M)→ <
, i.e., the Borel set
B
:
{ξ∈B}={γ∈Γ|ξ(γ)∈B}
is an event. Then the uncertain distribution is defined as Φ(x) = M{ξ≤x},Φ:< → [0, 1].
The expected value is defined as
E[ξ]=Z∞
0M{ξ≥r}dr −Z0
−∞M{ξ≤r}dr. (1)
and its variance is defined as
V[ξ] = E[(ξ−E[ξ])2]. (2)
If ξand ηare independent uncertain variables with finite expected values, then it holds that
E[aξ+bη] = aE[ξ] + bE[η], for∀a,b∈ <. (3)
An uncertain variable ξis named as the normal uncertain variable if it satisfies
Φ(x)=1+expπ(e−x)
√3σ−1
,x∈ <, (4)
which is denoted by
N(e,σ)
with the mean value
e
and standard variance
σ
. The inverse function can
be written as [13]
Φ−1(α) = e+√3σ
πln α
1−α, (5)
where
α∈(
0, 1
)
and
Φ−1(α)
is the inverse function of
Φ
. Furthermore, we can calculate, for a linear
uncertain variable ξ~L(a,b)[23], the expected value is (a+b)/2 and variance is (b−a)2/12.
2.2. Entropy Constraint
Numerous portfolio selections operate problematically in practice [
25
]. Then, to avoid putting
excessive weights on only a few assets and reduce the impact of estimation error associated with
parameters of the moments of security returns, several diversity constraints have been introduced
and added to previous portfolio selection models. For example, Philippatos and Wilson [
26
] first used
entropy as a measurement of the uncertainty in portfolio selection. Usta and Kantar [
3
] presented
a mean-variance-skewness entropy measure for a multi-objective portfolio selection. Lin [
27
] put
forward a canonical form for diversity entropy constraint. Zhou et al. [
28
] introduced the application
Entropy 2017,19, 80 4 of 12
of entropy in finance. Zhou et al. [
29
] established a mean-variance hybrid-entropy model. Huang [
30
]
developed an entropy method to solve the diversified fuzzy portfolio problem. These studies imply
that entropy is a more general measure of risk than variance and it can be calculated from non-metric
data for it has nothing to do with the assumption of symmetric probability distributions [
26
]. This paper
will introduce Shannon’s entropy [3] in the portfolio selection constraints as follows.
Suppose that investment proportion in the i-th securities is denoted by
xi
(
xi≥
0,
for i=1, 2, ··· ,n). Then
H(x)=−
n
∑
i=1
xiln xi(6)
is named as the proportion entropy. Furthermore, it is obvious to see that
(Hmax =ln n,i f xi≡1
n,f or i =1, 2, ···n,
Hmin =0, i f xi=1or xj=0, j6=i,f or j =1, 2, ···n,
and the larger the absolute value (the greater the value of proportion entropy), the more diversely the
assets can be allocated to the alternative securities.
3. Risk-Free Protection Index Model with Entropy for an Uncertain Portfolio
In this section, we present the RIM for an uncertain portfolio and the MVEM for a diversified
fuzzy portfolio in Section 3.1. Furthermore, we propose a risk-free protection index model with
entropy constraint in uncertainty environment in Section 3.2 and give an algorithm to solve the model
in Section 3.3.
Figure 3. The relationship between RFPI and the expected return rates.
Figure 4. The relationship between RFPI and VaRU.
possible correlation effect among i-th securities in today’s highly related markets and the return
rates of i-th securities don’t completely subject to normal uncertain distribution. Thus, we can take
both the co-variance of a pair of assets in the model and abnormal uncertain distribution or other
kinds of distributions into account in the future research. In addition, we can extend our portfolio
selection problem to multi-objective portfolio problems and also add the crisp forms of the proposed
model in the further study.
Acknowledgments: The author acknowledges the support from Natural Science Foundation of China under
Grant Nos. 71271083, 71671064, and Humanities and Social Science Fund Major Project of Beijing under Grant
No. 15ZDA19 and Fundamental Research Funds for Central Universities under Grant No. 2016XS70.
Author Contributions: Jianwei Gao and Huicheng Liu conceived and designed the experiments. Huicheng Liu
performed the experiments and contributed analysis tools. Jianwei Gao wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
References
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Berlin/Heidelberg, Germany, 2010.
3. Usta, I.; Kantar, Y.M. Mean-variance-skewness-entropy measures: A multi-objective approach for
portfolio selection. Entropy 2011, 13, 117–133.
4. Grootveld, H.; Hallerbach, W. Variance vs. downside risk: Is there really that much difference? Eur. J.
Oper. Res. 1999, 114, 304–319.
