Uncertain Random Portfolio Optimization Based on Skew Chance Distribution

Seyyed Hamed Abtahi

doi:10.5391/IJFIS.2023.23.1.44

Original Article

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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

Uncertain Random Portfolio Optimization Based on Skew Chance Distribution

Seyyed Hamed Abtahi

Bank Pasargad, Tehran, Iran

Correspondence to :
Seyyed Hamed Abtahi (dr_abtahi@yahoo.com)

Received: August 31, 2022; Revised: January 24, 2023; Accepted: March 7, 2023

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.

Abstract

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

This section reviews concepts of uncertainty theory and chance theory including the definition of uncertain variables, uncertainty distribution, uncertain random variables, and chance distribution.

2.1 Uncertainty Theory

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 L is a σ-algebra over Γ. Subsequently, (Γ, L) is called the measurable space. Each element ∧ in L is known as a measurable set. A measurable set can be considered an event in uncertainty theory. That is, a number M{∧} is assigned to each event ∧ to indicate the likelihood that ∧ will occur. To address the degree of belief, Liu [30] suggested the following axioms:

Axiom 1 (Normality). For the universal set Γ, M{Γ} = 1.
Axiom 2 (Duality). For any event, M{∧} +M{∧_c} = 1.
Axiom 3 (Subadditivity). For every countable sequence of events ∧₁, ∧₂, . . ., we have

M{∪i=1∞∧i}≤∑i=1∞M{∧i}.

Subsequently, (Γ, L, M) is called an uncertainty space.
Axiom 4 (Product). Suppose (Γ_i, ℒ_i,M_i) are the uncertainty spaces for i = 1, 2, . . . . Product uncertainty measure M is an uncertain measure that satisfies

M {∏i=1∞∧i}=⋀i=1∞Mi{∧i},

where ∧_i is an arbitrarily chosen event from L_i for i = 1, 2, . . . , respectively.

Definition 1 (Liu [26])

An uncertain variable is a function τ from an uncertainty space (Γ, L, M) to the set of real numbers such that {τ ∈ B} is an event for any Borel set B of real numbers.

Definition 2 (Liu [48])

For all Borel sets B₁, B₂, . . . , B_n of real numbers, the uncertain variables τ₁, τ₂, . . . , τ_n are independent if

M {∩i=1n(τi∈Bi)}=⋀i=1nM {τi∈Bi}.

Definition 3 (Abtahi et al. [35])

An uncertain variable τ is called skew-normal and denoted by SN(m, p, δ) if it has a skew-normal uncertainty distribution

γ(x)=(1+exp (π(m-x)3δ))-p,x∈R,

where m, δ (δ > 0) and p (p > 0) are real numbers.

Remark 1 (Abtahi et al. [35])

Parameter m specifies distribution location, parameter δ (δ > 0) specifies distribution spread and parameter p (p > 0) specifies distribution shape. There are three conditions for p as follows:

1. If p = 1, then the uncertain variable τ is a normal uncertainty distribution.
2. If p > 1, then the uncertain variable τ is a positive skew-normal uncertainty distribution.
3. If 0 < p < 1, then the uncertain variable τ is a negative skew-normal uncertainty distribution.

Theorem 1 (Liu [26])

A function γ⁻¹(r) is an inverse uncertainty distribution if and only if γ⁻¹(r) is a strictly increasing continuous function with respect to r.

Definition 4 (Abtahi et al. [35])

Let τ ∼ SN(m, p, δ); then, the inverse uncertainty distribution of the skew-normal uncertain variable τ is as follows:

γ-1(r)=m+3δπln (r1p1-r1p);0<r<1.

Theorem 2 (Liu [48])

Let τ be an uncertain variable with a regular uncertainty distribution γ(x). If the expected value of τ exists, then

E[τ]=∫01γ-1(r)dr,

where γ⁻¹(r) is the inverse uncertainty function of τ with respect to r.

Theorem 3 (Abtahi et al. [35])

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n; then the expected value of the skew-normal uncertain variable τ_i is

E[τi]=mi-3δiπpi-3δiπ∫01ln (1-u1pi) du.

Theorem 4

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n, then the variance of the skew-normal uncertain variable τ_i is

V [τi]=∫01(mi+3δiπln (r1pi1-r1pi)-(mi-3δiπpi-3δiπ∫01ln (1-u1pi)du))2dr.

Proof

This follows from the definition of variance for uncertain variables [26],

V[τ]=∫01(γ-1(r)-E[τ])2dr.

Theorem 5

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n; then, the skewness of the skew-normal uncertain variable τ_i is

S [τi]=∫01(mi+3δiπln (r1pi1-r1pi)-(mi-3δiπpi-3δiπ∫01ln (1-u1pi)du))3dr.

Proof

This follows from the definition of the K^th central moment of the uncertain variables for K = 3 [26],

S[τ]=∫01(γ-1(r)-E[τ])3dr.

Definition 5 (Liu [27])

Let τ be an uncertain variable with uncertainty distribution γ(x). Subsequently, the logarithmic entropy of the uncertain variable τ is

H[τ]=∫-∞+∞L (γ(x))dx,

where L(t) = −(t) ln(t) − (1 − t) ln(1 − t).

Theorem 6 (Dai and Chen [30])

Let τ be an uncertain variable with inverse uncertainty distribution γ⁻¹(r). Subsequently, the logarithmic entropy of τ is

H[τ]=∫01γ-1(r)ln (r1-r) dr.

Theorem 7 (Liu [32])

Suppose τ₁, τ₂, . . . , τ_n are independent uncertain variables with regular uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. If f(τ₁, τ₂, . . . , τ_n) is strictly increasing with respect to τ₁, τ₂, . . . , τ_m and strictly decreasing with respect to τ_m₊₁, τ_m₊₂, . . . , τ_n, then ξ = f(τ₁, τ₂, . . . , τ_n) is an uncertain variable with an inverse uncertainty distribution

ϕ-1(r)=f(γ1-1(r), …,γm-1(r), γm+1-1(1-r), …,γn-1(1-r)).

2.2 Chance Theory

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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, L, M) × (Ω, A, Pr), where (Γ, L, M) denotes an uncertainty space, and (Ω, A, Pr) denotes a probability space. The chance measure of an uncertain random event Θ = L × A is defined as follows:

Ch{θ}=∫01Pr{ω∈Ω∣M{γ∈Γ∣(γ,ω)∈Θ}≥r}dr.

Liu [36] demonstrated that a chance measure satisfies the following properties:

(i) (normality) Ch(Γ × Ω) = 1;
(ii) (duality) Ch{Θ} + Ch{Θ^c} = 1, for any event Θ;
(iii) (monotonicity) Ch{Θ₁} < Ch{Θ₂} for any real number set Θ₁ ⊂ Θ₂.

Furthermore, Hou [49] proved that, for a sequence of events Θ₁, Θ₂, . . . , a chance measure satisfies the following subadditivity:

Ch{∪i=1∞Θi}≤∑i=1∞Ch(Θi).

Definition 6 (Liu [36])

An uncertain random variable is a function ξ from a chance space (Γ, L, M) × (Ω, A, Pr) to the set of real numbers such that for any Borel set B of real numbers, {ξ ∈ B} is an event in L × A.

Definition 7 (Liu [36])

Suppose ξ is an uncertain random variable. Subsequently, the chance distribution of ξ for any x ∈ R is defined by

Φ(x)=Ch{ξ≤x}.

Definition 8 (Sheng et al. [38])

A chance distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ(x) < 1, and lim_x_→−∞Φ(x) = 0, lim_x_→+∞Φ(x) = 1.

Definition 9 (Liu [36])

Let ξ be an uncertain random variable. Subsequently, the expected value of ξ is defined as

E[ξ]=∫0+∞Ch{ξ≥x}dx-∫-∞0Ch{ξ≤x}dx,

provided that at least one of the two integrals is finite.

Definition 10 (Liu [36])

Let ξ be an uncertain random variable with a finite expected value E[ξ]. Subsequently, the variance of ξ is

V[ξ]=E(ξ-E[ξ])2.

Definition 11 (Liu [36])

Let ξ be an uncertain random variable with a finite expected value E[ξ]. Subsequently, the skewness of ξ is defined as follows:

S[ξ]=E(ξ-E[ξ])3.

Theorem 8 (Liu [50])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Subsequently, the uncertain random variable ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n) has a chance distribution:

Φ(x)=∫RmF(x; y1, …, ym)dΨ1(y1)…dΨm(ym),

where F(x; y₁, . . . , y_m) is the uncertainty distribution of the uncertain variable f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) for any real numbers y₁, y₂, . . . , y_m.

Theorem 9 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

E[ξ]=∫Rm∫01F-1(r, y1, …, ym)×drdΨ1(y1)…dΨm(ym),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 10 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

V[ξ]=∫Rm∫01(F-1(r, y1, …, ym)-E[ξ])2×drdΨ1(y1)…dΨm(ym),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 11 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

S[ξ]=∫Rm∫01(F-1(r, y1, …, ym)-E[ξ])3×drdΨ1(y1)…dΨm(ym),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Remark 2

It is obvious that F⁻¹(r, y₁, . . . , y_m) = γ⁻¹(r)+ y₁ + · · · + y_m.

Definition 12 (Sheng et al. [52])

Let ξ be an uncertain random variable with chance distribution Φ(x). Subsequently, the entropy of ξ is defined as

H[ξ]=∫-∞+∞S(Φ(x))dx,

where S(t) = −t ln t − (1 − t) ln(1 − t).

Theorem 12 (Ahmadzade et al. [39])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m and τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively, and let f be a measurable function. Additionally, let ξ = f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) be an uncertain random variable. Subsequently, ξ = f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) has partial entropy

PH[ξ]=∫Rm∫-∞+∞S(F(x, y1, …, ym))×dxdΨ1(y1)…dΨm(ym),

where S(t) = −t ln t− (1−t) ln(1−t) and F(x, y₁, . . . , y_m) is the uncertainty distribution of uncertain variable f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) for all real numbers y₁, y₂, . . . , y_m.

Theorem 13 (Ahmadzade et al. [39])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively, and let f be a measurable function. Subsequently, ξ = f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) has partial entropy

PH[ξ]=∫Rm∫-∞+∞F-1(r, y1, …, ym) lnr1-r×drdΨ1(y1)…dΨm(ym),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 14 (Ahmadzade et al. [5])

Let τ be an uncertain variable with uncertainty distribution function γ, and let η be a random variable with probability distribution function Ψ. If ξ = η + τ , then

PH[ξ]=H[τ].

Theorem 15 (Ahmadzade et al. [39])

Let η₁ and η₂ be independent random variables and let τ₁ and τ₂ be independent uncertain variables. Additionally, assume that ξ₁ = f(η₁, τ₁) and ξ₂ = f(η₂, τ₂). Subsequently, for any real numbers a and b, we have

PH[aξ1+bξ2]=∣a∣PH[ξ1]+∣b∣PH[ξ2].

3. Skew Chance Distributions

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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.

Theorem 16

Let η be a normal distribution denoted by N(μ, σ²) with the probability density function

f(y)=12πσexp (-(y-μ)22σ2),

and let τ be a skew-normal uncertain variable denoted by SN(m, p, δ) with an uncertainty distribution

γ(x)=(1+exp (π(m-x)3δ))-p;x∈R.

Then, for an uncertain random variable ξ = η + τ .

(i) The expected value of ξ is

E[ξ]=E[η]+E[τ]=μ+m-3δπp-3δπ∫01ln (1-u1p) du.
(ii) The variance of ξ is

V[ξ]=V[η]+V[τ]=σ2+∫01(m+3δπln (r1p1-r1p)-(m-3δπp-3δπ∫01ln (1-u1p)du))2dr.
(iii) The skewness of ξ is

S[ξ]=S[η]+S[τ]=∫01(m+3δπln (r1p1-r1p)-(m-3δπp-3δπ∫01ln (1-u1p)du))3dr.

Proof

(i) According to Theorems 9 and 2, the expected value of the uncertain random variable ξ is

E[ξ]=∫-∞+∞∫01F-1(r,y)drdΨ(y)=∫-∞+∞∫01(γ-1(r)+y) drdΨ(y)=∫-∞+∞∫01(m+3δπln (r1p1-r1p)+y)×(12πσexp (-(y-μ)22σ2)) drdy=∫-∞+∞∫01y (12πσexp (-(y-μ)22σ2)) drdy+∫-∞+∞∫01(m+3δπln (r1p1-r1p))×(12πσexp (-(y-μ)22σ2)) drdy=∫-∞+∞y (12πσexp (-(y-μ)22σ2)) dy+(∫-∞+∞(12πσexp (-(y-μ)22σ2)) dy)×(∫01(m+3δπln (r1p1-r1p)) dr)=E[η]+E[τ]=μ+m-3δπp-3δπ∫01ln (1-u1p) du.

(ii) According to Theorems 10 and 2, the variance of the uncertain random variable ξ is

V[ξ]=∫-∞+∞∫01(F-1(r,y)-E(ξ))2drdΨ(y)=∫-∞+∞∫01(γ-1(r)+y-E(ξ))2 drdΨ(y)=∫-∞+∞∫01(γ-1(r)+y-E[η]-E[τ])2 drdΨ(y)=∫-∞+∞∫01(m+3δπln (r1p1-r1p)+y-E[η]-E[τ])2×(12πσexp (-(y-μ)22σ2)) drdy=∫-∞+∞∫01(m+3δπln (r1p1-r1p)+y-μ-(m-3δπp-3δπ∫01ln (1-u1p) du))2×(12πσexp (-(y-μ)22σ2)) drdy=V[η]+V[τ]=σ2+∫01(m+3δπln (r1p1-r1p)-(m-3δπp-3δπ∫01ln (1-u1p)du))2dr.

(iii) According to Theorem 11 and Remark 2, the skewness of ξ is

S[ξ]=∫-∞+∞∫01(F-1(r,y)-E(ξ))3drdΨ(y)=∫-∞+∞∫01(γ-1(r)+y-E(ξ))3 drdΨ(y)=∫-∞+∞∫01(γ-1(r)+y-E[η]-E[τ])3 drdΨ(y)=∫-∞+∞∫01(m+3δπln (r1p1-r1p)+y-E[η]-E[τ])3×(12πσexp (-(y-μ)22σ2)) drdy=∫-∞+∞∫01(m+3δπln (r1p1-r1p)+y-μ-(m-3δπp-3δπ∫01ln (1-u1p) du))3×(12πσexp (-(y-μ)22σ2)) drdy=S[η]+S[τ]=0+∫01(m+3δπln (r1p1-r1p)-(m-3δπp-3δπ∫01ln (1-u1p)du))3dr.

The proof is completed.

Remark 3

By setting p = 1 in Theorem 16,

(i) the expected value of ξ is E[ξ] = μ + m;
(ii) the variance of ξ is V [ξ] = σ² + δ²;
(iii) the skewness of ξ is S[ξ] = 0.

4. Uncertain Random Portfolio Optimization

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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 n independent securities with uncertain random returns ξ_i = τ_i + η_i; i = 1, 2, . . . , n. Moreover, let x_i be the investment proportion in security i = 1, 2, . . . , n. To optimize the portfolio optimization problem, the meanentropy model is presented as follows:

(1)Min PH[x1ξ1+⋯+xnξn],S.t.E[x1ξ1+⋯+xnξn]≥C,x1+x2+⋯+xn=1,0≤xi≤1;∀i=1, 2, ⋯, n,

where the predetermined parameter C is designated by an investor.

By applying the expected value formula for uncertain random variables ξ_i = τ_i +η_i; i = 1, 2, . . . , n in Theorem 9, we obtain

E[x1ξ1+⋯+xnξn]=∫Rn∫01(x1F1-1(r,y1)+⋯+xnFn-1(r,yn))×drdΨ1(y1)…dΨn(yn)=∫Rn∫01(x1(γ1-1(r)+y1)+…+xn(γn-1(r)+yn))×drdΨ1(y1)…dΨn(yn)=x1(E(τ1)+E(η1))+…+xn(E(τn)+E(ηn)).

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 ξ_i = τ_i + η_i; i = 1, 2, . . . , n is obtained as follows:

PH[x1ξ1+⋯+xnξn]=PH[x1ξ1]+⋯+PH[xnξn]=x1PH[ξ1]+⋯+xnPH[ξn]=x1H[τ1]+⋯+xnH[τn],

where H[τ_i] is the logarithm entropy of uncertain variable τ_i; i = 1, 2, . . . , n.

Thus, Model (1) is equivalent to the following model,

(2)Min x1H[τ1]+⋯+xnH[τn],S.t.x1 (E(τ1)+E(η1))+…+xn(E(τn)+E(ηn))≥C,x1+x2+⋯+xn=1,0≤xi≤1;∀i=1, 2, …, n.

Example 1

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 ξ_i = τ_i + η_i, i = 1, 2, . . . , 5 depicted in Table 1. Moreover, parameter C in Model (2) is designated as 2.5 by an investor.

Model (2) is equivalent to the following model:

(3)Min 1.2697x1+0.7515x2+0.5641x3+2.1095x4+1.2661x5,S.t.2x1+3.2757x2+1.3308x3+2.1456x4+2.6178x5≥2.5,x1+x2+⋯+x5=1,0≤xi≤1;∀i=1, 2, …, 5.

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 δ and greater parameter p is higher than that in other securities.

5. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

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 δ and high p were greater than those in other securities.

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.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

No potential conflict of interest relevant to this article was reported.

Tables

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Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

Table. 1.

Table 1. Uncertain random returns.

No	Uncertain term	Random term
1
2
3
4
5

Table. 2.

Table 2. Investment proportion in securities.

No	1	2	3	4	5
Investment proportion	0.0323	0.1631	0.7950	0.0034	0.0062

References

Go to

Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

Markowitz, H (1952). Portfolio selection. The Journal of Finance. 7, 77-91. https://doi.org/10.2307/2975974
Sheikh, AZ, and Qiao, H (2010). Non-normality of market returns: a framework for asset allocation decision making. The Journal of Alternative Investments. 12, 8-35. https://doi.org/10.3905/JAI.2010.12.3.008
Mercurio, PJ, Wu, Y, and Xie, H (2020). An entropy-based approach to portfolio optimization. Entropy. 22. article no 332
Shannon, CE (1948). A mathematical theory of communication. The Bell System Technical Journal. 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Philippatos, GC, and Wilson, CJ (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics. 4, 209-220. https://doi.org/10.1080/00036847200000017
Rockafellar, RT, and Uryasev, S (2000). Optimization of conditional value-at-risk. Journal of Risk. 2, 21-42. https://doi.org/10.21314/JOR.2000.038
Rockafellar, RT, and Uryasev, S (2002). Conditional value-atrisk for general loss distributions. Journal of Banking & Finance. 26, 1443-1471. https://doi.org/10.1016/S0378-4266(02)00271-6
Artzner, P, Delbaen, F, Eber, JM, and Heath, D (1999). Coherent measures of risk. Mathematical Finance. 9, 203-228. https://doi.org/10.1111/1467-9965.00068
Jorion, P (1996). Risk2: measuring the risk in value at risk. Financial Analysts Journal. 52, 47-56. https://doi.org/10.2469/faj.v52.n6.2039
Ahmadi-Javid, A (2012). Entropic value-at-risk: a new coherent risk measure. Journal of Optimization Theory and Applications. 155, 1105-1123. https://doi.org/10.1007/s10957-011-9968-2
Ahmadi-Javid, A, and Fallah-Tafti, M (2019). Portfolio optimization with entropic value-at-risk. European Journal of Operational Research. 279, 225-241. https://doi.org/10.1016/j.ejor.2019.02.007
Arditti, FD (1967). Risk and the required return on equity. The Journal of Finance. 22, 19-36. https://doi.org/10.2307/2977297
Arditti, FD, and Levy, H (1975). Portfolio efficiency analysis in three moments: the multiperiod case. The Journal of Finance. 30, 797-809. https://doi.org/10.1111/j.1540-6261.1975.tb01851.x
Chunhachinda, P, Dandapani, K, Hamid, S, and Prakash, AJ (1997). Portfolio selection and skewness: Evidence from international stock markets. Journal of Banking & Finance. 21, 143-167. https://doi.org/10.1016/S0378-4266(96)00032-5
Keel, S, Herzog, F, Geering, H, and Mirjolet, N (). Optimal portfolios with skewed and heavy-tailed distributions, 42-48.
Konno, H, Shirakawa, H, and Yamazaki, H (1993). A mean-absolute deviation-skewness portfolio optimization model. Annals of Operations Research. 45, 205-220. https://doi.org/10.1007/BF02282050
Hu, W, and Kercheval, AN (2010). Portfolio optimization for student t and skewed t returns. Quantitative Finance. 10, 91-105. https://doi.org/10.1080/14697680902814225
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Liu, B, and Liu, YK (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems. 10, 445-450. https://doi.org/10.1109/TFUZZ.2002.800692
Kwakernaak, H (1978). Fuzzy random variables. I. Definitions and theorems. Information Sciences. 15, 1-29. https://doi.org/10.1016/0020-0255(78)90019-1
Kwakernaak, H (1979). Fuzzy random variables. II. Algorithms and examples for the discrete case. Information Sciences. 17, 253-278. https://doi.org/10.1016/0020-0255(79)90020-3
Deng, X, He, X, and Huang, C (2022). A new fuzzy random multi-objective portfolio model with different entropy measures using fuzzy programming based on artificial bee colony algorithm. Engineering Computations. 39, 627-649. https://doi.org/10.1108/EC-11-2020-0654
Khanjani Shiraz, R, Tavana, M, and Fukuyama, H (2020). A random-fuzzy portfolio selection DEA model using value-at-risk and conditional value-at-risk. Soft Computing. 24, 17167-17186. https://doi.org/10.1007/s00500-020-05010-7
Wu, X, Ralescu, DA, and Liu, Y (2020). A new quadratic deviation of fuzzy random variable and its application to portfolio optimization. Iranian Journal of Fuzzy Systems. 17, 1-18. https://doi.org/10.22111/IJFS.2020.5344
Liu, B (2012). Why is there a need for uncertainty theory. Journal of Uncertain Systems. 6, 3-10.
Liu, B (2007). Uncertainty Theory. Heidelberg, Germany: Springer
Liu, B (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems. 3, 3-10.
Chen, X, Kar, S, and Ralescu, DA (2012). Cross-entropy measure of uncertain variables. Information Sciences. 201, 53-60. https://doi.org/10.1016/j.ins.2012.02.049
Chen, X, and Dai, W (2011). Maximum entropy principle for uncertain variables. International Journal of Fuzzy Systems. 13, 232-236.
Dai, W, and Chen, X (2012). Entropy of function of uncertain variables. Mathematical and Computer Modelling. 55, 754-760. https://doi.org/10.1016/j.mcm.2011.08.052
Abtahi, SH, Yari, G, Lotfi, FH, and Farnoosh, R (2022). Pseudo-triangular entropy of uncertain variables: an entropy-based approach to uncertain portfolio optimization. International Journal of Mathematical Modelling & Computations. 12, 61-77.
Dai, W (2018). Quadratic entropy of uncertain variables. Soft Computing. 22, 5699-5706. https://doi.org/10.1007/s00500-017-2602-y
Tang, W, and Gao, W (2013). Triangular entropy of uncertain variables. Information (Tokyo). 16, 1279-1282.
Yao, K, Gao, J, and Dai, W (2013). Sine entropy for uncertain variables. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems. 21, 743-753. https://doi.org/10.1142/S0218488513500359
Abtahi, SH, Yari, G, Lotfi, FH, and Farnoosh, R (2021). Multi-objective portfolio selection based on skew-normal uncertainty distribution and asymmetric entropy. International Journal of Fuzzy Logic and Intelligent Systems. 21, 38-48. http://doi.org/10.5391/IJFIS.2021.21.1.38
Liu, Y (2013). Uncertain random variables: a mixture of uncertainty and randomness. Soft Computing. 17, 625-634. https://doi.org/10.1007/s00500-012-0935-0
Guo, H, and Wang, X (2014). Variance of uncertain random variables. Journal of Uncertainty Analysis and Applications. 2. article no 6
Sheng, Y, and Yao, K (2014). Some formulas of variance of uncertain random variable. Journal of Uncertainty Analysis and Applications. 2. article no 12
Ahmadzade, H, Gao, R, Dehghan, MH, and Sheng, Y (2017). Partial entropy of uncertain random variables. Journal of Intelligent & Fuzzy Systems. 33, 105-112. http://doi.org/10.3233/JIFS-161161
Ahmadzade, H, Gao, R, Dehghan, MH, and Ahmadi, R (2018). Partial triangular entropy of uncertain random variables and its application. Journal of Ambient Intelligence and Humanized Computing. 9, 1455-1464. https://doi.org/10.1007/s12652-017-0565-6
Chen, L, Gao, R, Bian, Y, and Di, H (2021). Elliptic entropy of uncertain random variables with application to portfolio selection. Soft Computing. 25, 1925-1939. https://doi.org/10.1007/s00500-020-05266-z
He, Z, Ahmadzade, H, Rezaei, K, Rezaei, H, and Naderi, H (2021). Tsallis entropy of uncertain random variables and its application. Soft Computing. 25, 11735-11743. https://doi.org/10.1007/s00500-021-06070-z
Liu, Y, and Ralescu, DA. (2017) . Value-at-risk in uncertain random risk analysis. Information Sciences. 391–392, 1-8. https://doi.org/10.1016/j.ins.2017.01.034
Qin, Z, Dai, Y, and Zheng, H (2017). Uncertain random portfolio optimization models based on value-at-risk. Journal of Intelligent & Fuzzy Systems. 32, 4523-4531. https://doi.org/10.3233/JIFS-169216
Liu, Y, Ralescu, DA, Xiao, C, and Lio, W (2020). Tail value-at-risk in uncertain random environment. Soft Computing. 24, 2495-2502. https://doi.org/10.1007/s00500-018-3492-3
Li, Q, Qin, Z, and Yan, Y (2022). Uncertain random portfolio optimization model with tail value-at-risk. Soft Computing. 26, 9385-9394. https://doi.org/10.1007/s00500-022-07249-8
Li, B, and Shu, Y (2022). The skewness for uncertain random variable and application to portfolio selection problem. Journal of Industrial & Management Optimization. 18, 457-467. https://doi.org/10.3934/jimo.2020163
Liu, B (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Heidelberg, Germany: Springer
Hou, Y (2014). Subadditivity of chance measure. Journal of Uncertainty Analysis and Applications. 2. article no 14
Liu, Y (2013). Uncertain random programming with applications. Fuzzy Optimization and Decision Making. 12, 153-169. https://doi.org/10.1007/s10700-012-9149-2
Ahmadzade, H, Sheng, Y, and Hassantabar Darzi, F (2017). Some results of moments of uncertain random variables. Iranian Journal of Fuzzy Systems. 14, 1-21. http://doi.org/10.22111/ijfs.2017.3131
Sheng, Y, Shi, G, and Ralescu, DA (2017). Entropy of uncertain random variables with application to minimum spanning tree problem. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems. 25, 497-514. https://doi.org/10.1142/S0218488517500210
Simonelli, MR (2005). Indeterminacy in portfolio selection. European Journal of Operational Research. 163, 170-176. https://doi.org/10.1016/j.ejor.2004.01.006

Biography

Go to

Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion
Conflict of Interest
Tables
References
Biography

Seyyed Hamed Abtahi is a risk analyst at Bank Pasargad. He received his Ph.D. degree in Statistics from Science and Research Branch, Islamic Azad University (SRBIAU) in 2022. He also received his M.Sc. in Mathematical Statistics from Allameh Tabataba’i University in 2014. His research interests include uncertainty theory, chance theory, and entropy with application to portfolio risk management.

E-mail: dr abtahi@yahoo.com

Article

Original Article

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

Uncertain Random Portfolio Optimization Based on Skew Chance Distribution

Seyyed Hamed Abtahi

Bank Pasargad, Tehran, Iran

Correspondence to:Seyyed Hamed Abtahi (dr_abtahi@yahoo.com)

Received: August 31, 2022; Revised: January 24, 2023; Accepted: March 7, 2023

Abstract

Keywords: Chance theory, Uncertain random variable, Skew chance distribution, Entropy, Portfolio optimization

1. Introduction

2. Preliminaries

This section reviews concepts of uncertainty theory and chance theory including the definition of uncertain variables, uncertainty distribution, uncertain random variables, and chance distribution.

2.1 Uncertainty Theory

Axiom 1 (Normality). For the universal set Γ, M{Γ} = 1.
Axiom 2 (Duality). For any event, M{∧} +M{∧_c} = 1.
Axiom 3 (Subadditivity). For every countable sequence of events ∧₁, ∧₂, . . ., we have

$M {\cup_{i = 1}^{\infty} \land_{i}} \leq \sum_{i = 1}^{\infty} M {\land_{i}} .$

Subsequently, (Γ, L, M) is called an uncertainty space.
Axiom 4 (Product). Suppose (Γ_i, ℒ_i,M_i) are the uncertainty spaces for i = 1, 2, . . . . Product uncertainty measure M is an uncertain measure that satisfies

$M {\prod_{i = 1}^{\infty} \land_{i}} = ⋀_{i = 1}^{\infty} M_{i} {\land_{i}},$

where ∧_i is an arbitrarily chosen event from L_i for i = 1, 2, . . . , respectively.

Definition 1 (Liu [26])

An uncertain variable is a function τ from an uncertainty space (Γ, L, M) to the set of real numbers such that {τ ∈ B} is an event for any Borel set B of real numbers.

Definition 2 (Liu [48])

For all Borel sets B₁, B₂, . . . , B_n of real numbers, the uncertain variables τ₁, τ₂, . . . , τ_n are independent if

M {\cap_{i = 1}^{n} (τ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {τ_{i} \in B_{i}} .

Definition 3 (Abtahi et al. [35])

An uncertain variable τ is called skew-normal and denoted by SN(m, p, δ) if it has a skew-normal uncertainty distribution

γ (x) = {(1 + exp (\frac{π (m - x)}{\sqrt{3} δ}))}^{- p}, x \in R,

where m, δ (δ > 0) and p (p > 0) are real numbers.

Remark 1 (Abtahi et al. [35])

Parameter m specifies distribution location, parameter δ (δ > 0) specifies distribution spread and parameter p (p > 0) specifies distribution shape. There are three conditions for p as follows:

1. If p = 1, then the uncertain variable τ is a normal uncertainty distribution.
2. If p > 1, then the uncertain variable τ is a positive skew-normal uncertainty distribution.
3. If 0 < p < 1, then the uncertain variable τ is a negative skew-normal uncertainty distribution.

Theorem 1 (Liu [26])

A function γ⁻¹(r) is an inverse uncertainty distribution if and only if γ⁻¹(r) is a strictly increasing continuous function with respect to r.

Definition 4 (Abtahi et al. [35])

Let τ ∼ SN(m, p, δ); then, the inverse uncertainty distribution of the skew-normal uncertain variable τ is as follows:

γ^{- 1} (r) = m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}); 0 < r < 1.

Theorem 2 (Liu [48])

Let τ be an uncertain variable with a regular uncertainty distribution γ(x). If the expected value of τ exists, then

E [τ] = \int_{0}^{1} γ^{- 1} (r) d r,

where γ⁻¹(r) is the inverse uncertainty function of τ with respect to r.

Theorem 3 (Abtahi et al. [35])

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n; then the expected value of the skew-normal uncertain variable τ_i is

E [τ_{i}] = m_{i} - \frac{\sqrt{3} δ_{i}}{π p_{i}} - \frac{\sqrt{3} δ_{i}}{π} \int_{0}^{1} l n (1 - u^{\frac{1}{p_{i}}}) d u .

Theorem 4

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n, then the variance of the skew-normal uncertain variable τ_i is

V [τ_{i}] = \int_{0}^{1} {(m_{i} + \frac{\sqrt{3} δ_{i}}{π} ln (\frac{r^{\frac{1}{p_{i}}}}{1 - r^{\frac{1}{p_{i}}}}) - (m_{i} - \frac{\sqrt{3} δ_{i}}{π p_{i}} - \frac{\sqrt{3} δ_{i}}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p_{i}}}) d u))}^{2} d r .

Proof

This follows from the definition of variance for uncertain variables [26],

V [τ] = \int_{0}^{1} {(γ^{- 1} (r) - E [τ])}^{2} d r .

Theorem 5

Let τ_i ∼ SN(m_i, p_i, δ_i); ∀i = 1, 2, . . . , n; then, the skewness of the skew-normal uncertain variable τ_i is

S [τ_{i}] = \int_{0}^{1} {(m_{i} + \frac{\sqrt{3} δ_{i}}{π} ln (\frac{r^{\frac{1}{p_{i}}}}{1 - r^{\frac{1}{p_{i}}}}) - (m_{i} - \frac{\sqrt{3} δ_{i}}{π p_{i}} - \frac{\sqrt{3} δ_{i}}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p_{i}}}) d u))}^{3} d r .

Proof

This follows from the definition of the K^th central moment of the uncertain variables for K = 3 [26],

S [τ] = \int_{0}^{1} {(γ^{- 1} (r) - E [τ])}^{3} d r .

Definition 5 (Liu [27])

Let τ be an uncertain variable with uncertainty distribution γ(x). Subsequently, the logarithmic entropy of the uncertain variable τ is

H [τ] = \int_{- \infty}^{+ \infty} L (γ (x)) d x,

where L(t) = −(t) ln(t) − (1 − t) ln(1 − t).

Theorem 6 (Dai and Chen [30])

Let τ be an uncertain variable with inverse uncertainty distribution γ⁻¹(r). Subsequently, the logarithmic entropy of τ is

H [τ] = \int_{0}^{1} γ^{- 1} (r) ln (\frac{r}{1 - r}) d r .

Theorem 7 (Liu [32])

ϕ^{- 1} (r) = f (γ_{1}^{- 1} (r), \dots, γ_{m}^{- 1} (r), γ_{m + 1}^{- 1} (1 - r), \dots, γ_{n}^{- 1} (1 - r)) .

2.2 Chance Theory

C h {θ} = \int_{0}^{1} P r {ω \in Ω ∣ M {γ \in Γ ∣ (γ, ω) \in Θ} \geq r} d r .

Liu [36] demonstrated that a chance measure satisfies the following properties:

(i) (normality) Ch(Γ × Ω) = 1;
(ii) (duality) Ch{Θ} + Ch{Θ^c} = 1, for any event Θ;
(iii) (monotonicity) Ch{Θ₁} < Ch{Θ₂} for any real number set Θ₁ ⊂ Θ₂.

Furthermore, Hou [49] proved that, for a sequence of events Θ₁, Θ₂, . . . , a chance measure satisfies the following subadditivity:

C h {\cup_{i = 1}^{\infty} Θ_{i}} \leq \sum_{i = 1}^{\infty} C h (Θ_{i}) .

Definition 6 (Liu [36])

An uncertain random variable is a function ξ from a chance space (Γ, L, M) × (Ω, A, Pr) to the set of real numbers such that for any Borel set B of real numbers, {ξ ∈ B} is an event in L × A.

Definition 7 (Liu [36])

Suppose ξ is an uncertain random variable. Subsequently, the chance distribution of ξ for any x ∈ R is defined by

Φ (x) = C h {ξ \leq x} .

Definition 8 (Sheng et al. [38])

Definition 9 (Liu [36])

Let ξ be an uncertain random variable. Subsequently, the expected value of ξ is defined as

E [ξ] = \int_{0}^{+ \infty} C h {ξ \geq x} d x - \int_{- \infty}^{0} C h {ξ \leq x} d x,

provided that at least one of the two integrals is finite.

Definition 10 (Liu [36])

Let ξ be an uncertain random variable with a finite expected value E[ξ]. Subsequently, the variance of ξ is

V [ξ] = E {(ξ - E [ξ])}^{2} .

Definition 11 (Liu [36])

Let ξ be an uncertain random variable with a finite expected value E[ξ]. Subsequently, the skewness of ξ is defined as follows:

S [ξ] = E {(ξ - E [ξ])}^{3} .

Theorem 8 (Liu [50])

Φ (x) = \int_{R^{m}} F (x; y_{1}, \dots, y_{m}) d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

where F(x; y₁, . . . , y_m) is the uncertainty distribution of the uncertain variable f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) for any real numbers y₁, y₂, . . . , y_m.

Theorem 9 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

E [ξ] = \int_{R^{m}} \int_{0}^{1} F^{- 1} (r, y_{1}, \dots, y_{m}) \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 10 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

V [ξ] = \int_{R^{m}} \int_{0}^{1} {(F^{- 1} (r, y_{1}, \dots, y_{m}) - E [ξ])}^{2} \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 11 (Ahmadzade et al. [51])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively. Suppose ξ = f(η₁, η₂, ..., η_m, τ₁, τ₂, ..., τ_n), then

S [ξ] = \int_{R^{m}} \int_{0}^{1} {(F^{- 1} (r, y_{1}, \dots, y_{m}) - E [ξ])}^{3} \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Remark 2

It is obvious that F⁻¹(r, y₁, . . . , y_m) = γ⁻¹(r)+ y₁ + · · · + y_m.

Definition 12 (Sheng et al. [52])

Let ξ be an uncertain random variable with chance distribution Φ(x). Subsequently, the entropy of ξ is defined as

H [ξ] = \int_{- \infty}^{+ \infty} S (Φ (x)) d x,

where S(t) = −t ln t − (1 − t) ln(1 − t).

Theorem 12 (Ahmadzade et al. [39])

P H [ξ] = \int_{R^{m}} \int_{- \infty}^{+ \infty} S (F (x, y_{1}, \dots, y_{m})) \times d x d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

Theorem 13 (Ahmadzade et al. [39])

Let η₁, η₂, . . . , η_m be independent random variables with probability distributions Ψ₁, Ψ₂, . . . , Ψ_m, and let τ₁, τ₂, . . . , τ_n be independent uncertain variables with uncertainty distributions γ₁, γ₂, . . . , γ_n, respectively, and let f be a measurable function. Subsequently, ξ = f(η₁, η₂, . . . , η_m, τ₁, τ₂, . . . , τ_n) has partial entropy

P H [ξ] = \int_{R^{m}} \int_{- \infty}^{+ \infty} F^{- 1} (r, y_{1}, \dots, y_{m}) ln \frac{r}{1 - r} \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{m} (y_{m}),

where F⁻¹(r, y₁, . . . , y_m) is the inverse uncertainty distribution of uncertain variable f(y₁, y₂, . . . , y_m, τ₁, τ₂, . . . , τ_m).

Theorem 14 (Ahmadzade et al. [5])

Let τ be an uncertain variable with uncertainty distribution function γ, and let η be a random variable with probability distribution function Ψ. If ξ = η + τ , then

P H [ξ] = H [τ] .

Theorem 15 (Ahmadzade et al. [39])

P H [a ξ_{1} + b ξ_{2}] = ∣ a ∣ P H [ξ_{1}] + ∣ b ∣ P H [ξ_{2}] .

3. Skew Chance Distributions

Theorem 16

Let η be a normal distribution denoted by N(μ, σ²) with the probability density function

f (y) = \frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}}),

and let τ be a skew-normal uncertain variable denoted by SN(m, p, δ) with an uncertainty distribution

γ (x) = {(1 + exp (\frac{π (m - x)}{\sqrt{3} δ}))}^{- p}; x \in R .

Then, for an uncertain random variable ξ = η + τ .

(i) The expected value of ξ is

$\begin{array}{l} E [ξ] = E [η] + E [τ] \\ = μ + m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u . \end{array}$
(ii) The variance of ξ is

$\begin{array}{l} V [ξ] = V [η] + V [τ] \\ = σ^{2} + \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) - (m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u))}^{2} d r . \end{array}$
(iii) The skewness of ξ is

$\begin{array}{l} S [ξ] = S [η] + S [τ] \\ = \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) - (m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u))}^{3} d r . \end{array}$

Proof

(i) According to Theorems 9 and 2, the expected value of the uncertain random variable ξ is

\begin{array}{l} E [ξ] = \int_{- \infty}^{+ \infty} \int_{0}^{1} F^{- 1} (r, y) d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} (γ^{- 1} (r) + y) d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} (m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) + y) \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} y (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ + \int_{- \infty}^{+ \infty} \int_{0}^{1} (m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}})) \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = \int_{- \infty}^{+ \infty} y (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d y \\ + (\int_{- \infty}^{+ \infty} (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d y) \\ \times (\int_{0}^{1} (m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}})) d r) \\ = E [η] + E [τ] \\ = μ + m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u . \end{array}

(ii) According to Theorems 10 and 2, the variance of the uncertain random variable ξ is

\begin{array}{l} V [ξ] = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(F^{- 1} (r, y) - E (ξ))}^{2} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(γ^{- 1} (r) + y - E (ξ))}^{2} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(γ^{- 1} (r) + y - E [η] - E [τ])}^{2} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) + y - E [η] - E [τ])}^{2} \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} (m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) \\ + y - μ - (m - \frac{\sqrt{3} δ}{π p} \\ - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln {(1 - u^{\frac{1}{p}}) d u))}^{2} \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = V [η] + V [τ] \\ = σ^{2} + \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) - (m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u))}^{2} d r . \end{array}

(iii) According to Theorem 11 and Remark 2, the skewness of ξ is

\begin{array}{l} S [ξ] = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(F^{- 1} (r, y) - E (ξ))}^{3} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(γ^{- 1} (r) + y - E (ξ))}^{3} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(γ^{- 1} (r) + y - E [η] - E [τ])}^{3} d r d Ψ (y) \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) + y - E [η] - E [τ])}^{3} \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = \int_{- \infty}^{+ \infty} \int_{0}^{1} (m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) \\ + y - μ - (m - \frac{\sqrt{3} δ}{π p} \\ - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln {(1 - u^{\frac{1}{p}}) d u))}^{3} \\ \times (\frac{1}{\sqrt{2 π} σ} exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}})) d r d y \\ = S [η] + S [τ] \\ = 0 + \int_{0}^{1} {(m + \frac{\sqrt{3} δ}{π} ln (\frac{r^{\frac{1}{p}}}{1 - r^{\frac{1}{p}}}) - (m - \frac{\sqrt{3} δ}{π p} - \frac{\sqrt{3} δ}{π} \int_{0}^{1} ln (1 - u^{\frac{1}{p}}) d u))}^{3} d r . \end{array}

The proof is completed.

Remark 3

By setting p = 1 in Theorem 16,

(i) the expected value of ξ is E[ξ] = μ + m;
(ii) the variance of ξ is V [ξ] = σ² + δ²;
(iii) the skewness of ξ is S[ξ] = 0.

4. Uncertain Random Portfolio Optimization

(1)

\begin{array}{l} M i n P H [x_{1} ξ_{1} + \dots + x_{n} ξ_{n}], \\ S.t. \\ E [x_{1} ξ_{1} + \dots + x_{n} ξ_{n}] \geq C, \\ x_{1} + x_{2} + \dots + x_{n} = 1, \\ 0 \leq x_{i} \leq 1; \forall i = 1, 2, \dots, n, \end{array}

where the predetermined parameter C is designated by an investor.

By applying the expected value formula for uncertain random variables ξ_i = τ_i +η_i; i = 1, 2, . . . , n in Theorem 9, we obtain

\begin{array}{l} E [x_{1} ξ_{1} + \dots + x_{n} ξ_{n}] = \int_{R^{n}} \int_{0}^{1} (x_{1} {F_{1}}^{- 1} (r, y_{1}) + \dots + x_{n} {F_{n}}^{- 1} (r, y_{n})) \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{n} (y_{n}) \\ = \int_{R^{n}} \int_{0}^{1} (x_{1} ({γ_{1}}^{- 1} (r) + y_{1}) + \dots + x_{n} ({γ_{n}}^{- 1} (r) + y_{n})) \times d r d Ψ_{1} (y_{1}) \dots d Ψ_{n} (y_{n}) \\ = x_{1} (E (τ_{1}) + E (η_{1})) + \dots + x_{n} (E (τ_{n}) + E (η_{n})) . \end{array}

According to Theorems 13 and 14, the partial entropy of uncertain random variables ξ_i = τ_i + η_i; i = 1, 2, . . . , n is obtained as follows:

\begin{array}{l} P H [x_{1} ξ_{1} + \dots + x_{n} ξ_{n}] = P H [x_{1} ξ_{1}] + \dots + P H [x_{n} ξ_{n}] \\ = x_{1} P H [ξ_{1}] + \dots + x_{n} P H [ξ_{n}] \\ = x_{1} H [τ_{1}] + \dots + x_{n} H [τ_{n}], \end{array}

where H[τ_i] is the logarithm entropy of uncertain variable τ_i; i = 1, 2, . . . , n.

Thus, Model (1) is equivalent to the following model,

(2)

\begin{array}{l} M i n x_{1} H [τ_{1}] + \dots + x_{n} H [τ_{n}], \\ S.t. \\ x_{1} (E (τ_{1}) + E (η_{1})) + \dots + x_{n} (E (τ_{n}) + E (η_{n})) \geq C, \\ x_{1} + x_{2} + \dots + x_{n} = 1, \\ 0 \leq x_{i} \leq 1; \forall i = 1, 2, \dots, n . \end{array}

Example 1

Model (2) is equivalent to the following model:

(3)

\begin{array}{l} M i n 1.2697 x_{1} + 0.7515 x_{2} + 0.5641 x_{3} + 2.1095 x_{4} + 1.2661 x_{5}, \\ S.t. \\ 2 x_{1} + 3.2757 x_{2} + 1.3308 x_{3} + 2.1456 x_{4} \\ + 2.6178 x_{5} \geq 2.5, \\ x_{1} + x_{2} + \dots + x_{5} = 1, \\ 0 \leq x_{i} \leq 1; \forall i = 1, 2, \dots, 5. \end{array}

5. Conclusion

Abstract
1. Introduction
2. Preliminaries
2.2 Chance Theory
3. Skew Chance Distributions
4. Uncertain Random Portfolio Optimization
5. Conclusion

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

References

Markowitz, H (1952). Portfolio selection. The Journal of Finance. 7, 77-91. https://doi.org/10.2307/2975974
Sheikh, AZ, and Qiao, H (2010). Non-normality of market returns: a framework for asset allocation decision making. The Journal of Alternative Investments. 12, 8-35. https://doi.org/10.3905/JAI.2010.12.3.008
Mercurio, PJ, Wu, Y, and Xie, H (2020). An entropy-based approach to portfolio optimization. Entropy. 22. article no 332
Shannon, CE (1948). A mathematical theory of communication. The Bell System Technical Journal. 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Philippatos, GC, and Wilson, CJ (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics. 4, 209-220. https://doi.org/10.1080/00036847200000017
Rockafellar, RT, and Uryasev, S (2000). Optimization of conditional value-at-risk. Journal of Risk. 2, 21-42. https://doi.org/10.21314/JOR.2000.038
Rockafellar, RT, and Uryasev, S (2002). Conditional value-atrisk for general loss distributions. Journal of Banking & Finance. 26, 1443-1471. https://doi.org/10.1016/S0378-4266(02)00271-6
Artzner, P, Delbaen, F, Eber, JM, and Heath, D (1999). Coherent measures of risk. Mathematical Finance. 9, 203-228. https://doi.org/10.1111/1467-9965.00068
Jorion, P (1996). Risk2: measuring the risk in value at risk. Financial Analysts Journal. 52, 47-56. https://doi.org/10.2469/faj.v52.n6.2039
Ahmadi-Javid, A (2012). Entropic value-at-risk: a new coherent risk measure. Journal of Optimization Theory and Applications. 155, 1105-1123. https://doi.org/10.1007/s10957-011-9968-2
Ahmadi-Javid, A, and Fallah-Tafti, M (2019). Portfolio optimization with entropic value-at-risk. European Journal of Operational Research. 279, 225-241. https://doi.org/10.1016/j.ejor.2019.02.007
Arditti, FD (1967). Risk and the required return on equity. The Journal of Finance. 22, 19-36. https://doi.org/10.2307/2977297
Arditti, FD, and Levy, H (1975). Portfolio efficiency analysis in three moments: the multiperiod case. The Journal of Finance. 30, 797-809. https://doi.org/10.1111/j.1540-6261.1975.tb01851.x
Chunhachinda, P, Dandapani, K, Hamid, S, and Prakash, AJ (1997). Portfolio selection and skewness: Evidence from international stock markets. Journal of Banking & Finance. 21, 143-167. https://doi.org/10.1016/S0378-4266(96)00032-5
Keel, S, Herzog, F, Geering, H, and Mirjolet, N (). Optimal portfolios with skewed and heavy-tailed distributions, 42-48.
Konno, H, Shirakawa, H, and Yamazaki, H (1993). A mean-absolute deviation-skewness portfolio optimization model. Annals of Operations Research. 45, 205-220. https://doi.org/10.1007/BF02282050
Hu, W, and Kercheval, AN (2010). Portfolio optimization for student t and skewed t returns. Quantitative Finance. 10, 91-105. https://doi.org/10.1080/14697680902814225
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Liu, B, and Liu, YK (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems. 10, 445-450. https://doi.org/10.1109/TFUZZ.2002.800692
Kwakernaak, H (1978). Fuzzy random variables. I. Definitions and theorems. Information Sciences. 15, 1-29. https://doi.org/10.1016/0020-0255(78)90019-1
Kwakernaak, H (1979). Fuzzy random variables. II. Algorithms and examples for the discrete case. Information Sciences. 17, 253-278. https://doi.org/10.1016/0020-0255(79)90020-3
Deng, X, He, X, and Huang, C (2022). A new fuzzy random multi-objective portfolio model with different entropy measures using fuzzy programming based on artificial bee colony algorithm. Engineering Computations. 39, 627-649. https://doi.org/10.1108/EC-11-2020-0654
Khanjani Shiraz, R, Tavana, M, and Fukuyama, H (2020). A random-fuzzy portfolio selection DEA model using value-at-risk and conditional value-at-risk. Soft Computing. 24, 17167-17186. https://doi.org/10.1007/s00500-020-05010-7
Wu, X, Ralescu, DA, and Liu, Y (2020). A new quadratic deviation of fuzzy random variable and its application to portfolio optimization. Iranian Journal of Fuzzy Systems. 17, 1-18. https://doi.org/10.22111/IJFS.2020.5344
Liu, B (2012). Why is there a need for uncertainty theory. Journal of Uncertain Systems. 6, 3-10.
Liu, B (2007). Uncertainty Theory. Heidelberg, Germany: Springer
Liu, B (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems. 3, 3-10.
Chen, X, Kar, S, and Ralescu, DA (2012). Cross-entropy measure of uncertain variables. Information Sciences. 201, 53-60. https://doi.org/10.1016/j.ins.2012.02.049
Chen, X, and Dai, W (2011). Maximum entropy principle for uncertain variables. International Journal of Fuzzy Systems. 13, 232-236.
Dai, W, and Chen, X (2012). Entropy of function of uncertain variables. Mathematical and Computer Modelling. 55, 754-760. https://doi.org/10.1016/j.mcm.2011.08.052
Abtahi, SH, Yari, G, Lotfi, FH, and Farnoosh, R (2022). Pseudo-triangular entropy of uncertain variables: an entropy-based approach to uncertain portfolio optimization. International Journal of Mathematical Modelling & Computations. 12, 61-77.
Dai, W (2018). Quadratic entropy of uncertain variables. Soft Computing. 22, 5699-5706. https://doi.org/10.1007/s00500-017-2602-y
Tang, W, and Gao, W (2013). Triangular entropy of uncertain variables. Information (Tokyo). 16, 1279-1282.
Yao, K, Gao, J, and Dai, W (2013). Sine entropy for uncertain variables. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems. 21, 743-753. https://doi.org/10.1142/S0218488513500359
Abtahi, SH, Yari, G, Lotfi, FH, and Farnoosh, R (2021). Multi-objective portfolio selection based on skew-normal uncertainty distribution and asymmetric entropy. International Journal of Fuzzy Logic and Intelligent Systems. 21, 38-48. http://doi.org/10.5391/IJFIS.2021.21.1.38
Liu, Y (2013). Uncertain random variables: a mixture of uncertainty and randomness. Soft Computing. 17, 625-634. https://doi.org/10.1007/s00500-012-0935-0
Guo, H, and Wang, X (2014). Variance of uncertain random variables. Journal of Uncertainty Analysis and Applications. 2. article no 6
Sheng, Y, and Yao, K (2014). Some formulas of variance of uncertain random variable. Journal of Uncertainty Analysis and Applications. 2. article no 12
Ahmadzade, H, Gao, R, Dehghan, MH, and Sheng, Y (2017). Partial entropy of uncertain random variables. Journal of Intelligent & Fuzzy Systems. 33, 105-112. http://doi.org/10.3233/JIFS-161161
Ahmadzade, H, Gao, R, Dehghan, MH, and Ahmadi, R (2018). Partial triangular entropy of uncertain random variables and its application. Journal of Ambient Intelligence and Humanized Computing. 9, 1455-1464. https://doi.org/10.1007/s12652-017-0565-6
Chen, L, Gao, R, Bian, Y, and Di, H (2021). Elliptic entropy of uncertain random variables with application to portfolio selection. Soft Computing. 25, 1925-1939. https://doi.org/10.1007/s00500-020-05266-z
He, Z, Ahmadzade, H, Rezaei, K, Rezaei, H, and Naderi, H (2021). Tsallis entropy of uncertain random variables and its application. Soft Computing. 25, 11735-11743. https://doi.org/10.1007/s00500-021-06070-z
Liu, Y, and Ralescu, DA. (2017) . Value-at-risk in uncertain random risk analysis. Information Sciences. 391–392, 1-8. https://doi.org/10.1016/j.ins.2017.01.034
Qin, Z, Dai, Y, and Zheng, H (2017). Uncertain random portfolio optimization models based on value-at-risk. Journal of Intelligent & Fuzzy Systems. 32, 4523-4531. https://doi.org/10.3233/JIFS-169216
Liu, Y, Ralescu, DA, Xiao, C, and Lio, W (2020). Tail value-at-risk in uncertain random environment. Soft Computing. 24, 2495-2502. https://doi.org/10.1007/s00500-018-3492-3
Li, Q, Qin, Z, and Yan, Y (2022). Uncertain random portfolio optimization model with tail value-at-risk. Soft Computing. 26, 9385-9394. https://doi.org/10.1007/s00500-022-07249-8
Li, B, and Shu, Y (2022). The skewness for uncertain random variable and application to portfolio selection problem. Journal of Industrial & Management Optimization. 18, 457-467. https://doi.org/10.3934/jimo.2020163
Liu, B (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Heidelberg, Germany: Springer
Hou, Y (2014). Subadditivity of chance measure. Journal of Uncertainty Analysis and Applications. 2. article no 14
Liu, Y (2013). Uncertain random programming with applications. Fuzzy Optimization and Decision Making. 12, 153-169. https://doi.org/10.1007/s10700-012-9149-2
Ahmadzade, H, Sheng, Y, and Hassantabar Darzi, F (2017). Some results of moments of uncertain random variables. Iranian Journal of Fuzzy Systems. 14, 1-21. http://doi.org/10.22111/ijfs.2017.3131
Sheng, Y, Shi, G, and Ralescu, DA (2017). Entropy of uncertain random variables with application to minimum spanning tree problem. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems. 25, 497-514. https://doi.org/10.1142/S0218488517500210
Simonelli, MR (2005). Indeterminacy in portfolio selection. European Journal of Operational Research. 163, 170-176. https://doi.org/10.1016/j.ejor.2004.01.006

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Uncertain Random Portfolio Optimization Based on Skew Chance Distribution

Abstract

1. Introduction

2. Preliminaries

2.1 Uncertainty Theory

Theorem 5

Definition 5 (Liu [27])

Theorem 6 (Dai and Chen [30])

Theorem 7 (Liu [32])

2.2 Chance Theory

Definition 6 (Liu [36])

Definition 7 (Liu [36])

Definition 8 (Sheng et al. [38])

Definition 9 (Liu [36])

Definition 10 (Liu [36])

Definition 11 (Liu [36])

Theorem 8 (Liu [50])

Theorem 9 (Ahmadzade et al. [51])

Theorem 10 (Ahmadzade et al. [51])

Theorem 11 (Ahmadzade et al. [51])

Remark 2

Definition 12 (Sheng et al. [52])

Theorem 12 (Ahmadzade et al. [39])

Theorem 13 (Ahmadzade et al. [39])

Theorem 14 (Ahmadzade et al. [5])

Theorem 15 (Ahmadzade et al. [39])

3. Skew Chance Distributions

Theorem 16

Remark 3

4. Uncertain Random Portfolio Optimization

Example 1

5. Conclusion

Conflict of Interest

Tables

References

Biography

Article

Original Article

Uncertain Random Portfolio Optimization Based on Skew Chance Distribution

Abstract

1. Introduction

2. Preliminaries

2.1 Uncertainty Theory

Theorem 5

Definition 5 (Liu [27])

Theorem 6 (Dai and Chen [30])

Theorem 7 (Liu [32])

2.2 Chance Theory

Definition 6 (Liu [36])

Definition 7 (Liu [36])

Definition 8 (Sheng et al. [38])

Definition 9 (Liu [36])

Definition 10 (Liu [36])

Definition 11 (Liu [36])

Theorem 8 (Liu [50])

Theorem 9 (Ahmadzade et al. [51])

Theorem 10 (Ahmadzade et al. [51])

Theorem 11 (Ahmadzade et al. [51])

Remark 2

Definition 12 (Sheng et al. [52])

Theorem 12 (Ahmadzade et al. [39])

Theorem 13 (Ahmadzade et al. [39])

Theorem 14 (Ahmadzade et al. [5])

Theorem 15 (Ahmadzade et al. [39])

3. Skew Chance Distributions

Theorem 16

Remark 3

4. Uncertain Random Portfolio Optimization

Example 1

5. Conclusion

References

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