International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 38-48
Published online March 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.1.38
© The Korean Institute of Intelligent Systems
Seyyed Hamed Abtahi^{1}, Gholamhossein Yari^{2}, Farhad Hosseinzadeh Lotfi^{1}, and Rahman Farnoosh^{2}
^{1}Department of Statistics, Science and Research Branch, Islamic Azad University, 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.
Empirical studies illustrate that in numerous cases the returns of securities are not normally distributed. In this paper, skew-normal uncertainty distribution is proposed to capture skewness in the portfolio selection problem. Furthermore, the concept of asymmetric entropy for uncertain variables as the quantifier of diversification is presented and its mathematical properties such as translation invariance and positive linearity are studied. To examine the effect of asymmetric entropy parameter on portfolio diversification, a mean-CVaR-entropy portfolio selection problem is presented based on asymmetric entropy with different parameter values and logarithm entropy. A non-dominated sorting genetic algorithm II (NSGA-II) is implemented in MATLAB to solve the corresponding problem. Numerical results show that asymmetric entropy for a specific parameter value will outperform logarithm entropy in portfolio diversification.
Keywords: Uncertain variable, Skew-normal uncertainty distribution, Asymmetric entropy, Mean-CVaR-entropy portfolio selection model, Non-dominated sorting genetic algorithms II (NSGA-II)
The financial crisis of 2007–2008 showed that rare and unpredictable events can challenge conventional ideas about portfolio construction. However, we do not know in advance when such events will occur and how traumatic they can be, we can modify current risk management frameworks to better manage these rare and dangerous events.
The prime reason for the occurrence of such anomalies lies in the conventional approach to applying mean-variance model which was introduced by Markowitz [1] in 1952. The basic assumption in the mean-variance model and many other models is that future returns will be independent and normally distributed. Sheikh and Qiao [2] declared that in many cases it can be empirically observed that returns are not normally distributed and under non-normality, variance becomes inefficient as the quantifier of portfolio risk.
Most of studies on portfolio selection have been down based on only the first two moments of return distributions. However, there has always been a contentious issue whether higher moments should be considered in portfolio selection. Arditti [3] and Arditti and Levy [4] declared that higher moments should be considered for non-normal returns.
Skewness as the third central moment was defined to measure asymmetry of return distributions. Chunhachinda et al. [5] studied international stock market and confirmed that higher moments cannot be neglected in the portfolio selection. Konno et al. [6] constructed a portfolio optimization model by applying skewness. Keel et al. [7] showed that presence of skewness has a significant influence on portfolio selection. Furthermore, to better capture skewness, some scholars applied different distributions in portfolio optimization. Hu and Kercheval [8] argued that t and skewed t distribution show a much better fit to real data than the normal distribution.
Several portfolio selection models are developed by considering more efficient risk measure such as value at risk (VaR), conditional value at risk (CVaR) and entropy. VaR better reflects extreme events, but it does not aggregate risk in the sense of being sub-additive on portfolios [9]. CVaR is a coherent risk measure which is a superior to VaR [10]. VaR and CVaR has been widely applied by researchers [11–16].
Entropy of random variables first proposed by Shannon [17] in logarithm form. The study carried out by Philippatos and Wilson [18] was the pioneering work to associate entropy with a measure of risk in portfolio selection. They argued that entropy is more general and better suited for the selection of portfolios than variance. Furthermore, Simonelli [19] showed that entropy as a measure of risk is better than variance in wealth allocation.
According to the knowledge of securities investment, when the investment portfolio is more diversified, the risk of investment will decline. Entropy as a measure of diversification, originally developed by Jacquemin and Berry [20]. Entropy is a widely accepted measure of diversity [21] and has major advantages in the analysis of portfolio diversification [20]. It is well-established that the greater the level of entropy, the higher the degree of portfolio diversification. Bera and Park [22] constructed a well-diversified portfolio based on maximum entropy principle.
In the mentioned literatures it is assumed that the security returns are random variable with probability distribution. The fundamental assumption for using probability theory in the portfolio selection is that the probability distribution of security returns is similar to the past one and close enough to frequencies. However, it is difficult to ensure this assumption. In financial businesses, sometimes we have historical data scarcity. Therefore, we ask domain experts to evaluate the belief degree that each event will happen. Using fuzzy set theory is a way to handle portfolio optimization problems with returns given by experts’ evaluations. Liu [23] presented a counterexample to show that modeling belief degree which uses subjective probability may lead to counterintuitive results. Furthermore, Liu [24] showed that fuzzy set theory is not self-consistent in mathematics and may lead to wrong results in practice. The main mistake of fuzzy set theory is based on the wrong assumption that the belief degree of a union of events is the maximum of the belief degrees of the individual events no matter if they are independent or not.
In order to better deal with belief degree, uncertainty theory was founded by Liu [25] in 2007. Uncertainty theory has been employed in the portfolio optimization researches. Zhai et al. [26] formulated an uncertain multi-objective portfolio optimization considering skewness. Bhattacharyya et al. [27] studied effects of skewness on the portfolio selection. Yan [28] presented mean-VaR optimization model under uncertainty. Ning et al. [29] proposed mean-TVaR model under uncertainty framework.
Liu [30] proposed a concept of entropy for uncertain variables in the form of logarithm function. Chen and Dia [31] proposed the maximum entropy principle for uncertain variables. Then, Dai and Chen [32] presented a formula for calculating the entropy of function of uncertain variables. Entropy as a quantitative estimate of diversity, has been applied in portfolio selection [33].
There are a large number of optimization problems that have more than one objective. In some cases, the optimization problem is formulated to maximize one objective while minimize the other one. Numerous multi-objective optimization problems have been proposed. Deb et al. [34] proposed non-dominated sorting genetic algorithm II (NSGA-II) for multi-objective optimization problem. Kennedy and Eberhart [35] proposed particle swarm optimization (PSO) algorithm which is used to simulate the social behaviors of animals. PSO lies somewhere between genetic and evolutionary algorithms. Fonseca and Fleming [36] introduced multi-objective genetic algorithms (MOGAs). Ziztler et al. [37] carried out comparison of several evolutionary algorithms and discussed some results. Sherinov and Unveren [38] proposed multi-objective imperialistic competitive algorithm (MOICA) which is multi-objective version of imperialist competitive algorithm (IAC) introduced by Atashpaz-Gargari and Lucas [39]. The authors concluded that ICA and GA (genetic algorism) algorithms reach the global solution in the same iterations unlike PSO algorithm which needs more.
In this research, after providing definition of key terms such as skew-normal uncertainty distribution and asymmetric entropy and stating their properties in Sections 2 and 3, mean-CVaR-entropy model based on skew-normal uncertain variables and asymmetric entropy with different parameter values will be discussed in Section 4. Then in Section 5, a numerical example is put forth and finally in Section 6, conclusions and suggestions are presented.
Uncertainty theory is a branch of mathematics and was founded by Liu [25] in 2007. Having sample scarcity, we should ask experts to evaluate the degree of belief in the occurrence of an event. Modeling belief degree which uses subjective probability or fuzzy set theory may lead to counterintuitive results. Therefore, we use uncertainty theory to model belief degree. This section comes with reviewing some necessary definitions and theorems.
Assume that Γ is a nonempty set and ℒ represents a
ℳ(Γ) = 1 for the universal set Γ.
ℳ{Λ} +ℳ{Λ
For every countable sequence of events Λ_{1}, Λ_{2}, . . ., we have
the triplet (Γ,ℒ, ℳ) is called an uncertainty space.
The product uncertain measureℳon the product
Let (Γ
where Λ
The uncertainty distribution of an uncertain variable
for any real number
A real-valued function Φ(
An uncertain variable
where
An uncertain variable
where
Suppose that
If
If
If 0
In order to estimate the parameter vector Θ = (
Let Φ(
Let
A function Φ^{−1}(
Let
Let
has an inverse uncertainty distribution
Let
Let
Let
The purpose of this section is to present a new type of entropy called asymmetric entropy for uncertain variables for regular distributions. We first recall the concept of logarithm entropy proposed by Liu [30].
Suppose that
where L(t) = −t ln t − (1 − t)ln(1 − t).
Suppose that
where
For function S(
If
If
If 0
Suppose that
where q
It is clear that
is a derivable function whose derivative has the form
Since,
we have:
It follows from Fubini theorem that
The theorem is verified.
Let
has an entropy
Since
By applying Theorem 7, the entropy formula is obtained.
Let
Suppose that
Let
And for any numbers
Suppose that
If
By Theorem 7 we have:
If
If
By Theorem 7 we have:
by changing the variable
if
Therefore, we have
it follows from Theorem 7 that
And for any real numbers
The theorem is proved.
Let
The total return for a portfolio is denoted by
according to Definition 4,
and it can be rewritten as follows:
denoting the distribution function of the total returns
According to Definition 5, CVaR of the loss function
and it can be rewritten as follows:
Now according to Theorem 5 we can obtain uncertainty distribution of portfolio returns.
Suppose that Φ_{1}, Φ_{2}, . . . , Φ
To maximize the investment return and to minimize the investment risk, an investor should select a portfolio which provides an optimal trade-off between expected returns and risk. If an investor is to maximize the expected return of portfolio and to minimize the investment risk simultaneously, given that entropy is greater than some preset value
Suppose that
By applying (
where
By using
According to
Now by using
Suppose an investor is to construct portfolio containing five securities. Expert’s evaluation of security returns is depicted in Table 1. The parameters
We solve the following mean-CVaR-entropy multi-objective portfolio selection problem for asymmetry entropy with different values of parameter
Table 2 is derived by solving Model (6) implementing NSGA-II in MATLAB (2015b) with setting the parameter
In this paper, portfolio selection problem was improved via introducing skew-normal uncertainty distribution and asymmetric entropy. To show the performance of asymmetric entropy in portfolio diversification, a numerical example was presented. The example illustrated that portfolio constructed based on asymmetric entropy with
Meanwhile, some major issues remain to be discussed. The outperformance of asymmetric entropy with
No potential conflict of interest relevant to this article was reported.
Table 1. Uncertain return of securities.
Security | Uncertain return |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
Table 2. Investment allocation of securities.
Entropy | Allocation |
---|---|
Logarithm | (0.5827, 0.0852, 0.3321, 0, 0) |
Asymmetric (q=0.5) | (0.0414, 0.5535, 0.4051, 0, 0) |
Asymmetric (q=1) | (0.5425, 0.0213, 0.4362, 0, 0) |
Asymmetric (q=2) | (0.2534, 0.2737, 0.4481, 0.0002, 0.0247) |
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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 38-48
Published online March 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.1.38
Copyright © The Korean Institute of Intelligent Systems.
Seyyed Hamed Abtahi^{1}, Gholamhossein Yari^{2}, Farhad Hosseinzadeh Lotfi^{1}, and Rahman Farnoosh^{2}
^{1}Department of Statistics, Science and Research Branch, Islamic Azad University, 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.
Empirical studies illustrate that in numerous cases the returns of securities are not normally distributed. In this paper, skew-normal uncertainty distribution is proposed to capture skewness in the portfolio selection problem. Furthermore, the concept of asymmetric entropy for uncertain variables as the quantifier of diversification is presented and its mathematical properties such as translation invariance and positive linearity are studied. To examine the effect of asymmetric entropy parameter on portfolio diversification, a mean-CVaR-entropy portfolio selection problem is presented based on asymmetric entropy with different parameter values and logarithm entropy. A non-dominated sorting genetic algorithm II (NSGA-II) is implemented in MATLAB to solve the corresponding problem. Numerical results show that asymmetric entropy for a specific parameter value will outperform logarithm entropy in portfolio diversification.
Keywords: Uncertain variable, Skew-normal uncertainty distribution, Asymmetric entropy, Mean-CVaR-entropy portfolio selection model, Non-dominated sorting genetic algorithms II (NSGA-II)
The financial crisis of 2007–2008 showed that rare and unpredictable events can challenge conventional ideas about portfolio construction. However, we do not know in advance when such events will occur and how traumatic they can be, we can modify current risk management frameworks to better manage these rare and dangerous events.
The prime reason for the occurrence of such anomalies lies in the conventional approach to applying mean-variance model which was introduced by Markowitz [1] in 1952. The basic assumption in the mean-variance model and many other models is that future returns will be independent and normally distributed. Sheikh and Qiao [2] declared that in many cases it can be empirically observed that returns are not normally distributed and under non-normality, variance becomes inefficient as the quantifier of portfolio risk.
Most of studies on portfolio selection have been down based on only the first two moments of return distributions. However, there has always been a contentious issue whether higher moments should be considered in portfolio selection. Arditti [3] and Arditti and Levy [4] declared that higher moments should be considered for non-normal returns.
Skewness as the third central moment was defined to measure asymmetry of return distributions. Chunhachinda et al. [5] studied international stock market and confirmed that higher moments cannot be neglected in the portfolio selection. Konno et al. [6] constructed a portfolio optimization model by applying skewness. Keel et al. [7] showed that presence of skewness has a significant influence on portfolio selection. Furthermore, to better capture skewness, some scholars applied different distributions in portfolio optimization. Hu and Kercheval [8] argued that t and skewed t distribution show a much better fit to real data than the normal distribution.
Several portfolio selection models are developed by considering more efficient risk measure such as value at risk (VaR), conditional value at risk (CVaR) and entropy. VaR better reflects extreme events, but it does not aggregate risk in the sense of being sub-additive on portfolios [9]. CVaR is a coherent risk measure which is a superior to VaR [10]. VaR and CVaR has been widely applied by researchers [11–16].
Entropy of random variables first proposed by Shannon [17] in logarithm form. The study carried out by Philippatos and Wilson [18] was the pioneering work to associate entropy with a measure of risk in portfolio selection. They argued that entropy is more general and better suited for the selection of portfolios than variance. Furthermore, Simonelli [19] showed that entropy as a measure of risk is better than variance in wealth allocation.
According to the knowledge of securities investment, when the investment portfolio is more diversified, the risk of investment will decline. Entropy as a measure of diversification, originally developed by Jacquemin and Berry [20]. Entropy is a widely accepted measure of diversity [21] and has major advantages in the analysis of portfolio diversification [20]. It is well-established that the greater the level of entropy, the higher the degree of portfolio diversification. Bera and Park [22] constructed a well-diversified portfolio based on maximum entropy principle.
In the mentioned literatures it is assumed that the security returns are random variable with probability distribution. The fundamental assumption for using probability theory in the portfolio selection is that the probability distribution of security returns is similar to the past one and close enough to frequencies. However, it is difficult to ensure this assumption. In financial businesses, sometimes we have historical data scarcity. Therefore, we ask domain experts to evaluate the belief degree that each event will happen. Using fuzzy set theory is a way to handle portfolio optimization problems with returns given by experts’ evaluations. Liu [23] presented a counterexample to show that modeling belief degree which uses subjective probability may lead to counterintuitive results. Furthermore, Liu [24] showed that fuzzy set theory is not self-consistent in mathematics and may lead to wrong results in practice. The main mistake of fuzzy set theory is based on the wrong assumption that the belief degree of a union of events is the maximum of the belief degrees of the individual events no matter if they are independent or not.
In order to better deal with belief degree, uncertainty theory was founded by Liu [25] in 2007. Uncertainty theory has been employed in the portfolio optimization researches. Zhai et al. [26] formulated an uncertain multi-objective portfolio optimization considering skewness. Bhattacharyya et al. [27] studied effects of skewness on the portfolio selection. Yan [28] presented mean-VaR optimization model under uncertainty. Ning et al. [29] proposed mean-TVaR model under uncertainty framework.
Liu [30] proposed a concept of entropy for uncertain variables in the form of logarithm function. Chen and Dia [31] proposed the maximum entropy principle for uncertain variables. Then, Dai and Chen [32] presented a formula for calculating the entropy of function of uncertain variables. Entropy as a quantitative estimate of diversity, has been applied in portfolio selection [33].
There are a large number of optimization problems that have more than one objective. In some cases, the optimization problem is formulated to maximize one objective while minimize the other one. Numerous multi-objective optimization problems have been proposed. Deb et al. [34] proposed non-dominated sorting genetic algorithm II (NSGA-II) for multi-objective optimization problem. Kennedy and Eberhart [35] proposed particle swarm optimization (PSO) algorithm which is used to simulate the social behaviors of animals. PSO lies somewhere between genetic and evolutionary algorithms. Fonseca and Fleming [36] introduced multi-objective genetic algorithms (MOGAs). Ziztler et al. [37] carried out comparison of several evolutionary algorithms and discussed some results. Sherinov and Unveren [38] proposed multi-objective imperialistic competitive algorithm (MOICA) which is multi-objective version of imperialist competitive algorithm (IAC) introduced by Atashpaz-Gargari and Lucas [39]. The authors concluded that ICA and GA (genetic algorism) algorithms reach the global solution in the same iterations unlike PSO algorithm which needs more.
In this research, after providing definition of key terms such as skew-normal uncertainty distribution and asymmetric entropy and stating their properties in Sections 2 and 3, mean-CVaR-entropy model based on skew-normal uncertain variables and asymmetric entropy with different parameter values will be discussed in Section 4. Then in Section 5, a numerical example is put forth and finally in Section 6, conclusions and suggestions are presented.
Uncertainty theory is a branch of mathematics and was founded by Liu [25] in 2007. Having sample scarcity, we should ask experts to evaluate the degree of belief in the occurrence of an event. Modeling belief degree which uses subjective probability or fuzzy set theory may lead to counterintuitive results. Therefore, we use uncertainty theory to model belief degree. This section comes with reviewing some necessary definitions and theorems.
Assume that Γ is a nonempty set and ℒ represents a
ℳ(Γ) = 1 for the universal set Γ.
ℳ{Λ} +ℳ{Λ
For every countable sequence of events Λ_{1}, Λ_{2}, . . ., we have
the triplet (Γ,ℒ, ℳ) is called an uncertainty space.
The product uncertain measureℳon the product
Let (Γ
where Λ
The uncertainty distribution of an uncertain variable
for any real number
A real-valued function Φ(
An uncertain variable
where
An uncertain variable
where
Suppose that
If
If
If 0
In order to estimate the parameter vector Θ = (
Let Φ(
Let
A function Φ^{−1}(
Let
Let
has an inverse uncertainty distribution
Let
Let
Let
The purpose of this section is to present a new type of entropy called asymmetric entropy for uncertain variables for regular distributions. We first recall the concept of logarithm entropy proposed by Liu [30].
Suppose that
where L(t) = −t ln t − (1 − t)ln(1 − t).
Suppose that
where
For function S(
If
If
If 0
Suppose that
where q
It is clear that
is a derivable function whose derivative has the form
Since,
we have:
It follows from Fubini theorem that
The theorem is verified.
Let
has an entropy
Since
By applying Theorem 7, the entropy formula is obtained.
Let
Suppose that
Let
And for any numbers
Suppose that
If
By Theorem 7 we have:
If
If
By Theorem 7 we have:
by changing the variable
if
Therefore, we have
it follows from Theorem 7 that
And for any real numbers
The theorem is proved.
Let
The total return for a portfolio is denoted by
according to Definition 4,
and it can be rewritten as follows:
denoting the distribution function of the total returns
According to Definition 5, CVaR of the loss function
and it can be rewritten as follows:
Now according to Theorem 5 we can obtain uncertainty distribution of portfolio returns.
Suppose that Φ_{1}, Φ_{2}, . . . , Φ
To maximize the investment return and to minimize the investment risk, an investor should select a portfolio which provides an optimal trade-off between expected returns and risk. If an investor is to maximize the expected return of portfolio and to minimize the investment risk simultaneously, given that entropy is greater than some preset value
Suppose that
By applying (
where
By using
According to
Now by using
Suppose an investor is to construct portfolio containing five securities. Expert’s evaluation of security returns is depicted in Table 1. The parameters
We solve the following mean-CVaR-entropy multi-objective portfolio selection problem for asymmetry entropy with different values of parameter
Table 2 is derived by solving Model (6) implementing NSGA-II in MATLAB (2015b) with setting the parameter
In this paper, portfolio selection problem was improved via introducing skew-normal uncertainty distribution and asymmetric entropy. To show the performance of asymmetric entropy in portfolio diversification, a numerical example was presented. The example illustrated that portfolio constructed based on asymmetric entropy with
Meanwhile, some major issues remain to be discussed. The outperformance of asymmetric entropy with
The function S(t) for different parameter q.
Table 1 . Uncertain return of securities.
Security | Uncertain return |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
Table 2 . Investment allocation of securities.
Entropy | Allocation |
---|---|
Logarithm | (0.5827, 0.0852, 0.3321, 0, 0) |
Asymmetric (q=0.5) | (0.0414, 0.5535, 0.4051, 0, 0) |
Asymmetric (q=1) | (0.5425, 0.0213, 0.4362, 0, 0) |
Asymmetric (q=2) | (0.2534, 0.2737, 0.4481, 0.0002, 0.0247) |
Alireza Sajedi and Gholamhossein Yari
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(1): 35-42 https://doi.org/10.5391/IJFIS.2020.20.1.35The function S(t) for different parameter q.