# Selecting the Best of Portfolio Using OWA Operator Weights in Cross Efficiency-Evaluation

Authors: Masoud Sanei; Shokoofeh Banihashemi
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

The present study is an attempt toward evaluating the performance of portfolios and asset selection using cross-efficiency evaluation. Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). The most widely used approach is to evaluate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and consider it as the overall performance measurement of the DMU. This paper focuses on the evaluation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency evaluation. The OWA operator weights are generated by the minimax disparity approach and allow the decision maker (DM) or investor to select the best assets that are characterized by an orness degree. The problem consists of choosing an optimal set of assets in order to minimize the risk and maximize return. This method is illustrated by application in mutual funds and  weights are obtained via OWA operator for making the best portfolio. The finding could be used for constructing the best portfolio in stock companies, in various finance organization, and public and private sector companies.


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## Body

## 1. Introduction

In financial literature, a portfolio is an appropriate mix of investments held by an institution or private individuals. Evaluation of portfolio performance has created a large interest among employees also academic researchers because of huge amount of money being invested in financial markets. The theory of mean-variance, Markowitz [1] is considered the basis of many current models and this theory is widely used to select portfolios. This model is due to the nature of the variance in quadratic form. Other problem in Markowitz model is that increasing the number of assets will develop the covariance matrix of asset returns and will be added to the content calculation. Due to these problems sharp one-factor model is proposed by Sharpe [2]. This method reduces the number of calculations requiring information for the decision. Data envelopment analysis (DEA) has proved the efficiency for assessing the relative efficiency of decision making units (DMUs) employing multiple inputs to produce multiple outputs [3]. M. R. Morey and R. C. Morey [4] proposed mean-variance framework based on Data Envelopment Analysis, which the variance of the portfolios is used as an input to the DEA and expected return is the output. Joro and Na [5] introduced mean-variance-skewness framework and skewness of returns are also considered as an output. The portfolio optimization problem is a well-known problem in financial real world. The investor’s objective is to get the maximum possible return on an investment with the minimum possible risk. Since there are a large number of assets to invest in, this objective leads to select the best assets via cross-efficiency matrix by using OWA weighted. Cross-efficiency evaluation, proposed by Sexton et al., [6] is the effective way of ranking decision making units (DMUs). It allows the overall efficiencies of the DMUs to be evaluated through self- and peer-evaluations. The self-evaluation allows the efficiencies of the DMUs to be evaluated with the most favorable weights so that each of them can achieve its best possible relative efficiency, whereas the peer-evaluation requires the efficiency of each DMU to be evaluated with the weights determined by the other DMUs. The self-evaluated efficiency and peer-evaluated efficiencies of each DMU are then averaged as the overall efficiency of the DMU. Since, its remarkable discrimination power, the cross-efficiency evaluation has found significant number of applications in a wide variety of areas such as preference voting and project ranking [7, 8], economic-environmental performance assessment [9, 10], and Olympic ranking and benchmarking [11–13]. Besides a large number of applications, theoretical research has also been conducted on the cross-efficiency evaluation. For example, Doyle and Green [14, 15] presented mathematical formulations for possible implementations of aggressive and benevolent cross-efficiencies. Liang et al. [16] suggested the concept of game cross-efficiency and developed a game cross-efficiency model which treats each DMU as a player that seeks to maximize own efficiency under the condition that the cross-efficiency of each of the other DMUs dose not deteriorate. Wu et al. [13] extended the game cross-efficiency model to variable returns to scale later. In our work, the use of equal weights for cross-efficiency model has a significant problem. That is, self-evaluated efficiencies are much less weighted than peer-evaluated efficiencies. This is because each DMU has only one self-evaluated efficiency value, but multiple peer-evaluated efficiency values. When they are simply averaged together, the weight assigned to the self-evaluated efficiency is only 1/n if there are n DMUs to be evaluated, whereas the remaining weights (n-1)/n are all given to those peer-evaluated efficiencies. To overcome this problem, the use of ordered weighted averaging (OWA) operator weights is stated for assets cross-efficiencies. The use of OWA operator weights for the assets cross-efficiency allows the weights to be reasonably allocated between self- and peer-evaluated efficiencies by investor’s control [17]. The OWA operator weights are generated by the minimax disparity approach and allow the decision maker (DM) or investors to select the best assets that are characterized by an orness degree [18]. The method consists of choosing an optimal set of assets in order to minimize the risk and maximize return in cross-efficiency using OWA operator. Since there are a large number of assets to invest in, the best assets are chosen via cross-efficiency evaluation by using OWA weighted by control investors.The rest of the paper is organized as follows: Section2 briefly reviews the portfolio performance literature, OWA operators, and their weight determination methods, thus the cross-efficiency evaluation in DEA. Section 3 develops a proposed method for selecting the best of the portfolio. Section 4 presents computational results using mutual funds data and finally conclusions are given in Section 5.


## 2. Background

### 2.1. Portfolio Performance Literature

Portfolio theory to investing is published by Markowitz (1952). This approach starts by assuming that an investor has a given sum of money to invest at the present time. This money will be invested for a time as the investor’s holding period. At the end of the holding period, the investor will sell all of the assets that were bought at the beginning of the period and then either consume or reinvest. Since portfolio is a collection of assets, it is better to select an optimal portfolio from a set of possible portfolios. Hence, the investor should recognize the returns (and portfolio returns), expected (mean) return, and standard deviation of return. This means that the investor wants to both maximize expected return and minimize uncertainty (risk). Rate of return (or simply the return) of the investor’s wealth from the beginning to the end of the period is calculated as follows:(1)Return=(end-of-periodwealth)-(beginning-of-periodwealth)beginning-of-periodwealth.
Since Portfolio is a collection of assets, its return rp can be calculated in a similar manner. Thus, according to Markowitz, the investor should view the rate of return associated with any one of these portfolios as what is called in statistics a random variable. These variables can be described as the expected return (min or r-p) and standard deviation of return. Expected return and deviation standard of return are calculated as follows:
(2)r-p=∑i=1nλir-i,σp=[∑i=1n∑j=1nλiλjσij]1/2,
where n is the number of assets in the portfolio, r-p is the expected return of the portfolio, λi is the proportion of the portfolio’s initial value invested in asset i, r-i is the expected return of asset i, σp is the deviation standard of the portfolio, and σij is the covariance of the returns between asset i and asset j.In the above, optimal portfolio from the set of portfolios will be chosen as maximum expected return for varying levels of risk and minimum risk for varying levels of expected return [19]. Data Envelopment Analysis is a nonparametric method for evaluating the efficiency of systems with multiple inputs and multiple outputs. In this section, we present some basic definitions, models, and concepts that will be used in other sections in DEA. They will not be discussed in details. Consider DMUj, (j=1,…,n), where each DMU consumes m inputs to produce s outputs. Suppose that the observed input and output vectors of DMUj are Xj=(x1j,…,xmj) and Yj=(y1j,…,ysj), respectively, and let Xj≥0 and Xj≠0, Yj≥0, and Yj≠0. A basic DEA formulation in input orientation is as follows:
(3)min⁡⁡θ-ε(∑r=1ssr++∑i=1msi-)s.t.∑j=1nλjxij+si-=θxioi=1,…,m,∑j=1nλjyrj+sr+=yror=1,…,sλ∈Λ,s+,s-≥0,ε≥0,
where λis an n-vector of λvariables, s+ as-vector of output slacks, s- an m-vector of input slacks and set Λ is defined as follows:
(4)Λ={{λ∈R+n}withconstantreturnstoscale,{λ∈R+n,1λ≤1}withnonincreasingreturnstoscale,{λ∈R+n,1λ=1}withvariablereturnstoscale.
Note that subscript “o” refers to the unit under the evaluation. A DMU is efficient if and only if θ=1 and all slack variables s-,s+ equal zero; otherwise, it is inefficient [20]. In the DEA formulation above, the left-hand sides in the constraints define an efficient portfolio. θ is a multiplier which defines the distance from the efficient frontier. The slack variables are used to ensure that the efficient point is fully efficient. This model is used for asset selection. The portfolio performance evaluation literature is vast. In recent years, these models have been used to evaluate the portfolio efficiency. Also, in the Markowitz theory, it is required to characterize the whole efficient frontier but the proposed models by Joro and Na do not need to characterize the whole efficient frontier but only the projection points. The distance between the asset and its projection which means the ratio between the variance of the projection point and the variance of the asset is considered as an efficiency measure (θ) [5].


### 2.2. OWA Operators and Their Weight Determination Methods

An OWA operator of dimensionn is a mapping F:ℜn→ℜ with an associated weight vector W=(w1,…,wn)T such that
(5)w1+⋯+wn=1,o≤wi≤1,i=1,…,n,F(a1,…,an)=∑i=1nwibi,
where bi is the ith largest of a1,…,an.OWA operators, introduced by Yager [21], provide a unified framework for decision making under uncertainty, where different decision criteria such as maximax (optimistic), maximin (pessimistic), and equally likely (Laplace), and Hurwicz criteria are characterized by different OWA operator weights.For different weight selections, they are distinguished by the following orness degree [21]:
(6)orness(W)=1n-1∑i=1n(n-i)wi.
The orness degree can be regarded as a measure of the optimism level of the DM.To apply OWA operators for decision making, it is essential to determine the weights of OWA operators. The following models (7) and (8) are two important approaches for determining OWA operator weights under a given orness degree:
(7)MaximizeDisp(W)=-∑i=1nwiln⁡wi,Subjecttoorness(W)=α=1n-1∑i=1n(n-i)wi,0≤α≤1,∑i=1nwi=1,wi≥0,i=1,…,n,(8)MinimizeδSubjecttoorness(W)=α=1n-1∑i=1n(n-i)wi,0≤α≤1,∑i=1nwi=1,wi-wi+1-δ≤0,i=1,…,n-1,wi-wi+1-δ≥0,i=1,…,n-1,wi≥0,i=1,…,n.
Model (7), suggested by O’Hagan [22], maximizes the entropy of weight distribution and is thus referred to as the maximum entropy method, whereas model (8) that was proposed by Wang and Parkan [18] minimizes the maximum disparity between two adjacent weights and is thus called the minimax disparity approach.The OWA operator weights determined by the above models have the following characteristics.The weights are ordered. That is,w1≥w2≥⋯≥wn≥0 if the orness degree α>0.5 and 0≤w1≤w2≤⋯≤wn if α≤0.5.The weights have nothing to do with the magnitudes of the aggregatesb1~bn but depend upon their ranking orders and the DM’s optimism level (orness degree).Considerw1=1 and wj=0(j=1) if α=1, which means that the DM or investor is purely optimistic and considers only the biggest value b1=maxi(ai) in decision analysis.Considerwn=1 and wj=0(j≠n) if α=0, which represents that the DM or investor is purely pessimistic and is only concerned with the most conservative value bn=mini(ai) when making decision.Considerw1=⋯=wn=(1/n) if α=0.5, which stands for that the DM or investor is neutral and makes use of all the aggregates b1~bn equally in decision making.Considerw1,…,wn determined by model (7) vary in the form of geometric progression, that is wi+1/wi≡q for i=1,…,n-1, where q>0, while w1,…,wn determined by model (8) vary in the form of arithmetical progression; namely, wi-wi+1=d for i=1,…,K(K≤n) or wi+1-wi=d for i=K,…,n(K≥1), where d>0.


### 2.3. The Cross-Efficiency Evaluation

Considern DMUs that are to be evaluated with m inputs and s output. Denote by xij(1,…,m) and yrj(r=1,…,s) the input and output values of DMUj(j=1,…,n). The efficiencies of the n DMUs can then be computed by solving the following CCR model for each of the n DMUs, respectively [3]:
(9)max⁡θkk=∑r=1surkyrk,subjectto∑i=1mvikxik=1,∑r=1surkyrj-∑r=1svikxij≤0,j=1,…,n,urk,vik≥0,r=1,…,s,i=1,…,m,
where DMUk is the DMU under evaluation and vik(i=1,…,m) and urk(r=1,…,s) are input and output weights. Let urk*(r=1,…,s) and vik*(i=1,…,m) be the optimal solution to the above CCR model. Then, θkk*=∑r=1surk*yrk is referred to as the CCR-efficiency of DMUk, which is the best relative efficiency of DMUk by self-evaluation. If θkk*=1, DMUk is said to be CCR-efficient; otherwise, it is said to be non-CCR-efficient. θjk=∑r=1surk*yrj/∑i=1mvik*xij is referred to as the cross-efficiency of DMUk to DMUj by peer-evaluation; where j=1,…,n;j≠k.Model (9) is solved n times, each time for one particular DMU. As a result, we can get one CCR-efficiency value and (n-1) cross-efficiency values for each DMU. The n efficiency values constitute a cross-efficiency matrix, as shown in Table 1, where θkk(k=1,…,n) are the CCR-efficiency values of the n DMUs; that is, θkk=θkk*. The n efficiency values of each DMU are then simply averaged as its overall performance, which is called average cross-efficiency value. Based on these overall performance values, the n DMUs can be compared or fully ranked.Table 1
Cross-efficiency matrix forn DMUs.
DMU
Target DMU
Average cross-efficiencies
1
2
⋯
n
1
θ
11
θ
12
⋯
θ
1
n
(
1
n
)
∑
k
=
1
n
θ
1
k
2
θ
21
θ
22
⋯
θ
2
n
(
1
n
)
∑
k
=
1
n
θ
2
k
⋮
⋮
⋮
⋮
⋮
n
θ
n
1
θ
n
2
⋯
θ
n
n
(
1
n
)
∑
k
=
1
n
θ
n
kThe above approach about cross-efficiency value in CCR efficiencies or constant returns to scale (CRS) DEA model was extended to the variable returns to scale (VRS) DEA model [13]. The VRS DEA model can generate negative cross-efficiency scores.The VRS DEA model is as follows [23]:
(10)max⁡∑r=1surkyrk-u0s.t.∑r=1surkyrj-∑i=1mvikxij-u0≤0,j=1,…,n∑i=1mvikxik=1urk≥0,r=1,…,svik≥0,i=1,…,mu0≥0.
For each DMUk(k=1,…,n) under evaluation in model (10), we obtain a set of optimal weights (urk*,vrk*). Using this set of weights, the DMUk-based cross-efficiency for any DMUj(j=1,…,n) is calculated as
(11)Ekj=∑r=1surk*yrj-u0∑i=1mvik*xijk,j=1,2,…,n.The average of allEkj(k=1,…,n)(12)E-j=1n∑k=1nEkj
is used as the cross-efficiency score for DMUj(j=1,…,n).Note that the cross-efficiency score obtained in the above manner can be negative. This subject is presented by a simple numerical example involving five DMUs, with two input and single output [13].The negative VRS cross-efficiency score is due to the fact that∑r=1surkyrj-u0<0 for some DMUj; that is, some DMUj will have negative efficiency ratios when they use a set of optimal weights obtained when DMUk is under evaluation. Naturally, we want every output-input efficiency ratio to be positive regardless of the chosen weights. Therefore, adding ∑r=1surkyrj-u0≥0 into the VRS model is proposed when calculating the cross-efficiency scores [13]. This will also guarantee nonnegativity of both VRS cross-efficiency scores and VRS efficiency ratios.Therefore, the following modified VRS DEA model is used for model (10) development and application:
(13)max⁡∑r=1surkyrk-u0s.t.∑r=1surkyrj-∑i=1mvikxij-u0≤0,j=1,…,n∑i=1mvikxik=1∑r=1surkyrj-u0≥0,j=1,2,…,nurk≥0,r=1,…,svik≥0,i=1,…,mu0≥0.


## 2.1. Portfolio Performance Literature

Portfolio theory to investing is published by Markowitz (1952). This approach starts by assuming that an investor has a given sum of money to invest at the present time. This money will be invested for a time as the investor’s holding period. At the end of the holding period, the investor will sell all of the assets that were bought at the beginning of the period and then either consume or reinvest. Since portfolio is a collection of assets, it is better to select an optimal portfolio from a set of possible portfolios. Hence, the investor should recognize the returns (and portfolio returns), expected (mean) return, and standard deviation of return. This means that the investor wants to both maximize expected return and minimize uncertainty (risk). Rate of return (or simply the return) of the investor’s wealth from the beginning to the end of the period is calculated as follows:(1)Return=(end-of-periodwealth)-(beginning-of-periodwealth)beginning-of-periodwealth.
Since Portfolio is a collection of assets, its return rp can be calculated in a similar manner. Thus, according to Markowitz, the investor should view the rate of return associated with any one of these portfolios as what is called in statistics a random variable. These variables can be described as the expected return (min or r-p) and standard deviation of return. Expected return and deviation standard of return are calculated as follows:
(2)r-p=∑i=1nλir-i,σp=[∑i=1n∑j=1nλiλjσij]1/2,
where n is the number of assets in the portfolio, r-p is the expected return of the portfolio, λi is the proportion of the portfolio’s initial value invested in asset i, r-i is the expected return of asset i, σp is the deviation standard of the portfolio, and σij is the covariance of the returns between asset i and asset j.In the above, optimal portfolio from the set of portfolios will be chosen as maximum expected return for varying levels of risk and minimum risk for varying levels of expected return [19]. Data Envelopment Analysis is a nonparametric method for evaluating the efficiency of systems with multiple inputs and multiple outputs. In this section, we present some basic definitions, models, and concepts that will be used in other sections in DEA. They will not be discussed in details. Consider DMUj, (j=1,…,n), where each DMU consumes m inputs to produce s outputs. Suppose that the observed input and output vectors of DMUj are Xj=(x1j,…,xmj) and Yj=(y1j,…,ysj), respectively, and let Xj≥0 and Xj≠0, Yj≥0, and Yj≠0. A basic DEA formulation in input orientation is as follows:
(3)min⁡⁡θ-ε(∑r=1ssr++∑i=1msi-)s.t.∑j=1nλjxij+si-=θxioi=1,…,m,∑j=1nλjyrj+sr+=yror=1,…,sλ∈Λ,s+,s-≥0,ε≥0,
where λis an n-vector of λvariables, s+ as-vector of output slacks, s- an m-vector of input slacks and set Λ is defined as follows:
(4)Λ={{λ∈R+n}withconstantreturnstoscale,{λ∈R+n,1λ≤1}withnonincreasingreturnstoscale,{λ∈R+n,1λ=1}withvariablereturnstoscale.
Note that subscript “o” refers to the unit under the evaluation. A DMU is efficient if and only if θ=1 and all slack variables s-,s+ equal zero; otherwise, it is inefficient [20]. In the DEA formulation above, the left-hand sides in the constraints define an efficient portfolio. θ is a multiplier which defines the distance from the efficient frontier. The slack variables are used to ensure that the efficient point is fully efficient. This model is used for asset selection. The portfolio performance evaluation literature is vast. In recent years, these models have been used to evaluate the portfolio efficiency. Also, in the Markowitz theory, it is required to characterize the whole efficient frontier but the proposed models by Joro and Na do not need to characterize the whole efficient frontier but only the projection points. The distance between the asset and its projection which means the ratio between the variance of the projection point and the variance of the asset is considered as an efficiency measure (θ) [5].


## 2.2. OWA Operators and Their Weight Determination Methods

An OWA operator of dimensionn is a mapping F:ℜn→ℜ with an associated weight vector W=(w1,…,wn)T such that
(5)w1+⋯+wn=1,o≤wi≤1,i=1,…,n,F(a1,…,an)=∑i=1nwibi,
where bi is the ith largest of a1,…,an.OWA operators, introduced by Yager [21], provide a unified framework for decision making under uncertainty, where different decision criteria such as maximax (optimistic), maximin (pessimistic), and equally likely (Laplace), and Hurwicz criteria are characterized by different OWA operator weights.For different weight selections, they are distinguished by the following orness degree [21]:
(6)orness(W)=1n-1∑i=1n(n-i)wi.
The orness degree can be regarded as a measure of the optimism level of the DM.To apply OWA operators for decision making, it is essential to determine the weights of OWA operators. The following models (7) and (8) are two important approaches for determining OWA operator weights under a given orness degree:
(7)MaximizeDisp(W)=-∑i=1nwiln⁡wi,Subjecttoorness(W)=α=1n-1∑i=1n(n-i)wi,0≤α≤1,∑i=1nwi=1,wi≥0,i=1,…,n,(8)MinimizeδSubjecttoorness(W)=α=1n-1∑i=1n(n-i)wi,0≤α≤1,∑i=1nwi=1,wi-wi+1-δ≤0,i=1,…,n-1,wi-wi+1-δ≥0,i=1,…,n-1,wi≥0,i=1,…,n.
Model (7), suggested by O’Hagan [22], maximizes the entropy of weight distribution and is thus referred to as the maximum entropy method, whereas model (8) that was proposed by Wang and Parkan [18] minimizes the maximum disparity between two adjacent weights and is thus called the minimax disparity approach.The OWA operator weights determined by the above models have the following characteristics.The weights are ordered. That is,w1≥w2≥⋯≥wn≥0 if the orness degree α>0.5 and 0≤w1≤w2≤⋯≤wn if α≤0.5.The weights have nothing to do with the magnitudes of the aggregatesb1~bn but depend upon their ranking orders and the DM’s optimism level (orness degree).Considerw1=1 and wj=0(j=1) if α=1, which means that the DM or investor is purely optimistic and considers only the biggest value b1=maxi(ai) in decision analysis.Considerwn=1 and wj=0(j≠n) if α=0, which represents that the DM or investor is purely pessimistic and is only concerned with the most conservative value bn=mini(ai) when making decision.Considerw1=⋯=wn=(1/n) if α=0.5, which stands for that the DM or investor is neutral and makes use of all the aggregates b1~bn equally in decision making.Considerw1,…,wn determined by model (7) vary in the form of geometric progression, that is wi+1/wi≡q for i=1,…,n-1, where q>0, while w1,…,wn determined by model (8) vary in the form of arithmetical progression; namely, wi-wi+1=d for i=1,…,K(K≤n) or wi+1-wi=d for i=K,…,n(K≥1), where d>0.


## 2.3. The Cross-Efficiency Evaluation

Considern DMUs that are to be evaluated with m inputs and s output. Denote by xij(1,…,m) and yrj(r=1,…,s) the input and output values of DMUj(j=1,…,n). The efficiencies of the n DMUs can then be computed by solving the following CCR model for each of the n DMUs, respectively [3]:
(9)max⁡θkk=∑r=1surkyrk,subjectto∑i=1mvikxik=1,∑r=1surkyrj-∑r=1svikxij≤0,j=1,…,n,urk,vik≥0,r=1,…,s,i=1,…,m,
where DMUk is the DMU under evaluation and vik(i=1,…,m) and urk(r=1,…,s) are input and output weights. Let urk*(r=1,…,s) and vik*(i=1,…,m) be the optimal solution to the above CCR model. Then, θkk*=∑r=1surk*yrk is referred to as the CCR-efficiency of DMUk, which is the best relative efficiency of DMUk by self-evaluation. If θkk*=1, DMUk is said to be CCR-efficient; otherwise, it is said to be non-CCR-efficient. θjk=∑r=1surk*yrj/∑i=1mvik*xij is referred to as the cross-efficiency of DMUk to DMUj by peer-evaluation; where j=1,…,n;j≠k.Model (9) is solved n times, each time for one particular DMU. As a result, we can get one CCR-efficiency value and (n-1) cross-efficiency values for each DMU. The n efficiency values constitute a cross-efficiency matrix, as shown in Table 1, where θkk(k=1,…,n) are the CCR-efficiency values of the n DMUs; that is, θkk=θkk*. The n efficiency values of each DMU are then simply averaged as its overall performance, which is called average cross-efficiency value. Based on these overall performance values, the n DMUs can be compared or fully ranked.Table 1
Cross-efficiency matrix forn DMUs.
DMU
Target DMU
Average cross-efficiencies
1
2
⋯
n
1
θ
11
θ
12
⋯
θ
1
n
(
1
n
)
∑
k
=
1
n
θ
1
k
2
θ
21
θ
22
⋯
θ
2
n
(
1
n
)
∑
k
=
1
n
θ
2
k
⋮
⋮
⋮
⋮
⋮
n
θ
n
1
θ
n
2
⋯
θ
n
n
(
1
n
)
∑
k
=
1
n
θ
n
kThe above approach about cross-efficiency value in CCR efficiencies or constant returns to scale (CRS) DEA model was extended to the variable returns to scale (VRS) DEA model [13]. The VRS DEA model can generate negative cross-efficiency scores.The VRS DEA model is as follows [23]:
(10)max⁡∑r=1surkyrk-u0s.t.∑r=1surkyrj-∑i=1mvikxij-u0≤0,j=1,…,n∑i=1mvikxik=1urk≥0,r=1,…,svik≥0,i=1,…,mu0≥0.
For each DMUk(k=1,…,n) under evaluation in model (10), we obtain a set of optimal weights (urk*,vrk*). Using this set of weights, the DMUk-based cross-efficiency for any DMUj(j=1,…,n) is calculated as
(11)Ekj=∑r=1surk*yrj-u0∑i=1mvik*xijk,j=1,2,…,n.The average of allEkj(k=1,…,n)(12)E-j=1n∑k=1nEkj
is used as the cross-efficiency score for DMUj(j=1,…,n).Note that the cross-efficiency score obtained in the above manner can be negative. This subject is presented by a simple numerical example involving five DMUs, with two input and single output [13].The negative VRS cross-efficiency score is due to the fact that∑r=1surkyrj-u0<0 for some DMUj; that is, some DMUj will have negative efficiency ratios when they use a set of optimal weights obtained when DMUk is under evaluation. Naturally, we want every output-input efficiency ratio to be positive regardless of the chosen weights. Therefore, adding ∑r=1surkyrj-u0≥0 into the VRS model is proposed when calculating the cross-efficiency scores [13]. This will also guarantee nonnegativity of both VRS cross-efficiency scores and VRS efficiency ratios.Therefore, the following modified VRS DEA model is used for model (10) development and application:
(13)max⁡∑r=1surkyrk-u0s.t.∑r=1surkyrj-∑i=1mvikxij-u0≤0,j=1,…,n∑i=1mvikxik=1∑r=1surkyrj-u0≥0,j=1,2,…,nurk≥0,r=1,…,svik≥0,i=1,…,mu0≥0.


## 3. Methodology

Return of assets consists of money which we receive among period plus difference of buying and selling. Return is not definitely usually obvious. This is uncertain in rate of expected return defined as deviation of return. Deviation of return is called risk. The investor’s objective is to get the maximum possible return on an investment with the minimum possible risk. In this regard, mean-variance model Markowitz expected return is treated as output and deviation as input. The methodology in this paper starts with asset selection via cross-efficiency evaluation using OWA operator weights. The data used for this methodology is from 57 mutual funds [5]. In many cases similar to this example, there are a lot of assets. It is better that starts with asset selection. The choice of the asset can be random or discrete. The random choice of assets is usually biased and does not promise an optimum portfolio; hence, it is more rational to have an objective choice while selecting the assets to be included in the portfolio. Among many evaluation methods, Data Envelopment Analysis (DEA) is one of the best ways for assessing the relative efficiency a group of homogenous decision making units (DMUs) that use multiple inputs to produce multiple outputs, originated from the work by Charnes et al. [3]. Selection of assets to be included in portfolio is followed by using cross-efficiency in DEA. The variable returns to scale (VRS) DEA model is used for efficiency evaluation. In the analysis, the variance of the assets is used as an input to the DEA and expected return is used as an output. Because the VRS DEA model can generate negative cross-efficiency score, thus model (13) is proposed so that the cross-efficiency scores are nonnegative. Traditional approaches for the cross-efficiency evaluation do not differentiate between self-evaluated and peer-evaluated efficiencies. A significant problem with these approaches is that the weight assigned to the self-evaluated efficiency of each DMU is fixed and has no way of incorporating the DM’s or investor’s subjective preferences in to the evaluation. For example, the investors may wish self-evaluated efficiencies to account for 20% or play a leading role in the final overall efficiency assessment. Obviously, equal evaluation has no method to obtain this purpose. To show the investor’s subjective preferences on different efficiencies, the use of OWA operator weights is stated for cross-efficiency evaluation (see Table 4). This requires the reordering of the efficiencies, both self-evaluated and peer-evaluated, of each DMU, as shown in Table 2, where w1,…,wn are OWA operator weights and ϑij(i,j=1,…,n) are reordered efficiencies of each DMU from the biggest to the smallest. Obviously, self-evaluated efficiencies are always ranked in the first place; that is, ϑi1=θii*,i=1,…,n. In order to determine the weights of OWA operator, it is necessary for the investor to provide his/her preferences on different efficiencies or optimism level towards the best relative efficiencies. For example, if the investor wants the self-evaluated efficiencies to account for 20% in the final overall efficiency assessment, then w1 should take 0.2, whereas the other weights can be designated minimax disparity approach. With regard to the minimax disparity OWA operator weights, the following theorems are existed [17].Table 2
Reordered cross-efficiency matrix of then DMUs.
DMU
Reordered efficienciesin descending order
Weightedaverage cross-efficiency
1st
2nd
⋯
nth
w
1
w
2
⋯
w
n
1
ϑ
11
ϑ
12
⋯
ϑ
1
n
∑
k
=
1
n
w
k
ϑ
1
k
2
ϑ
21
ϑ
22
⋯
ϑ
2
n
∑
k
=
1
n
w
k
ϑ
2
k
⋮
⋮
⋮
⋮
⋮
n
ϑ
n
1
ϑ
n
2
⋯
ϑ
n
n
∑
k
=
1
n
w
k
ϑ
n
kTheorem 1.
For a givenw1, there exists an integer k≤n such that wi=w1-(i-1)d≥0for i=1,…,k and wi=0 for i=k+1,…,n, where k and d are determined by
(14)k=min⁡(n,INT[2w1]),d=2(kw1-1)k(k-1).Theorem 2.
For a given orness degreeα∈(0.5,1), there exists an integer k≤n such that wi=w1-(i-1)d≥0for i=1,…,k and wi=0for i=k+1,…,n, where k,w1, and d are determined by
(15)w1=4(k+1)-6n+6α(n-1)k(k+1),d=2(kw1-1)k(k-1).In this paper, the OWA operator weights can be determined by using the minimax disparity approach. In the following, some very special OWA operator weights for the cross-efficiency evaluation are given.Considerw1=1 and wj=0(j≠1). In this case, orness(W) = 1 and θ-i=∑k=1nwkϑik=ϑi1=θii* for i=1,…,n. The investor considers only self-evaluated efficiencies in the final overall efficiency assessment and is purely optimistic.Considerwn=1 and wj=0(j≠1). In this case, orness(W) = 0 and θ-i=∑k=1nwkϑik=ϑin=mink(θik) for i=1,…,n. The investor chooses the least efficiency value of each DMU as its overall efficiency and is extremely pessimistic.Considerw1,…,wn=1/n. In this case, orness(W) = 0.5 and θ-i=∑k=1nwkϑik=(1/n)∑k=1nθik for i=1,…,n, which is the average cross-efficiency value in the traditional cross-efficiency evaluation.The orness degree can be regarded as a measure of the optimism level of the investor. If the investor wants self-evaluated to be more influenced, it should be used of orness > 0.5. And if investor wants peer-evaluated to be more influenced, it should be used of orness < 0.5. Obviously, the best selection of mutual funds is not fixed. It varies with the investor’s optimism level or subjective performance.


## 4. Application in Mutual Funds

We illustrate our approach using OWA operator weights in cross-efficiency evaluation for a dataset of 57 mutual funds. A list of funds used is provided in Table3. In this report, there is expected return and variance of mutual funds which expected return is considered as output and variance is as input. The example is received from Joro and Na [5] and is about portfolio performance evaluation in a mean-variance framework. Four mutual funds are evaluated as efficient in model [1] which portfolio can be composed with them. But it is better to use cross-efficiency to choose the best portfolio. Because the model (10) can generate negative cross-efficiency score, thus model (13) is used so that the cross-efficiency scores are nonnegative. Traditional approaches for the cross-efficiency evaluation do not differentiate between self-evaluated and peer-evaluated efficiencies. A main problem with these approaches is that the weight assigned to the self-evaluated efficiency of each DMU is fixed and has no way of incorporating the investor’s subjective preferences into the evaluation. Obviously, equal evaluation has no way to obtain this goal. To show the investor’s subjective preferences on different efficiencies, the use of OWA operator weights is stated for cross-efficiency evaluation. This requires the reordering of the efficiencies. The orness degree can be regarded as a measure of the optimism level of the investor. If the investor wants self-evaluated to be more influenced, it should be used of orness > 0.5. And if investor wants peer-evaluated to be more influenced, it should be used of orness < 0.5. In the traditional equal of cross-efficiencies, the weight assigned to the self-evaluated efficiencies is only 0.017% =(1/57). For an optimistic investor, he/she may wish the self-evaluated efficiencies to play a more role in the final overall efficiency assessment. For example, the investor may wish the weight for the self-evaluated efficiencies to account for 20% rather than 0.017% in the final overall efficiency assessment. By Theorem 1, k=min⁡(57,INT[2/0.2])=10 and d=2(kw1-1)/k(k-1)=2(10×0.2-1)/(10×9)=0.022 are obtained. As a result, the weights for cross-efficiency are computed as w1=0.2,  w2=w1-d=0.178,   w3=w1-2d=0.156,w4=w1-3d=0.134,w5=w1-4d=0.112,  w6=w1-5d=0.09,  w7=w1-6d=0.068,  w8=w1-7d=0.046,  w9=w1-8d=0.024,  w10=w1-9d=0.002, and w11=⋯=w57=0. The investor’s optimism level is measured as orness(W) = (1/(n-1))∑i=1n(n-i)wi=0.9613. The weighted average cross-efficiency values of the 57 mutual funds are computed by θ-i=∑k=1nwkϑik for i=1,…,n and the results are presented in the fifth column of the Table 5. As is seen in Tables 5 and 6, ranks are not the same. We calculated these ranks for α=0.8 and α=0.1. Some of the best ranks are designated according to investor. We consider ten of the best ranks. Five of the best ranks become the same, in this example,incidentally. Selecting of mutual funds to be included in portfolio is followed by ten of the best ranks in Tables 7 and 8 for α≥0.5,α≤0.4, respectively.Table 3
Descriptive statistics of the mutual funds.
Mutual fund
Expected return
Variance
ACEFX
2.671
64.173
ACEGX
2.734
64.254
ACEMX
2.668
64.098
AELAX
1.241
22.976
AELGX
1.287
22.970
AGTHX
1.902
23.445
ARCGX
2.017
30.697
AVLFX
1.615
19.817
BJBIX
1.506
33.203
CUCAX
1.990
24.858
FAGIX
0.533
5.471
FAIAX
1.404
39.099
FAICX
1.420
39.095
FDEGX
2.445
53.059
FEURX
1.696
30.655
FIUIX
1.398
13.813
FSUTX
1.697
18.829
GEGTX
1.727
19.483
HRCPX
2.376
34.856
IDETX
2.356
35.331
IDEUX
1.874
24.913
IGLBX
1.858
25.178
IGLCX
2.488
39.982
JAMRX
1.991
35.740
JAOSX
1.870
23.264
JAVLX
2.237
30.771
JAWWX
1.946
20.705
LMVTX
1.735
19.463
MAFGX
1.849
20.490
MBFGX
1.923
20.626
MCFGX
2.463
60.247
MCOBX
1.959
19.729
MCOFX
1.875
18.821
MDFGX
1.942
18.833
MGCAX
1.789
31.720
MSEQX
1.842
31.348
NAWCX
1.318
24.631
NAWGX
1.384
24.676
POVBX
2.579
149.234
POVSX
1.619
21.647
RYOSX
2.690
42.951
SCGEX
1.656
18.530
SRGFX
2.224
40.412
SSGWX
2.044
26.771
TALGX
1.368
30.664
TRGEX
1.786
25.739
TWIEX
1.773
23.208
UMBIX
1.823
12.379
USBOX
2.093
24.542
VGHCX
1.334
26.919
VPMCX
1.513
20.243
WAGEX
1.625
17.619
WBIGX
0.979
25.015
FMAGX
1.681
14.225
JANSX
2.643
35.453
VFINX
1.690
27.731
VWNDX
1.889
21.968Table 4
OWA operator weights for cross efficiency evaluation.
α
=
1
α
=
0.9613
α
=
0.9
α
=
0.8
α
=
0.7
α
=
0.6
α
=
0.5
α
=
0.4
α
=
0.3
α
=
0.2
α
=
0.1
1
0.2
0.106
0.056
0.038
0.028
0.018
0.007
0
0
0
0
0.178
0.1
0.055
0.037
0.027
0.018
0.008
0
0
0
0
0.156
0.094
0.053
0.037
0.027
0.018
0.008
0
0
0
0
0.134
0.088
0.051
0.036
0.027
0.018
0.008
0
0
0
0
0.112
0.082
0.05
0.035
0.026
0.018
0.009
0
0
0
0
0.09
0.077
0.048
0.034
0.026
0.018
0.009
3.00
E
-
04
0
0
0
0.068
0.071
0.046
0.034
0.026
0.018
0.01
0.001
0
0
0
0.046
0.065
0.045
0.033
0.025
0.018
0.01
0.002
0
0
0
0.024
0.059
0.043
0.032
0.025
0.018
0.01
0.003
0
0
0
0.002
0.053
0.042
0.031
0.024
0.018
0.011
0.003
0
0
0
0
0.047
0.04
0.031
0.024
0.018
0.011
0.004
0
0
0
0
0.041
0.038
0.03
0.024
0.018
0.011
0.005
0
0
0
0
0.035
0.037
0.029
0.023
0.018
0.012
0.005
0
0
0
0
0.029
0.035
0.029
0.023
0.018
0.012
0.006
0
0
0
0
0.023
0.033
0.028
0.023
0.018
0.012
0.007
0
0
0
0
0.017
0.032
0.027
0.022
0.018
0.013
0.008
0
0
0
0
0.011
0.03
0.026
0.022
0.018
0.013
0.008
0
0
0
0
0.005
0.029
0.026
0.022
0.018
0.014
0.009
0
0
0
0
0
0.027
0.025
0.021
0.018
0.014
0.01
0
0
0
0
0
0.025
0.024
0.021
0.018
0.014
0.011
0
0
0
0
0
0.024
0.023
0.02
0.018
0.015
0.011
0
0
0
0
0
0.022
0.023
0.02
0.018
0.015
0.012
0
0
0
0
0
0.02
0.022
0.02
0.018
0.015
0.013
0
0
0
0
0
0.019
0.021
0.019
0.018
0.016
0.014
0.003
0
0
0
0
0.017
0.02
0.019
0.018
0.016
0.014
0.004
0
0
0
0
0.016
0.02
0.019
0.018
0.016
0.015
0.006
0
0
0
0
0.014
0.019
0.018
0.018
0.017
0.016
0.007
0
0
0
0
0.012
0.018
0.018
0.018
0.017
0.017
0.009
0
0
0
0
0.011
0.017
0.018
0.018
0.018
0.017
0.011
0
0
0
0
0.009
0.017
0.017
0.018
0.018
0.018
0.012
0
0
0
0
0.007
0.016
0.017
0.018
0.018
0.019
0.014
0
0
0
0
0.006
0.015
0.016
0.018
0.019
0.02
0.016
0
0
0
0
0.004
0.014
0.016
0.018
0.019
0.02
0.017
0
0
0
0
0.003
0.014
0.016
0.018
0.019
0.021
0.019
0
0
0
0
0
0.013
0.015
0.018
0.02
0.022
0.02
0
0
0
0
0
0.012
0.015
0.018
0.02
0.023
0.022
0
0
0
0
0
0.011
0.015
0.018
0.02
0.023
0.024
0
0
0
0
0
0.011
0.014
0.018
0.021
0.024
0.025
0
0
0
0
0
0.01
0.014
0.018
0.021
0.025
0.027
0
0
0
0
0
0.009
0.014
0.018
0.022
0.026
0.029
0.005
0
0
0
0
0.008
0.013
0.018
0.022
0.026
0.03
0.011
0
0
0
0
0.008
0.013
0.018
0.022
0.027
0.032
0.017
0
0
0
0
0.007
0.012
0.018
0.023
0.028
0.033
0.023
0
0
0
0
0.006
0.012
0.018
0.023
0.029
0.035
0.029
0
0
0
0
0.005
0.012
0.018
0.023
0.029
0.037
0.035
0
0
0
0
0.005
0.011
0.018
0.024
0.03
0.038
0.041
0
0
0
0
0.004
0.011
0.018
0.024
0.031
0.04
0.047
0
0
0
0
0.003
0.011
0.018
0.024
0.031
0.042
0.053
0
0
0
0
0.003
0.01
0.018
0.025
0.032
0.043
0.059
0
0
0
0
0.002
0.01
0.018
0.025
0.033
0.045
0.065
0
0
0
0
0.001
0.01
0.018
0.026
0.034
0.046
0.071
0
0
0
0
3.00
E
-
04
0.009
0.018
0.026
0.034
0.048
0.077
0
0
0
0
0
0.009
0.018
0.026
0.035
0.05
0.082
0
0
0
0
0
0.008
0.018
0.027
0.036
0.051
0.088
0
0
0
0
0
0.008
0.018
0.027
0.037
0.053
0.094
0
0
0
0
0
0.008
0.018
0.027
0.037
0.055
0.1
0
0
0
0
0
0.007
0.018
0.028
0.038
0.056
0.106Table 5
Cross efficiency by optimism level of the investor forα≥0.5.
DMU
Mutual funds
VRS efficiency
α
=
1
α
=
0.9613
α
=
0.9
α
=
0.8
α
=
0.7
α
=
0.6
α
=
0.5
Rankingα=0.8
1
ACEFX
0.32
0.324
0.323
0.323
0.319
0.312
0.306
0.308
50
2
ACEGX
0.33
0.334
0.333
0.333
0.328
0.322
0.315
0.317
48
3
ACEMX
0.32
0.324
0.287
0.323
0.319
0.312
0.306
0.308
49
4
AELAX
0.37
0.370
0.373
0.370
0.363
0.350
0.3392
0.336
44
5
AELGX
0.38
0.377
0.371
0.383
0.377
0.365
0.3550
0.353
43
6
AGTHX
0.56
0.569
0.572
0.566
0.560
0.558
0.556
0.568
21
7
ARCGX
0.46
0.472
0.454
0.468
0.463
0.459
0.456
0.465
34
8
AVLFX
0.55
0.560
0.563
0.558
0.552
0.546
0.541
0.549
22
9
BJBIX
0.31
0.311
0.313
0.310
0.307
0.301
0.297
0.300
51
10
CUCAX
0.56
0.572
0.564
0.568
0.561
0.558
0.554
0.565
20
11
FAGIX
0.66
0.669
0.667
0.670
0.627
0.509
0.410
0.320
9
12
FAIAX
0.24
0.246
0.248
0.246
0.242
0.236
0.232
0.233
55
13
FAICX
0.25
0.249
0.251
0.249
0.245
0.239
0.235
0.236
54
14
FDEGX
0.35
0.351
0.352
0.349
0.345
0.339
0.334
0.337
46
15
FEURX
0.38
0.379
0.382
0.378
0.3751
0.372
0.370
0.377
42
16
FIUIX
0.69
0.694
0.693
0.693
0.683
0.667
0.653
0.655
6
17
FSUTX
0.61
0.618
0.615
0.620
0.611
0.607
0.603
0.615
13
18
GEGTX
0.60
0.608
0.605
0.607
0.601
0.598
0.595
0.607
16
19
HRCPX
0.51
0.516
0.518
0.512
0.506
0.498
0.491
0.496
27
20
IDETX
0.50
0.503
0.505
0.500
0.494
0.486
0.480
0.485
31
21
IDEUX
0.52
0.525
0.527
0.521
0.517
0.515
0.513
0.525
26
22
IGLBX
0.50
0.513
0.515
0.510
0.506
0.504
0.503
0.514
28
23
IGLCX
0.47
0.477
0.478
0.474
0.468
0.459
0.452
0.456
34
24
JAMRX
0.39
0.398
0.399
0.395
0.390
0.388
0.386
0.393
39
25
JAOSX
0.55
0.561
0.562
0.557
0.552
0.550
0.548
0.560
19
26
JAVLX
0.53
0.540
0.542
0.536
0.530
0.523
0.517
0.524
24
27
JAWWX
0.65
0.666
0.668
0.661
0.654
0.650
0.647
0.661
7
28
LMVTX
0.61
0.611
0.604
0.610
0.605
0.602
0.599
0.611
15
29
MAFGX
0.62
0.627
0.628
0.622
0.618
0.616
0.614
0.628
10
30
MBFGX
0.65
0.657
0.659
0.653
0.646
0.643
0.640
0.654
8
31
MCFGX
0.31
0.312
0.314
0.310
0.306
0.301
0.299
0.282
52
32
MCOBX
0.69
0.705
0.707
0.700
0.693
0.688
0.685
0.699
4
33
MCOFX
0.68
0.696
0.698
0.691
0.685
0.682
0.680
0.695
5
34
MDFGX
0.72
0.730
0.732
0.725
0.717
0.713
0.710
0.725
3
35
MGCAX
0. 38
0.387
0.387
0.386
0.383
0.382
0.381
0.389
40
36
MSEQX
0.40
0.407
0.409
0.405
0.402
0.400
0.399
0.408
36
37
NAWCX
0.36
0.367
0.312
0.366
0.360
0.350
0.340
0.340
45
38
NAWGX
0.38
0.385
0.337
0.384
0.378
0.369
0.361
0.362
41
39
POVBX
0.13
0.134
0.134
0.133
0.131
0.128
0.126
0.127
57
40
POVSX
0.51
0.513
0.511
0.512
0.506
0.501
0.497
0.504
30
41
RYOSX
0.48
0.490
0.492
0.486
0.480
0.470
0.462
0.465
32
42
SCGEX
0.61
0.613
0.593
0.612
0.606
0.600
0.596
0.606
14
43
SRGFX
0.40
0.408
0.410
0.405
0.401
0.395
0.391
0.396
38
44
SSGWX
0.54
0.551
0.552
0.547
0.541
0.536
0.532
0.542
23
45
TALGX
0.30
0.306
0.266
0.305
0.301
0.293
0.286
0.287
53
46
TRGEX
0.47
0.476
0.476
0.475
0.472
0.470
0.469
0.479
33
47
TWIEX
0.52
0.524
0.523
0.523
0.519
0.517
0.515
0.526
25
48
UMBIX
1.00
1.000
1.020
1.012
1.005
1.002
1.021
0.988
1
49
USBOX
0.61
0.620
0.622
0.616
0.609
0.603
0.598
0.608
12
50
VGHCX
0.34
0.342
0.339
0.342
0.336
0.327
0.3191
0.319
47
51
VPMCX
0.51
0.513
0.473
0.512
0.505
0.497
0.490
0.495
29
52
WAGEX
0.63
0.633
0.627
0.631
0.624
0.618
0.613
0.623
10
53
WBIGX
0.27
0.174
0.174
0.172
0.172
0.187
0.201
0.220
56
54
FMAGX
0.80
0.810
0.790
0.808
0.801
0.795
0.790
0.804
2
55
JANSX
0.57
0.580
0.583
0.576
0.569
0.558
0.548
0.552
18
56
VFINX
0.41
0.418
0.408
0.417
0.413
0.410
0.408
0.415
36
57
VWNDX
0.59
0.602
0.583
0.597
0.592
0.590
0.588
0.601
17Table 6
Cross efficiency by OWA operator weights forα≤0.4.
DMU
Mutual funds
VRS efficiency
α
=
0.4
α
=
0.3
α
=
0.2
α
=
0.1
Rankingα=0.1
1
ACEFX
0.32
0.295
0.288
0.282
0.282
48
2
ACEGX
0.33
0.302
0.296
0.289
0.287
46
3
ACEMX
0.32
0.295
0.288
0.282
0.281
49
4
AELAX
0.37
0.316
0.305
0.293
0.291
45
5
AELGX
0.38
0.334
0.323
0.312
0.310
43
6
AGTHX
0.56
0.551
0.548
0.545
0.545
17
7
ARCGX
0.46
0.450
0.446
0.443
0.443
32
8
AVLFX
0.55
0.530
0.524
0.518
0.515
20
9
BJBIX
0.31
0.288
0.283
0.278
0.276
49
10
CUCAX
0.56
0.547
0.543
0.540
0.539
18
11
FAGIX
0.66
0.213
0.115
0.011
0
57
12
FAIAX
0.24
0.222
0.217
0.211
0.210
55
13
FAICX
0.25
0.225
0.220
0.215
0.214
54
14
FDEGX
0.35
0.323
0.318
0.312
0.311
42
15
FEURX
0.38
0.365
0.362
0.359
0.358
40
16
FIUIX
0.69
0.625
0.610
0.595
0.591
9
17
FSUTX
0.61
0.595
0.590
0.586
0.584
11
18
GEGTX
0.60
0.588
0.585
0.581
0.578
13
19
HRCPX
0.51
0.477
0.469
0.461
0.460
28
20
IDETX
0.50
0.466
0.459
0.451
0.450
31
21
IDEUX
0.52
0.509
0.507
0.505
0.504
23
22
IGLBX
0.50
0.499
0.497
0.494
0.494
25
23
IGLCX
0.47
0.437
0.430
0.421
0.420
34
24
JAMRX
0.39
0.381
0.378
0.376
0.375
35
25
JAOSX
0.55
0.544
0.542
0.539
0.539
19
26
JAVLX
0.53
0.505
0.498
0.492
0.490
26
27
JAWWX
0.65
0.640
0.637
0.633
0.633
6
28
LMVTX
0.61
0.592
0.589
0.585
0.583
12
29
MAFGX
0.62
0.609
0.607
0.604
0.603
8
30
MBFGX
0.65
0.634
0.631
0.627
0.627
7
31
MCFGX
0.31
0.287
0.282
0.277
0.276
51
32
MCOBX
0.69
0.677
0.673
0.669
0.669
4
33
MCOFX
0.68
0.675
0.672
0.669
0.668
5
34
MDFGX
0.72
0.702
0.698
0.695
0.694
3
35
MGCAX
0. 38
0.378
0.376
0.374
0.373
38
36
MSEQX
0.40
0.396
0.395
0.393
0.392
36
37
NAWCX
0.36
0.322
0.313
0.303
0.300
44
38
NAWGX
0.38
0.345
0.336
0.327
0.325
41
39
POVBX
0.13
0.122
0.119
0.117
0.116
56
40
POVSX
0.51
0.486
0.481
0.475
0.473
27
41
RYOSX
0.48
0.444
0.435
0.425
0.422
33
42
SCGEX
0.61
0.586
0.580
0.574
0.572
16
43
SRGFX
0.40
0.382
0.377
0.372
0.371
39
44
SSGWX
0.54
0.524
0.519
0.515
0.515
21
45
TALGX
0.30
0.273
0.266
0.258
0.257
53
46
TRGEX
0.47
0.465
0.463
0.460
0.459
29
47
TWIEX
0.52
0.511
0.508
0.505
0.504
22
48
UMBIX
1.00
0.992
0.988
0.983
0.981
1
49
USBOX
0.61
0.587
0.581
0.576
0.575
15
50
VGHCX
0.34
0.303
0.295
0.286
0.284
47
51
VPMCX
0.51
0.475
0.467
0.459
0.457
30
52
WAGEX
0.63
0.601
0.594
0.588
0.585
10
53
WBIGX
0.27
0.228
0.242
0.259
0.264
52
54
FMAGX
0.80
0.778
0.772
0.765
0.761
2
55
JANSX
0.57
0.527
0.517
0.505
0.503
24
56
VFINX
0.41
0.402
0.398
0.395
0.394
35
57
VWNDX
0.59
0.583
0.580
0.578
0.577
14Table 7
Selecting the best assets for making portfolio forα≥0.5.
Expected return
Variance
Ranking
48
UMBIX
1.833
12.379
1
54
FMAGX
1.681
14.225
2
34
MDFGX
1.942
18.833
3
32
MCOBX
1.959
19.729
4
33
MCOFX
1.875
18.821
5
16
FIUIX
1.398
13.813
6
27
JAWWX
1.946
20.705
7
30
MBFGX
1.923
20.626
8
11
FAGIX
0.533
5.471
9
29
MAFGX
1.849
20.490
10Table 8
Selecting the best assets for making portfolio forα≤0.4.
Expected return
Variance
Ranking
48
UMBIX
1.833
12.379
1
54
FMAGX
1.681
14.225
2
34
MDFGX
1.942
18.833
3
32
MCOBX
1.959
19.729
4
33
MCOFX
1.875
18.821
5
27
JAWWX
1.946
20.705
6
30
MBFGX
1.923
20.626
7
29
MAFGX
1.849
20.490
8
16
FIUIX
1.398
13.813
9
52
WAGEX
1.625
17.619
10


## 5. Conclusion

In this paper, a new method is suggested for selecting the best of portfolio with one input (variance) and one output (expected return) in the DEA context. As an advanced management decision tool, DEA has been widely used for performance evaluation [24, 25], productivity analysis [26–28], resource allocation [29], and so on. The cross-efficiency evaluation is an important method for ranking DMUs in DEA. Traditional approaches for the cross-efficiency evaluation do not differentiate between self-evaluated and peer-evaluated efficiencies. A main problem with these approaches is that the weight assigned to the self-evaluated efficiency of each DMU is fixed and has no way of incorporating the investor’s subjective preferences in to the evaluation. To show the investor’s subjective preferences on different efficiencies, the use of OWA operator weights is stated for cross-efficiency evaluation. In this case, if the investor wants self-evaluated to be more influenced, it should be used of orness > 0.5. Thus, if investor wants peer-evaluated to be more influenced, it should be used of orness < 0.5. In Tables 7 and 8, rankings have been designated for ten of the best mutual funds via OWA operator weights in cross-efficiency evaluation. Since there are a large number of assets to invest in, this objective leads to two investment problems. First, the assets are selected for making portfolio and, second, the proportion or weights are determined to be allocated to the selected assets. Selection of assets to be included in portfolio is followed by using cross-efficiency evaluation. Model (13) is used for this purpose. In this regard, this model is used to analyze the given 57 mutual funds and ten of the best mutual funds are obtained. The other methods of ranking can be used for ten of the best mutual funds for asset allocation in the future.



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*Source: 978314-2014-03-19.xml*