Research ArticleUsage of Cholesky Decomposition in order to Decreasethe Nonlinear Complexities of Some Nonlinear andDiversification Models and Present a Model in Framework ofMean-Semivariance for Portfolio Performance Evaluation

H Siaby-Serajehlo1 M Rostamy-Malkhalifeh1 F Hosseinzadeh Lotfi1 and M H Behzadi2

1Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran2Department of Statistics Science and Research Branch Islamic Azad University Tehran Iran

Correspondence should be addressed to M Rostamy-Malkhalifeh mohsen rostamyyahoocom

Received 22 October 2015 Revised 12 February 2016 Accepted 14 February 2016

Academic Editor Shey-Huei Sheu

Copyright copy 2016 H Siaby-Serajehlo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In order to get efficiency frontier and performance evaluation of portfolio nonlinear models and DEA nonlinear (diversification)models are mostly used One of the most fundamental problems of usage of nonlinear and diversification models is theircomputational complexityTherefore in this paper a method is presented in order to decrease nonlinear complexities and simplifycalculations of nonlinear and diversification models used from variance and covariance matrix For this purpose we use a lineartransformation which is obtained from the Cholesky decomposition of covariance matrix and eliminate linear correlation amongfinancial assets In the following variance is an appropriate criterion for the risk when distribution of stock returns is to be normaland symmetric as such a thing does not occur in reality On the other hand investors of the financial markets do not have an equalreaction to positive and negative exchanges of the stocks and show more desirability towards the positive exchanges and highersensitivity to the negative exchanges Therefore we present a diversification model in the mean-semivariance framework which isbased on the desirability or sensitivity of investor to positive and negative exchanges and rate of this desirability or sensitivity canbe controlled by use of a coefficient

1 Introduction

Despite the thousands of investment mutual funds for therendering services to the investors and high volume of finan-cial transactions accomplished in the these mutual funds [1]and with regard to the plentiful studies performed in thisfield no discussions can be raised that portfolio performanceevaluation and investment mutual fundsrsquo performance eval-uation play an effective role in the decision making of theinvestors in order to invest in the financial markets Variousinvestors use different definitions of performance evaluationdepending on their perspective and outlook in order topredict financial markets

One of the most important viewpoints in the portfolioperformance evaluation is to use portfolio efficiency frontierIn this method distance of the desired portfolio from

efficiency frontier is the considered a criterion for portfolioperformance evaluation Markowitz [2] presented a modelin the mean-variance framework for portfolio performanceevaluation using this criterion In this model existence ofthe linear correlation among financial assets has led toexistence of nonlinear complexities in the model Duringrecent years the researches carried out in this field havebeen concentrated on the results of this theory Sharpe[3] presented a model in order to decrease the volume ofcovariance matrix calculations (linear correlation among thefinancial assets) using return market index In this methodlinear correlation among the financial assets is considered tobe zero Instead using regression return of each financialassets is related to return market index which is referred toas single-index model Usage of Markowitz model involvesexistence of enough data regarding mean variance and

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016 Article ID 7828071 9 pageshttpdxdoiorg10115520167828071

2 Advances in Operations Research

covariance between each two pairs of financial assets whichinclude (119899(119899 + 3)2) of data Now if the sharpe single-index model is used required data will decrease to (3119899 minus2) On the other hand in the single-index model it wasassumed that return of one asset is only affected by an index(market index) but due to available weaknesses and lack ofignoring of all indexes affecting the performance evaluationsome researchers created multi-index models via additionof the indexes and nonmarket factors (inflation rate foreignexchange rate oil price industry index etc) [4] M RMoreyandR CMorey presented a diversificationmodel inspired byDEA [5] Also in this model existence of linear correlationamong financial assets has been led to existence of thenonlinear complexities in themodel Briec et al [6] presenteda diversification model in the mean-variance frameworkthrough introduction of efficiency improvement possibilityfunction (shortage function) For the sake of investorsrsquo incli-nation towards the positive skewness [7ndash9] Joro and Na [10]offered a model in the mean-variance-skewness frameworkby consideration of the variance as input and skewness andmean as output In some studies semivariance has beenused instead of variance because variance is counted as anappropriate criterion for the risk when the stocks returndistribution is to be in normal and symmetric form whichit is not in such a manner most of times [11 12]

One of our basic problems is complexity of the calcu-lations of diversification and nonlinear models We usu-ally use Lagrange method KKT method [13] and variousheuristic methods [14] in order to solve these models incase of having the required conditions For example inthe Lagrange or KKT method we solve the problem bytransformation of optimization problem into linear systemsIn order to solve linear systems there are various methodswhich for example can be referred to the direct methodssuch as Gaussian and Gaussian-Jordan elimination methodsand matrix decomposition methods such as Lu decomposi-tion orthogonal matrixes and Cholesky decomposition [15]Cholesky decomposition can be applied for the matrixeswhich are positive definite and symmetric Solving linearsystems is one of the principal applications of the Choleskydecomposition Another one of its application is data pro-duction for dependent variable using simulation [16 17] Inthis paper we at first present a linear transformation basedon Cholesky decomposition In continuation we present adiversification model in the mean-semivariance frameworkand using the mentioned linear transformation decreasenonlinear computational complexities of the Markowitz MRMorey and R CMorey Briec et al and suggestivemodels

This paper has been classified as follows in Section 2 wepresent a brief description in connection with the Choleskydecomposition positive definite matrix covariance matrixand their properties and finish the section through introduc-tion of a linear transformation and financial interpretation ofthe Cholesky decomposition In Section 3 we apply the men-tioned linear transformation for Markowitz M R Morey RC Morey and Briec et al models and express advantagesof the obtained models In Section 4 we present a model inthe mean-semivariance framework and apply the mentionedlinear transformation for that and at the end the potential

integration of higher moments briefly is discussed Section 5includes numerical examples for explanation of the methodsand suggestive models We finish the paper with conclusionin Section 6

2 Required Concepts

In this section we take a look at the required concepts brieflyand deal with their properties by presentation of a lineartransformation

Definition 1 119860119899times119899

(square matrix) is known as positivedefinite matrix whenever

forall119909 (119909 isin R119899

119909 = 0) 997904rArr 119909119879

119860119909 gt 0 (1)

Definition 2 (Cholesky decomposition) If119860119899times119899

is symmetricand positive definite matrix then there exists a unique lowertriangular matrix 119871

119899times119899with positive diagonal element such

that

119860 = 119871119871119879

(2)

Consider the vector random variable 119883 = 1199091 119909

119899

which has 119864(119883) (mean vector) and var(119883) (covariancematrix) var(119883) is a positive and symmetrical definite matrixTherefore we will use Cholesky decomposition as follows

var (119883) = [cov (119909119894 119909119895)]119899times119899

= 119871119871119879

119871 =

[[[[[[

[

119871110 sdot sdot sdot 0

1198712111987122sdot sdot sdot 0

11987111989911198711198992sdot sdot sdot 119871

119899119899

]]]]]]

]

(3)

Consider the linear transformation presented in the followingrelation

119882 = 119871minus1

119883 (4)

We prove that the linear correlation between the randomvariables of119882 is equal to zero

119871119871119879

= var (119883) = var (119871119882)

= 119864 [(119871119882 minus 119871119864 (119882)) (119871119882 minus 119871119864 (119882))119879

]

= 119871 var (119882) 119871119879

(5)

Therefore

var (119882) = 119868 997904rArr

cov (119908119894 119908119895)119894 =119895

= 0(6)

21 Financial Interpretation of the Cholesky DecompositionCholesky decomposition technique is one of the availablemethods for data production frommultivariate distributions

Advances in Operations Research 3

because we require data production and simulation in someproblems due to historical insufficiencies lack of enoughknowledge concerning future process of variables and soforth use statistical properties of historical data and presentrandom variable 119883 with a specified distribution Now if thisrandom variable 119883 is independent we will deal with dataproduction using simulation easily but if we confront witha few variables or a series of random observations havingdependence action cannot be taken any more to simulateeach one of the random variables without consideration oftheir dependence on other variables In order to solve thisproblem we use the following three-step process

(1) Transfer of data to a space in which there is not anydependence among the variables

(2) Production and simulation of data in this new space(3) Transfer of produced data to the initial space

One of the main methods to perform this job is Choleskydecomposition

For other cases of Cholesky decomposition usage refer tothe multivariate options evaluation [18] Cholesky decompo-sition plays an important role in analysis of the risk factorsputting price on financial markets and risk management forexample refer to the VAR models [19] and copula models[20]

With regard to Cholesky decomposition properties wetransfer financial assets from initial space to a new spaceconsidering financial assets as random variables and usingthe linear transformation presented in this paper In the newspace linear correlation among financial assets becomes zeroand standard deviation of financial assets becomes equalto one Therefore nonlinear complexities of the mentionedmodels are decreased in the new spaces

3 Reduction of Nonlinear Complexities ofthe Markowitz M R Morey R C Moreyand Briec et al Models

In this section we implement linear transformation ofrelation (4) on the Markowitz M R Morey and R CMorey models and compare the models obtained from lineartransformation with principal models Before beginning thediscussion we get the relations using the mentioned lineartransformation and use themuntil the end of the paper Let ussuppose that we possess 119899 financial assets or 119899 portfolios likerandom variable 119909

119894 We consider the following assumptions

119864 (119909119894) = 120583119894

119864 ((119909119894minus 120583119894)2

) = 1205902

119894

cov (119909119894 119909119895) = 120590119894119895

119864 (119908119894) = 120583119894

119864 ((119908119894minus 120583119894)2

) = 1205902

119894

cov (119908119894 119908119895) = 120590119894119895

(7)

Now using linear transformation of119882 = 119871minus1119883 and relationof 119861 = 119871minus1 we have the following

119899

sum119895=1

120582119895119909119895=

119899

sum119895=1

120572119895119908119895 (8)

where 120572119895is defined as follows

[1205821 120582

119899] 119871 = [120572

1 120572

119899] (9)

Of relation (9) we will have

[1205821 120582

119899] = [120572

1 120572

119899] 119861 (10)

The following relation can be concluded from relation (10)119899

sum119895=1

120582119895= 1 997904rArr

119899

sum119895=1

119861119895120572119895= 1

(11)

where 119861119895is sum total entries of the 119895th row in matrix 119871minus1 In

continuation using relation (8) we have

119864 (119909119901) = 119864(

119901

sum119895=1

119871119901119895119908119895) =

119901

sum119895=1

119871119901119895120583119895 (12)

119864(

119899

sum119895=1

120582119895119909119895) = 119864(

119899

sum119895=1

120572119895119908119895) =

119899

sum119895=1

120572119895120583119895 (13)

Using relations (6) (7) (8) and (12) the followingrelations are achieved

119864((119909119901minus 120583119901)2

) =

119901

sum119895=1

(119871119901119895)2

(14)

119864((

119899

sum119895=1

120582119895(119909119894minus 120583119894))

2

)

= 119864((

119899

sum119895=1

120572119895(119908119895minus 120583119895))

2

) =

119899

sum119895=1

1205722

119895

(15)

31 Markowitz Model In the Markowitz model we considerthe mean to be constant in the specified levels and makethe variance to be minimum in order to obtain portfolioefficiency frontier

min119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

st119899

sum119895=1

120582119895120583119895= 119877expected

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(16)

4 Advances in Operations Research

In continuation through implementation of the linear trans-formation (4) and placement of relations (6) (10) (11) (13)and (15) in the model (16) we reach the following model

min119899

sum119895=1

1205722

119895

st119899

sum119895=1

120572119895120583119895= 119877expected

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(17)

where 119861119895119894is the same as 119871minus1

119895119894 It is evident that models (16)

and (17) are equivalent We consider objective function ofmodel (16) which has (1198992 + 119899)2 nonlinear combinationsof variables in form of 1205822

119895and 120582

119894120582119895 but in the objective

function of model (17) nonlinear combinations of variablesdecrease up to 119899 which is in the form of 1205722

119895 In fact through

implementation of the introduced linear transformationwe decrease the nonlinear complexities of model (16) andpoint out that while solving model (17) by use of commonmethods of solving nonlinear models calculations will beless and easier With regard to solving the majority of thenonlinear problems using heuristic methods (the obtainedsolution is approximate) we can solve the problem quicklyand accurately using model (17) while facing a great volumeof data in a portfolio optimization problem It is required tomention that while solving model (16) and model (17) withKKTmethod size of Hessian matrix in model (16) is equal to1198992 and size of Hessian matrix in model (17) is equal to 119899

32 M R Morey and R C Morey Model M R Morey andR C Morey used the following model in order to evaluateportfolio performance In this model they consider themeanas output and variance as input

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205791205902

119901

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(18)

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in the model (18) we get the following model

min 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895

119899

sum119895=1

1205722

119895le 120579

119901

sum119895=1

(119871119901119895)2

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(19)

It is evident that models (18) and (19) are equivalent Thesecond constraint of model (18) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in the

second constraint of model (19) nonlinear combinations ofvariables decrease to 119899 which is in form of 1205722

119895 Also here

through implementation of the introduced linear transforma-tion the nonlinear complexities of model (18) are decreasedand the examples with high data number can be worked onmore easily due to simplicity of the calculations Also size ofHessian matrix in model (18) and model (19) is equal to 1198992and 119899 respectively

33 Briec et al Model For portfolio performance evaluationBriec et al [6] used the following model In this model theyconsidered mean as output and variance as input Efficiencyimprovement possibility function (shortage function) in thismodel searches for improvement in direction of vector 119892 (119892 =(119892119881 119892119864) ge 0) and simultaneously seeks for increase of the

mean and decrease of variance

max 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901+ 120579119892119864

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205902

119901minus 120579119892119881

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(20)

Advances in Operations Research 5

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in model (20) we get the following model

max 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895+ 120579119892119864

119899

sum119895=1

1205722

119895le

119901

sum119895=1

(119871119901119895)2

minus 120579119892119881

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(21)

It is obvious that models (20) and (21) are equivalent Thesecond constraint of model (20) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in

the second constraint of model (21) nonlinear combinationsof variables decrease up to 119899 which is in form of 1205722

119895

Also here through implementation of the introduced lineartransformation the nonlinear complexities of model (20) aredecreased Also size of Hessian matrix in model (20) andmodel (21) is equal to 1198992 and 119899 respectively

4 Presentation of the Diversification Model inthe Mean-Semivariance Framework

Consideration of variance criterion instead of risk criterionis counted as an appropriate criterion in case distributionof random variable 119883 that is in symmetric and normalform but such a thing does not occur in reality Investorsof financial markets do not react to the positive and negativeexchanges of the stocks equally and show more sensitivity tothe negative exchanges and always intend to increase positiveexchanges and decrease negative exchanges For this purposewe present the following model in the mean-semivarianceframework We define favorable variance and unfavorablevariance respectively as follows

1205902

up = 119864((sum119895isin119860

120582119895(119909119894minus 120583119894))

2

)

119860 = 119895 | 119909119894minus 120583119894ge 0 119895 = 1 119899

1205902

down = 119864((sum119895isin119861

120582119895(119909119894minus 120583119894))

2

)

119861 = 119895 | 119909119894minus 120583119894lt 0 119895 = 1 119899

(22)

We present the following diversification model in themean-semivariance framework which we consider as outputand combination of the favorable and unfavorable variance asinput

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119864((

119899

sum119895=1

120573119910119895120582119895(119909119895minus 120583119895) +

119899

sum119895=1

(1 minus 120573) (1 minus 119910119895) 120582119895(119909119895minus 120583119895))

2

)

le 120579119864 ((120573119910119901(119909119901minus 120583119901) + (1 minus 120573) (1 minus 119910

119901) (119909119901minus 120583119901))2

)

119899

sum119895=1

120582119895= 1

119910119895(119909119895minus 120583119895) ge 0 119895 = 1 119899

(1 minus 119910119895) (119909119895minus 120583119895) le 0 119895 = 1 119899

120582119895ge 0 119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(23)

Scalar of 0 le 120573 le 1 is specified by the investor As muchas 120573 gets close to one it suggests more priority of the positive

exchanges over negative exchanges (with an optimistic visioninvestor deals with portfolio performance evaluation) and

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Operations Research

covariance between each two pairs of financial assets whichinclude (119899(119899 + 3)2) of data Now if the sharpe single-index model is used required data will decrease to (3119899 minus2) On the other hand in the single-index model it wasassumed that return of one asset is only affected by an index(market index) but due to available weaknesses and lack ofignoring of all indexes affecting the performance evaluationsome researchers created multi-index models via additionof the indexes and nonmarket factors (inflation rate foreignexchange rate oil price industry index etc) [4] M RMoreyandR CMorey presented a diversificationmodel inspired byDEA [5] Also in this model existence of linear correlationamong financial assets has been led to existence of thenonlinear complexities in themodel Briec et al [6] presenteda diversification model in the mean-variance frameworkthrough introduction of efficiency improvement possibilityfunction (shortage function) For the sake of investorsrsquo incli-nation towards the positive skewness [7ndash9] Joro and Na [10]offered a model in the mean-variance-skewness frameworkby consideration of the variance as input and skewness andmean as output In some studies semivariance has beenused instead of variance because variance is counted as anappropriate criterion for the risk when the stocks returndistribution is to be in normal and symmetric form whichit is not in such a manner most of times [11 12]

One of our basic problems is complexity of the calcu-lations of diversification and nonlinear models We usu-ally use Lagrange method KKT method [13] and variousheuristic methods [14] in order to solve these models incase of having the required conditions For example inthe Lagrange or KKT method we solve the problem bytransformation of optimization problem into linear systemsIn order to solve linear systems there are various methodswhich for example can be referred to the direct methodssuch as Gaussian and Gaussian-Jordan elimination methodsand matrix decomposition methods such as Lu decomposi-tion orthogonal matrixes and Cholesky decomposition [15]Cholesky decomposition can be applied for the matrixeswhich are positive definite and symmetric Solving linearsystems is one of the principal applications of the Choleskydecomposition Another one of its application is data pro-duction for dependent variable using simulation [16 17] Inthis paper we at first present a linear transformation basedon Cholesky decomposition In continuation we present adiversification model in the mean-semivariance frameworkand using the mentioned linear transformation decreasenonlinear computational complexities of the Markowitz MRMorey and R CMorey Briec et al and suggestivemodels

This paper has been classified as follows in Section 2 wepresent a brief description in connection with the Choleskydecomposition positive definite matrix covariance matrixand their properties and finish the section through introduc-tion of a linear transformation and financial interpretation ofthe Cholesky decomposition In Section 3 we apply the men-tioned linear transformation for Markowitz M R Morey RC Morey and Briec et al models and express advantagesof the obtained models In Section 4 we present a model inthe mean-semivariance framework and apply the mentionedlinear transformation for that and at the end the potential

integration of higher moments briefly is discussed Section 5includes numerical examples for explanation of the methodsand suggestive models We finish the paper with conclusionin Section 6

2 Required Concepts

In this section we take a look at the required concepts brieflyand deal with their properties by presentation of a lineartransformation

Definition 1 119860119899times119899

(square matrix) is known as positivedefinite matrix whenever

forall119909 (119909 isin R119899

119909 = 0) 997904rArr 119909119879

119860119909 gt 0 (1)

Definition 2 (Cholesky decomposition) If119860119899times119899

is symmetricand positive definite matrix then there exists a unique lowertriangular matrix 119871

119899times119899with positive diagonal element such

that

119860 = 119871119871119879

(2)

Consider the vector random variable 119883 = 1199091 119909

119899

which has 119864(119883) (mean vector) and var(119883) (covariancematrix) var(119883) is a positive and symmetrical definite matrixTherefore we will use Cholesky decomposition as follows

var (119883) = [cov (119909119894 119909119895)]119899times119899

= 119871119871119879

119871 =

[[[[[[

[

119871110 sdot sdot sdot 0

1198712111987122sdot sdot sdot 0

11987111989911198711198992sdot sdot sdot 119871

119899119899

]]]]]]

]

(3)

Consider the linear transformation presented in the followingrelation

119882 = 119871minus1

119883 (4)

We prove that the linear correlation between the randomvariables of119882 is equal to zero

119871119871119879

= var (119883) = var (119871119882)

= 119864 [(119871119882 minus 119871119864 (119882)) (119871119882 minus 119871119864 (119882))119879

]

= 119871 var (119882) 119871119879

(5)

Therefore

var (119882) = 119868 997904rArr

cov (119908119894 119908119895)119894 =119895

= 0(6)

21 Financial Interpretation of the Cholesky DecompositionCholesky decomposition technique is one of the availablemethods for data production frommultivariate distributions

Advances in Operations Research 3

because we require data production and simulation in someproblems due to historical insufficiencies lack of enoughknowledge concerning future process of variables and soforth use statistical properties of historical data and presentrandom variable 119883 with a specified distribution Now if thisrandom variable 119883 is independent we will deal with dataproduction using simulation easily but if we confront witha few variables or a series of random observations havingdependence action cannot be taken any more to simulateeach one of the random variables without consideration oftheir dependence on other variables In order to solve thisproblem we use the following three-step process

(1) Transfer of data to a space in which there is not anydependence among the variables

(2) Production and simulation of data in this new space(3) Transfer of produced data to the initial space

One of the main methods to perform this job is Choleskydecomposition

For other cases of Cholesky decomposition usage refer tothe multivariate options evaluation [18] Cholesky decompo-sition plays an important role in analysis of the risk factorsputting price on financial markets and risk management forexample refer to the VAR models [19] and copula models[20]

With regard to Cholesky decomposition properties wetransfer financial assets from initial space to a new spaceconsidering financial assets as random variables and usingthe linear transformation presented in this paper In the newspace linear correlation among financial assets becomes zeroand standard deviation of financial assets becomes equalto one Therefore nonlinear complexities of the mentionedmodels are decreased in the new spaces

3 Reduction of Nonlinear Complexities ofthe Markowitz M R Morey R C Moreyand Briec et al Models

In this section we implement linear transformation ofrelation (4) on the Markowitz M R Morey and R CMorey models and compare the models obtained from lineartransformation with principal models Before beginning thediscussion we get the relations using the mentioned lineartransformation and use themuntil the end of the paper Let ussuppose that we possess 119899 financial assets or 119899 portfolios likerandom variable 119909

119894 We consider the following assumptions

119864 (119909119894) = 120583119894

119864 ((119909119894minus 120583119894)2

) = 1205902

119894

cov (119909119894 119909119895) = 120590119894119895

119864 (119908119894) = 120583119894

119864 ((119908119894minus 120583119894)2

) = 1205902

119894

cov (119908119894 119908119895) = 120590119894119895

(7)

Now using linear transformation of119882 = 119871minus1119883 and relationof 119861 = 119871minus1 we have the following

119899

sum119895=1

120582119895119909119895=

119899

sum119895=1

120572119895119908119895 (8)

where 120572119895is defined as follows

[1205821 120582

119899] 119871 = [120572

1 120572

119899] (9)

Of relation (9) we will have

[1205821 120582

119899] = [120572

1 120572

119899] 119861 (10)

The following relation can be concluded from relation (10)119899

sum119895=1

120582119895= 1 997904rArr

119899

sum119895=1

119861119895120572119895= 1

(11)

where 119861119895is sum total entries of the 119895th row in matrix 119871minus1 In

continuation using relation (8) we have

119864 (119909119901) = 119864(

119901

sum119895=1

119871119901119895119908119895) =

119901

sum119895=1

119871119901119895120583119895 (12)

119864(

119899

sum119895=1

120582119895119909119895) = 119864(

119899

sum119895=1

120572119895119908119895) =

119899

sum119895=1

120572119895120583119895 (13)

Using relations (6) (7) (8) and (12) the followingrelations are achieved

119864((119909119901minus 120583119901)2

) =

119901

sum119895=1

(119871119901119895)2

(14)

119864((

119899

sum119895=1

120582119895(119909119894minus 120583119894))

2

)

= 119864((

119899

sum119895=1

120572119895(119908119895minus 120583119895))

2

) =

119899

sum119895=1

1205722

119895

(15)

31 Markowitz Model In the Markowitz model we considerthe mean to be constant in the specified levels and makethe variance to be minimum in order to obtain portfolioefficiency frontier

min119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

st119899

sum119895=1

120582119895120583119895= 119877expected

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(16)

4 Advances in Operations Research

In continuation through implementation of the linear trans-formation (4) and placement of relations (6) (10) (11) (13)and (15) in the model (16) we reach the following model

min119899

sum119895=1

1205722

119895

st119899

sum119895=1

120572119895120583119895= 119877expected

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(17)

where 119861119895119894is the same as 119871minus1

119895119894 It is evident that models (16)

and (17) are equivalent We consider objective function ofmodel (16) which has (1198992 + 119899)2 nonlinear combinationsof variables in form of 1205822

119895and 120582

119894120582119895 but in the objective

function of model (17) nonlinear combinations of variablesdecrease up to 119899 which is in the form of 1205722

119895 In fact through

implementation of the introduced linear transformationwe decrease the nonlinear complexities of model (16) andpoint out that while solving model (17) by use of commonmethods of solving nonlinear models calculations will beless and easier With regard to solving the majority of thenonlinear problems using heuristic methods (the obtainedsolution is approximate) we can solve the problem quicklyand accurately using model (17) while facing a great volumeof data in a portfolio optimization problem It is required tomention that while solving model (16) and model (17) withKKTmethod size of Hessian matrix in model (16) is equal to1198992 and size of Hessian matrix in model (17) is equal to 119899

32 M R Morey and R C Morey Model M R Morey andR C Morey used the following model in order to evaluateportfolio performance In this model they consider themeanas output and variance as input

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205791205902

119901

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(18)

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in the model (18) we get the following model

min 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895

119899

sum119895=1

1205722

119895le 120579

119901

sum119895=1

(119871119901119895)2

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(19)

It is evident that models (18) and (19) are equivalent Thesecond constraint of model (18) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in the

second constraint of model (19) nonlinear combinations ofvariables decrease to 119899 which is in form of 1205722

119895 Also here

through implementation of the introduced linear transforma-tion the nonlinear complexities of model (18) are decreasedand the examples with high data number can be worked onmore easily due to simplicity of the calculations Also size ofHessian matrix in model (18) and model (19) is equal to 1198992and 119899 respectively

33 Briec et al Model For portfolio performance evaluationBriec et al [6] used the following model In this model theyconsidered mean as output and variance as input Efficiencyimprovement possibility function (shortage function) in thismodel searches for improvement in direction of vector 119892 (119892 =(119892119881 119892119864) ge 0) and simultaneously seeks for increase of the

mean and decrease of variance

max 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901+ 120579119892119864

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205902

119901minus 120579119892119881

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(20)

Advances in Operations Research 5

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in model (20) we get the following model

max 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895+ 120579119892119864

119899

sum119895=1

1205722

119895le

119901

sum119895=1

(119871119901119895)2

minus 120579119892119881

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(21)

It is obvious that models (20) and (21) are equivalent Thesecond constraint of model (20) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in

the second constraint of model (21) nonlinear combinationsof variables decrease up to 119899 which is in form of 1205722

119895

Also here through implementation of the introduced lineartransformation the nonlinear complexities of model (20) aredecreased Also size of Hessian matrix in model (20) andmodel (21) is equal to 1198992 and 119899 respectively

4 Presentation of the Diversification Model inthe Mean-Semivariance Framework

Consideration of variance criterion instead of risk criterionis counted as an appropriate criterion in case distributionof random variable 119883 that is in symmetric and normalform but such a thing does not occur in reality Investorsof financial markets do not react to the positive and negativeexchanges of the stocks equally and show more sensitivity tothe negative exchanges and always intend to increase positiveexchanges and decrease negative exchanges For this purposewe present the following model in the mean-semivarianceframework We define favorable variance and unfavorablevariance respectively as follows

1205902

up = 119864((sum119895isin119860

120582119895(119909119894minus 120583119894))

2

)

119860 = 119895 | 119909119894minus 120583119894ge 0 119895 = 1 119899

1205902

down = 119864((sum119895isin119861

120582119895(119909119894minus 120583119894))

2

)

119861 = 119895 | 119909119894minus 120583119894lt 0 119895 = 1 119899

(22)

We present the following diversification model in themean-semivariance framework which we consider as outputand combination of the favorable and unfavorable variance asinput

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119864((

119899

sum119895=1

120573119910119895120582119895(119909119895minus 120583119895) +

119899

sum119895=1

(1 minus 120573) (1 minus 119910119895) 120582119895(119909119895minus 120583119895))

2

)

le 120579119864 ((120573119910119901(119909119901minus 120583119901) + (1 minus 120573) (1 minus 119910

119901) (119909119901minus 120583119901))2

)

119899

sum119895=1

120582119895= 1

119910119895(119909119895minus 120583119895) ge 0 119895 = 1 119899

(1 minus 119910119895) (119909119895minus 120583119895) le 0 119895 = 1 119899

120582119895ge 0 119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(23)

Scalar of 0 le 120573 le 1 is specified by the investor As muchas 120573 gets close to one it suggests more priority of the positive

exchanges over negative exchanges (with an optimistic visioninvestor deals with portfolio performance evaluation) and

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 3

because we require data production and simulation in someproblems due to historical insufficiencies lack of enoughknowledge concerning future process of variables and soforth use statistical properties of historical data and presentrandom variable 119883 with a specified distribution Now if thisrandom variable 119883 is independent we will deal with dataproduction using simulation easily but if we confront witha few variables or a series of random observations havingdependence action cannot be taken any more to simulateeach one of the random variables without consideration oftheir dependence on other variables In order to solve thisproblem we use the following three-step process

(1) Transfer of data to a space in which there is not anydependence among the variables

(2) Production and simulation of data in this new space(3) Transfer of produced data to the initial space

One of the main methods to perform this job is Choleskydecomposition

For other cases of Cholesky decomposition usage refer tothe multivariate options evaluation [18] Cholesky decompo-sition plays an important role in analysis of the risk factorsputting price on financial markets and risk management forexample refer to the VAR models [19] and copula models[20]

With regard to Cholesky decomposition properties wetransfer financial assets from initial space to a new spaceconsidering financial assets as random variables and usingthe linear transformation presented in this paper In the newspace linear correlation among financial assets becomes zeroand standard deviation of financial assets becomes equalto one Therefore nonlinear complexities of the mentionedmodels are decreased in the new spaces

3 Reduction of Nonlinear Complexities ofthe Markowitz M R Morey R C Moreyand Briec et al Models

In this section we implement linear transformation ofrelation (4) on the Markowitz M R Morey and R CMorey models and compare the models obtained from lineartransformation with principal models Before beginning thediscussion we get the relations using the mentioned lineartransformation and use themuntil the end of the paper Let ussuppose that we possess 119899 financial assets or 119899 portfolios likerandom variable 119909

119894 We consider the following assumptions

119864 (119909119894) = 120583119894

119864 ((119909119894minus 120583119894)2

) = 1205902

119894

cov (119909119894 119909119895) = 120590119894119895

119864 (119908119894) = 120583119894

119864 ((119908119894minus 120583119894)2

) = 1205902

119894

cov (119908119894 119908119895) = 120590119894119895

(7)

Now using linear transformation of119882 = 119871minus1119883 and relationof 119861 = 119871minus1 we have the following

119899

sum119895=1

120582119895119909119895=

119899

sum119895=1

120572119895119908119895 (8)

where 120572119895is defined as follows

[1205821 120582

119899] 119871 = [120572

1 120572

119899] (9)

Of relation (9) we will have

[1205821 120582

119899] = [120572

1 120572

119899] 119861 (10)

The following relation can be concluded from relation (10)119899

sum119895=1

120582119895= 1 997904rArr

119899

sum119895=1

119861119895120572119895= 1

(11)

where 119861119895is sum total entries of the 119895th row in matrix 119871minus1 In

continuation using relation (8) we have

119864 (119909119901) = 119864(

119901

sum119895=1

119871119901119895119908119895) =

119901

sum119895=1

119871119901119895120583119895 (12)

119864(

119899

sum119895=1

120582119895119909119895) = 119864(

119899

sum119895=1

120572119895119908119895) =

119899

sum119895=1

120572119895120583119895 (13)

Using relations (6) (7) (8) and (12) the followingrelations are achieved

119864((119909119901minus 120583119901)2

) =

119901

sum119895=1

(119871119901119895)2

(14)

119864((

119899

sum119895=1

120582119895(119909119894minus 120583119894))

2

)

= 119864((

119899

sum119895=1

120572119895(119908119895minus 120583119895))

2

) =

119899

sum119895=1

1205722

119895

(15)

31 Markowitz Model In the Markowitz model we considerthe mean to be constant in the specified levels and makethe variance to be minimum in order to obtain portfolioefficiency frontier

min119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

st119899

sum119895=1

120582119895120583119895= 119877expected

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(16)

4 Advances in Operations Research

In continuation through implementation of the linear trans-formation (4) and placement of relations (6) (10) (11) (13)and (15) in the model (16) we reach the following model

min119899

sum119895=1

1205722

119895

st119899

sum119895=1

120572119895120583119895= 119877expected

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(17)

where 119861119895119894is the same as 119871minus1

119895119894 It is evident that models (16)

and (17) are equivalent We consider objective function ofmodel (16) which has (1198992 + 119899)2 nonlinear combinationsof variables in form of 1205822

119895and 120582

119894120582119895 but in the objective

function of model (17) nonlinear combinations of variablesdecrease up to 119899 which is in the form of 1205722

119895 In fact through

implementation of the introduced linear transformationwe decrease the nonlinear complexities of model (16) andpoint out that while solving model (17) by use of commonmethods of solving nonlinear models calculations will beless and easier With regard to solving the majority of thenonlinear problems using heuristic methods (the obtainedsolution is approximate) we can solve the problem quicklyand accurately using model (17) while facing a great volumeof data in a portfolio optimization problem It is required tomention that while solving model (16) and model (17) withKKTmethod size of Hessian matrix in model (16) is equal to1198992 and size of Hessian matrix in model (17) is equal to 119899

32 M R Morey and R C Morey Model M R Morey andR C Morey used the following model in order to evaluateportfolio performance In this model they consider themeanas output and variance as input

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205791205902

119901

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(18)

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in the model (18) we get the following model

min 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895

119899

sum119895=1

1205722

119895le 120579

119901

sum119895=1

(119871119901119895)2

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(19)

It is evident that models (18) and (19) are equivalent Thesecond constraint of model (18) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in the

second constraint of model (19) nonlinear combinations ofvariables decrease to 119899 which is in form of 1205722

119895 Also here

through implementation of the introduced linear transforma-tion the nonlinear complexities of model (18) are decreasedand the examples with high data number can be worked onmore easily due to simplicity of the calculations Also size ofHessian matrix in model (18) and model (19) is equal to 1198992and 119899 respectively

33 Briec et al Model For portfolio performance evaluationBriec et al [6] used the following model In this model theyconsidered mean as output and variance as input Efficiencyimprovement possibility function (shortage function) in thismodel searches for improvement in direction of vector 119892 (119892 =(119892119881 119892119864) ge 0) and simultaneously seeks for increase of the

mean and decrease of variance

max 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901+ 120579119892119864

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205902

119901minus 120579119892119881

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(20)

Advances in Operations Research 5

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in model (20) we get the following model

max 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895+ 120579119892119864

119899

sum119895=1

1205722

119895le

119901

sum119895=1

(119871119901119895)2

minus 120579119892119881

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(21)

It is obvious that models (20) and (21) are equivalent Thesecond constraint of model (20) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in

the second constraint of model (21) nonlinear combinationsof variables decrease up to 119899 which is in form of 1205722

119895

Also here through implementation of the introduced lineartransformation the nonlinear complexities of model (20) aredecreased Also size of Hessian matrix in model (20) andmodel (21) is equal to 1198992 and 119899 respectively

4 Presentation of the Diversification Model inthe Mean-Semivariance Framework

Consideration of variance criterion instead of risk criterionis counted as an appropriate criterion in case distributionof random variable 119883 that is in symmetric and normalform but such a thing does not occur in reality Investorsof financial markets do not react to the positive and negativeexchanges of the stocks equally and show more sensitivity tothe negative exchanges and always intend to increase positiveexchanges and decrease negative exchanges For this purposewe present the following model in the mean-semivarianceframework We define favorable variance and unfavorablevariance respectively as follows

1205902

up = 119864((sum119895isin119860

120582119895(119909119894minus 120583119894))

2

)

119860 = 119895 | 119909119894minus 120583119894ge 0 119895 = 1 119899

1205902

down = 119864((sum119895isin119861

120582119895(119909119894minus 120583119894))

2

)

119861 = 119895 | 119909119894minus 120583119894lt 0 119895 = 1 119899

(22)

We present the following diversification model in themean-semivariance framework which we consider as outputand combination of the favorable and unfavorable variance asinput

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119864((

119899

sum119895=1

120573119910119895120582119895(119909119895minus 120583119895) +

119899

sum119895=1

(1 minus 120573) (1 minus 119910119895) 120582119895(119909119895minus 120583119895))

2

)

le 120579119864 ((120573119910119901(119909119901minus 120583119901) + (1 minus 120573) (1 minus 119910

119901) (119909119901minus 120583119901))2

)

119899

sum119895=1

120582119895= 1

119910119895(119909119895minus 120583119895) ge 0 119895 = 1 119899

(1 minus 119910119895) (119909119895minus 120583119895) le 0 119895 = 1 119899

120582119895ge 0 119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(23)

Scalar of 0 le 120573 le 1 is specified by the investor As muchas 120573 gets close to one it suggests more priority of the positive

exchanges over negative exchanges (with an optimistic visioninvestor deals with portfolio performance evaluation) and

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Operations Research

In continuation through implementation of the linear trans-formation (4) and placement of relations (6) (10) (11) (13)and (15) in the model (16) we reach the following model

min119899

sum119895=1

1205722

119895

st119899

sum119895=1

120572119895120583119895= 119877expected

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(17)

where 119861119895119894is the same as 119871minus1

119895119894 It is evident that models (16)

and (17) are equivalent We consider objective function ofmodel (16) which has (1198992 + 119899)2 nonlinear combinationsof variables in form of 1205822

119895and 120582

119894120582119895 but in the objective

function of model (17) nonlinear combinations of variablesdecrease up to 119899 which is in the form of 1205722

119895 In fact through

implementation of the introduced linear transformationwe decrease the nonlinear complexities of model (16) andpoint out that while solving model (17) by use of commonmethods of solving nonlinear models calculations will beless and easier With regard to solving the majority of thenonlinear problems using heuristic methods (the obtainedsolution is approximate) we can solve the problem quicklyand accurately using model (17) while facing a great volumeof data in a portfolio optimization problem It is required tomention that while solving model (16) and model (17) withKKTmethod size of Hessian matrix in model (16) is equal to1198992 and size of Hessian matrix in model (17) is equal to 119899

32 M R Morey and R C Morey Model M R Morey andR C Morey used the following model in order to evaluateportfolio performance In this model they consider themeanas output and variance as input

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205791205902

119901

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(18)

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in the model (18) we get the following model

min 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895

119899

sum119895=1

1205722

119895le 120579

119901

sum119895=1

(119871119901119895)2

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(19)

It is evident that models (18) and (19) are equivalent Thesecond constraint of model (18) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in the

second constraint of model (19) nonlinear combinations ofvariables decrease to 119899 which is in form of 1205722

119895 Also here

through implementation of the introduced linear transforma-tion the nonlinear complexities of model (18) are decreasedand the examples with high data number can be worked onmore easily due to simplicity of the calculations Also size ofHessian matrix in model (18) and model (19) is equal to 1198992and 119899 respectively

33 Briec et al Model For portfolio performance evaluationBriec et al [6] used the following model In this model theyconsidered mean as output and variance as input Efficiencyimprovement possibility function (shortage function) in thismodel searches for improvement in direction of vector 119892 (119892 =(119892119881 119892119864) ge 0) and simultaneously seeks for increase of the

mean and decrease of variance

max 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901+ 120579119892119864

119899

sum119895=1

1205822

1198951205902

119895+ (

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895)

119894 =119895

le 1205902

119901minus 120579119892119881

119899

sum119895=1

120582119895= 1

120582119895ge 0 119895 = 1 119899

(20)

Advances in Operations Research 5

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in model (20) we get the following model

max 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895+ 120579119892119864

119899

sum119895=1

1205722

119895le

119901

sum119895=1

(119871119901119895)2

minus 120579119892119881

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(21)

It is obvious that models (20) and (21) are equivalent Thesecond constraint of model (20) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in

the second constraint of model (21) nonlinear combinationsof variables decrease up to 119899 which is in form of 1205722

119895

Also here through implementation of the introduced lineartransformation the nonlinear complexities of model (20) aredecreased Also size of Hessian matrix in model (20) andmodel (21) is equal to 1198992 and 119899 respectively

4 Presentation of the Diversification Model inthe Mean-Semivariance Framework

Consideration of variance criterion instead of risk criterionis counted as an appropriate criterion in case distributionof random variable 119883 that is in symmetric and normalform but such a thing does not occur in reality Investorsof financial markets do not react to the positive and negativeexchanges of the stocks equally and show more sensitivity tothe negative exchanges and always intend to increase positiveexchanges and decrease negative exchanges For this purposewe present the following model in the mean-semivarianceframework We define favorable variance and unfavorablevariance respectively as follows

1205902

up = 119864((sum119895isin119860

120582119895(119909119894minus 120583119894))

2

)

119860 = 119895 | 119909119894minus 120583119894ge 0 119895 = 1 119899

1205902

down = 119864((sum119895isin119861

120582119895(119909119894minus 120583119894))

2

)

119861 = 119895 | 119909119894minus 120583119894lt 0 119895 = 1 119899

(22)

We present the following diversification model in themean-semivariance framework which we consider as outputand combination of the favorable and unfavorable variance asinput

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119864((

119899

sum119895=1

120573119910119895120582119895(119909119895minus 120583119895) +

119899

sum119895=1

(1 minus 120573) (1 minus 119910119895) 120582119895(119909119895minus 120583119895))

2

)

le 120579119864 ((120573119910119901(119909119901minus 120583119901) + (1 minus 120573) (1 minus 119910

119901) (119909119901minus 120583119901))2

)

119899

sum119895=1

120582119895= 1

119910119895(119909119895minus 120583119895) ge 0 119895 = 1 119899

(1 minus 119910119895) (119909119895minus 120583119895) le 0 119895 = 1 119899

120582119895ge 0 119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(23)

Scalar of 0 le 120573 le 1 is specified by the investor As muchas 120573 gets close to one it suggests more priority of the positive

exchanges over negative exchanges (with an optimistic visioninvestor deals with portfolio performance evaluation) and

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 5

By implementation of the linear transformation (4) andplacement of relations (6) (10) (11) (12) (13) (14) and (15)in model (20) we get the following model

max 120579

st119899

sum119895=1

120572119895120583119895ge

119901

sum119895=1

119871119901119895120583119895+ 120579119892119864

119899

sum119895=1

1205722

119895le

119901

sum119895=1

(119871119901119895)2

minus 120579119892119881

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(21)

It is obvious that models (20) and (21) are equivalent Thesecond constraint of model (20) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1205822

119895and 120582

119894120582119895 but in

the second constraint of model (21) nonlinear combinationsof variables decrease up to 119899 which is in form of 1205722

119895

Also here through implementation of the introduced lineartransformation the nonlinear complexities of model (20) aredecreased Also size of Hessian matrix in model (20) andmodel (21) is equal to 1198992 and 119899 respectively

4 Presentation of the Diversification Model inthe Mean-Semivariance Framework

Consideration of variance criterion instead of risk criterionis counted as an appropriate criterion in case distributionof random variable 119883 that is in symmetric and normalform but such a thing does not occur in reality Investorsof financial markets do not react to the positive and negativeexchanges of the stocks equally and show more sensitivity tothe negative exchanges and always intend to increase positiveexchanges and decrease negative exchanges For this purposewe present the following model in the mean-semivarianceframework We define favorable variance and unfavorablevariance respectively as follows

1205902

up = 119864((sum119895isin119860

120582119895(119909119894minus 120583119894))

2

)

119860 = 119895 | 119909119894minus 120583119894ge 0 119895 = 1 119899

1205902

down = 119864((sum119895isin119861

120582119895(119909119894minus 120583119894))

2

)

119861 = 119895 | 119909119894minus 120583119894lt 0 119895 = 1 119899

(22)

We present the following diversification model in themean-semivariance framework which we consider as outputand combination of the favorable and unfavorable variance asinput

min 120579

st119899

sum119895=1

120582119895120583119895ge 120583119901

119864((

119899

sum119895=1

120573119910119895120582119895(119909119895minus 120583119895) +

119899

sum119895=1

(1 minus 120573) (1 minus 119910119895) 120582119895(119909119895minus 120583119895))

2

)

le 120579119864 ((120573119910119901(119909119901minus 120583119901) + (1 minus 120573) (1 minus 119910

119901) (119909119901minus 120583119901))2

)

119899

sum119895=1

120582119895= 1

119910119895(119909119895minus 120583119895) ge 0 119895 = 1 119899

(1 minus 119910119895) (119909119895minus 120583119895) le 0 119895 = 1 119899

120582119895ge 0 119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(23)

Scalar of 0 le 120573 le 1 is specified by the investor As muchas 120573 gets close to one it suggests more priority of the positive

exchanges over negative exchanges (with an optimistic visioninvestor deals with portfolio performance evaluation) and

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Operations Research

anywhere it gets close to zero it suggests higher sensitivityof the investor to negative exchanges (with a pessimisticvision investor deals with portfolio performance evaluation)Model (23) is general state of model (18) (it is sufficient toplace 120573 = 05) A portfolio has a better performance whichpossesses a higher 120579 Excellence of the presented model isconsideration of the linear correlation among the financialassets because linear correlation among financial assets hasbeen considered to be zero in most of models which haveused semivariance Now using linear transformation (4) andimplementing relations (6) (10) (11) (12) (13) (14) and (15)in the model (23) the following model can be presented

min 120579

st (1 minus 120573)2

119899

sum119895=1

1205722

119895+ (2120573 minus 1)

119899

sum119895=1

1199101198951205722

119895

le 120579

119901

sum119895=1

(119871119901119895)2

((1 minus 120573)2

+ (2120573 minus 1) 119910119901)

119899

sum119895=1

119861119895120572119895= 1

119899

sum119895=1

119861119895119894120572119895ge 0 119894 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894) 119910119895) ge 0 119895 = 1 119899

(

119895

sum119894=1

119871119895119894(119908119894minus 120583119894)) (1 minus 119910

119895) le 0

119895 = 1 119899

119910119895isin 0 1 119895 = 1 119899

(24)

It is evident that models (23) and (24) are equivalent Thesecond constraint of model (23) has (1198992 + 119899)2 nonlinearcombinations of variables in form of 1199102

1198951205822119895and 119910

119894119910119895120582119894120582119895

but in the second constraint of model (24) nonlinearcombinations of variables have been reduced to 119899 appearingin form of 119910

1198951205722119895 Also here model (24) has less nonlinear

complexities than model (23) and the examples with highdata number can be worked on easily due to simplicity of thecalculations

At the end we are concerned with reviewing a sum-mary of higher moments As it was stated timing mean-variance framework is construed as an appropriate criterionfor portfolio performance evaluation in which distributionof random variables 119883 (financial assets and portfolio) is insymmetrical and normal form But Mandelbrot [21] showedthat such a thing does not occur in reality In order to analyzethese problems Popova et al [22] and Davies et al [23] used

higher moments for example refer to skewness and kurtosis[24] as follows

119864 (119877) = 119864(

119899

sum119895=1

120582119895119909119895) =

119899

sum119895=1

120582119895120583119895 (25)

1205752

= 119864 [(119877 minus 119864 (119877))2

] =

119899

sum119894=1

119899

sum119895=1

120582119894120582119895120590119894119895 (26)

1198783

= 119864 (119877 minus 119864 (119877))3

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

120582119894120582119895120582119896119878119894119895119896 (27)

1205814

= 119864 (119877 minus 119864 (119877))4

=

119899

sum119894=1

119899

sum119895=1

119899

sum119896=1

119899

sum119897=1

120582119894120582119895120582119896120582119897120581119894119895119896119897

(28)

Relations (25) and (26) are the same as mean returnand variance and considered to be the main frameworkdiscussed by this paper Relation (27) suggests skewnessor the third central moment In fact rate of deviationfrom symmetry of a distribution is known as skewnessOf the models presented in the mean-variance skewness(MVS) framework as an example it can be referred to [1025] We were reminded earlier that investors are interestedhighly in the positive skewness and consider the skewnessas the output in the diversification models Relation (28)represents kurtosis or the fourth central moment Kurtosisshows the same amount of climax or higher distributioncompared to normal distribution Of themodels presented inthe mean-variance-skewness-kurtosis (MVSK) frame workas an example it can be referred to [26ndash28] Consideringthe conducted researches investors are interested in lowkurtosis and therefore consider the kurtosis is input in thediversification models At the end we refer to this subjectthat linear transformation presented in this paper can beused to decrease nonlinear complexities of the nonlinearand diversification models in MVS and MVSK frameworksbut perhaps this decrease is not tangible so much yet thisdecrease is tangible in MV framework completely

5 Numerical Example

In this section we deal with presentation of three numericalexamples in order to describe the suggestive methods andmodels We point out that the examples have been solved byuse ofGAMS (2412) SoftwareMATLAB (2012) Software andWindows (81) Oprating System

Example 1 Consider the data related to 26 financial assetswith mean and covariance matrix presented in Table 3 [5]In this example we are going to obtain portfolios efficiencyfrontier achieved from the mentioned financial assets usingmodel (16) and model (17) For this purpose it is sufficientto solve model (16) and model (17) for a number of the fixedmean levels The results can be seen in Table 1

With regard to equality (analytical) of models (16) and(17) the obtained results for both models must be equal but

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 7

Table 1 Solution of models (16) and (17) for a number of the fixed mean levels

119877expected 1737 1074 1543 1791 1033 1463 1368 1367 0985 1165 1303 1349 1411 1114 1385Model (16) 40314 20048 30678 4558 19467 283447 25854 25829 1899 21538 24322 25391 26944 20669 26277Model (17) 40299 20040 30664 45562 19459 28331 25842 25817 18987 21528 24311 25380 26931 20660 26265

in reality answers of these two models are equal up to onedecimal digit and answers ofmodel (17) are less than or equalto answers of the model (16) As a result due to minimizationof the problem it can be concluded that answers of model(17) have a better numerical approximation In fact whenwe use KKT method in order to solve models (16) and (17)size of Hessian matrix used for model (16) will be equal to676 and for model (17) will be equal to 26 Therefore GAMSsoftware solves model (17) within a less time span comparedto model (16) Now if we have 1000 financial assets instead of26 financial assets this time difference will be remarkable

Example 2 Let us suppose that we consider the data relatedto 26 financial assets with mean and covariance matrixgiven in Table 3 as portfolios In this example we dealwith evaluation of the mentioned portfolios performanceConsidering equality (analytical) of models (18) and (19)the achieved results for both models must be equal but inreality answers of these two models are equal up to twodecimal digits and answers of model (19) are less than orequal to answers of model (18) (Table 2) Therefore due tominimization of the problem it can be concluded that answerof model (19) has a better numerical approximation In factwhen we use KKT method in order to solve models (18) and(19) size of Hessian matrix used for model (18) will be equalto 676 and for model (19) will be equal to 26 Thus GAMSsoftware solves model (19) within a less time span comparedto model (18)

The results of Example 2 can be considered for models(20) and (21) Therefore due to similar results we avoid toprovide any additional example

Example 3 Consider the data related to 26 portfolios ofExample 2 In this example wewant to evaluate the portfoliosperformance using model (24) For simplicity let us supposethat odd portfolios have favorable semivariance and evenportfolios have unfavorable semivariance Therefore

1199101= 1199103= sdot sdot sdot = 119910

25= 1

1199102= 1199104= sdot sdot sdot = 119910

26= 0

(29)

We solve model (24) for various 120573 (Table 2) If we consider120573 = 05 portfolios number 4 and number 9 will have the bestperformance With an impartial vision performance of thesetwo portfolios is close to each other and they have no priorityover one another but if we consider 120573 = 04 (Pessimisticvision) portfolio number 9 will have the best performanceand if we consider 120573 = 06 (Optimistic vision) portfolionumber 4 will have the best performance So the selectiveportfolio changes via change of 120573 On the other hand GAMSsoftware solves model (24) due to the same reasons stated inExample 2 within a less time span compared to model (23)

Table 2 Obtained efficiency for models (18) (19) and (24)

Portfolio Model (18) Model (19) Model (21)120573 = 04 120573 = 05 120573 = 06

1 07423 07422 07946 07422 071892 03552 03551 01568 03251 070073 06933 06932 0733 06932 067444 1 09996 09027 09996 215545 05055 05054 0551 05054 048276 06201 062 0292 062 135657 06062 06061 06448 06061 058868 08138 08136 03847 08136 177789 1 09998 1093 09998 0958410 07211 07209 03451 07209 1563311 06462 06461 06895 06461 0626412 08188 08187 03874 08187 1788113 06213 06212 06596 06212 0603714 06976 06974 03552 06974 1507615 0645 06449 06855 06449 0626516 09803 09803 04655 09803 2124117 09718 09716 10283 09716 0945118 04413 04412 02144 04412 0951619 06857 06855 07394 06855 0659920 06387 06387 03007 06387 1397621 08097 08097 08631 08097 0780622 0566 05659 02708 05659 1227523 04953 04952 05414 04952 0474724 08223 08221 03921 08221 1787425 0524 05236 05724 05236 0501926 08581 08579 04113 08579 18582

6 Conclusion

In this paper we at first consider 119899 random variables(119899 financial assets) which have linear correlation Thenusing Cholesky decomposition of the covariance matrixwe introduced a linear transformation and converted therandom variables 119883 into random variables 119882 by use ofthis linear transformation in which the random variables119882 have zero linear correlation in the new space Later weimplemented this change of variable in the Markowitz M RMorey R C Morey and Briec et al models and decreasedthe nonlinear complexities of this model In continuationsince distribution of random variable 119883 is not normal andsymmetric mostly we present a new model inspired by MR Morey and R C Morey model using favorable and unfa-vorable semivariance which is more general state of M RMorey and R C Morey model as well The mentioned modelincludes a coefficient like 120573 showing favorability or sensitivity

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Operations Research

Table3Datafor

exam

ples

Mean

Covariance

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

1737

1074

1543

1791

1033

1463

1368

1367

0985

1165

1303

1349

1411

1114

1385

1659

151

0735

1145

1479

1619

1169

0979

1215

085

1145

12

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

15433

4582

452

4719

4119

4519

4206

3758

2788

3613

4237

3779

4637

3426

434

4077

3794

4159

3722

458

4445

4158

4207

3537

4106

3094

24582

6166

40774269

39414286

3751

3348

2545

3269

3939

343

3951

327

3546

3406

34473649

3308

42013893

3776

3642

3214

3798

3158

3452

407744

25

4232

3825

4151

3772

3273

2473

3214

3833

3241

417

3122

367

373

3406

3942

3329

4108

3953

3751

38413057

3734

285

44719

4269

4232

4558

3922

4263

4038

3533

2674

3476

3959

3492

4282

3269

3949

3789

3587

4023

3597

4251

4101

3971

3972

3342

3822

2991

54119

3941

3825

3922

3851

399

3565

3193

2407

309

351

3097

3788

3064

3477

3381

3213

3654

3149

388

3684

3538

34952906

3462

2847

64519

4286

4151

4263

399

4571

384

3473

25813348

3863

3325

4133

327

3834

3694

3486

4031

3453

4212

4008

3882

3851

3146

3739

3001

74206

3751

3772

4038

3565

384

4265

3199

2387

3289

3495

3087

38713053

3554

3438

3263

3553

3268

3768

3721

3586

3625

29513423

2729

83758

3348

3273

3533

3193

3473

3199

3174

222

2818

3144

2809

3419

2724

3144

3057

2921

322945

3393

3254

3172

3159

2729

307

2465

92788

2545

2473

2674

2407

2581

2387

222

1899

2116

2388

2194

2577

2134

2274

2278

2199

2438

2189

2545

2544

2366

2379

2111

2286

1914

103613

3269

3214

3476

309

3348

3289

2818

2116

2987

305

2734

336

2662

3144

294

2849

30812837

3334

3236

3094

3166

2633

2983

2367

114237

3939

3833

3959

351

3863

3495

3144

2388

305

3764

3126

3814

2931

346

3409

319

3653

3135

383

3652

3482

3532

2974

34672679

123779

343

32413492

3097

3325

3087

2809

2194

2734

3126

3101

3406

2627

3095

2992

2827

30722804

3313

3296

2999

3058

2732

2997

2352

1346

373951

417

4282

3788

4133

3871

3419

2577

336

3814

3406

4337

3192

3889

3754

3456

3923

3443

4107

4081

3789

386

3193

3746

2866

143426

327

3122

3269

3064

327

3053

2724

2134

2662

2931

2627

3192

2963

2827

2784

2705

3115

2713

3149

3121

2839

2979

2529

2889

246

15434

3546

367

3949

3477

3834

3554

3144

2274

3144

346

3095

3889

2827

4074

3347

3195

3497

3176

3865

3777

3509

3529

2964

3358

2534

1640

773406

373

3789

33813694

3438

3057

2278

294

3409

2992

3754

2784

3347

3633

3061

3451

30213597

363356

342842

3374

2567

173794

34473406

3587

3213

3486

3263

2921

2199

2849

319

2827

3456

2705

3195

30613055

32012945

3499

3321

3227

3244

273

3137

2454

184159

3649

3942

40233654

4031

3553

322438

30813653

3072

3923

3115

3497

3451

32014303

3242

3835

3737

3597

3637

2922

3457

2706

193722

3308

3329

3597

3149

3453

3268

2945

2189

2837

3135

2804

3443

2713

3176

30212945

3242

309

3405

3269

3202

3178

2719

30612438

20458

42014108

4251

388

4212

3768

3393

2545

3334

383

3313

4107

3149

3865

3597

3499

3835

34054508

3949

3805

3838

3164

3738

2868

2144

45

3893

3953

41013684

4008

3721

3254

2544

3236

3652

3296

4081

3121

3777

363321

3737

3269

3949

4148

3627

3711

30913546

2802

224158

3776

3751

3971

3538

3882

3586

3172

2366

3094

3482

2999

3789

2839

3509

3356

3227

3597

3202

3805

3627

3818

3571

2918

3412

2646

234207

3642

3841

3972

3495

3851

3625

3159

2379

3166

3532

3058

386

2979

3529

343244

3637

3178

3838

3711

35713834

2957

3432

2637

243537

3214

3057

3342

2906

3146

2951

2729

2111

2633

2974

2732

3193

2529

2964

2842

2731

2922

2719

3164

3091

2918

2957

2733

2868

228

254106

3798

3734

3822

3462

3739

3423

307

2286

2983

3467

2997

3746

2889

3358

3374

3137

3457

3061

3738

3546

3412

3432

28683626

2634

263094

3158

285

29912847

3001

2729

2465

1914

2367

2679

2352

2866

246

2534

2567

2454

2706

2438

2868

2802

2646

2637

228

2634

2469

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 9

of investor towards favorable and unfavorable semivarianceIn continuation using change of the mentioned variable wedecreased nonlinear complexities of the suggestive model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Time Magazine vol 148 no 16 1996[2] H Markowitz ldquoPortfolio selectionrdquoThe Journal of Finance vol

7 no 1 pp 77ndash91 1952[3] W F Sharpe ldquoCapital asset prices a theory of market equilib-

rium under conditions of riskrdquo The Journal of Finance vol 19no 3 pp 425ndash442 1964

[4] M J Lynge and J K Zumwalt ldquoAn empirical study of theinterest rate sensitivity of commercial bank returns a multi-index approachrdquo Journal of Financial and Quantitative Analysisvol 15 no 3 pp 731ndash742 1980

[5] M R Morey and R C Morey ldquoMutual fund performanceappraisals a multi-horizon perspective with endogenousbenchmarkingrdquo Omega vol 27 no 2 pp 241ndash258 1999

[6] W Briec K Kerstens and J B Lesourd ldquoSingle-periodMarkowitz portfolio selection performance gauging and dual-ity a variation on the Luenberger shortage functionrdquo Journal ofOptimization Theory and Applications vol 120 no 1 pp 1ndash272004

[7] A Kane ldquoSkewness preference and portfolio choicerdquo TheJournal of Financial and Quantitative Analysis vol 17 no 1 pp15ndash25 1982

[8] F D Arditti ldquoSkewness and investorsrsquo decisions a replyrdquo TheJournal of Financial and Quantitative Analysis vol 10 no 1 pp173ndash176 1975

[9] Y K Ho and Y L Cheung ldquoBehavior of intra-daily stock returnon an Asian emerging market-Hong Kongrdquo Applied Economicsvol 23 no 5 pp 957ndash966 1991

[10] T Joro and P Na ldquoPortfolio performance evaluation in ameanndashvariancendashskewness frameworkrdquo European Journal of Opera-tional Research vol 175 no 1 pp 446ndash461 2006

[11] P B Hazell ldquoA linear alternative to quadratic and semivarianceprogramming for farm planning under uncertaintyrdquo AmericanJournal of Agricultural Economics vol 53 no 1 pp 53ndash62 1971

[12] J Estrada ldquoMean-semivariance behavior downside risk andcapital asset pricingrdquo International Review of Economics ampFinance vol 16 no 2 pp 169ndash185 2007

[13] M S Bazaraa H D Sherali and C M Shetty NonlinearProgramming Theory and Algorithms JohnWiley amp Sons NewYork NY USA 2013

[14] C Buchanan Techniques for solving nonlinear programmingproblems with emphasis on interior point methods and optimalcontrol problems [Master of Philosophy] Department of Mathe-matics and Statistics University of Edinburgh 2008

[15] K E Atkinson An Introduction to Numerical Analysis JohnWiley amp Sons New York NY USA 2008

[16] MW Davis ldquoProduction of conditional simulations via the LUtriangular decomposition of the covariance matrixrdquoMathemat-ical Geology vol 19 no 2 pp 91ndash98 1987

[17] D M Hawkins and W J R Eplett ldquoThe Cholesky factorizationof the inverse correlation or covariance matrix in multipleregressionrdquo Technometrics vol 24 no 3 pp 191ndash198 1982

[18] C HassoldThe Valuation of Multivariate Options diplom De2004

[19] D Duffie and J Pan ldquoAn overview of value at riskrdquoThe Journalof Derivatives vol 4 no 3 pp 7ndash49 1997

[20] C Genest and A-C Favre ldquoEverything you always wanted toknow about copula modeling but were afraid to askrdquo Journal ofHydrologic Engineering vol 12 no 4 pp 347ndash368 2007

[21] B Mandelbrot ldquoThe variation of certain speculative pricesrdquoJournal of Business vol 36 no 4 pp 394ndash419 1963

[22] I Popova E Popova D Morton and J Yau ldquoOptimal hedgefund allocationwith asymmetric preferences and distributionsrdquoSSRN 900012 2006

[23] R J DaviesHMKat and S Lu ldquoFund of hedge funds portfolioselection a multiple-objective approachrdquo Journal of Derivativesamp Hedge Funds vol 15 no 2 pp 91ndash115 2009

[24] L T DeCarlo ldquoOn the meaning and use of kurtosisrdquo Psycholog-ical Methods vol 2 no 3 pp 292ndash307 1997

[25] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[26] A S Taylan and H Tatlıdil ldquoPortfolio optimization withshort function and higher order moments an application inISE-30rdquo in Proceedings of the 24th Mini EURO ConferenceContinuous Optimization and Information-Based Technologiesin the Financial Sector (MEC EurOPT rsquo10) Izmir Turkey June2010

[27] K K Lai L Yu and S Wang ldquoMean-variance-skewness-kurtosis-based portfolio optimizationrdquo in Proceedings of the 1stInternational Multi-Symposiums on Computer and Computa-tional Sciences (IMSCCS rsquo06) vol 2 pp 292ndash297 HanzhouChina April 2006

[28] B Aracioglu F Demircan and H Soyuer ldquoMean-variance-skewness-kurtosis approach to portfolio optimization an appli-cation in Istanbul stock exchange IMKB Uygulamasirdquo EgeAkademik Bakis vol 11 pp 9ndash17 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of