International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 44-55
Published online March 25, 2023
https://doi.org/10.5391/IJFIS.2023.23.1.44
© The Korean Institute of Intelligent Systems
Bank Pasargad, Tehran, Iran
Correspondence to :
Seyyed Hamed Abtahi (dr_abtahi@yahoo.com)
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.
The financial market is a complex system owing to its inherent dynamics and indeterminacy. It is well known that uncertainty and randomness are two types of indeterminacy. When there are sufficient historical data, random variables are considered, and when the data are lacking, uncertain variables are considered. In numerous complex systems, such as financial markets, uncertainty and randomness often appear simultaneously. To model the indeterminacy associated with complex systems, including uncertainty and randomness, uncertain random variables are used. Massive shifts in financial market prices can cause considerable discrepancies in the normality assumption of security returns. To overcome the restriction imposed by the normality assumption and capture the non-normality of security returns, this study proposes the concept of skew chance distribution based on a skew-normal uncertainty distribution. A mean-entropy model is presented as an application for portfolio optimization problems. Moreover, a genetic algorithm is implemented in MATLAB to obtain the numerical results.
Keywords: Chance theory, Uncertain random variable, Skew chance distribution, Entropy, Portfolio optimization
The global financial crisis of 2007–2008 demonstrated that unexpected incidents could severely challenge conventional ideas regarding risk management. However, we cannot predict when unexpected events will transpire; thus, we must optimize our risk management frameworks to address such incidents. The 2007–2008 stock market crash was initially triggered by the collapse of the United States mortgage-backed securities in the housing sector. An exploration of the primary cause of stock market crashes should be conducted using traditional portfolio risk models, such as Markowitz’s mean-variance portfolio optimization [1]. The fundamental assumption in the mean-variance portfolio optimization model and in many similar models is that future returns are independent and normally distributed. In numerous cases, it can be empirically observed that returns are not normally distributed, and under non-normality, variance becomes inefficient as a quantifier of portfolio risk [2]. In addition, there are other limitations in employing mean-variance portfolio optimization models. In particular, assigning greater asset allocation weights to high-risk assets and disturbance in the assets’ dependence structure [3].
Entropy is a quantitative measurement of the indeterminacy associated with a variable. The entropy of random variables was first proposed in logarithmic form by Shannon [4]. A pioneering study conducted by scholars to associate entropy with a measure of risk in portfolio optimization problems illustrated that entropy is more common and better suited to portfolio optimization than is variance [5]. Moreover, several studies have demonstrated that entropy as a measure of risk is more effective than is variance in wealth allocation, and that by using entropy rather than variance in portfolio optimization problems, all major difficulties with Markowitz’s mean-variance portfolio optimization model can be eliminated [3].
Several portfolio selection models have been developed that consider other types of risk measures, such as value-at-risk (VaR) and conditional value-at-risk (CVaR) [6, 7]. VaR accurately captures extreme events, but it does not provide a comprehensive view of risk across multiple portfolios, as it is not subadditive [8]. VaR and CVaR have been widely used by researchers [6, 7, 9]. Recently, a novel coherent risk measure called the entropic value-at-risk (EVaR) was introduced by Ahmadi-Javid [10], and it is an upper bound for both VaR and CVaR. Ahmadi-Javid and Fallah-Tafti [11] applied EVaR to portfolio optimization problems. They concluded that the EVaR-based model generally performed on par with the CVaR-based model. Additionally, EVaR outperformed CVaR as the sample size increased.
Most studies on portfolio optimization problems have been based on only the first two central moments of return distributions. However, whether higher moments should be included in portfolio selection problems has always been controversial. Several scholars have declared that higher moments should be considered for nonnormal returns [12, 13]. Skewness is a third central moment that measures the asymmetry of return distributions. Studying the international stock market illustrated that higher moments cannot be disregarded in portfolio optimization problems [14]. Several researchers have confirmed the significant influence of skewness on portfolio selection and its application in constructing portfolio optimization models [15, 16]. Furthermore, to better capture skewness, several researchers have applied different types of skewed distributions to portfolio optimization problems [17].
In the aforementioned studies, indeterminacy was considered under the probability theory. Although probability theory is a conventional tool for analyzing phenomena over time, numerous phenomena do not accommodate randomness. To address problems associated with nonrandom phenomena, Zadeh [18] proposed the fuzzy set theory. As an improvement, Liu and Liu [19] presented a self-dual-credibility measure for fuzzy events. In developing fuzzy set theory, Kwakernakk [20, 21] introduced a fuzzy random variable for modeling fuzzy stochastic phenomena. Scholars have widely studied fuzzy random variables in relation to portfolio optimization problems [22–24]. Despite the widespread use of fuzzy set theory, Liu [25] confirmed that using fuzzy set theory or subjective probability to model human uncertainty may lead to inaccurate results.
To effectively address non-random phenomena, particularly human uncertainty, Liu [26] proposed uncertainty theory. Liu [26] also presented an uncertain set for modeling concepts such as “tall”, “warm”, “young”, “large”, and “most”. Currently, uncertainty theory is a mathematical methodology for addressing indeterminate phenomena involving uncertainty.
The entropy of uncertain variables was first proposed by Liu [27] in logarithmic form. Subsequently, several scholars have investigated entropy under uncertainty theory. Chen et al. [28] proposed the concept of cross-entropy to measure the degree of divergence of uncertain variables and presented the minimum cross-entropy principle. Chen and Dai [29] proposed the maximum-entropy principle for uncertain variables. Moreover, Dai and Chen [30] presented a formula for calculating the entropy of uncertain variables. As a supplement to logarithmic entropy, several types of entropy for uncertain variables have been investigated by scholars [31–34]. Moreover, to better capture skewness in uncertain portfolio optimization problems, Abtahi et al. [35] proposed asymmetric entropy and the skew-normal uncertainty distribution.
In numerous complex systems, uncertainty and randomness may coexist in phenomena. Under these circumstances, the concepts of uncertain random variables and chance theory are used to model such phenomena. To describe such phenomena, Liu [36] proposed the use of uncertain random variables. Liu [36] also discussed the concepts of chance distribution, expected value, and the variance of uncertain random variables. Subsequently, Guo and Wang [37] presented a formula to calculate the variance of the uncertain random variables. Sheng et al. [38] proposed the concept of entropy for uncertain random variables in logarithmic form. Ahmadzade et al. [39] defined partial entropy for uncertain random variables and derived several properties. Further applications of entropy for uncertain random variables in portfolio optimization problems have been investigated by several scholars [40–42]. Liu ad Ralescu [43] presented the concept of VaR for uncertain random variables, and Qin et al. [44] optimized portfolio selection problems of uncertain random returns based on VaR models. Liu et al. [45] proposed the concept of a tail vaule-at-risk (TVaR) for uncertain random variables and applied it to series systems, parallel systems, k-out-of-n systems, standby systems, and structural systems. Li et al. [46] demonstrated certain mathematical properties of the TVaR for uncertain random variables and formulated several mean-TVaR hybrid portfolio optimization models. Li and Shu [47] proposed skewness for uncertain random variables and applied it to portfolio selection problems.
Chance distributions that cover nonnormal returns have not been used in any similar study. Therefore, to better capture skewness in uncertain random environments, the concept of the skew chance distribution is investigated. The remainder of this paper is organized as follows: in Section 2, the concepts of uncertainty theory and chance theory are reviewed. In Section 3, the concept of skew chance distribution and its mathematical properties are explored. In Section 4, a portfolio optimization problem based on the skew chance distribution is optimized via a mean-entropy model. Finally, the conclusions are presented in Section 5.
This section reviews concepts of uncertainty theory and chance theory including the definition of uncertain variables, uncertainty distribution, uncertain random variables, and chance distribution.
Uncertainty theory was proposed by Liu [26] in 2007 to model human uncertainty. Owing to a lack of samples, we should solicit expert opinion to gauge the probability of an event’s occurrence. This section reviews some necessary definitions and theorems in uncertainty theory.
Assume that Γ is a nonempty set and
Subsequently, (Γ,
where ∧
An uncertain variable is a function
For all Borel sets
An uncertain variable
where
Parameter
1. If
2. If
3. If 0
A function
Let
Let
where
Let
Let
This follows from the definition of variance for uncertain variables [26],
Let
This follows from the definition of the K
Let
where
Let
Suppose
To handle both uncertainty and randomness, chance theory was proposed by Liu [48]. In chance theory, the chance space refers to the product of (Γ,
Liu [36] demonstrated that a chance measure satisfies the following properties:
(i) (normality)
(ii) (duality)
(iii) (monotonicity)
Furthermore, Hou [49] proved that, for a sequence of events Θ1, Θ2, . . . , a chance measure satisfies the following subadditivity:
An uncertain random variable is a function
Suppose
A chance distribution Φ(
Let
provided that at least one of the two integrals is finite.
Let
Let
Let
where
Let
where
Let
where
Let
where
It is obvious that
Let
where
Let
where
Let
where
Let
Let
This section investigates the concept of skew chance distribution and its mathematical properties. In numerous cases, uncertainty and randomness appear simultaneously in complex systems, such as security markets. Furthermore, in several cases, it can be empirically observed that security returns are not normally distributed. Therefore, to overcome the restriction imposed by the normality assumption and capture the non-normality of security returns under uncertain random environments, the concept of skew chance distribution is presented.
To create a skew chance distribution, a skew-normal uncertainty distribution proposed by Abtahi et al. [35] in conjunction with a normal distribution is considered. The expected value, variance, and skewness of the skew chance distribution are obtained to better describe the concept of skew chance distribution.
Let
and let
Then, for an uncertain random variable
(i) The expected value of
(ii) The variance of
(iii) The skewness of
(i) According to Theorems 9 and 2, the expected value of the uncertain random variable
(ii) According to Theorems 10 and 2, the variance of the uncertain random variable
(iii) According to Theorem 11 and Remark 2, the skewness of
The proof is completed.
By setting
(i) the expected value of
(ii) the variance of
(iii) the skewness of
In this section, a mean-entropy optimization model is proposed to solve the portfolio optimization problem of uncertain random variables. Both historical and recently issued stocks have coexisted in security markets since the markets began. From historical stock data, a probability distribution can be obtained. For recently issued stocks, estimations by domain experts can be used to estimate stock returns. Thus, chance theory can be employed to investigate uncertain random portfolio optimization problems.
In conventional portfolio optimization models, such as Markowitz’s mean-variance model, returns are quantified as expected values and risk as the variance. However, several studies have confirmed that entropy as a measure of risk outperforms variance [3, 5, 53]. Thus, in this study, the partial entropy of uncertain random variables was applied to portfolio optimization problems.
Suppose that there are
where the predetermined parameter
By applying the expected value formula for uncertain random variables
According to Ahmadzade et al. [39], a random variable has no term for an uncertain variable. When the uncertain random variable degenerates into a random variable, the partial entropy is zero. When the uncertain random variable degenerates into an uncertain variable, the partial entropy becomes the entropy of the uncertain variables.
According to Theorems 13 and 14, the partial entropy of uncertain random variables
where
Thus, Model (
Suppose that an investment portfolio containing five securities exists. According to the expert’s evaluation and data from the Tehran Stock Exchange, five securities are assumed to be uncertain random variables with
Model (
Optimal solutions were obtained by implementing a genetic algorithm (GA) in MATLAB. The investment proportions of the securities are listed in Table 2. The objective value was set to 0.6252. The numerical results indicate that the investment proportion in securities having a smaller parameter
In this study, the concept of skew chance distribution for uncertain random variables was proposed by leveraging the skew-normal uncertainty distribution. Furthermore, certain mathematical properties of the skewed chance distribution were investigated. As an application of the skewed chance distribution, a portfolio optimization problem was optimized via a mean-entropy model. To address this problem, a GA was implemented in MATLAB. The numerical results showed that investments in securities with a low
However, several issues need to be explored further. In future research, we aim to propose a novel type of entropy for uncertain random variables, called partial pseudo-triangular entropy, and to investigate its possible applications in uncertain random portfolio optimization problems. Moreover, we aim to optimize uncertain random portfolio optimization problems via various types of coherent risk measures, such as CVaR and EVaR.
No potential conflict of interest relevant to this article was reported.
E-mail: dr abtahi@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 44-55
Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.44
Copyright © The Korean Institute of Intelligent Systems.
Bank Pasargad, Tehran, Iran
Correspondence to:Seyyed Hamed Abtahi (dr_abtahi@yahoo.com)
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.
The financial market is a complex system owing to its inherent dynamics and indeterminacy. It is well known that uncertainty and randomness are two types of indeterminacy. When there are sufficient historical data, random variables are considered, and when the data are lacking, uncertain variables are considered. In numerous complex systems, such as financial markets, uncertainty and randomness often appear simultaneously. To model the indeterminacy associated with complex systems, including uncertainty and randomness, uncertain random variables are used. Massive shifts in financial market prices can cause considerable discrepancies in the normality assumption of security returns. To overcome the restriction imposed by the normality assumption and capture the non-normality of security returns, this study proposes the concept of skew chance distribution based on a skew-normal uncertainty distribution. A mean-entropy model is presented as an application for portfolio optimization problems. Moreover, a genetic algorithm is implemented in MATLAB to obtain the numerical results.
Keywords: Chance theory, Uncertain random variable, Skew chance distribution, Entropy, Portfolio optimization
The global financial crisis of 2007–2008 demonstrated that unexpected incidents could severely challenge conventional ideas regarding risk management. However, we cannot predict when unexpected events will transpire; thus, we must optimize our risk management frameworks to address such incidents. The 2007–2008 stock market crash was initially triggered by the collapse of the United States mortgage-backed securities in the housing sector. An exploration of the primary cause of stock market crashes should be conducted using traditional portfolio risk models, such as Markowitz’s mean-variance portfolio optimization [1]. The fundamental assumption in the mean-variance portfolio optimization model and in many similar models is that future returns are independent and normally distributed. In numerous cases, it can be empirically observed that returns are not normally distributed, and under non-normality, variance becomes inefficient as a quantifier of portfolio risk [2]. In addition, there are other limitations in employing mean-variance portfolio optimization models. In particular, assigning greater asset allocation weights to high-risk assets and disturbance in the assets’ dependence structure [3].
Entropy is a quantitative measurement of the indeterminacy associated with a variable. The entropy of random variables was first proposed in logarithmic form by Shannon [4]. A pioneering study conducted by scholars to associate entropy with a measure of risk in portfolio optimization problems illustrated that entropy is more common and better suited to portfolio optimization than is variance [5]. Moreover, several studies have demonstrated that entropy as a measure of risk is more effective than is variance in wealth allocation, and that by using entropy rather than variance in portfolio optimization problems, all major difficulties with Markowitz’s mean-variance portfolio optimization model can be eliminated [3].
Several portfolio selection models have been developed that consider other types of risk measures, such as value-at-risk (VaR) and conditional value-at-risk (CVaR) [6, 7]. VaR accurately captures extreme events, but it does not provide a comprehensive view of risk across multiple portfolios, as it is not subadditive [8]. VaR and CVaR have been widely used by researchers [6, 7, 9]. Recently, a novel coherent risk measure called the entropic value-at-risk (EVaR) was introduced by Ahmadi-Javid [10], and it is an upper bound for both VaR and CVaR. Ahmadi-Javid and Fallah-Tafti [11] applied EVaR to portfolio optimization problems. They concluded that the EVaR-based model generally performed on par with the CVaR-based model. Additionally, EVaR outperformed CVaR as the sample size increased.
Most studies on portfolio optimization problems have been based on only the first two central moments of return distributions. However, whether higher moments should be included in portfolio selection problems has always been controversial. Several scholars have declared that higher moments should be considered for nonnormal returns [12, 13]. Skewness is a third central moment that measures the asymmetry of return distributions. Studying the international stock market illustrated that higher moments cannot be disregarded in portfolio optimization problems [14]. Several researchers have confirmed the significant influence of skewness on portfolio selection and its application in constructing portfolio optimization models [15, 16]. Furthermore, to better capture skewness, several researchers have applied different types of skewed distributions to portfolio optimization problems [17].
In the aforementioned studies, indeterminacy was considered under the probability theory. Although probability theory is a conventional tool for analyzing phenomena over time, numerous phenomena do not accommodate randomness. To address problems associated with nonrandom phenomena, Zadeh [18] proposed the fuzzy set theory. As an improvement, Liu and Liu [19] presented a self-dual-credibility measure for fuzzy events. In developing fuzzy set theory, Kwakernakk [20, 21] introduced a fuzzy random variable for modeling fuzzy stochastic phenomena. Scholars have widely studied fuzzy random variables in relation to portfolio optimization problems [22–24]. Despite the widespread use of fuzzy set theory, Liu [25] confirmed that using fuzzy set theory or subjective probability to model human uncertainty may lead to inaccurate results.
To effectively address non-random phenomena, particularly human uncertainty, Liu [26] proposed uncertainty theory. Liu [26] also presented an uncertain set for modeling concepts such as “tall”, “warm”, “young”, “large”, and “most”. Currently, uncertainty theory is a mathematical methodology for addressing indeterminate phenomena involving uncertainty.
The entropy of uncertain variables was first proposed by Liu [27] in logarithmic form. Subsequently, several scholars have investigated entropy under uncertainty theory. Chen et al. [28] proposed the concept of cross-entropy to measure the degree of divergence of uncertain variables and presented the minimum cross-entropy principle. Chen and Dai [29] proposed the maximum-entropy principle for uncertain variables. Moreover, Dai and Chen [30] presented a formula for calculating the entropy of uncertain variables. As a supplement to logarithmic entropy, several types of entropy for uncertain variables have been investigated by scholars [31–34]. Moreover, to better capture skewness in uncertain portfolio optimization problems, Abtahi et al. [35] proposed asymmetric entropy and the skew-normal uncertainty distribution.
In numerous complex systems, uncertainty and randomness may coexist in phenomena. Under these circumstances, the concepts of uncertain random variables and chance theory are used to model such phenomena. To describe such phenomena, Liu [36] proposed the use of uncertain random variables. Liu [36] also discussed the concepts of chance distribution, expected value, and the variance of uncertain random variables. Subsequently, Guo and Wang [37] presented a formula to calculate the variance of the uncertain random variables. Sheng et al. [38] proposed the concept of entropy for uncertain random variables in logarithmic form. Ahmadzade et al. [39] defined partial entropy for uncertain random variables and derived several properties. Further applications of entropy for uncertain random variables in portfolio optimization problems have been investigated by several scholars [40–42]. Liu ad Ralescu [43] presented the concept of VaR for uncertain random variables, and Qin et al. [44] optimized portfolio selection problems of uncertain random returns based on VaR models. Liu et al. [45] proposed the concept of a tail vaule-at-risk (TVaR) for uncertain random variables and applied it to series systems, parallel systems, k-out-of-n systems, standby systems, and structural systems. Li et al. [46] demonstrated certain mathematical properties of the TVaR for uncertain random variables and formulated several mean-TVaR hybrid portfolio optimization models. Li and Shu [47] proposed skewness for uncertain random variables and applied it to portfolio selection problems.
Chance distributions that cover nonnormal returns have not been used in any similar study. Therefore, to better capture skewness in uncertain random environments, the concept of the skew chance distribution is investigated. The remainder of this paper is organized as follows: in Section 2, the concepts of uncertainty theory and chance theory are reviewed. In Section 3, the concept of skew chance distribution and its mathematical properties are explored. In Section 4, a portfolio optimization problem based on the skew chance distribution is optimized via a mean-entropy model. Finally, the conclusions are presented in Section 5.
This section reviews concepts of uncertainty theory and chance theory including the definition of uncertain variables, uncertainty distribution, uncertain random variables, and chance distribution.
Uncertainty theory was proposed by Liu [26] in 2007 to model human uncertainty. Owing to a lack of samples, we should solicit expert opinion to gauge the probability of an event’s occurrence. This section reviews some necessary definitions and theorems in uncertainty theory.
Assume that Γ is a nonempty set and
Subsequently, (Γ,
where ∧
An uncertain variable is a function
For all Borel sets
An uncertain variable
where
Parameter
1. If
2. If
3. If 0
A function
Let
Let
where
Let
Let
This follows from the definition of variance for uncertain variables [26],
Let
This follows from the definition of the K
Let
where
Let
Suppose
To handle both uncertainty and randomness, chance theory was proposed by Liu [48]. In chance theory, the chance space refers to the product of (Γ,
Liu [36] demonstrated that a chance measure satisfies the following properties:
(i) (normality)
(ii) (duality)
(iii) (monotonicity)
Furthermore, Hou [49] proved that, for a sequence of events Θ1, Θ2, . . . , a chance measure satisfies the following subadditivity:
An uncertain random variable is a function
Suppose
A chance distribution Φ(
Let
provided that at least one of the two integrals is finite.
Let
Let
Let
where
Let
where
Let
where
Let
where
It is obvious that
Let
where
Let
where
Let
where
Let
Let
This section investigates the concept of skew chance distribution and its mathematical properties. In numerous cases, uncertainty and randomness appear simultaneously in complex systems, such as security markets. Furthermore, in several cases, it can be empirically observed that security returns are not normally distributed. Therefore, to overcome the restriction imposed by the normality assumption and capture the non-normality of security returns under uncertain random environments, the concept of skew chance distribution is presented.
To create a skew chance distribution, a skew-normal uncertainty distribution proposed by Abtahi et al. [35] in conjunction with a normal distribution is considered. The expected value, variance, and skewness of the skew chance distribution are obtained to better describe the concept of skew chance distribution.
Let
and let
Then, for an uncertain random variable
(i) The expected value of
(ii) The variance of
(iii) The skewness of
(i) According to Theorems 9 and 2, the expected value of the uncertain random variable
(ii) According to Theorems 10 and 2, the variance of the uncertain random variable
(iii) According to Theorem 11 and Remark 2, the skewness of
The proof is completed.
By setting
(i) the expected value of
(ii) the variance of
(iii) the skewness of
In this section, a mean-entropy optimization model is proposed to solve the portfolio optimization problem of uncertain random variables. Both historical and recently issued stocks have coexisted in security markets since the markets began. From historical stock data, a probability distribution can be obtained. For recently issued stocks, estimations by domain experts can be used to estimate stock returns. Thus, chance theory can be employed to investigate uncertain random portfolio optimization problems.
In conventional portfolio optimization models, such as Markowitz’s mean-variance model, returns are quantified as expected values and risk as the variance. However, several studies have confirmed that entropy as a measure of risk outperforms variance [3, 5, 53]. Thus, in this study, the partial entropy of uncertain random variables was applied to portfolio optimization problems.
Suppose that there are
where the predetermined parameter
By applying the expected value formula for uncertain random variables
According to Ahmadzade et al. [39], a random variable has no term for an uncertain variable. When the uncertain random variable degenerates into a random variable, the partial entropy is zero. When the uncertain random variable degenerates into an uncertain variable, the partial entropy becomes the entropy of the uncertain variables.
According to Theorems 13 and 14, the partial entropy of uncertain random variables
where
Thus, Model (
Suppose that an investment portfolio containing five securities exists. According to the expert’s evaluation and data from the Tehran Stock Exchange, five securities are assumed to be uncertain random variables with
Model (
Optimal solutions were obtained by implementing a genetic algorithm (GA) in MATLAB. The investment proportions of the securities are listed in Table 2. The objective value was set to 0.6252. The numerical results indicate that the investment proportion in securities having a smaller parameter
In this study, the concept of skew chance distribution for uncertain random variables was proposed by leveraging the skew-normal uncertainty distribution. Furthermore, certain mathematical properties of the skewed chance distribution were investigated. As an application of the skewed chance distribution, a portfolio optimization problem was optimized via a mean-entropy model. To address this problem, a GA was implemented in MATLAB. The numerical results showed that investments in securities with a low
However, several issues need to be explored further. In future research, we aim to propose a novel type of entropy for uncertain random variables, called partial pseudo-triangular entropy, and to investigate its possible applications in uncertain random portfolio optimization problems. Moreover, we aim to optimize uncertain random portfolio optimization problems via various types of coherent risk measures, such as CVaR and EVaR.
Table 1 . Uncertain random returns.
No | Uncertain term | Random term |
---|---|---|
1 | ||
2 | ||
3 | ||
4 | ||
5 |
Table 2 . Investment proportion in securities.
No | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Investment proportion | 0.0323 | 0.1631 | 0.7950 | 0.0034 | 0.0062 |
Young-Seok Choi
Int. J. Fuzzy Log. Intell. Syst. 2017; 17(3): 194-201 https://doi.org/10.5391/IJFIS.2017.17.3.194Jooyoung Park , Jungdong Lim , Wonbu Lee , Seunghyun Ji , Keehoon Sung , and Kyungwook Park
International Journal of Fuzzy Logic and Intelligent Systems 2014; 14(2): 73-83 https://doi.org/10.5391/IJFIS.2014.14.2.73