Portfolio Selection Models Based on Interval-Valued Conditional Value-at-Risk (ICVaR) and Case Study on the Data from Stock Markets

: Risk management is very important for individual investors or companies. There are several ways to measure the risk of investment. Prices of risky assets vary rapidly and randomly due to the complexity of ﬁnance market. Random interval is a good tool to describe uncertainty including both randomness and imprecision. Considering the uncertainty of ﬁnancial market, we employ random intervals to describe returns of a risk asset and deﬁne an interval-valued risk measurement, which considers the tail risk. It is called the interval-valued conditional value-at-risk (ICVaR, for short). Similar to the classical conditional value-at-risk, ICVaR satisﬁes the sub-additivity. Under the new risk measure ICVaR, as a manner similar to the classical Mean-CVaR portfolio model, two optimal interval-valued portfolio selection models are built. The sub-additivity of ICVaR guarantees the global optimal solution to the Mean-ICVaR portfolio model. Based on the real data from mainland Chinese stock markets and international stock markets, the case study shows that our models are interpretable and consistent with the practical scenarios.


Introduction
A portfolio is a collection of stocks, bonds, and financial derivatives held by investors or financial institutions to spread risk. Since Markowitz published the pioneering paper on Mean-Variance model in 1952, there have been a large number of papers on the applications and extension of the classical models. It is known that one major drawback of Markowitz's model is that the solutions are sensitive to the underlying parameters, i.e., mean and covariance matrix of assets. In order to overcome the sensitivity, some robust portfolio models are proposed. For example, Blanchet et al. [1] (2021) studied a distributionally robust version of Markowitz's Mean-Variance model, in which the Wasserstein distance is used to describe the discrepancy between probability measures. For more research on the robustness of portfolio selection models, readers can refer to some references in [1].
Risk measurement is very important for investors or financial analysts and must be considered in portfolio selection models. Besides variance, there are some other ways to measure the risk of assets. For example, Konnor and Yamazaki [2] (1991) studied the optimal portfolio problem based on absolute deviation or semi-variance, which is easier to calculate than variance of portfolio. Value at risk (VaR, for short) is a financial risk measurement that has been developed and widely used (see, e.g., [3][4][5]). Hamel et al. [6] extended real-valued VaR to a set-valued case. As a risk measure, VaR is convenient and easy to calculate. However, VaR has no nice mathematical properties such as convexity and sub-additivity, which are crucial to solve the optimal problem. The Mean-VaR portfolio model may not have a global optimal solution since VaR is not a convex risk measurement. In order to overcome the drawback of VaR, the conditional value-at-risk (CVaR, for short) was proposed,

Preliminaries
In this section, we list the knowledge of interval-valued variables and interval-valued programming needed later.

Operations and Order of Intervals
Let R be the set of all real numbers, A, B real closed intervals.
If a L = a U = a, the interval A = [a, a] degenerates to a one-element set A = {a}.
For any intervals A, B and any real number λ, the Minkowski addition and scalarmultiplication are defined as follows (see e.g., [21]): Let m(A) := 1 2 (a L + a U ), w(A) := 1 2 (a U − a L ), which are called the midpoint and half-width (or simply be termed as 'width') of the interval A, respectively. Then, A can also be determined by m(A) and w(A).
Ranking intervals is a crucial step to study interval-valued optimization problems. There are infinitely many ways to rank intervals [22]. The popular and natural way is using the endpoints or their functions such as midpoint and width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating the IVaR and ICVaR. Multiple intervals can be ranked well according to this rule.

Interval-Valued Optimization
We shall employ the method introduced in [20] (2001) to solve interval-valued optimization portfolio. In [20], at first, the authors defined an index Γ to describe the acceptability for ranking two intervals. Let w(A) := 1 2 (a U − a L ), which are called the midpoint and half-width ( or simply be terval A, respectively. Then A also can be determined by m(A) and w(A). a crucial step to study interval-valued optimization problems. There are infinitely ls [22]. The popular and natural way is using the endpoints or their functions such as s paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating for two intervals ization method introduced in [20] (2001) to solve interval-valued optimization portfolio. In fined an index Γ to describe the acceptability for ranking two intervals. Let be a or any closed intervals A and B, define The value of function Γ(A B), denoted by γ, may be interpreted as the grade of al A is less than the interval B'. bility of A B may be classified and interpreted further on the basis of comparative idth of interval B with respect to those of interval A as follows: premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter ess than B' with different grades of satisfaction ranging from zero to one (excluding ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
degenerates to a one-element and any real number λ, the Minkowski addition and scalar-multiplication are defined , which are called the midpoint and half-width ( or simply be terval A, respectively. Then A also can be determined by m(A) and w(A). a crucial step to study interval-valued optimization problems. There are infinitely ls [22]. The popular and natural way is using the endpoints or their functions such as is paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating . for two intervals ization method introduced in [20] (2001) to solve interval-valued optimization portfolio. In efined an index Γ to describe the acceptability for ranking two intervals. Let be a For any closed intervals A and B, define The value of function Γ(A B), denoted by γ, may be interpreted as the grade of al A is less than the interval B'. bility of A B may be classified and interpreted further on the basis of comparative idth of interval B with respect to those of interval A as follows: premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter less than B' with different grades of satisfaction ranging from zero to one (excluding ) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
where w(B) + w(A) = 0. The value of function Γ(A degenerates to a one-element {a}. or any intervals A, B and any real number λ, the Minkowski addition and scalar-multiplication are defined ows (see e.g. [11]) , which are called the midpoint and half-width ( or simply be as 'width')of the interval A, respectively. Then A also can be determined by m(A) and w(A). anking intervals is a crucial step to study interval-valued optimization problems. There are infinitely ays to rank intervals [22]. The popular and natural way is using the endpoints or their functions such as int and width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating R and ICVaR. E.g. for two intervals terval-valued optimization e shall employ the method introduced in [20] (2001) to solve interval-valued optimization portfolio. In t first the authors defined an index Γ to describe the acceptability for ranking two intervals. Let be a order of intervals. For any closed intervals A and B, define w(B) + w(A) 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of ability of 'the interval A is less than the interval B'. he grade of acceptability of A B may be classified and interpreted further on the basis of comparative n of midpoint and width of interval B with respect to those of interval A as follows: B) = 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter B), denoted by γ, may be interpreted as the grade of acceptability of 'the interval A is less than the interval B'.
The grade of acceptability of A degenerates to a one-element A, B and any real number λ, the Minkowski addition and scalar-multiplication are defined ]) , which are called the midpoint and half-width ( or simply be e interval A, respectively. Then A also can be determined by m(A) and w(A). s is a crucial step to study interval-valued optimization problems. There are infinitely rvals [22]. The popular and natural way is using the endpoints or their functions such as this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating .g. for two intervals 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of terval A is less than the interval B'. ptability of A B may be classified and interpreted further on the basis of comparative d width of interval B with respect to those of interval A as follows: B may be classified and interpreted further on the basis of comparative position of midpoint and width of interval B with respect to those of interval A as follows: , which are called the midpoint and half-width ( or simply be y. Then A also can be determined by m(A) and w(A). dy interval-valued optimization problems. There are infinitely and natural way is using the endpoints or their functions such as ule "midpoint-first, left-second" to rank intervals for calculating on Γ(A B), denoted by γ, may be interpreted as the grade of interval B'. be classified and interpreted further on the basis of comparative B), denoted by γ, may be interpreted as the grade of '. ed and interpreted further on the basis of comparative t to those of interval A as follows: (2) ot accepted. If 0 < Γ(A B) < 1, then the interpreter es of satisfaction ranging from zero to one (excluding tely satisfied with the premise 'A B'.
negative singleton variable. In order to solve linear pta et al. [20] proposed a satisfactory crisp equivalent B.
B) = 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A midpoint and width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating   [20], at first the authors defined an index Γ to describe the acceptability for ranking two intervals. Let be a 97 partial order of intervals. For any closed intervals A and B, define where w(B) + w(A) 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of 99 acceptability of 'the interval A is less than the interval B'.

100
The grade of acceptability of A B may be classified and interpreted further on the basis of comparative 101 position of midpoint and width of interval B with respect to those of interval A as follows: If Γ(A B) = 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter 103 accepts the premise 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding 104 zero and one). If Γ(A B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear 107 programming problems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent 108 form for the interval inequality relations Ax ≤ B or Ax ≥ B.

109
B) < 1, then the interpreter accepts the premise 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding zero and one). If Γ(A Ranking intervals is a crucial step to study interval-valued optimization problems. There are infinitely 89 many ways to rank intervals [22]. The popular and natural way is using the endpoints or their functions such as 90 midpoint and width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating 91 the IVaR and ICVaR. E.g. for two intervals   [20], at first the authors defined an index Γ to describe the acceptability for ranking two intervals. Let be a 97 partial order of intervals. For any closed intervals A and B, define where w(B) + w(A) 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of 99 acceptability of 'the interval A is less than the interval B'.

100
The grade of acceptability of A B may be classified and interpreted further on the basis of comparative 101 position of midpoint and width of interval B with respect to those of interval A as follows: If Γ(A B) = 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter 103 accepts the premise 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding 104 zero and one). If Γ(A B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear 107 programming problems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent 108 form for the interval inequality relations Ax ≤ B or Ax ≥ B.

109
B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A f the interval A, respectively. Then A also can be determined by m(A) and w(A). vals is a crucial step to study interval-valued optimization problems. There are infinitely intervals [22]. The popular and natural way is using the endpoints or their functions such as . In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating R. E.g. for two intervals optimization loy the method introduced in [20] (2001) to solve interval-valued optimization portfolio. In hors defined an index Γ to describe the acceptability for ranking two intervals. Let be a rvals. For any closed intervals A and B, define ) 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of interval A is less than the interval B'. cceptability of A B may be classified and interpreted further on the basis of comparative t and width of interval B with respect to those of interval A as follows: en the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding A B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear lems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent l inequality relations Ax ≤ B or Ax ≥ B.
x be a nonnegative singleton variable. In order to solve linear programming problems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent form for the interval inequality relations Ax ≤ B or Ax ≥ B.
The satisfactory crisp equivalent form of Ax ≤ B is given as follows ( [20]): ntervals A, B and any real number λ, the Minkowski addition and scalar-multiplication are defined e.g. [11]) , which are called the midpoint and half-width ( or simply be th')of the interval A, respectively. Then A also can be determined by m(A) and w(A). intervals is a crucial step to study interval-valued optimization problems. There are infinitely rank intervals [22]. The popular and natural way is using the endpoints or their functions such as width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating w(A) 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of f 'the interval A is less than the interval B'. e of acceptability of A B may be classified and interpreted further on the basis of comparative dpoint and width of interval B with respect to those of interval A as follows: 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter mise 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding If Γ(A B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear problems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent terval inequality relations Ax ≤ B or Ax ≥ B.
where γ may be interpreted as an optimistic threshold given and fixed by the decision maker. Similarly, for Ax ≥ B, the satisfactory crisp equivalent form is: degenerates to a one-element intervals A, B and any real number λ, the Minkowski addition and scalar-multiplication are defined ee e.g. [11]) , which are called the midpoint and half-width ( or simply be idth')of the interval A, respectively. Then A also can be determined by m(A) and w(A). g intervals is a crucial step to study interval-valued optimization problems. There are infinitely o rank intervals [22]. The popular and natural way is using the endpoints or their functions such as width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating , denoted by γ, may be interpreted as the grade of of 'the interval A is less than the interval B'. de of acceptability of A B may be classified and interpreted further on the basis of comparative idpoint and width of interval B with respect to those of interval A as follows: = 0, then the premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter remise 'A is less than B' with different grades of satisfaction ranging from zero to one (excluding ). If Γ(A B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear g problems with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent interval inequality relations Ax ≤ B or Ax ≥ B.
Now, we consider the maximizing interval-valued linear programming problem: Maximize list the knowledge of interval-valued variables and interval-valued programming needed er of intervals all real numbers, A, B real closed intervals.
degenerates to a one-element , B and any real number λ, the Minkowski addition and scalar-multiplication are defined ]) , which are called the midpoint and half-width ( or simply be interval A, respectively. Then A also can be determined by m(A) and w(A). is a crucial step to study interval-valued optimization problems. There are infinitely rvals [22]. The popular and natural way is using the endpoints or their functions such as this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating .g. for two intervals 0. The value of function Γ(A B), denoted by γ, may be interpreted as the grade of erval A is less than the interval B'. ptability of A B may be classified and interpreted further on the basis of comparative d width of interval B with respect to those of interval A as follows: he premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter is less than B' with different grades of satisfaction ranging from zero to one (excluding B) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'.
] and x be a nonnegative singleton variable. In order to solve linear with interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent quality relations Ax ≤ B or Ax ≥ B.
According to [20], the principle of γ-index indicates that, for the maximization (minimization) problem, an interval with a higher midpoint is superior (inferior) to an interval with a lower mid-value. Therefore, in order to obtain the max (min) of the interval-valued objective function, considering the midpoint is our primary concern. We reduce the interval objective function its central value and use conventional LP (Linear Programming) techniques for its solution. We also consider the width but as a secondary attribute, only to confirm whether it is within the acceptable limit of the decision maker. If it is not, one has to reduce the extent of width (uncertainty) according to his satisfaction and thus to obtain a less wide interval from the non-dominated alternatives accordingly. Thus, by the satisfactory crisp equivalent form (3), the following LP problem (6) is the necessary equivalent form of the model (5): Similarly, the minimizing interval linear programming can be constructed as follows:

Minimize
degenerates to a one-element ny real number λ, the Minkowski addition and scalar-multiplication are defined , which are called the midpoint and half-width ( or simply be A, respectively. Then A also can be determined by m(A) and w(A). ial step to study interval-valued optimization problems. There are infinitely . The popular and natural way is using the endpoints or their functions such as r, we use the rule "midpoint-first, left-second" to rank intervals for calculating , f or j = 1, · · · , k x i ≥ 0, i = 1, · · · , n. According to (4), the interval-valued programming problem (7) can be transformed to the following real valued linear programming model: (8): For any intervals A, B and any real number λ, the Minkowski addition and scalar-multiplication are defined as follows (see e.g. [11]) Let m(A) := 1 2 (a L + a U ), w(A) := 1 2 (a U − a L ), which are called the midpoint and half-width ( or simply be 87 termed as 'width')of the interval A, respectively. Then A also can be determined by m(A) and w(A).

88
Ranking intervals is a crucial step to study interval-valued optimization problems. There are infinitely 89 many ways to rank intervals [22]. The popular and natural way is using the endpoints or their functions such as 90 midpoint and width. In this paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating If Γ(A B) = 0, then the premise 'A is less than B' is not accepted.  . If a = a = a, the interval A = [a, a] degenerates to a one-element umber λ, the Minkowski addition and scalar-multiplication are defined U − a L ), which are called the midpoint and half-width ( or simply be ctively. Then A also can be determined by m(A) and w(A). to study interval-valued optimization problems. There are infinitely pular and natural way is using the endpoints or their functions such as the rule "midpoint-first, left-second" to rank intervals for calculating unction Γ(A B), denoted by γ, may be interpreted as the grade of the interval B'. may be classified and interpreted further on the basis of comparative l B with respect to those of interval A as follows: ess than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter h different grades of satisfaction ranging from zero to one (excluding reter is absolutely satisfied with the premise 'A B'.
of Ax ≤ B and x be a nonnegative singleton variable. In order to solve linear fficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent x ≤ B or Ax ≥ B.

Portfolio Selection Based on Icvar
We consider the portfolio problem with n risky assets. Due to the volatility and complexity of financial market, the profit is uncertain. There are several tools to describe uncertainty, for example, random variable, fuzzy variable, set-valued variable, etc. Here, the return is considered as a random interval, which includes not only randomness but also imprecision.

Interval-Valued Random Variable
Let (Ω, F , P) be a complete probability space and R be the real space equipped with the Borel sigma-algebra B(R). K(R) (K c (R)) denotes the family of all nonempty closed (nonempty, closed and convex, respectively) subsets of R . The mapping V : Ω → K(R) is called measurable set-valued mapping or a random set if, for any element B ∈ B(R), {ω ∈ Ω : V is called an interval-valued random variable or random interval if V is measurable and takes a value in K c (R).
For any X ∈ K c (R), the distribution function of V is defined by where ≺ is the partial order according to the rule 'midpoint-first, left-second'. V is called integrably bounded if Ω sup x∈V(ω) |x|dP < ∞. The real valued random variable f is called an integrable selection of V if f (ω) ∈ V(ω) a.s. and Ω f dP < ∞.
For an integrably bounded random set V, where L 1 (Ω, R) denotes the family of all integrable real valued random variables, the closure cl is taken in L 1 (Ω, R). For two random sets V and U, it is well known that V = U a.s. if and only if S 1 V = S 1 U ( [21]). The expectation of V is defined by which is closed in R. A random set V is an interval-valued random variable if and only if V = [ f , g], where both f and g are real valued random variables (see, e.g., [23]).

IVaR and ICVaR
In a manner similar to the usual value-at-risk and conditional value-at-risk, we define the interval-valued value-at-risk (IVaR) and interval-valued conditional value-at-risk (ICVaR) as follows: Definition 1. Let (Ω, F , P) be a complete probability space. R is the random interval-valued return of a risky asset. Given confidence level 1 − α (0 < α < 1), the value-at-risk is defined by where inf is taken according to the order ≺. Definition 2. Let (Ω, F , P) be a complete probability space. R is the random interval-valued return of a risky asset. Given confidence level 1 − α (0 < α < 1), IVaR is the value-at-risk. The conditional value-at-risk is defined by Similar to the real-valued CVaR, ICVaR describes the tail risk.

Remark 1.
The expectation of a random interval is linear for positive coefficients (c.f. [21]). As a manner similar to the real-valued CVaR, it can be proved that the risk measure ICVaR satisfies the sub-additivity.

Portfolio Models
Let the rate of return of asset R = [r L , r U ] be an interval-valued random variable, R ij = [r L ij , r U ij ] be the rate of return of the i-th security at the j-th period, and i = 1, · · · , n, j = 1, · · · , k.
Given the confidence level 1 − α, IVaR ij (ICVaR ij ) is the value-at-risk (conditional value-at-risk, resp.) of the i-th risky asset at the j-th period, i = 1, · · · , n, j = 1, · · · , k. Let x i (i = 1, · · · , 10) be the proportion of the i-th asset in the portfolio. By Remark 1, the value ICVaR of a portfolio is less than the linear composition of ICVaR i with weights x i ≥ 0 (i = 1, · · · , n), which consists with the rule that the portfolio can spread the risk. Now, we build two models of portfolio selection as follows: • Model 1 We consider the optimal portfolio problem under a given acceptable maximum of risk level:

Maximize
degenerates to a one-element y real number λ, the Minkowski addition and scalar-multiplication are defined if λ < 0.
:= 1 2 (a U − a L ), which are called the midpoint and half-width ( or simply be , respectively. Then A also can be determined by m(A) and w(A). al step to study interval-valued optimization problems. There are infinitely The popular and natural way is using the endpoints or their functions such as , we use the rule "midpoint-first, left-second" to rank intervals for calculating introduced in [20] (2001) to solve interval-valued optimization portfolio. In n index Γ to describe the acceptability for ranking two intervals. Let be a closed intervals A and B, define lue of function Γ(A B), denoted by γ, may be interpreted as the grade of ess than the interval B'. f A B may be classified and interpreted further on the basis of comparative interval B with respect to those of interval A as follows: e 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter B' with different grades of satisfaction ranging from zero to one (excluding e interpreter is absolutely satisfied with the premise 'A B'.
t system of Ax ≤ B L , b U ] and x be a nonnegative singleton variable. In order to solve linear where E(R i ) is the expectation return of the i-th asset, ICVaR 0j = [ICVaR L 0j , ICVaR U 0j ] is the acceptable maximum risk in the j-period, j = 1, · · · , k, which are subjective and given by the investors. By (6), the corresponding real-valued linear programming problem of (9) is represented as (10) below: where m(A) is the midpoint of interval A and w(A) the semi-width of A. γ ∈ (0, 1) is a given index in advance, which can describe the degree of risk appetite of investors. The larger γ, the lower the risk aversion: Minimize ll real numbers, A, B real closed intervals. Let A = [a L , a U ] = {a ∈ R : a L ≤ a ≤ a U }, b L ≤ b ≤ b U }. If a L = a U = a, the interval A = [a, a] degenerates to a one-element and any real number λ, the Minkowski addition and scalar-multiplication are defined , w(A) := 1 2 (a U − a L ), which are called the midpoint and half-width ( or simply be terval A, respectively. Then A also can be determined by m (A) and w(A). a crucial step to study interval-valued optimization problems. There are infinitely ls [22]. The popular and natural way is using the endpoints or their functions such as is paper, we use the rule "midpoint-first, left-second" to rank intervals for calculating . for two intervals A = [a L , a U ] and ization method introduced in [20] (2001) to solve interval-valued optimization portfolio. In efined an index Γ to describe the acceptability for ranking two intervals. Let be a or any closed intervals A and B, define The value of function Γ(A B), denoted by γ, may be interpreted as the grade of al A is less than the interval B'. bility of A B may be classified and interpreted further on the basis of comparative idth of interval B with respect to those of interval A as follows: premise 'A is less than B' is not accepted. If 0 < Γ(A B) < 1, then the interpreter less than B' with different grades of satisfaction ranging from zero to one (excluding ) ≥ 1, the interpreter is absolutely satisfied with the premise 'A B'. quivalent system of Ax ≤ B = [ b L , b U ] and x be a nonnegative singleton variable. In order to solve linear ith interval coefficients, Sengupta et al. [20] proposed a satisfactory crisp equivalent ality relations Ax ≤ B or Ax ≥ B.
Correspondingly, by (8), the interval-valued linear programming problem (11) is represented as the following real-valued linear programming model (12) below where m, w and γ have the same meaning as that in (10).

Remark 2.
Due to the complexity of practical market and random intervals, it is difficult to obtain the distribution function of the return of a risk asset. Therefore, it is almost impossible to calculate the exact values of E(R ij ), E(R i ), IVaR i , ICVaR i , etc. In the following case study, we use historical data to estimate the related values. For mathematical expectation, the moment estimator (13) is employed, which is consistent. For example, let r t , t = 1, · · · , T be the given observations of random variable R, and the estimation of E(R) is given as follows:

Case Study
This section presents two case studies based on the data from different stock markets. We use a historical simulation method to estimate the expectation, IVaR and ICVaR without any special assumption about the financial market. Therefore, there is nothing technically difficult to replicate the study for different data sets. The software R is employed to make the empirical analysis. The data used in the case studies are stock's daily closing price, daily highest price, and daily lowest price during the research period. The closing price of shares on day j is denoted by S j . S U j denotes the highest price on day j, and S L j denotes the lowest price on day j. According to the daily closing price, the highest price, and the lowest price of the stock, we calculate the exact value and the interval value of the return rate. Denote the real valued returns by r j := ln S j − ln S j−1 , r L j := ln S L j − ln S j−1 , r U j := ln S U j − ln S j−1 , R j := [r L j , r U j ]. R j is the interval of return on day j. As usual, the real valued return does not obey normal distribution. Then, it is not suitable to compute the real valued VaR by using the variance-covariance method. By using a historical simulation method, we estimate the values of real-valued VaR, CVaR and IVaR, ICVaR based on historical data. There are many methods to rank intervals. Here, the "midpoint-first, left-second" of the intervals ranking method is used to estimate IVaR and ICVaR under the confidence level 1 − α = 0.95. By the satisfactory crisp equivalent system, interval-valued programming problems are transformed into real-valued ones. Then, the optimal solutions of the corresponding portfolio models are obtained. •

Case 1
In the case study 1, we selected the stocks of ten listed companies of financial markets on the Chinese mainland as samples. The listed companies include oil, bank, coal, liquor, security, medicine, technology industry, and so on. The ticker symbols of stocks are listed in Table 1. Daily prices were collected in the period from January 2016 to September 2020 (Data source: http://www.finance.sina.cn (accessed on 1 October 2021)). In Table 2, you can find that returns of all assets are not normally distributed. We will use historical simulation method to estimate expectation of return. In Table 3, when the maximum risk level ICVaR 0 is different, the selection of assets is different. When taking the lower acceptable risk level ICVaR j0 = [0.003, 0.07], the investor prefers the stock ST05 (CCB, China Construction Bank) and ST08 (GF Securities Co., Ltd) since these two stocks have stable returns and lower risk compared with the stock ST06 and ST09, which implies that the investor is more conservative than one who takes ICVaR 0j = [0.008, 0.08]. Table 3 also shows that the less γ, the more allocation to assets with lower risk. Correspondingly, the weights of ST06 (Mediea Group) and ST09 (Wuliangye Group) increase with respect to γ. The result implies that ST06 and ST09 have higher returns but accompanied with higher risk, which is consistent with the actual situation of the market.
In Table 4, given the minimum return constraint R 0j = [−0.025, 0.025], and [−0.02, 0.02], to minimize the risk, it is obvious that ST05 has the highest proportion in each scenario, very different from the case in Model 1. Its lower risk property explains this result. In addition, one can figure out that the weight of stock ST05 increases with respect to γ. However, the proportion of ST07 behaves on the contrary. In a word, the solution is sensitive to the constraints and also depends on the index γ. One can also figure out the law from Tables 5 and 6. Table 3. Portfolio, Model 1 with the same ICVaR 0j , j = 1, · · · , 5.     Table 7 is the detailed information of six stocks from international markets. In order to visually present the daily returns of six stocks, we give the time series of return as Figure 1. The blue (red) curve represents the highest (lowest, respectively) daily return of the underlying stock. The green curve is the daily return based on daily closing prices. It can be seen that the first stock ST01(Barclays Bank PLC.) is very special. It is almost a straight line except some sharp jumps (−11.9% on 19 March 2021, −9.86% on 14 April 2021, −7.01% on 30 April 2021, −4.87% on 3 March 2022, −7.19% on 4 March 2022, and −9.35% on 7 March 2022). Maybe there was something important that happened. We will see that there is no allocation to stock ST01 (Barclays Bank PLC) according to the results of Model 1 and Model 2 because smaller volatility means less opportunity for big profits. According to the result of the Jarque-Bera test in Table 8, none of the daily returns are normally distributed.  Table 9 presents the daily return r j (j = 1, · · · , 6), the lower daily return r L j (j = 1, · · · , 6), the higher daily return r U j (j = 1, · · · , 6), and the midpoint r L j +r U j 2 (j = 1, · · · , 6). As you can see, the stock ST04 (Microsoft Corporation) has the best return, followed by stock ST06 (Toyota Motor Corporation). The stock ST01 (Barclays Bank PLC.) has the worst return. Table 10 shows the values of the lower bound of ICVaR, the upper bound of ICVaR, midpoint, and width. One can see that stock ST01 has the biggest ICVaR since there were several sharp drops. ICVaR is computed in terms of the quantile given level α. The stock ST02 (Hang Seng Bank Ltd.) has the smallest ICVaR, followed by ST06 and then ST04. That is why you will find that the investment shares are allocated to these three stocks according to Model 1 and Model 2.  Table 11 presents the result of Model 1 (9) under given constraint CVaR 0 and γ. Here, we take j = 1 since it is a small sample size. That is, all data from January 2022 to August 2022 are considered in one investment period. Then, one constraint condition is needed. In order to show the validity of Model 1, we solve the problem under serval different constraints ICVaR 0 with two levels of γ for comparison. Table 11 shows that the solution is sensitive to the constraint CVaR 0 . When taking CVaR 0 = [0.008, 0.08] (wide interval), all investment shares are allocated to ST02 since ST02 has lower risk and better return. If taking CVaR 0 = [0.001, 0.045] (less tolerance for risk), then the allocation is dispersed to ST02, ST04, and ST06. Among them, ST06 accounted for the main investment share since compared with ST02, it has much better return. In addition, compared with ST04, ST06 has much lower ICVaR.
Similar to Model 1, for Model 2 (11), we also take j = 1. The constraint condition is denoted by R 0 . For comparison, we take three different constraints. Solutions of Model 2 under three different constraints with five levels of γ are given in Table 12. Given the smallest return acceptable, investors select ST02 and ST06. It is reasonable since ST02 has the lowest ICVaR and is followed by that of ST06, which is consistent with minimizing the objective function. As the value of γ decreases, the investment share of ST02 decreases, while the investment share of ST06 and the optimal value of objective function increase. It can be explained that smaller γ means that the investors have less tolerance of lower than expected portfolio returns. High returns are often accompanied by high risks. Therefore, less γ leads to higher optimal ICVaR of the portfolio. The size of γ indicates the degree of risk preference of investors. From the results of the case study, such as in Tables 3, 4 and 12, it is seen that the investment share allocation changes significantly with different values of γ under appropriate constraint conditions. This means that a different risk preference has different weights for portfolio selection. In the process of solving the model, we need to give appropriate constraints, which must be based on the relevant returns and values of ICVaR calculated by historical data. For example, in Table 11, the constraint ICVaR 0 = [0.008, 0.08], it almost covers the ICVaR of six stocks, which leads to the entire investment in ST04. A similar thing will happen if