Mean-Gini portfolio selection with uncertain returns

1 Introduction

Mean-Gini portfolio selection model with random returns was proposed by Yitzhaki [29] and Shalit and Yitzhaki [22]. In fact, in this approach, Gini coefficient was considered as a risk measure rather than variance in mean-variance Markowitz model [18]. After that, several authors devoted their studies to the case of portfolio selection via considering other risk measures. Konno and Yamazaki [14] optimized portfolio selection problems of random returns via mean-absolute deviation models. Favre and Galeano [7] discussed about mean-modified value at risk optimization with hedge funds. By invoking value at risk, Agouram and Lakhnati [1] solved portfolio selection of random returns.

However, in an indeterminate economic environment, it is not possible to predict accurately the future returns and risks associated with the various financial assets, based exclusively on historical data. Fuzzy set theory can deal with indeterminacy and vagueness which are usual features of any stock market. Fuzzy logic [30] can express vague knowledge making it suitable for representing the inherent indeterminate nature of the portfolio optimization problem. Some scholars devoted their works to the case of portfolio selection problem, for instance [5, 24]. Furthermore, Sadjadi et al. [15] introduced different rates for borrowing and lending into the multi-period portfolio optimization model with fuzzy returns. Bertsimas and Pachamanova [4] handled the fuzzy multi-period portfolio optimization problem with the assistance of a computationally efficient robust optimization approach. Zhang et al. [32] proposed a mean semi-variance-entropy model for fuzzy multi-period portfolio selection. Qin et al. [20] introduced the concept of cross entropy for fuzzy variables and applied it to portfolio optimization model with fuzzy returns. Also, Wu et al. [25] studied multi-criteria group decision-making for portfolio allocation with consensus reaching process under interval type-2 fuzzy environment.

As described in [17], some authors have noted that human uncertainty is not adequately described by fuzziness. This fact was one of the reasons why Baoding Liu founded uncertainty theory. In fact, in many situations, we face with lack of data to derive empirical distribution functions. In order to model in these situations, Liu [16] proposed the concept of uncertain variables as a main devise in uncertainty theory rather than fuzzy variables in fuzzy set theory. Now, we recall some related papers in the case of portfolio selection for uncertain returns. Qin et al. [21] optimized the portfolio selection problems for uncertain returns via mean-variance models. By invoking risk curve, Huang [11] solved portfolio selection problems for uncertain returns via mean- risk models. After that, Huang [12] proposed the concept of risk index for uncertain returns and applied it to portfolio optimization. Kar et al. [13] optimized portfolio selection problems with uncertain returns by considering value at risk and sharp ration as the risk measures. By minimizing semi-absolute deviation, Qin et al. [19] solved portfolio problem for uncertain returns. Furthermore, Zhou et al. [33] obtained optimal portfolio-proportions of uncertain returns via mean-entropy models. Bhattacharyya et al. [3] applied mean-entropy-skewness to uncertain portfolio problems with transaction costs.

It is mentioned some risk measures such as variance and absolute deviation are strongly dependent to expected value(mean). Also, some other risk measures such as semi-variance, semi-entropy and semi absolute deviation are down risk measures and consider the values less than mean. Furthermore, other measures such as skewness is affected by symmetry of distributions. But, Gini coefficient can well reflect the dispersive degree of the portfolio return no matter if the uncertainty distributions are symmetric or not. In addition, in mean-Gini model, we encounter with linear programming rather than quadratic programming in mean-variance model.

In this paper, we present the concept of Gini coefficient for uncertain variables and obtain a formula via inverse uncertainty distributions. Also, we explain Gini coefficient of uncertain variables as an expectation of random variables and apply this property to compute Gini coefficient via Monte-Carlo simulation. As an application of Gini coefficient, we optimize portfolio selection problems with uncertain returns via mean-Gini models.

This paper is organized as follows. Some preliminaries about uncertain measure, uncertain variable, expected value, variance and covariance are recalled in Section 2. Definition of Gini coefficient for uncertain variables is proposed and some properties of this concept is investigated in Section 3. As an application of Gini coefficient, portfolio selection problem is optimized via mean-Gini models in Section 4. Finally, some conclusions are derived in Section 5.

2 Preliminaries

In this section, we recall some preliminaries about uncertain measure, uncertain variable, uncertainty distribution, operational law, expected value and so on.

Definition 1. (Liu [16]) Let Γ be a nonempty set, and L be a σ- algebra over L . An uncertain measure M is a set function defined on L satisfying the following four axioms:

Axiom 1: (Normality Axiom) M {Γ} =1.

Axiom 2: (Normality Axiom) M {A} + M {A ^c } =1 for any event A ∈ L .

Axiom 3: (Subadditivity Axiom) For every countable sequence of events A₁, A₂, ⋯ , we have M { ⋃ n = 1 ∞ A n } ≤ ∑ n = 1 ∞ M { A n } .

Axiom 4: (Product Measure Axiom) Let Γ _k be nonempty sets on which _Mk are uncertain measures, k = 1, 2, ⋯ , n respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L = L 1 × L 2 × ⋯ × L n satisfying M { ∏ i = 1 n A i } = min 1 ≤ i ≤ n M { A i } .

Definition 2. (Liu [16]) An uncertain variable τ is a function 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 3. (Liu [16]) The uncertainty distribution Φ of an uncertain variable τ is defined by Φ ( x ) = M { τ ≤ x } for any real number x.

Definition 4. (Liu [16]) An uncertainty distribution Φ (x) is said to be regular if it is a continuous and strictly increasing function with respect to 0 < Φ (x) <1, and lim x → - ∞ Φ ( x ) = 0 , lim x → ∞ Φ ( x ) = 1 .

It is clear that a regular uncertainty distribution Φ (x) has an inverse function on the range of x with 0 < Φ (x) <1, and the inverse function Φ^-1 (α) exists on the open interval (0, 1).

Definition 5. (Liu [16]) Let τ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of τ.

Definition 6. (Liu [16]) The uncertain variables τ₁, τ₂, ⋯ , τ _n are said independent if M { ⋂ i = 1 n ( τ i ∈ B i ) } = ⋀ i = 1 n M { τ i ∈ B i } for any Borel sets B₁, B₂, ⋯ , B _n of real numbers.

The following theorem provides an operational law for calculating the uncertainty distributions of strictly increasing function.

Theorem 1. (Liu [16]) Let τ₁, τ₂, ⋯ , τ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f is a continuous and strictly increasing function, then τ = f (τ₁, τ₂, ⋯ , τ _n ) has an inverse uncertainty distribution Φ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( α ) , ⋯ , Φ n - 1 ( α ) ) . In particular, τ = τ₁ + τ₂ + ⋯ + τ _n has the inverse uncertainty distribution Φ - 1 ( α ) = Φ 1 - 1 ( α ) + Φ 2 - 1 ( α ) + ⋯ + Φ n - 1 ( α ) .

Expected value is the average value of uncertain variable in the sense of uncertain measure.

Definition 7. (Liu [16]) Let τ be an uncertain variable. Then the expected value of τ is defined by E [ τ ] = ∫ 0 ∞ M { τ ≥ x } d x - ∫ - ∞ 0 M { τ ≤ x } d x , provided that at least one of the two integrals is finite.

Theorem 2. (Liu [16]) Let τ be an uncertain variable with regular uncertainty distribution Φ. Then E [ τ ] = ∫ 0 1 Φ - 1 ( α ) d α .

The variance of uncertain variable provides a degree of the spread of the distribution around its expected value.

Definition 8. (Liu [16]) Let τ be an uncertain variable with finite expected value e. Then the variance of τ is V [ τ ] = E [ ( τ - e ) 2 ] .

Theorem 3. (Yao [28]) Let τ be an uncertain variable with regular uncertainty distribution Φ and finite expected value e. Then V [ τ ] = ∫ 0 1 ( Φ - 1 ( α ) - e ) 2 d x .

Definition 9. (Zhao et al. [31]) Let τ₁ and τ₂ be two uncertain variables. The covariance of τ₁ and τ₂ is defined by Cov [ τ 1 , τ 2 ] = E [ ( τ 1 - μ 1 ) ( τ 2 - μ 2 ) ] , where μ₁ and μ₂ are the expected values of τ₁ and τ₂, respectively.

Theorem 4. (Zhao et al. [31]) Let τ₁ and τ₂ be two regular uncertain variables with distributions Φ₁ and Φ₂ and finite expected values μ₁ and μ₂, respectively. Then the covariance of τ₁ and τ₂ is Cov [ τ 1 , τ 2 ] = ∫ 0 1 ( Φ 1 - 1 ( α ) - μ 1 ) ( Φ 2 - 1 ( α ) - μ 2 ) d α .

Theorem 5. (Zhao et al. [31]) Suppose τ₁ and τ₂ are independent regular uncertain variables with expected values μ₁ and μ₂, respectively. Then, V [ τ 1 + τ 2 ] = V [ τ 1 ] + V [ τ 2 ] + 2 Cov [ τ 1 , τ 2 ] .

3 Gini coefficient for uncertain variables

In this section, we present the concept of Gini coefficient for uncertain variables. Also, a formula for calculating Gini coefficient of uncertain variables is obtained via inverse uncertainty distributions. It is mentioned that Gini coefficient is a single number that demonstrates a degree of inequality in a distribution of income. As similar to variance, Gini coefficient measures dispersion of income variables.

First, we recall the concept of Gini coefficient for random variables which presented by Dorfman [6].

Definition 10. (Dorfman [6]) Suppose X is a random variable with probability distribution function G. Gini coefficient of random variable X is defined by

Γ ( X ) = 2 Cov ( X , G ( X ) ) .(1)

By inception of relation (1), we propose the concept of Gini coefficient for uncertain variables.

Definition 11. Suppose τ is an uncertain variable with uncertainty distribution Φ. Gini coefficient of uncertain variable τ is defined by

Γ ( τ ) = 2 Cov ( τ , Φ ( τ ) ) .(2)

Theorem 6. Let τ be an uncertain variable with uncertainty distributions Φ. Then τ has Gini coefficient Γ ( τ ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α

Proof. Since uncertainty distribution Φ is increasing, operational law implies the inverse uncertainty distribution of uncertain variable Φ (τ) is equal to Φ^-1 (Φ (α)) = α. Also, expected value of uncertain variable Φ (τ) is E [ Φ ( τ ) ] = ∫ 0 1 Φ - 1 ( Φ ( α ) ) d α = ∫ 0 1 α d α = 1 2 . By invoking Theorem 2, we have Γ ( τ ) = 2 Cov ( τ , Φ ( τ ) ) = 2 ∫ 0 1 ( Φ - 1 ( α ) - E [ τ ] ) ( Φ - 1 ( Φ ( α ) ) - E [ Φ ( τ ) ] ) d α = 2 ∫ 0 1 ( Φ - 1 ( α ) - E [ τ ] ) ( α - 1 2 ) d α = 2 ∫ 0 1 ( Φ - 1 ( α ) - E [ τ ] ) α d α - 1 2 ∫ 0 1 ( Φ - 1 ( α ) - E [ τ ] ) d α = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α , where, μ = E [τ].

Remark 1. By using Theorem 6, we can calculate Gini coefficient of uncertain variables via Monte-Carlo simulation. In fact, Gini coefficient can be explained as follows Γ ( τ ) = E Pr [ U ( Φ - 1 ( U ) - μ ) ] , where U is a standard uniform random variable. Thus, by invoking Monte-Carlo simulation and strong law of large numbers, the sample mean is an approximation for Gini coefficient of uncertain variables. In fact, we can consider the following mean as an approximation of Gini coefficient. 1 n ∑ i = 1 n u i ( Φ - 1 ( u i ) - μ ) , where, u₁, ⋯ , u _n are a sample generated from the standard uniform distribution.

Theorem 7. Let τ be an uncertain variable with uncertainty distribution Φ. Then, we have Γ ( τ ) ≤ 2 Var ( τ ) 3 .

Proof. By using Theorem 6 and Cauchy-Schwarz inequality, we have Γ ( τ ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α ≤ 2 ∫ 0 1 ( Φ - 1 ( α ) - μ ) 2 d α ∫ 0 1 α 2 d α = 2 Var ( τ ) 3 .

Example 1. Suppose τ ∼ N ( 0 , 1 ) is a normal uncertain variable. Then Gini coefficient of τ is Γ ( τ ) = 2 ∫ 0 1 α ( 3 π ln α 1 - α ) d α = 0.5513289 . It is easy to see that Γ ( τ ) = 0.5513289 ≤ 1.154701 = 2 3 3 = 2 Var ( τ ) 3 .

Example 2. Suppose τ ∼ LOGN ( 0 , 1 ) is a normal uncertain variable. Then Gini coefficient of τ is Γ ( τ ) = 2 ∫ 0 1 α . exp ( 3 π ln α 1 - α ) d α = 1.154701 .

Example 3. Suppose τ ∼ L (a, b) is a normal uncertain variable. Then Gini coefficient of τ is Γ ( τ ) = 2 ∫ 0 1 α ( a ( 1 - α ) + b α - a + b 2 ) d α = 2 ( a ∫ 0 1 α d α - a ∫ 0 1 α 2 d α + b ∫ 0 1 α 2 d α - a + b 2 ∫ 0 1 α d α ) = b - a 6 .

Example 4. Suppose τ₁ and τ₂ are independent uncertain variables with uncertainty distributions N ( 0 , 1 ) and N ( 1 , 2 ) , respectively. Consider τ = τ₁ τ₂ as an uncertain variable. Γ ( τ ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α , By invoking operational law, we have Φ - 1 ( α ) = Φ 1 - 1 ( α ) Φ 2 - 1 ( α ) . Also, we have E [ τ ] = E [ τ 1 τ 2 ] = ∫ 0 1 Φ 1 - 1 ( α ) Φ 2 - 1 ( α ) d α = ∫ 0 1 ( 3 π ln ( α 1 - α ) ) ( 1 + 2 3 π ln ( α 1 - α ) ) d α = 2 . Thus, Γ ( τ ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α , = 2 ∫ 0 1 α ( ( 3 π ln ( α 1 - α ) ) ( 1 + 2 3 π ln ( α 1 - α ) ) - 2 ) d α = 0.5512 .

Example 5. Suppose τ₁ and τ₂ are independent uncertain variables with uncertainty distributions N ( 0 , 1 ) and LOGN ( 0 , 1 ) , respectively. Consider τ = τ 1 τ 2 as an uncertain variable. μ = E [ τ ] = E [ τ 1 τ 2 ] = ∫ 0 1 Φ 1 - 1 ( α ) Φ 2 - 1 ( 1 - α ) d α = ∫ 0 1 ( 3 π ln ( α 1 - α ) ) exp ( 3 π ln ( 1 - α α ) ) d α = 2.24919 . Then, Γ ( τ ) = ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α = 2 ∫ 0 1 α ( Φ 1 - 1 Φ 2 - 1 ( 1 - α ) - μ ) d α = 2 ∫ 0 1 α ( ( 3 π ln ( α 1 - α ) ) exp ( 3 π ln ( 1 - α α ) ) - 2.24919 ) d α = 2.20759 .

Theorem 8. Suppose τ₁ and τ₂ are independent uncertain variables with uncertainty distributions Φ₁ and Φ₂, respectively. Consider τ = τ₁ + τ₂ as an uncertain variable. Then Γ (τ₁ + τ₂) = Γ (τ₁) + Γ (τ₂) .

Proof. By invoking operational law, we have Φ - 1 ( α ) = Φ 1 - 1 ( α ) + Φ 2 - 1 ( α ) . Also, linearity property of expected value for independent uncertain variables implies that

E [ τ ] = E [ τ 1 + τ 2 ] = E [ τ 1 ] + E [ τ 2 ] . Then Γ ( τ 1 + τ 2 ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - μ ) d α = 2 ∫ 0 1 α ( Φ 1 - 1 ( α ) + Φ 2 - 1 ( α ) - E [ τ 1 ] - E [ τ 2 ] ) d α = 2 ∫ 0 1 α ( Φ 1 - 1 ( α ) - E [ τ 1 ] + Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α = 2 ∫ 0 1 α ( Φ 1 - 1 ( α ) - E [ τ 1 ] ) d α + 2 ∫ 0 1 α ( Φ 2 - 1 ( α ) - E [ τ 2 ] ) d α = Γ ( τ 1 ) + Γ ( τ 2 ) .

In many situations, we encounter with some new markets and we have not enough frequency to estimate empirical distribution function. Therefore, we assign uncertain variables to the securities. Also, we invoke uncertainty distribution functions rather probability distribution functions.

Suppose we have n securities with uncertain returns τ₁, τ₂, ⋯ , τ _n , respectively. Also, consider x _i ’s as investment proportions in security i, i = 1, 2, . . . , n . We consider R p = ∑ i = 1 n x i τ i as the return of portfolio. In order to find the portfolio proportions, we present two following mean-Gini models.

{ max x i E [ R p ] subject to : Γ ( R p ) ≤ λ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , where, λ is predetermined parameters. In another model, the investor likes to minimize risk via entropy under constrained expected value as follows: { min x i Γ ( R p ) subject to : E [ R p ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , By using operational law, we have inverse uncertainty distribution of uncertain variable R _p is equal to Φ - 1 ( α ) = ∑ i = 1 n x i Φ i - 1 ( α ) . Also, linearity property of expected value for independent uncertain variables implies that E [ R p ] = E [ x 1 τ 1 + x 2 τ 2 + ⋯ + x n τ n ] = ∑ i = 1 n x i E [ τ i ] . Therefore, Gini coefficient for return of portfolio R _p is equal to

Γ ( R p ) = 2 Cov ( R p , Φ ( R p ) ) = 2 ∫ 0 1 α ( Φ - 1 ( α ) - E [ R p ] ) = 2 ∫ 0 1 α ( ∑ i = 1 n x i Φ i - 1 ( α ) - ∑ i = 1 n x i E [ τ i ] ) d α = 2 ∫ 0 1 α ∑ i = 1 n ( Φ i - 1 ( α ) - E [ τ i ] ) d α = 2 ∑ i = 1 n ∫ 0 1 α ( Φ i - 1 ( α ) - E [ τ i ] ) d α = ∑ i = 1 n x i Γ ( τ i ) .

Therefore, the above mean-Gini models can be converted to the following optimization problems.

{ max x i ∑ i = 1 n x i E [ τ i ] subject to : ( 1 ) ∑ i = 1 n x i Γ ( τ i ) ≤ λ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , Another model is reduced via minimizing Gini coefficient with constrained expected value.

{ min x i ∑ i = 1 n x i Γ ( τ i ) subject to : ( 2 ) ∑ i = 1 n x i E [ τ i ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , Consider five securities with uncertain returns τ _i , i = 1, 2, ⋯ , 5 shown in Table 1.

Now, we want to optimize portfolio selection of uncertain returns via maximizing mean under constrained Gini coefficient. Thus, model (1) can be converted to the following crisp model. Optimal solutions are displayed in Table 2.

{ max x i 0.07 * x 1 + 0.06 * x 2 + 0.09 * x 3 + 0.22 * x 4 + 0.13 * x 5 subject to : ( 3 ) 0.017 * x 1 + 0.066 * x 2 + 0.01 * x 3 + 0.033 * x 4 + 0.017 * x 5 ≤ λ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n ,

Here, we want to optimize portfolio selection of uncertain returns via minimizing Gini coefficient under constrained expected value. Thus, model (2) can be converted to the following crisp model. Optimal solutions are displayed in Table 3.

{ min x i 0.017 * x 1 + 0.066 * x 2 + 0.01 * x 3 + 0.033 * x 4 + 0.017 * x 5 subject to : ( 4 ) 0.07 * x 1 + 0.06 * x 2 + 0.09 * x 3 + 0.22 * x 4 + 0.13 * x 5 ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n ,

Furthermore, we want to compare above model with mean-variance model. Thus, we consider portfolio selection problem via minimizing variance under constrained expected value.

{ min x i V [ R p ] subject to : E [ R p ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n ,

By using linearity property of expected value and Theorem 2, above model converted to the following quadratic model.

{ min x i ∑ i = 1 n ∑ i = 1 n x i 2 V [ τ i ] + ∑ i = 1 n ∑ j = 1 n x i x j Cov [ τ i , τ j ] subject to : ∑ i = 1 n x i E [ τ i ] ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n , Thus, for this five securities, above model reduced to the following crisp one.

{ min x i 0.0001 12 * x 1 + 0.0016 12 * x 2 + 0.0036 12 * x 3 + 0.04 12 * x 4 + 0.01 12 * x 5 + 0.0004 12 * x 1 * x 2 + 0.006 12 x 1 x 3 + 0.02 12 * x 1 * x 4 + 0.0024 12 * x 2 * x 3 + 0.008 12 * x 2 * x 4 + 0.004 12 * x 2 * x 5 + 0.012 12 * x 3 * x 4 + 0.006 12 * x 3 * x 5 + 0.02 12 * x 4 * x 5 subject to : ( 5 ) 0.07 * x 1 + 0.06 * x 2 + 0.09 * x 3 + 0.22 * x 4 + 0.13 * x 5 ≥ δ x 1 + x 2 + . . . + x n = 1 , x i ≥ 0 , i = 1 , 2 , . . . , n ,

It is mentioned that several authors invoked performance ratio for compared their proposed model with other models, for instance [2, 27]. Therefore, for comparing mean-variance and mean-Gini models with together, we recall the performance ratio as follows: PR = E [ R p ] V [ R p ] , performance ration is denoted by PR.

For same fixed δ in model (4) and (5), we obtain optimal proportions and its related performance ratio in model (4) and (5). Also, we iterate this procedure for the same value of δ in model (4) and (5). In fact, we consider the pair (PRMG, PRMV) for any fixed δ, where, PRMG and PRMV are performance ratio in model (4) and (5), respectively. Finally, we invoke Wilcoxon-nonparametric test for related samples in SPSS software for detecting significant difference between performance ratio in model (4) and (5). Figure 1 implies there is not significant difference between performance ratio in model (4) and (5). It is mentioned that we encounter with linear programming in mean-Gini model rather than quadratic programming in mean-variance model. Also, variance of total return has n + n ( n - 1 ) 2 terms, whiles Gini coefficient of total return has n terms. Thus, we prefer mean-Gini model rather than mean-variance model for uncertain returns.

OPEN IN VIEWER

Fig. 1

Comparative analysis of difference between performance ratio for same values of δ in model (14) and (15).

This paper proposed the concept of Gini coefficient for uncertain variables as a risk measure, and investigated its properties. Also, Monte-Carlo simulation was presented for calculating Gini coefficient of uncertain variables. For better illustration of main results, we showed how portfolio selection of uncertain returns can be solved by invoking mean- Gini model, and provided several examples. In order to compare mean-Gini model with mean-variance one, Wilcoxon nonparametric test was applied to the pair of performance ratio in mean-Gini and mean-variance model. Although, there was not significant difference between performance ratio in mean-Gini and mean-variance models. Since, Gini coefficient of total return has n terms, whiles, variance of total return has n + n ( n - 1 ) 2 terms, we preferred Mean-Gini linear programming rather than mean-variance quadratic programming model. Considering mean-Gini portfolio selection involving new listed markets and historical ones as a mixture of uncertainty and randomness maybe a potential work for future research. In the real world, we encounter with dependent phenomena, invoking dependence structure in portfolio selection problems via copula modeling is another future research.