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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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