Portfolio Selection Based on Distance between Fuzzy Variables

This paper researches portfolio selection problem in fuzzy environment. We introduce a new simple method in which the distance between fuzzy variables is used to measure the divergence of fuzzy investment return from a prior one. Firstly, two new mathematical models are proposed by expressing divergence as distance, investment return as expected value, and risk as variance and semivariance, respectively. Secondly, the crisp forms of the new models are also provided for different types of fuzzy variables. Finally, several numerical examples are given to illustrate the effectiveness of the proposed approach.


Introduction
Portfolio selection is to select a combination of securities among a large number of candidate securities which is the best to meet the investors' goal.Markowitz [1] applied probability theory to portfolio selection problem and proposed the famous mean-variance model, in which expected return and variance were used to describe investment return and risk, respectively.Since then, variance has been widely accepted as a risk measure, and a great number of extensions have been proposed [2][3][4][5].However, the mean-variance model has limited generality since variance considers high returns as equally undesirable as low returns.Variance becomes a deficient measure of risk when security returns are asymmetrical.Thus, Markowitz proposed semivariance as an improvement measure of risk, and numerous models have been developed based on semivariance such as models proposed in [6][7][8][9][10].
Generally speaking, the extension of Markowitz model is defined by minimizing the risk and maximizing the investment return.However, Kapur and Kesavan [11] introduced an entropy maximization model and a cross entropy minimization model.The objective of the entropy model is to maximize the uncertainty of random return and that of the cross entropy model is to minimize the divergence of the random return from a prior one.After that, many scholars accepted and explored these new models [12][13][14][15].
In the above literatures, security returns are considered as random variables.Since the security market is complex, in many cases, security returns are hard to be well reflected by historical data.Therefore, many researches argued that we should find another theory to solve the portfolio selection problem in this situation.With the introduction of fuzzy set theory and credibility theory, many scholars began to employ them to describe and study fuzzy portfolio selection problems.Numerous models containing fuzzy variables are proposed.For example, Bilbao-Terol et al. [16], Gupta et al. [17], and Zhang et al. [18] extended the mean-variance model from different angles.Huang [19] developed fuzzy mean-variance models and further proposed fuzzy meansemivariance portfolio selection models [20].Li et al. [21] employed skewness to describe asymmetry of fuzzy returns and further established fuzzy mean-variance-skewness models.Bhattacharyya et al. [22] proposed fuzzy mean-varianceskewness portfolio selection models by interval analysis.Huang [23] has used the entropy method to fuzzy environment to provide the fuzzy diversification models.In 2008, Li and Liu [24] proposed a concept of fuzzy entropy for measuring the uncertainty of fuzzy variables.Based on this concept, Huang [25] researched Kapur entropy maximization model in fuzzy environment.Moreover, the fuzzy cross entropy was given in [26] for measuring the divergence of fuzzy variables from a prior one.According to [26], Qin et al. [27] extended the Kapur cross entropy minimization model to fuzzy environment.These models in [27] were solved by using a hybrid intelligent algorithm which is designed 2 Mathematical Problems in Engineering by integrating numerical integration, fuzzy simulation, and genetic algorithm.
Distance between fuzzy variables is an important concept in fuzzy theory.Many scholars gave different definitions of distance between fuzzy variables, such as Hamming distance, Euclidean distance, and Minkowski distance.Recently, Tang et al. [28] gave a kind of definition of distance based on expected value operator of fuzzy variable.We define a new distance between fuzzy variables based on distance measure for interval numbers in this paper.Comparing to the distance measure of [28], the proposed distance measure can be calculated more easily.
In this paper, our motivation is that the divergence of fuzzy investment return from a prior one is measured by using the proposed distance between fuzzy variables.Based on this idea, we establish two distance minimization models by defining investment return as expected value and risk as variance and semivariance, respectively.In addition, several crisp and simple equivalents of the optimization models are also proposed for different types of fuzzy variables.Finally, we compare our method with the methods presented by Chen et al. [29] and Wu and Liu [30] to demonstrate the effectiveness of the proposed approach.
The remainder of the paper is organized as follows.Some preliminary concepts of credibility theory are briefly recalled in Section 2. The concept of distance between fuzzy variables is introduced in Section 3. In Section 4, we will propose two new models by minimizing distance between fuzzy variables.In Section 5, the crisp forms of the new models will be presented.Section 6 gives several numerical examples to illustrate availability of the proposed approach.Finally, a brief summary is given in Section 7.

Necessary Knowledge about Credibility Theory
After Zadeh [31] initiated the concept of fuzzy set by membership function in 1965, he further indicated possibility theory [32].Many research scholars, such as Dubois and Prade [33,34], made their great contribution to its development.In 2002, B. Liu and Y.-K.Liu [35] defined a credibility measure to describe a fuzzy event.In order to develop an axiom system similar to the theory of probability, Liu founded the credibility theory in [36], which is a branch of mathematics for studying fuzzy phenomena.Further developments can be found in [37,38].
Let  be a fuzzy variable with membership function .The credibility measure is defined as [35] Cr { ∈ } = 1 2 (sub for any set  of real numbers.It is easy to see that credibility is self-dual.
In order to make a more general definition of expected value of a fuzzy variable, according to the credibility measure, B. Liu and Y.-K.Liu [35] defined the expected value of  as provided that at least one of the two integrals is finite.If the fuzzy variable  has a finite expected value, then its variance is defined as [35] Let  be a fuzzy variable with finite expected value.Then the semivariance of  is defined as [20] SV where Generally speaking, expected value is used to measure the return and variance or semivariance is used to reflect the risk in portfolio selection problem.
(16) According to the definition of the distance between interval numbers, we can obtain the definition of the distance between two fuzzy variables.

Mathematical Problems in Engineering
Definition 7. Suppose that  and  are fuzzy variables with membership functions () and ](), and the -cut of  and ] for all  ∈ [0, 1], respectively.Then the distance measure between  and  can be defined by Theorem 8. Suppose that , , and  are fuzzy variables.Let (⋅, ⋅) be distance; then the (⋅, ⋅) satisfies the following properties of a distance metric: Proof.Based on the properties (i) and (ii) of the interval numbers distance, the parts (a) and (b) follow immediately from Definition 7. Now we prove the part (c).Suppose that membership function of  is (); then the -cut of  is Based on the inequality ( 16), we have According to Minkowski inequality, we can obtain Form the above inequalities, we can get Thus The theorem is proved.

Distance Minimization Models
Let   be the investment proportions in securities  and   the fuzzy returns of the th securities,  = 1, 2, . . ., , respectively.Suppose that  is a prior fuzzy investment return for an investor, and his/her objective is to minimize the divergence of the fuzzy investment return from .In addition, the return remains above the minimum return level and the risk remains below the maximum risk level.In this paper, we use the distance to measure the degree of divergence and use the expected value to reflect the return.The main problem is how to measure the risk.If the fuzzy security returns   are symmetrical, we use variance to measure risk; then we have the following model: The first constraint ensures the expected return is no less than some given value , and the second one assures that risk does not exceed some given level  the investor can bear.The last two constraints imply that all capital will be invested in  securities.
Remark 9. Suppose that the security returns   ( = 1, 2, . . ., ) are fuzzy variables.It follows from Extension Principle of Zadeh that  =  1  1 +  2  2 + ⋅ ⋅ ⋅ +     , which is also a fuzzy variable.Let the membership functions of  and  be () and ](), respectively.Then the objective of model ( 22) can be calculated as Though it is usually adopted that the security returns are symmetrical, there do exist empirical evidences [41] indicating that many security returns are not symmetrically distributed.In the case where the fuzzy security returns are asymmetrical, we can use semivariance to replace variance.The semivariance is more suitable to measure risk because it only punishes the investment return below the expected value; thus we have the model as follows: where  and  are the predetermined confidence levels accepted by the investor.
Theorem 10.Assume that each security return is the symmetrical triangular fuzzy variable denoted by   = (  ,   ,   ) ( = 1, 2, . . ., ).Let the prior fuzzy investment return  = (  ,   ,   ) be a triangular fuzzy variable; then the model (22) can be transformed into the following crisp form: Proof.Since   = (  ,   ,   ) ( = 1, 2, . . ., ) are all symmetrical triangular fuzzy variables, it follows from Extension Principle of Zadeh that , which is also a symmetrical triangular fuzzy variable.According to (2) and (3), we have In addition, according to Example 5, for any real number  ∈ [0, 1], we have the -cut of and -cut of Thus, it is known from Definition 7 that The proof is completed.
When the investment return  = (, , ) is an asymmetrical triangular fuzzy variable, the investors focus on the case  −  <  − .Therefore, we only consider this situation in this paper.
According to above proof, we can also obtain Theorems 12-15 for different types of fuzzy variables.

Numerical Examples
In this section, some numerical examples are given to illustrate the availability of the two new models.Examples 1-3 consider the case in which there are 10 or 30 securities from different industries.Let   be the return of the th security determined as   = (   +   −   )/  , where    is the estimated closing price of the th security in the next period,   the closing price of the th security at present, and   the estimated dividends of the th security during the next period.It is clear that    and   are unknown at present.In other words, the predictions of security returns have to be given mainly based on expert's judgments and estimations.
Example 1. Assume that each security return is the symmetrical triangular fuzzy variable denoted by   = (  ,   ,   ) ( = 1, 2, . . ., 10), where the parameters   ,   , and   are determined based on the estimated values of financial experts.The data set is given in Table 1.Suppose that the minimum expected return the investor can accept is 1.95 and the bearable maximum risk is 1.0.In addition, the prior fuzzy investment return is  = (−0.2,1.9, 4.0).From the model (25), we can obtain a simple and crisp optimization model and employ fmincon in MATLAB 7.1 to solve this model.The numerical results are given in Table 2.
In order to obtain the minimized distance of the investment return from  when the portfolio satisfies the return and risk constraints, the investor should allocate his or her money according to Table 2.The corresponding objective value is 0.0150; the expected return and variance of the portfolio are 1.951 and 0.739, respectively.Furthermore, the investment return is  = (−0.15,1.95, 4.05).The graphic comparison of the obtained investment return  and the prior one  is shown in Figure 1.
From Figure 1, we see that the obtained investment return  is close to the prior one .It shows that the new approach is feasible.
Example 2. In this example, all data is from [27].The asymmetrical fuzzy returns of 10 securities are shown in Table 3.The maximum risk level and the minimum return level are 0.7 and 2.25, respectively.The prior fuzzy return is  = (−0.2,2.3, 4).From the model (30), we can obtain a simple and crisp optimization model and employ fmincon in MATLAB 7.1 to solve this model.The numerical results are given in Table 4.
In order to obtain the minimized distance of the investment return from the prior return  when the portfolio satisfies the return and risk constraints, the investor should assign his or her capital according to Table 4.The corresponding objective value is 0.165; the expected return and semivariance of the portfolio are 2.253 and 0.612, respectively.In addition, the investment return is  = (−0.18,2.58, 4.03). = (−0.20,2.63, 3.95) denotes the investment return calculated by [27].The graphic comparison of the investment returns ,  and the prior one  is shown in Figure 2.
From Figure 2, we can see that the investment returns obtained by our model and model of [27] are similar.However, solving our model is easier than solving the model of [27].The data set from [30] is shown in Table 5.The maximum risk level and the minimum return level are 0.000475 and 1.0732, respectively.In addition, the prior fuzzy return is  = (1.040,1.075, 1.095, 1.135) for an investor.From the model (34), we can obtain a simple and crisp optimization model and use gravitation search algorithm (GSA) [42] to solve this model.The numerical results are given in Table 6.
The results show that among 30 securities, satisfying the constraints, in order to minimize distance of the investment return from the prior return , the investor should allocate his or her money according to Table 6.The corresponding objective value is 9.7857 − 006.In addition, the investment return is  = (1.0424,1.0754, 1.0950, 1.1333).
In order to examine the sensitivity of the predetermined confidence level, we adjust the  value and do the experiment.The results are shown in Table 7.It is seen that as maximum risk level increases, the optimal objective will decrease.
In addition, we also examine the sensitivity of the return level  to optimal objective in the same way.The results are given in Table 8.The results indicate that as expected return level increases, the minimal distance will increase.
Furthermore, in order to examine the availability of the new approach, we compare the proposed method with the methods of [29,30].Based on the data in Table 5, we use GSA to solve the models (18) and (23) of [29] for different return level  0 and risk level  0 .The numerical results are given in Tables 9 and 10.In addition, Tables 11 and 12 show the results of models (21) and (22) of [30], which are from [30].
From Tables 7 to 12, it is seen that the computational results about optimal allocation proportion to 30 securities are different, and the optimal portfolios produced by our model are more diversified than the optimal portfolio produced by models of [29,30].

Conclusions
In this paper, a concept of distance between fuzzy variables was introduced for measuring the divergence of fuzzy investment returns from a prior one.By defining the risk as variance and semivariance, two distance minimization models were proposed.In addition, crisp equivalents of the optimization models have also been provided.Finally, the results of the numerical examples illustrated the availability of the new method.

Figure 1 :Figure 2 :
Figure 1: Comparison of investment return  and the prior one .

Table 1 :
The symmetrical fuzzy returns of 10 securities.

Table 3 :
The asymmetrical fuzzy returns of 10 securities.

Table 5 :
The trapezoidal fuzzy returns of 30 securities.

Table 8 :
Investment proportion of 30 securities under different return level constraints with  = 0.000475 (%).