Portfolio selection models based on Cross-entropy of uncertain variables

Abstract

This paper researches the uncertain portfolio selection problem when security returns are hard to be well reflected by previous data. We take security returns as uncertain variables. Firstly, three new portfolio selection models are presented based on the cross-entropy of uncertain variables. Secondly, the crisp form of the new model is also provided when security returns are linear uncertain variables. In addition, the gravitation search algorithm and numerical integration are introduced to solve the proposed models. Finally, several numerical examples are presented to illustrate the effectiveness of the algorithm and the application of the mathematical models.

Keywords

Uncertain measure portfolio selection problem gravitation search algorithm cross-entropy

1 Introduction

The key problem to solve the portfolio selection problems is balance of benefits and risks. Markowitz [1] used a quantitative method to solve portfolio selection problems. This method was called mean-variance model which used the mean and variance to measure the return and risk of portfolio, respectively. So far, variance has been widely accepted as a risk measure, and many extension models have been proposed [2, 3]. However, these mean-variance models have large limited generality. Since variance considers high return as equally undesirable as low return, it will become an unreasonable risk measure when security returns are asymmetrical. Thus, Markowitz proposed semivariance to replace the variance as an improvement measure of risk, and numerous models have been presented based on semivariance such as models proposed in [4 –6]. In addition, many researchers presented some other risk measures such as the chance of a bad event. This method was also accepted by many researchers such as [7, 8].

Different from the variation of the Markowitz model, Kapur and Kesavan [9] presented an entropy maximization model and a cross entropy minimization model. The goal of the entropy model is to maximize the uncertainty of random return and the object of the cross entropy model is to minimize the divergence of the random investment return from a prior one. After that, many scholars accepted and presented these new models [10 –12].

In the above literatures, security returns are considered as random variables. In many cases, because of the complexity of securities market, the future performance is difficult to be predicted by using past security returns. When security returns are predicted, a lot of subjective factors should be considered such as macroeconomic factors and industry policy factors. All these factors not only impact on security returns at random, but also make the securities returns often exist fuzzy randomness.

Since fuzzy set theory and credibility theory were put forward, many scholars began to use them to research portfolio problems under uncertainty. For example, Bilbao-Terol et al. [13], Gupta et al. [14] extended the mean-variance model from different angles based on fuzzy set theory. Huang [15] developed fuzzy mean-variance models and further proposed fuzzy mean semivariance portfolio selection models based on credibility theory [16]. Li et al. [17] employed skewness to reflect asymmetry of fuzzy returns and further proposed fuzzy mean-variance-skewness models. Bhattacharyya et al. [18] proposed fuzzy mean-variance-skewness portfolio selection models by interval analysis. Qian and Yin [19] presented two new mathematical models by describing divergence as distance, investment return as expected value, and risk as variance and semivariance, respectively. Huang [20] proposed the fuzzy portfolio selection models by using the entropy method in fuzzy environment. Qin et al. [21] proposed three portfolio selection models based on fuzzy cross-entropy.

However, Liu [22] found a paradox when fuzzy variables are employed to describe subjective uncertain phenomena. Huang [23, 24] proved the existence of the paradox by using the credibility theory. In 2007, Liu [22] proposed uncertain measure and further founded uncertainty theory which can be used to handle subjective imprecise quantity. In 2010, Huang [25] employed the uncertain variable to describe the investment return firstly, and further established a series of uncertain portfolio selection models. Since then, many scholars began to employ the uncertainty theory to study portfolio selection problems. For example, Nazemi et al. [26] proposed a neural network model for solving two models for uncertain portfolio selection. Bhattacharyya and Samarjit [27] developed a mean-variance-skewness model under uncertain environment. Huang [23] has defined a risk curve and has given a new selection method for uncertain portfolio selection. Huang [24] presented the mean-semivariance and mean-variance portfolio selection models in uncertain environment.

In reality, there is usually a prior portfolio return as a benchmark for an investor, and her/his objective is to keep the divergence of the investment return from the prior return as small as possible. However, in many real portfolio selection problems, except for partial information, the distribution function of the portfolio return is unavailable. For example, distribution function of prior investment return may be obtained based on intuition or experience, and the prior return will be described as uncertain variable. In order to deal with the divergence of two given uncertain distributions, Chen et al. [28] extended the Kapur cross-entropy measure to uncertain environment.

In this paper, the Kapur cross-entropy minimization model for portfolio selection problem is discussed under uncertain environment. The uncertaincross-entropy will be employed to reflect the divergence of uncertain portfolio return from prior uncertain return. Three mathematical models are proposed by minimizing uncertain cross-entropy and expressing risk as variance, semivariance and degree of bad outcome, respectively.

In addition, it is also a key problem that how to solve the proposed models effectively. There are some existing efficient algorithms such as [29 –32]. In recently, Rashedi et al. [33] proposed the gravitation search algorithm (GSA) which is a new heuristic optimization algorithm. GSA has the advantages of easy implementation, fast convergence and low computational cost. In the paper, we employ the GSA and numerical integration technique to solve the proposed models.

The rest of this paper is organized as follows: the section “preliminaries” reviews some basic and necessary concepts of uncertainty theory. In the section “cross-entropy measure of uncertain variables”, we will discuss the concept of Cross-entropy measure of uncertain variables and give two examples at the same time. Three models based on cross-entropy will be proposed in section “cross-entropy minimization models”. We will introduce the gravitation search algorithm and numerical integration technique for solving the proposed models in section “Numerical integration and Gravitation search algorithm”. Several numerical experiments are used to illustrate the effectiveness and correctness of the proposed models in section “numerical experiment”. Finally, the conclusions are given in the last section.

2 Preliminaries

In this section, we will describe some preliminary concepts and theorems about uncertainty theory as follows.

Let Γ be a nonempty set, and L be a σ-algebra over Γ. Each element of Λ ∈ L is called an event. A set function M{ Λ } is called an uncertain measure if and only if it satisfies the followings:

A xiom 1.M { Γ } = 1,

which is called normality.

Axiom 2.M { Λ } + M { Λ^C} = 1, for ∀Λ ∈ L,

which is called self-duality.

Axiom 3. For every event sequence {Λ_i}, one has $M {\cup_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}},$ (1) which is called countable subadditivity.

The triplet (Γ, L, M) is called an uncertainty space if M is an uncertain measure.

Axiom 4. Let (Γ_k, L_k, M_k)(k = 1, 2, ⋯ , n) be uncertainty spaces. The product uncertain measure is defined as follows: $M {\prod_{k = 1}^{n} Λ_{k}} = min_{1 \leq k \leq n} M_{k} {Λ_{k}} .$ (2) which is called product measure.

If ξ is an uncertain variable, then its uncertainty distribution is defined as follows: $Φ (x) = M {ξ \leq x},$ (3) where x ∈ R.

For example, by a linear uncertain variable, we mean the variable that has the following linear uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if & x \leq a \\ (x - a) / - (b - a), & if & a \leq x \leq b \\ 1, & if & x > b . \end{matrix}$ (4)

It is denoted by L (a, b) where a and b are real numbers with a < b.

We call an uncertain variable the zigzag uncertain variable if it has the following zigzag uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if & x \leq a \\ (x - a) / - 2 (b - a), & if & a \leq x \leq b \\ (x + c - 2 b) / - 2 (c - b), & if & b \leq x \leq c \\ 1, & if & x \geq c . \end{matrix}$ (5)

The zigzag uncertain variable is denoted by Z (a, b, c) where a, b, c are real numbers with a < b < c.

Definition 1. [22] Let ξ be an uncertain variable, then its expected value is defined as follow: $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} dx - \int_{- \infty}^{0} M {ξ \leq x} dx,$ (6) provided that at least one of the two integrals is finite.

It can be calculated that the expected value of the zigzag uncertain variable ξ ∼ Z (a, b, c) is E [ξ] = (a + 2b + c)/ - 4, and the expected value of the linear uncertain variable ξ ∼ L (a, b) is E [ξ] = (a + b)/ - 2.

If the uncertain variable ξ has a finite expected value, then its variance is defined as [22] $V [ξ] = E [{(ξ - E [ξ])}^{2}]$ (7)

Let ξ be an uncertain variable with finite expected value. Then the semivariance of ξ is defined as [24] $SV [ξ] = E [{({(ξ - E [ξ])}^{-})}^{2}]$ (8) where ${(ξ - E [ξ])}^{-} = {\begin{matrix} ξ - E [ξ], & if ξ \leq E [ξ] \\ 0, & if ξ > E [ξ] . \end{matrix}$

According to the (8), it can be proved that for a zigzag uncertain variable its semivariance is $SV [ξ] = {\begin{matrix} \frac{{(e - a)}^{3}}{6 (b - a)}, & if b - a \geq c - b \\ \frac{(a - b) (c - b) (a + 2 b - 3 e)}{6 (c - b)} + \\ \frac{{(b - e)}^{2} (3 c - 4 b + e)}{6 (c - b)}, & if b - a < c - b . \end{matrix}$ where e is the expected value of the uncertain variable ξ.

If ξ is a zigzag uncertain variable with b - a = c - b, then E [ξ] = b and $V [ξ] = \frac{{(c - a)}^{2}}{24}$ .

3 Cross-entropy measure of uncertain variables

Cross-entropy is a measure of the difference between two distribution functions. In order to deal with the divergence of uncertain variables via uncertainty distributions, Chen et al. [28] proposed the concept of cross-entropy for uncertain variables based on uncertainty theory. For better understanding of the cross-entropy measure and the cross-entropy minimization models, we recall the definition of the uncertain cross-entropy in this section.

Definition 2. [28] Let ξ and η be two uncertain variables. Then the cross-entropy of ξ from η is $D [ξ, η] = \int_{- \infty}^{+ \infty} T (M {ξ \leq x}, M {η \leq x}) dx,$ (9) where $T (s, t) = s ln (\frac{s}{t}) + (1 - s) ln (\frac{1 - s}{1 - t})$ .

If the distribution functions of the ξ and η are given, then the cross-entropy can be calculated by using the following definition.

Definition 3. [28] Suppose that Φ_ξ and Φ_η are the distribution functions of uncertain variables ξ and η, respectively. The cross-entropy of ξ and η can be written as: $\begin{matrix} D [ξ, η] = \int_{- \infty}^{+ \infty} (Φ_{ξ} (x) ln (\frac{Φ_{ξ} (x)}{Φ_{η} (x)}) \\ + (1 - Φ_{ξ} (x)) ln (\frac{1 - Φ_{ξ} (x)}{1 - Φ_{η} (x)})) dx . \end{matrix}$

Obviously, the cross-entropy depends only on the number of values and their uncertainties and does not lie on the actual values that the uncertain variables ξ and η take.

Example 1. Suppose that ξ is linear uncertain variable with L (a, b). Let η be linear uncertain variable with L (c, d), where c ≤ a < b ≤ d. Then the cross-entropy of ξ from η is given as follows: $\begin{matrix} D [ξ, η] = \int_{a}^{b} (\frac{x - a}{b - a} ln \frac{(x - a) (d - c)}{(b - a) (x - c)} \\ + \frac{b - x}{b - a} ln \frac{(b - x) (d - c)}{(b - a) (d - x)}) dx \\ + \int_{c}^{a} ln \frac{d - c}{d - x} dx + \int_{b}^{d} ln \frac{d - c}{x - c} dx . \end{matrix}$

Example 2. Suppose that ξ and η are zigzaguncertain variable with Z (a, b, c) and Z (l, m, n)(l ≤ a < b = m < c ≤ n), respectively. Then the cross-entropy of ξ from η is given as follows: $\begin{matrix} D [ξ, η] = \int_{l}^{a} ln \frac{2 (m - l)}{2 m + l - x} dx + \\ \int_{a}^{b} (\frac{x - a}{2 (b - a)} ln \frac{(x - a) (m - l)}{(x - l) (b - a)} + \\ \frac{2 b - a - x}{2 (b - a)} ln \frac{(2 b - a - x) (m - l)}{(2 m - l - x) (b - a)}) dx + \\ \int_{b}^{c} (\frac{(x + c - 2 b)}{2 (c - b)} ln \frac{(x + c - 2 b) (n - m)}{(x + n - 2 m) (c - b)} + \\ \frac{c - x}{2 (c - b)} ln \frac{(c - x) (n - m)}{(n - x) (c - b)}) dx + \\ \int_{c}^{n} ln \frac{2 (m - l)}{(x - l)} dx \end{matrix}$

4 The cross-entropy minimization models

Let x_i and ξ_i (i = 1, 2, ⋯ , n) be the investment proportions and the uncertain returns of the ith securities, respectively. Suppose that η is a prior uncertain investment return for an investor, and his/her purpose is to minimize the divergence of the uncertain investment return from η when the return and risk are greater than the minimum return level and less than the maximum risk level. Since the cross-entropy is a common method for measuring the degree of divergence of uncertain variables, we use the cross-entropy as the objective function to reflect the degree of divergence of uncertain return and prior one in this paper. In addition, expected value is used to measure the return. If use variance to measure the risk, we can give the mathematical model as follows: $\begin{matrix} minimize D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}, η] \\ subject to E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \geq α \\ V [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \leq β \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, n, \end{matrix}$ (10) where α and β are the predetermined confidence levels adopted by the investor.

If the uncertain security returns are asymmetrical, we will use semivariance to replace variance. The semivariance is more suitable to reflect risk because it only punishes the investment return below the expected value, thus we can obtain the mathematical model as follows: $\begin{matrix} minimize D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}, η] \\ subject to E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \geq α \\ SV [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \leq β \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, n . \end{matrix}$ (11)

Here, α and β are the predetermined confidence levels accepted by the investor.

In addition, if there is a based target return T, the risk can be measured by the belief degree of bad outcome. That is, the portfolio return is less than T. Thus, we can obtain the model as follows: $\begin{matrix} minimize D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}, η] \\ subject to E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \geq α \\ M {x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n} \leq T} \leq β \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, n, \end{matrix}$ (12) where α and β are the confidence levels accepted by the investor.

Furthermore, we suppose that the uncertain returns of the securities are all linear uncertain variables, the crisp form of the proposed model is given further.

Theorem 1.Suppose that each security return is the linear uncertain variable denoted byξ_i ∼ L (a_i, b_i) (i = 1, 2, ⋯ , n). Let the prior investment return beη = (c, d). Then the model (10) can be transformed into the following crisp form: $\begin{matrix} minimize \frac{(\sum_{i = 1}^{n} b_{i} x_{i} - d) (\sum_{i = 1}^{n} b_{i} x_{i} + d)}{2 (\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})} \\ ln (d - \sum_{i = 1}^{n} b_{i} x_{i}) \\ - \frac{{(c - \sum_{i = 1}^{n} a_{i} x_{i})}^{2}}{2 (\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})} ln (\sum_{i = 1}^{n} a_{i} x_{i} - c) \\ + \frac{{(\sum_{i = 1}^{n} b_{i} x_{i} - c)}^{2}}{2 (\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})} ln (\sum_{i = 1}^{n} b_{i} x_{i} - c) \\ + \frac{(d - \sum_{i = 1}^{n} a_{i} x_{i}) (d - 2 \sum_{i = 1}^{n} b_{i} x_{i} + 3 \sum_{i = 1}^{n} a_{i} x_{i})}{2 (\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})} \\ ln (d - \sum_{i = 1}^{n} a_{i} x_{i}) \\ + (d + 2 c - 2 \sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i}) \\ + (d - c) ln (d - c) \\ subject to (\sum_{i = 1}^{n} a_{i} x_{i} + \sum_{i = 1}^{n} b_{i} x_{i}) \geq 2 α \\ {(\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})}^{2} \leq 12 β \\ c \leq \sum_{i = 1}^{n} a_{i} x_{i} < \sum_{i = 1}^{n} b_{i} x_{i} \leq d \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, n . \end{matrix}$ Proof. Since ξ_i = (a_i, b_i) (i = 1, 2, ⋯ , n) are all linear uncertain variables, by uncertain arithmetic, the total return of the portfolio is $ξ = \sum_{i = 1}^{n} x_{i} ξ_{i} = (\sum_{i = 1}^{n} a_{i} x_{i}, \sum_{i = 1}^{n} b_{i} x_{i})$ which is also a linear uncertain variable. In addition, the expected value, variance and cross-entropy can be calculated by using the Equations (6), (7) and (9), respectively. Thus, it is easy to prove this theorem.

In addition, the investors focus on the case $c \leq \sum_{i = 1}^{n} a_{i} x_{i} < \sum_{i = 1}^{n} b_{i} x_{i} \leq d$ , because the value of $\frac{1}{\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i}}$ is the largest when $c \leq \sum_{i = 1}^{n} a_{i} x_{i} < \sum_{i = 1}^{n} b_{i} x_{i} \leq d$ . Thus, we only consider this situation in the theorem 1.

Remark 1. According to the Equations (3) and (8), it can be proved that, for an linear uncertain investment return $ξ = (\sum_{i = 1}^{n} a_{i} x_{i}, \sum_{i = 1}^{n} b_{i} x_{i})$ , the semivariance and the belief degree of bad outcome are $SV [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] = \frac{{(\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i})}^{2}}{24}$ and $M {x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n} \leq T} = \frac{T - \sum_{i = 1}^{n} a_{i} x_{i}}{\sum_{i = 1}^{n} b_{i} x_{i} - \sum_{i = 1}^{n} a_{i} x_{i}},$ respectively. Based on the results, the models (11) and (12) are also converted into deterministic mathematical programming problems by using a similar way to Theorem 1.

5 Numerical integration and Gravitation search algorithm

5.1 Numerical integration

If the security returns are general uncertain variables, then the crisp forms of the proposed models are difficult to be given. Considering the complexity of the integrand, the numerical integration technique will be employed. The cross-entropy of ξ from η can be calculated by using the Simpson formula. For example, suppose that ξ = (0, 60, 150) and η = (0, 100, 200) are two zigzag uncertain variables. We divide the integral interval [0, 200] into 2n equal parts, then the cross-entropy of ξ from η can becalculated as follows: $\begin{matrix} D [ξ, η] = \int_{0}^{200} f (x) dx \approx \frac{200 - 0}{6 n} \\ [\begin{matrix} f (0) + f (200) + 4 (\begin{matrix} f (0 + Δ x) + f (0 + 3 Δ x) \\ + \dots + f (0 + (2 n - 1) Δ x) \end{matrix}) \\ + 2 (\begin{matrix} f (0 + 2 Δ x) + f (0 + 4 Δ x) \\ + \dots + f (0 + (2 n - 2) Δ x) \end{matrix}) \end{matrix}] \end{matrix}$ where $f (x) = (Φ_{ξ} (x) ln (\frac{Φ_{ξ} (x)}{Φ_{η} (x)}) + (1 - Φ_{ξ} (x))$ $ln (\frac{1 - Φ_{ξ} (x)}{1 - Φ_{η} (x)}))$ and $Δ x = \frac{200 - 0}{2 n}$ .

Thus, we can employ the Simpson formula to calculate the cross-entropy, and then integrate the results into the Gravitation search algorithm (GSA) to solve the proposed models when the securities are general uncertain variables.

5.2 Gravitation search algorithm

The gravitation search algorithm (GSA) is introduced by E. Rashedi [33] which is a new heuristic optimization algorithm. GSA can be described as follows.

Suppose a swarm consists of N agents moving in search region Ω, the ith agent at the tth iteration can be represented by $\begin{matrix} x_{i} (t) & = & (x_{i}^{1} (t), \cdot \cdot \cdot, x_{i}^{d} (t), \cdot \cdot \cdot, x_{i}^{D} (t)) \\ for i = 1, 2, \cdot \cdot \cdot, N \end{matrix}$ where $x_{i}^{d} (t)$ is the position of the ith agent in the dth dimension at the tth iteration and D is the space dimension.

The mass of the agent i is defined as follows: $M_{i} (t) = \frac{m_{i} (t)}{\sum_{j = 1}^{N} m_{j} (t)},$ (13) where $m_{i} (t) = \frac{{fit}_{i} (t) - worst (t)}{best (t) - worst (t)},$ where fit_i (t) denotes the fitness value of the agent i at time t, and best (t) and worst (t) are calculated as follows: $\begin{matrix} best (t) = min_{i \in {1, \dots, N}} {fit}_{i} (t), \\ worst (t) = max_{i \in {1, \dots, N}} {fit}_{i} (t) . \end{matrix}$

The force acting on agent i from agent j is calculated as follows: $F_{ij}^{d} (t) = G (t) \frac{M_{i} (t) \times M_{j} (t)}{R_{ij} (t) + ɛ} (x_{j}^{d} (t) - x_{i}^{d} (t)),$ where R_ij represents the Euclidian distance between agents i and j, ɛ is a small constant, and G (t) is a gravitational constant which is defined as follows: $G (t) = G_{0} e^{- α \frac{t}{T}},$ where G₀ is the initial value, α is a constant, t is current iteration and T is the maximum number of iteration.

The total force that acts on agent i in a dimension d is given as follows: $F_{i}^{d} (t) = \sum_{j \in k_{best} (t), j \neq i} {rand}_{j} \times F_{ij}^{d} (t),$ (14) where rand_j is a uniform random variable in the interval [1]. k_best is the set of first K agents with the best fitness value and biggest mass. K will decrease linearly with time.

The acceleration of the agent i at time t and in a direction d is defined as follows: $\begin{matrix} a_{i}^{d} (t) = \frac{F_{i}^{d} (t)}{M_{i} (t)} \\ = \sum_{j \in k_{best} (t), j \neq i} {rand}_{j} G (t) \frac{M_{j} (t)}{{∥ x_{i} (t), x_{j} (t) ∥}_{2} + ɛ} \\ (x_{d}^{j} (t) - x_{d}^{i} (t)) . \end{matrix}$ The velocity and the position of agent are updated as follows: $v_{i}^{d} (t + 1) = {rand}_{j} \times v_{i}^{d} (t) + a_{i}^{d},$ (15) $x_{i}^{d} (t + 1) = x_{i}^{d} (t) + v_{i}^{d} (t + 1) .$ (16) In order to solve the proposed models, we take Ω = [0, 1]ⁿ, and the fitness value function of the agent i at time t is defined as follows: ${fit}_{i} (t) = - f (x_{i} (t)) + γ \cdot [\begin{matrix} {max {0, g_{1} (x_{i} (t))}}^{2} + \\ {max {0, g_{2} (x_{i} (t))}}^{2} + \\ {h (x_{i} (t))}^{2} \end{matrix}],$ (17) where $\begin{matrix} f (.) = D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}, η] \\ g_{1} (.) = α - E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \\ g_{2} (.) = V [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] - β, \\ for model (10) \\ g_{2} (.) = SV [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] - β, \\ for model (11) \\ g_{2} (.) = M {x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n} \leq T} - β, \\ for model (12) \\ h (.) = x_{1} + x_{2} + \dots + x_{n} - 1 . \end{matrix}$

The procedure of the gravitation search algorithm is shown as follows:

Step 1. The initialization parameters setting: including the population size N, the maximum number of iteration T, the initial gravitational constant G₀ and constant α, ɛ and γ.

Step 2. Initialization: generate N uniformly distributed random agents in Ω, calculate the fitness of each agent, and set initial velocity 0.

Step 3. Calculate the mass of each agent using Equation (12).

Step 4. Compute the total force that acts on agent i using Equation (14).

Step 5. Compute acceleration of the agent i.

Step 6. Update the velocity and the position of agent i using Equations (15) and (16).

Step 7. If $x_{i}^{d} (t + 1)$ is less than 0 or greater than 1, then it is assigned to be $x_{i}^{d} (t + 1) = rand$ , where rand is a uniform random variable in the interval [0, 1].

Step 8. Compute the fitness value function of the agent i using Equation (17).

Step 9. Repeat: steps from 3 to 8 are repeated until the iterations reach the maximum iteration limit.

Step 10. Report the best solution.

Remark 2. If the security returns are linear uncertain variables, then we use GSA to solve the crisp forms of three proposed models. When the security returns are general uncertain variables, three mathematical models are solved by using the GSA and Numerical integration technique. We define the f (x_i (t)) of the fit_i (t) as D [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n, η], the g₁ (x_i (t)) as α - E [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n], the g₂ (x_i (t)) as V [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n] - β, SV [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n] - β and M { x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n ≤ T } - β, respectively. WhereE [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n], V [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n], SV [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n], M { x₁ξ₁ + x₂ξ₂+ ⋯ + x_nξ_n ≤ T } can be calculated by usingthe equations (3), (6), (7) and (8), respectively.Furthermore, the cross-entropy D [x₁ξ₁ + x₂ξ₂ + ⋯ + x_nξ_n, η] is calculated by employing the Simpson formula, and further integrate the results into the Gravitation search algorithm (GSA) to solve the above three models.

6 Numerical experiments

In order to evaluate effective and correct of the proposed models, we propose several numerical examples. In addition, suppose that ξ_i is the returns of the ith securities determined as $ξ_{i} = (p_{i}^{'} + d_{i} - p_{i}) / p_{i}$ , where $p_{i}^{'}$ is the estimated closing prices of the securities i in the next period, p_i the closing prices of the securities i at present, and d_i the estimated dividends of the securities i during the period. It is clear that $p_{i}^{'}$ and d_i are unknown at present. In other words, the predictions of security returns have to be given mainly based on expert’s judgments and estimations.

The internal parameters are set as follows: the population size is set to 50, maximum iteration is 500, G₀ = 100, α = 20, and γ = 20000.

Example 3. Suppose that there are 10 securities, we look each security return as the linear uncertain variable denoted by ξ_i = Z (a_i, b_i), i = 1, 2, ⋯⋯10. The Table 1 shows the uncertain returns of 10 securities. Assume that the prior target return η = (- 0.2, 4.0), the minimum return level α = 1.9, and the maximum risk level β = 1.4. If the investor considers variance as risk measure, the crisp form of the model (10) can be written as follows:

$\begin{matrix} minimize \frac{(\sum_{i = 1}^{10} b_{i} x_{i} - 4) (\sum_{i = 1}^{10} b_{i} x_{i} + 4)}{2 (\sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i})} \\ ln (4 - \sum_{i = 1}^{10} b_{i} x_{i}) \\ - \frac{{(- 0.2 - \sum_{i = 1}^{10} a_{i} x_{i})}^{2}}{2 (\sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i})} ln (\sum_{i = 1}^{10} a_{i} x_{i} + 0.2) \\ + \frac{{(\sum_{i = 1}^{10} b_{i} x_{i} + 0.2)}^{2}}{2 (\sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i})} ln (\sum_{i = 1}^{10} b_{i} x_{i} + 0.2) \\ + \frac{(4 - \sum_{i = 1}^{10} a_{i} x_{i}) (4 - 2 \sum_{i = 1}^{10} b_{i} x_{i} + 3 \sum_{i = 1}^{10} a_{i} x_{i})}{2 (\sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i})} \\ ln (4 - \sum_{i = 1}^{10} a_{i} x_{i}) \\ - 2 \sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i} \\ subject to (\sum_{i = 1}^{10} a_{i} x_{i} + \sum_{i = 1}^{10} b_{i} x_{i}) \geq 3.8 \\ {(\sum_{i = 1}^{10} b_{i} x_{i} - \sum_{i = 1}^{10} a_{i} x_{i})}^{2} \leq 16.8 \\ - 0.2 \leq \sum_{i = 1}^{10} a_{i} x_{i} < \sum_{i = 1}^{10} b_{i} x_{i} \leq 4 \\ x_{1} + x_{2} + \dots + x_{10} = 1 \end{matrix}$ (18)

The model (18) is solved by GSA. The numerical result shows that in order to obtain the minimize cross-entropy when the portfolio satisfies the risk and return constraints, the investor should allocate his or her money according to Table 2. The corresponding objective value is 0.0135. Furthermore, the investment return ξ = (- 0.15, 3.95).

In addition, in order to examine the effectiveness of numerical integration technique, we use the Simpson formula to calculate the objective function of model (10) based on the data of the Example 3, and then employ the results to replace the objective function of the model (18) to produce an approximate model of the model (10).

We compare the exact model and approximate model of the model (10) through computational experiments. According to the data in Example 3, we use GSA to solve the exact model and approximate model for different return level α. The numerical results are given in Table 3.

Example 4. Suppose there are 20 securities, we look each security return as the Zigzag uncertain variable denoted by ξ_i = Z (a_i, b_i, c_i), i = 1, 2, ……20. The Table 4 shows the uncertain returns of 20 securities. Assume that the prior target return η = (-0.15, 2.35, 4.05), the minimum return level α = 2.25, and the maximum risk level β = 0.75. If the investor considers semivariance as risk measure, the following model can be employed to construct his/her portfolio,

$\begin{matrix} minimize D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20}, η] \\ subject to E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20}] \geq 2.25 \\ SV [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20}] \leq 0.75 \\ x_{1} + x_{2} + \dots + x_{20} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, 20 . \end{matrix}$ (19)

We use the Simpson formula to calculate the cross-entropy, and then integrate the results into the Gravitation search algorithm (GSA) to solve this model. The numerical results are shown in Table 5.

Based on the results, we can conclude that the investors should distribute their money consistent with Table 5 for minimizing the cross-entropy between the investment return ξ and the prior return η. According to the computation, 0.0185 and ξ = (-0.12, 2.65, 4.30) are the corresponding objective value and the investment return, respectively.

Furthermore, in order to examine the performance of the gravitation search algorithm, we compare GSA with classic GA and DE by the numerical experiments. The algorithms are coded in Matlab 7.0 and run on an Inter i7-3370 computer. Based on the data of the Example 4, we use GSA, GA and DE to solve the above model, respectively. Each algorithm is running 20 times for each return level α. The numerical results and comparisons are given in Table 6. In Table 6, α, best, worst, Ave. and time denote return level, the best objective value, the worst objective value, the average objective value and the average CPU time, respectively.

From the Table 6, it can be seen that the GSA outperforms the classic GA and DE under different return level α. The GSA can obtain better solutions. Thus, the results can demonstrate the GSA is effective for solving the proposed models.

In addition, there are some later algorithms such as algorithms proposed in [34 –36]. In order to further examine the effectiveness of the GSA, we compare GSA with the MPQGA of [36] by using the following numerical experiments.

Example 5. In this example, all data is from Example 4. The uncertain returns of 20 securities are shown in Table 4. If the risk is measured by the degree of bad outcome event, and the investor set the predetermined investment return as 0.75 and accept 0.25 as bearable maximum risk level, then we have the model as follows: $\begin{matrix} minimize D [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20}, η] \\ subject to E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20}] \geq 2.25 \\ M {x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{20} ξ_{20} \leq 0.75} \leq 0.25 \\ x_{1} + x_{2} + \dots + x_{20} = 1 \\ x_{i} \geq 0, i = 1, 2 \dots, 20 . \end{matrix}$

A run of gravitation search algorithm (500 generations) shows that the allocation of money should be based on Table 7. According to the computation, 0.0157 and ξ = (-0.15, 2.60, 4.04) are the corresponding objective value and the investment return, respectively.

In addition, we also use the algorithm of [36] to solve the above model for different target return T.

The numerical results and comparisons are given in Table 8. In Table 8, T, best, worst, Ave. and time denote target return, the best objective value, the worst objective value, the average objective value and the average CPU time, respectively.

According to the Table 8, it is seen that the optimal objective values produced by GSA and MPQGA are similar for different target return T. However, we can also see that when the base target return T decreases, the average computation time produced by using GSA is shorter than the computation time produced by using the MPQGA. Thus, the results indicated that GSA is effective for solving the proposed models.

7 Conclusions

In this paper, we have discussed the portfolio selection problem when security returns are hard to be well reflected by historical data. We extended three portfolio selection models to uncertain environment, and introduced uncertain cross-entropy into three models. In addition, crisp form of the optimization model has also been provided when all investment returns are linear uncertain variables. The gravitation search algorithm and numerical integration technique are introduced for solving the proposed models. Finally, several numerical experiments are given to illustrate the effectiveness of the algorithm and the proposed models.

Footnotes

Acknowledgments

The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions, which improve this paper greatly. This work is partly supported by the National Natural Science Foundation of China (11371071) and Scientific Research Foundation of Liaoning Province Educational Department (L2013426).

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