International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(1): 35-42
Published online March 25, 2020
https://doi.org/10.5391/IJFIS.2020.20.1.35
© The Korean Institute of Intelligent Systems
Alireza Sajedi^{1} and Gholamhossein Yari^{2}
^{1}Department of Statistics, Islamic Azad University, Science and Research Branch, Tehran, Iran
^{2}Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
Correspondence to :
Gholamhossein Yari (Yari@iust.ac.ir)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, we proposed definition of order
Keywords: Uncertain variable, Order 𝑣 entropy, Order 𝑣 cross entropy, Portfolio selection
The purpose of portfolio selection is to allocate optimal assets to purchase stocks. In general, to achieve this goal, returns are considered as random variables. The uncertainty variables are used as returns in the absence of historical data. Accordingly, the use of entropy in portfolio selection is considered for measuring the uncertainty of the uncertain variables. As an application of this concept, we can consider the generalized entropy maximization model and cross entropy minimization model under the uncertainty environment.
Portfolio selection is a very extensive topic in finance, which has been introduced to this field in the 1950’s, and the interest in it does not seem to be subsiding. Markowitz [1] was the first scientist to introduce the modern portfolio selection theory. Later, many researchers revised or developed the model with new methods or elements to improve the results. A typical extension was suggested by Philippatos and Wilson [2], who employed entropy to measure the risk of the portfolio selection. After that, several important works were published in this topic, for example see Zhou et al. [3] and Yu et al. [4].
Fuzzy entropy is a way to characterize the uncertainty on the possible values of fuzzy variables, which has been studied by many researchers such as Bhandari and Pal [5], De Luca and Termini [6], and Liu [7]. Within the framework of credibility theory, Li and Liu [8] presented an entropy for fuzzy variable. Li and Liu [9] proposed the maximum entropy. Based on the concept of fuzzy entropy, Li et al. [10] proposed the maximum optimization model by minimizing the uncertainty of the fuzzy objective under certain expected constraints. Further, Qin et al. [11] established credibilistic cross-entropy minimization models for portfolio optimization with fuzzy returns in the framework of credibility theory. Bhattacharyya et al. [12] introduced the cross-entropy, mean, variance, skewness model. Cross entropy was used to quantify the level of dispersion for the fuzzy return. Yari et al. [13] presented the Renyi entropy-mean-variance maximization and Renyi cross entropy-mean-variance minimization models for portfolio selection with fuzzy return under the credibility theory framework. Kar et al. [14] introduced a fuzzy bi-objectives portfolio model with objectives “fuzzy VaR ratio” and “fuzzy Sharpe ratio”. They tested the performance of the model with different evolutionary algorithms.
In the situations where historical data is not available, another feasible way is to estimate returns using expert opinion based on their subjective evaluation under the uncertainty theory (Liu [15]). Recently, portfolio selection problems have been studied under uncertainty conditions. By using the uncertainty theory, several researchers, including Qin et al. [11], who formulated the uncertain counterpart of mean-variance model, Liu and Qin [16], Huang and Qiao [17], Yao and Ji [18], took the security returns as uncertain variables. In order to measure the uncertainty of a variable, entropy was provided by Liu [19] under the uncertainty theory. Subsequently, some properties of entropy for uncertain variables were investigated by Dai and Chen [20]. In order to address the divergence of uncertain variables via uncertain distribution, Chen et al. [21] proposed cross entropy of uncertain variables. Bhattacharyya et al. [22] showed that their developed mean-entropy-skewness model in an uncertain environment is more effective in comparison with other earlier proposed models.
This paper focuses on the portfolio optimization with uncertain returns subject to expert evaluation. Section 2 reviews some important concepts and measures of uncertain variables and explain their implications in portfolio selection. Section 3 proposes order
The uncertainty theory, introduced by Liu [15] is a branch of mathematics that studies the behavior of human uncertainty. In this section, we review some basic concepts about uncertain measures and uncertain variables, which are related with this paper.
Let Γ be a nonempty set and be a
If
where
In uncertainty theory, Liu and Liu [24] defined the expected value and variance of
Provided that at least one of the two integrals is finite.
If
where
Let
where
Let
where
where Φ
The entropy concept was introduced by Shanon [25] and Renyi [26] defined the order
Suppose that
where
The order
Let
Considering
By the
Cross entropy was first proposed by Kullback and Leibler [27] to measure the difference between two probability distribution. Renyi [26] defined the order
Let
where
It is easy to verify that
The order
Let
Considering
In particular if the uncertainty distributions of
In this section, Kapur and Kesavan [28] entropy maximization and cross-entropy minimization model is extended to the portfolio optimization with uncertain returns. Let
where
Suppose that each security return is a zigzag variable denoted by
Note that in uncertain environment,
Since the cross entropy is a common method for measuring the degree of divergence of uncertain variables, we formulate different cross entropy minimization model for portfolio optimization. Suppose that
Suppose that each security return is a zigzag variable denoted by
The theorem can be easily proved taking into account the relations used in proving Theorem 3. Solving the integral using numeric methods is possible for different values of
Tables 1
In the present study, we compared the applicability of two models, order
No potential conflict of interest relevant to this article was reported.
Table 1. Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Table 2. Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Table 3. Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
E-mail: alireza.sajedi@srbiau.ac.ir
E-mail: Yari@iust.ac.ir
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(1): 35-42
Published online March 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.1.35
Copyright © The Korean Institute of Intelligent Systems.
Alireza Sajedi^{1} and Gholamhossein Yari^{2}
^{1}Department of Statistics, Islamic Azad University, Science and Research Branch, Tehran, Iran
^{2}Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
Correspondence to:Gholamhossein Yari (Yari@iust.ac.ir)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, we proposed definition of order
Keywords: Uncertain variable, Order 𝑣, entropy, Order 𝑣, cross entropy, Portfolio selection
The purpose of portfolio selection is to allocate optimal assets to purchase stocks. In general, to achieve this goal, returns are considered as random variables. The uncertainty variables are used as returns in the absence of historical data. Accordingly, the use of entropy in portfolio selection is considered for measuring the uncertainty of the uncertain variables. As an application of this concept, we can consider the generalized entropy maximization model and cross entropy minimization model under the uncertainty environment.
Portfolio selection is a very extensive topic in finance, which has been introduced to this field in the 1950’s, and the interest in it does not seem to be subsiding. Markowitz [1] was the first scientist to introduce the modern portfolio selection theory. Later, many researchers revised or developed the model with new methods or elements to improve the results. A typical extension was suggested by Philippatos and Wilson [2], who employed entropy to measure the risk of the portfolio selection. After that, several important works were published in this topic, for example see Zhou et al. [3] and Yu et al. [4].
Fuzzy entropy is a way to characterize the uncertainty on the possible values of fuzzy variables, which has been studied by many researchers such as Bhandari and Pal [5], De Luca and Termini [6], and Liu [7]. Within the framework of credibility theory, Li and Liu [8] presented an entropy for fuzzy variable. Li and Liu [9] proposed the maximum entropy. Based on the concept of fuzzy entropy, Li et al. [10] proposed the maximum optimization model by minimizing the uncertainty of the fuzzy objective under certain expected constraints. Further, Qin et al. [11] established credibilistic cross-entropy minimization models for portfolio optimization with fuzzy returns in the framework of credibility theory. Bhattacharyya et al. [12] introduced the cross-entropy, mean, variance, skewness model. Cross entropy was used to quantify the level of dispersion for the fuzzy return. Yari et al. [13] presented the Renyi entropy-mean-variance maximization and Renyi cross entropy-mean-variance minimization models for portfolio selection with fuzzy return under the credibility theory framework. Kar et al. [14] introduced a fuzzy bi-objectives portfolio model with objectives “fuzzy VaR ratio” and “fuzzy Sharpe ratio”. They tested the performance of the model with different evolutionary algorithms.
In the situations where historical data is not available, another feasible way is to estimate returns using expert opinion based on their subjective evaluation under the uncertainty theory (Liu [15]). Recently, portfolio selection problems have been studied under uncertainty conditions. By using the uncertainty theory, several researchers, including Qin et al. [11], who formulated the uncertain counterpart of mean-variance model, Liu and Qin [16], Huang and Qiao [17], Yao and Ji [18], took the security returns as uncertain variables. In order to measure the uncertainty of a variable, entropy was provided by Liu [19] under the uncertainty theory. Subsequently, some properties of entropy for uncertain variables were investigated by Dai and Chen [20]. In order to address the divergence of uncertain variables via uncertain distribution, Chen et al. [21] proposed cross entropy of uncertain variables. Bhattacharyya et al. [22] showed that their developed mean-entropy-skewness model in an uncertain environment is more effective in comparison with other earlier proposed models.
This paper focuses on the portfolio optimization with uncertain returns subject to expert evaluation. Section 2 reviews some important concepts and measures of uncertain variables and explain their implications in portfolio selection. Section 3 proposes order
The uncertainty theory, introduced by Liu [15] is a branch of mathematics that studies the behavior of human uncertainty. In this section, we review some basic concepts about uncertain measures and uncertain variables, which are related with this paper.
Let Γ be a nonempty set and be a
If
where
In uncertainty theory, Liu and Liu [24] defined the expected value and variance of
Provided that at least one of the two integrals is finite.
If
where
Let
where
Let
where
where Φ
The entropy concept was introduced by Shanon [25] and Renyi [26] defined the order
Suppose that
where
The order
Let
Considering
By the
Cross entropy was first proposed by Kullback and Leibler [27] to measure the difference between two probability distribution. Renyi [26] defined the order
Let
where
It is easy to verify that
The order
Let
Considering
In particular if the uncertainty distributions of
In this section, Kapur and Kesavan [28] entropy maximization and cross-entropy minimization model is extended to the portfolio optimization with uncertain returns. Let
where
Suppose that each security return is a zigzag variable denoted by
Note that in uncertain environment,
Since the cross entropy is a common method for measuring the degree of divergence of uncertain variables, we formulate different cross entropy minimization model for portfolio optimization. Suppose that
Suppose that each security return is a zigzag variable denoted by
The theorem can be easily proved taking into account the relations used in proving Theorem 3. Solving the integral using numeric methods is possible for different values of
Tables 1
In the present study, we compared the applicability of two models, order
No potential conflict of interest relevant to this article was reported.
Table 1 . Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Table 2 . Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Table 3 . Investment proportion of 5 securities (%) with
No. | Return | Allocation |
---|---|---|
(−0.2,0.5,0.9) | ||
(−0.3,0.6,1.0) | ||
(−0.1,0.3,0.8) | ||
(−0.2,0.3.1.0) | ||
(−0.3,0.5,0.7) |
Seyyed Hamed Abtahi, Gholamhossein Yari, Farhad Hosseinzadeh Lotfi, and Rahman Farnoosh
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 38-48 https://doi.org/10.5391/IJFIS.2021.21.1.38Gholanhossein Yari, Alireza Sajedi, and Mohamadtaghi Rahimi
Int. J. Fuzzy Log. Intell. Syst. 2018; 18(1): 78-83 https://doi.org/10.5391/IJFIS.2018.18.1.78