Portfolio Selection with Higher Moments: A Polynomial Goal Programming Approach to ISE-30 Index

Abstract

The aim of this paper is to propose a portfolio selection model which takes into account the investors preferences for higher return moments such as skewness and kurtosis. In the presence of skewness and kurtosis, the portfolio selection problem can be characterized with multiple conflicting and competing objective functions such as maximizing expected return and skewness, and minimizing risk and kurtosis, simultaneously. By constructing polynomial goal programming, in which investor preferences for skewness and kurtosis incorporated, a Turkish Stock Market example will be presented for the period from January 2005 to December 2010.

Full text

PORTFOLIO SELECTION WITH HIGHER
MOMENTS: A POLYNOMIAL GOAL
PROGRAMMING APPROACH TO ISE–30 INDEX
Gülder KEMALBAY* C. Murat ÖZKUT** Ceki FRANKO***
Abstract
Keywords: Mean-Variance-Skewness-Kurtosis Portfolio Model, Polynomial Goal Programming, Risk
Preference.
Jel Classification: C44, G11
Özet
Anahtar Kelimeler: Ortalama-Varyans-Çarpıklık-Basıklık Portföy Modeli, Polinomsal Hedef Programlama,
Risk Tercihi.
Jel Sınıflaması: C44, G11
*Arş. Grv., Yıldız Teknik Üniversitesi, Fen-Edebiyat Fakültesi, İstatistik Bölümü, E-Mail:
kemalbay@yildiz.edu.tr
** Arş.Grv., İzmir Ekonomi Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, E-Mail:
murat.ozkut@ieu.edu.tr
*** Arş.Grv., İzmir Ekonomi Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, E-Mail:
ceki.franko@ieu.edu.tr
The aim of this paper is to propose a portfolio selection model which takes into account the investors
preferences for higher return moments such as skewness and kurtosis. In the presence of skewness and
kurtosis, the portfolio selection problem can be characterized with multiple conflicting and competing
objective functions such as maximizing expected return and skewness, and minimizing risk and kurtosis,
simultaneously. By constructing polynomial goal programming, in which investor preferences for skewness
and kurtosis incorporated, a Turkish Stock Market example will be presented for the period from January 2005
to December 2010.
Bu makalenin amacı, çarpıklık ve basıklık gibi yüksek getiri momentleri için yatırımcının tercihlerini göz
önüne alan bir portföy seçimi modeli önermektir. Çarpıklık ve basıklığın varlığında, portföy seçimi problemi,
eş zamanlı olarak beklenen getiri ve çarpıklığın maksimizasyonu ile risk ve basıklığın minimize edilmesi gibi
birbiri ile çelişen ve rekabet eden amaç fonksiyonları ile karakterize edilir. Polinomsal hedef programlama
oluşturarak, Ocak 2005-Aralık 2010 periyodu için Türk Borsası’nda bir örnek sunulacaktır.
İSTANBUL ÜNİVERSİTESİ
İKTİSAT FAKÜLTESİ
EKONOMETRİ VE İSTATİSTİK
DERGİSİ
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42
1. INTRODUCTION
In the modern portfolio theory, the mean-variance model which is minimizing risk for
a given level of expected return, or equivalently, maximizing expected return for a given level
of risk originally introduced by Markowitz (1952) and has gained widespread acceptance as a
practical tool for portfolio optimization. Since the seminal work of Markowitz, most
contributions to portfolio selection are based only first two moments of return distribution.
In Markowitz’s framework, it is assumed that asset returns follow multivariate normal
distribution. This means that the distribution of asset return can be completely described by
the expected value and variance. However empirical finance has shown that the distribution of
individual asset returns sampled at a daily, weekly or monthly frequency exhibit negative
skewness and excess kurtosis so is not well described by a normal distribution. In the
presence of negative skewness, negative return has higher probability than positive return. In
addition, if a distribution of portfolio return is positively skewed, it indicates that poor returns
occur frequently but losses are small, whereas very high returns occur less frequently but are
more extreme. Furthermore, the kurtosis can reflect the probability of extreme events. Excess
positive kurtosis, or leptokurtosis indicates that a distribution of return has fatter tails than a
normal distribution, i.e., it indicates a higher probability of very high and very low returns
would be expected than the normal case. This departure from normality means that higher
moments of the return distribution are necessary to describe portfolio behavior. When the
skewness and kurtosis are significant, if we look at only the mean and variance under the
normality assumption for the return distribution, we may underestimate the risk and this leads
to obtain an inefficient portfolio. Thus the mean-variance model proposed by Markowitz is
inadequate for optimal portfolio selection problem and higher moments can not be neglected.
One of the problems of extending the mean-variance framework to higher moments
for portfolio selection is that it is not easy to find a trade-off between the four objectives
because in the presence of skewness and kurtosis, the problem turns into a nonconvex and
nonsmooth multiobjective optimization problem. Thus, many researches on portfolio
selection largely concentrate on the first three moments and kurtosis is neglected by most of
researchers. In addition, most of models only consider the distribution of asset return but other
factors, such as investor’s risk preferences and trading strategies, are not taken into account.

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To tackle these problems, many approaches have been proposed. One of the efficient
ways to solve this task is polynomial goal programming method. An important feature of
polynomial goal programming problem is the existence of optimal solution since feasible
solution always exists. The other important features of this method are its flexibility of
incorporating investor preferences and its simplicity of computational requirements. As a
result, this study extends the work of Lai et al. (2006) by utilizing polynomial goal
programming, which incorporates investor preferences for skewness and kurtosis.
In summary, the main focus of this study is to propose a mean-variance-skewness-
kurtosis model for portfolio selection problem based on investor’s risk preferences by
constructing polynomial goal program. The paper is organized as follows: Section 2 provides
a brief review of literature. In section 3, the theoretical framework of the portfolio selection
problem with higher moments is discussed. The numerical results are illustrated in section 4.
The final section concludes the study while some computational details are relegated to an
appendix.
2. LITERATURE REVIEW
Since Markowitz (1952, 1959) proposed the mean-variance portfolio model, numerous
studies of portfolio selection have focused on the first two moments of return distributions for
portfolio decisions. In his framework, return distribution is assumed to be normal or utility
function only depends on first two moments, i.e., utility function is quadratic. It is well known
that financial series are non-normal. Also many empirical evidences suggest that asset returns
tend to be asymmetric and leptokurtic, that is, more peaked and fatter tailed than the normal
distribution: See Mandelbrot (1963), Fama (1965), Blattberg ve Gonedes (1974), Kon (1984),
Mills (1995), Campbell (1997), Peiro (1999), Harvey and Siddique (1999, 2000), Premaratne
and Bera (2000). However, many researchers argued that the higher moments can not be
neglected unless there is a reason to believe that the asset returns are normally distributed or
the utility function is quadratic, or that the higher moments are irrevelant to the investor’s
decision: See Samuelson (1970), Arditti (1971), Rubinstein (1973), Scott and Horvath (1980),
Lai (1991), Konno and Suzuki (1995), Chunhachinda et al. (1997), Prakash et al. (2003), Lai
et al. (2006).

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44
Moreover, Hanoch and Levy (1970) pointed out that that the quadratic utility function
implies increasing absolute risk aversion which is contrary to the normal assumption of
decreasing absolute risk aversion. Levy and Sarnat (1972) also shows that the assumption of a
quadratic utility function is appropriate only for relatively low returns (Chunhachinda et
al.,1997).
Furthermore, when the investment decision is restricted to a finite-time interval,
Samuelson (1970) showed that the mean-variance efficiency becomes inadequate and higher
moments become relevant to the portfolio selection (Lai, 1991).
In general, investors will prefer high values for odd moments and low ones for even
moments. The former can be seen as a way to decrease extreme values on the side of losses
and increase them on the gains’ (Athayde and Flores, 2004). Scott and Horvath (1980),
investigated the use of higher moments in portfolio analysis by determining direction of
preference of moments. They showed that preference is positive (negative) for positive values
of every odd central moment and negative for every even central moment for investor who is
consistent in direction of preference of moments.
As a result, in some recent studies the concept of mean-variance framework has been
extented to include the skewness and kurtosis of return in portfolio selection (Yu et al., 2008).
The importance of skewness in return distribution is introduced by Arditti (1967,
1971) in the pricing stocks. Kraus and Litzenberger (1976), came up with three parameter
capital asset pricing model (Premaratne and Bera 2000). Lai (1991), Chunhachinda (1997)
and Prakash et al. (2003) showed that the incorporation of skewness into the investor’s
portfolio decision causes a major change in the constructing of the optimal portfolio.
In fact, kurtosis which reflects the probability of extreme events is neglected for a long
time by most researchers. As the dimensionality of the portfolio selection problem increases,
then it becomes difficult to develop geometric interpretation of the quartic portfolio efficient
frontier and to select the most preferred portfolio among boundary points (Jurczenko et al.,
2006). Mandelbrot (1963), was probably the first to take into account excess kurtosis in
financial data as he noted that the price changes were too peaked and thick-tailed than normal

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distribution (Premaratne and Bera 2000). In spite of the considerable empirical literature now
taking into account this fact, financial theory has been reluctant in incorporating higher
moments such that kurtosis in its developments (Athayde and Flores, 2004). Jean (1971),
extends the portfolio analysis to three or more parameters and derives the risk premium for
higher moments (Chunhachinda et al.,1997). Fang and Lai (1997), first introduced kurtosis to
develop capital asset pricing model as well as skewness. Jondeau et al. (2006), introduced the
kurtosis into the portfolio selection problem through utility function (Qi-fa et al., 2007).
Also, there are some researches look for the analytical solution in the mean-variance-
skewness-kurtosis space: See Athayde and Flores (2004), Adcock (2005), Jurczenko and
Maillet (2005b). Furthermore, Jondeau and Rockinger (2003, 2005), Jurczenko and Maillet
(2005a) used Taylor series expansion of the investors’ objective functions to determine
optimal portfolio (Jurczenko et al., 2006).
In the presence of skewness and kurtosis, the portfolio selection problem turns into a
nonconvex and nonsmooth optimization problem which can be characterized with multiple
conflicting and competing objective functions such as maximizing expected return and
skewness, and minimizing risk and kurtosis, simultaneously. To solve this complicated task,
different approaches have been proposed in the literature and one of the efficient way is
applying polynomial goal programming (PGP) which investment strategies and the investor’s
preferences should be included.
PGP was first introduced by Tayi and Leonard (1988). After, Lai (1991) applied PGP
to portfolio selection and explored incorporation of investor’s preferences in the construction
of a portfolio with skewness. Similarly, Leung et al. (2001) provided PGP to solve mean-
variance-skewness model with the aid of the general Minkovski distance. In the mean-
variance-skewness framework, also Chunhachinda et al. (1997), Wang and Xia (2002), Sun
and Yan (2003), Prakash et al. (2003) used PGP to construct optimal portfolio. Lai et al.
(2006) augmented the dimension of portfolio selection in PGP from mean-variance-skewness
to mean-variance-skewness-kurtosis. More recently, incorporating higher moments such as
skewness and kurtosis, PGP has subsequently been used as an efficient way by Qi-fa et al.
(2007), Taylan and Tatlıdil (2010), Mhiri and Prigent (2010) for efficient portfolio and also
Davies et al. (2009) and Berenyi (2005) for effiecient funds of hedge funds.

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3. PORTFOLIO SELECTION WITH HIGHER MOMENTS
In this section, we consider the problem of an investor who selects optimal portfolio
from n risky assets in the presence of skewness and kurtosis of return distribution which is a
trade-off between competing and conflicting objectives, i.e., maximizing expected return and
skewness, while minimizing variance and kurtosis, simultaneously. As Lai (1991) notes that
there are some standard assumptions in portfolio selection, we accept these assumptions
except some minor points such that:
i) investors are risk-averse individuals who maximize the expected utility of their end-of-
period wealth,
ii) there are n risky asset and investor does not have access to a riskless asset implying that
the portfolio weights must sum to one,
iii) all asset are marketable, perfectly divisible,
iv) the capital market is perfect, there are no taxes and transaction costs,
v) short selling is not allowed, implying that portfolio weights must be positive.
Our major interest is to determine the investment strategy of the investor among
different preferences and the investment weight of each asset which should be included within
the mean-variance-skewness-kurtosis framework.
Let’s denote portfolio return by
p
R
,


1 2
, ,...,
R
n
R R R
is the return vector,
i
R
is the
rate of return of ith asset. Wealthes are allocated to n assets by the weights


1 2
, ...,
X
n
x x x
,
i
x
is the proportion invested in the ith asset when the best trade-off is found. The mean,
variance, skewness and kurtosis of the rate of return
i
R
on asset i are assumed to exist for all
risky assets i, i=1,2,…,n and denoted by
i
R
,
2
i

,
3
i
s
,
4
i
k
; respectively. Then, the first four
moments of portfolio return
p
R
can be computed as:
Mean
n
i i
i 1
E( )
p
R x R


  

X R (1)
Variance 2 2
i 1 1 1
( )
n n n
2
p i i i j ij
i j
R x x x
  

       
 
X X ,
( )
i j

(2)
Skewness 3 3
( ) E( ( ))
p
=S R 
  
X R R

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3 3 2 2
1 1 1 1
3 ,( )
n n n n
i i i j iij i j ijj
i i j j
x s x x s x x s i j
   
 
  
 
 
    (3)
Kurtosis 4 4
( ) E( ( ))
p
K R 
   
X R R
4 4 3 3 2 2
1 1 1 1 1 1
4 6
n n n n n n
i i i j iiij i j ijjj i j iijj
i i j j i j
x k x x k x x k x x k
     
 
  
 
 
     ,
( )
i j

(4)
where
ij

is variance-covariance matrix;
iij
s
,
ijj
s
are skewness-coskewness (which measure
curvelinear relationship);
iiij
k
,
ijjj
k
,
iijj
k
are kurtosis-cokurtosis matrices of the joint
distribution of risky asset returns
i
R
and
j
R
and they are defined as follows:
 
 
1 1
1n n
ij i i j j
i j
R R R -R
t 
  
 (5)
 
 
2
1 1
1n n
iij i i j j
i j
s R R R -R
t 
 
 ,
 
 
2
1 1
1n n
ijj i i j j
i j
s R R R -R
t 
 
 (6)
 
 
3
1 1
1n n
iiij i i j j
i j
k R R R -R
t 
 
 ,
 
 
3
1 1
1n n
ijjj i i j j
i j
k R R R -R
t 
 

 
 
2
2
1 1
1n n
iijj i i j j
i j
k R R R -R
t 
 
 , (7)
(where t is the number of periods).
Then, the portfolio selection problem with higher moments can be formulated with
following competing and conflicting objective functions:
2
3 3
4 4
Max E( )
Min ( )
Max ( ) E( ( ))
(P1)
Min ( ) E( ( ))
s.t. 1
0, 1, 2, ,
p
p
p
p
i
R
R
S R
K R
x i ... n.



  


 


  


 

X R
X X
X R R
X R R
X I
(8)

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48
A general way to solve the multiobjective problem is to consolidate the various
objectives into a single objective function. Because of the contradiction and possible
incommensurability of the objective functions such as risk and return, it’s often not possible
to find a single solution where every objective function attains its optimum simultaneously.
Generally, instead of single solution, a set of nondominated solutions is considered. In this
case, subjective judgements of investor come into prominence.
As a result, the multiobjective problem involves two step procedures: First, a set of
nondominated solutions which is independent from investor’s preferences is developed. After,
investor selects the most preferable solution among the given set of solutions. The second step
can be accomplished by incorporating investor’s preferences for objective functions into the
construction of a polynomial goal programming. Consequently, portfolio selection with
higher moments is a solution of PGP.
3.1 Solving Polynomial Goal Programming
A solution depending on investor preferences for objectives can be determined by
constructing of a polynomial goal programming into which the specified investor’s personal
objectives are incorporated. Thus, we use this approach to combine our objectives into single
one, and to solve (P1).
The main interest of polynomial goal programming can be defined as the minimization
of deviations from ideal scenario set by aspired levels. The aspired level indicates the best
scenario for a particular objective without considering other objectives. Hence , the aspired
levels,
* * * *
, , ,
M V S K
, can be determined by solving four independent subproblems, using
linear and nonlinear programming:
Max E( )
(SP1) s.t. 1
0, 1, 2, ,
p
i
R
x i ... n.




 

X R
X I
2
Min ( )
(SP2) s.t. 1
0, 1, 2, ,
p
i
R
x i ... n.
   


 

X X
X I
3 3
Max ( ) E( ( ))
(SP3) s.t. 1
0, 1, 2, ,
p
i
S R
x i ... n.
 
 


 

X R R
X I
4 4
Min ( ) E( ( ))
(SP4) s.t. 1
0, 1, 2, ,
p
i
K R
x i ... n.
 
 


 

X R R
X I

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Let
1 2 3 4
, , ,
d d d d
be the nonnegative goal variables which account for the deviations
of expected return, variance, skewness and kurtosis from the aspired levels,
* * * *
, , ,
M V S K
,
respectively. In other words, the goal variables denote the amount of underachievement with
respect to the best scenario. To minimize objective function, general Minkovski distance is
often used in finance and economics. The computational form of Minkovski distance is:
1/
1
p
p
n
i
ii
d
ZZ

 
 

 
 
 
 (9)
where
i
Z
is the basis for normalizing the ith variable. To incorporate investor’s different
preferences towards to the mean, variance, skewness, kurtosis of portfolio return into model,
we introduce four parameters
1 2 3 4
,
, ,
   
, respectively. Since
i

parameters represent the
investor’s subjective degree of preferences, the greater
i

, the more important corresponding
moment of portfolio return is to the investor.
In PGP, the objective function contains deviational variables between goals and what
can be achieved and does not contain choice variables. Given the investor preferences, the
multiobjective portfolio selection problem (P1) turns into single-objective problem by
constructing the PGP model (P2) whose objective is to minimize deviations from ideal
scenario set by aspired levels as follows:
3
1 2 4
31 2 4
* * * *
*
1
*
2
3 *
3
4 *
4
Min
s.t.
(P2)
E( ( ))
E( ( ))
1
0, 1,2, ,
i i
dd d d
Z M V S K
d M
d V
d S
d K
x d 0, i ... n

  
   



 
  

  
  

  
X R
X X
X R R
X R R
X I









(10)

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50
The set of efficient portfolio consists of solutions of problem (P2) for various
combinations of
i

. In this study, we also obtained efficient portfolio for the mean-variance,
and mean-variance-skewness case and compared to those of the mean-variance-skewness-
kurtosis efficient portfolio.
4. EMPIRICAL RESULTS
In the light of earlier description, our analysis is based on two purposes:
i) to demonstrate the formulation of the polynomial goal programming for portfolio
selection problem in four-moment space,
ii) to illustrate how portfolio selection will vary for investors with different investment
strategies.
The sample data consists of monthly rates of return for 26 stocks from Istanbul Stock
Exchange-30 Index in Turkish Stock Market for the period January 2005 through December
2010. The historical data are used to estimate the expected return, covariance and central
comoments.
The empirical experiment employed in this study can be summarized in two main
stages: First, the distributional properties are computed and normality test results are
represented in Table 1. In addition, in Table 2, the stocks are ranked based on the coefficient
of variation to provide some preliminary information. Secondly, the aspired levels are found
by solving (SP1)-(SP4), as shown in Table 3. Then, by solving (P2) with PGP approach,
optimal objective values and the trade-off between them are shown in Table 4. Moreover, the
optimal weights of the stocks which should be included in portfolio are presented for the
given investor’s different preferences including also MV and MVS case in Table 5. All of the
results are calculated on GAMS program.
For preliminary analysis, Table 1 lists the descriptive statistics of the rate of return of
26 stocks. Interestingly, while DENİZ has the highest value of mean and skewness, i.e., return,
it also has the highest value of variance and kurtosis, that is, risk. The results of the normality
of return distributions using the Jarque-Bera test are also provided in the last column. Since

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test results supports non-normality of return distribution, there is an evidence to construct
portfolio including skewness and kurtosis.
Table 1. Descriptive statistics and normality test for rate of return distribution
Stock Variable
Mean
Variance Skewness
Kurtosis
J-B
Statistic*
p
-
value
AKBNK
1
x
1,882 189,614 0,806 4,868 15,23 0,00
ARCLK
2
x
1,563 226,188 0,398 4,287 5,727 0,06
DENİZ
3
x
3,733 609,699 3,498 18,687 737,597 0,00
DOAS
4
x
2,661 271,465 -0,349 2,79 1,329 0,51
DOHOL
5
x
0,739 170,357 0,554 3,656 4,143 0,13
DYHOL
6
x
0,452 363,36 0,53 4,196 6,385 0,04
EREGL
7
x
2,379 162,764 -0,232 2,73 0,718 0,70
FINBN
8
x
1,745 102,412 1,158 7,247 58,495 0,00
GARAN
9
x
2,682 208,699 0,395 3,012 1,561 0,46
HURGZ
10
x
0,014 276,257 0,424 3,499 2,418 0,30
ISCTR
11
x
0,98 141,844 0,4 2,906 1,618 0,45
ISGYO
12
x
0,657 148,502 -0,087 3,547 0,823 0,66
KCHOL
13
x
2,369 167,71 0,113 3,323 0,388 0,82
PETKM
14
x
1,353 137,457 -0,077 3,379 0,418 0,81
PTOFS
15
x
1,862 181,953 -0,168 4,138 3,516 0,17
SAHOL
16
x
1,758 202,707 0,582 3,818 5,064 0,08
SISE
17
x
1,404 140,435 -0,163 2,941 0,273 0,87
SKBNK
18
x
2,697 322,028 0,405 4,361 6,266 0,04
TCELL
19
x
1,558 99,753 -0,211 3,411 0,867 0,65
THYAO
20
x
3,114 170,82 0,04 2,705 0,233 0,89
TOASO
21
x
3,172 233,212 -0,178 4,296 4,518 0,10
TSKB
22
x
2,055 171,299 -0,199 2,592 0,812 0,67
TUPRS
23
x
1,975 105,238 -0,063 2,543 0,561 0,76
ULKER
24
x
1,239 143,948 -0,025 3,779 1,523 0,47
VESTL
25
x
0,174 263,331 1,563 9,732 137,757 0,00
YKBNK
26
x
2,207 170,699 0,429 3,857 3,672 0,16
J-B* represents Jarque-Bera Statistic:


2 2
n / 6 Skewness (Kurtosis 3) / 4
  . If the p-value is less than 0.05, the
null hypothesis of normality cannot be supported at the %5 significance level. Values in bold font signify the
highest value for mean and skewness and the lowest value for variance and kurtosis.
Table 2 list the mean, the standard deviation and the coefficient of variation of rate of
return of the each stock in ISE-30 index. Coefficient of variation shows the risk per unit
return. The ranking of coefficient of variation may provide some preliminary information,

Port. Sel. With Higher Mom.: A Poly. Goal Prog. App. To ISE-30 Ind.
52
with regard to potential candidacy for inclusion in the optimal portfolio. Ranking of C.V.
reveals that THYAO ranks at the top of the list, providing the least risk per unit of return,
whereas HURGZ ranks at the bottom of list , providing the highest risk per unit of return.
Furthermore, if we consider the coefficient of variation as a risk measure, it can be
failed to capture fully the true risk of distribution of the stock return. In this case, the role of
higher moments becomes important because true risk should be a multidimensional concept.
Table 2. Coefficient of variation rankings of stocks
Stock
Mean
Std.Dev. C.V.*
Rank
Stock
Mean
Std.Dev.
C.V.*
Rank
AKBNK
1,882 13,77 7,32 15
PETKM
1,353 11,724 8,67 18
ARCLK
1,563 15,04 9,62 19 PTOFS 1,862 13,489 7,24 14
DENİZ 3,733 24,692 6,61 12 SAHOL
1,758 14,238 8,1 16
DOAS 2,661 16,476 6,19 9 SISE 1,404 11,851 8,44 17
DOHOL
0,739 13,052 17,66 22 SKBNK
2,697 17,945 6,65 13
DYHOL
0,452 19,062 42,17 24 TCELL
1,558 9,988 6,41 11
EREGL
2,379 12,758 5,36 4 THYAO
3,114 13,07 4,2 1
FINBN 1,745 10,12 5,8 7 TOASO
3,172 15,271 4,81 2
GARAN
2,682 14,446 5,39 5 TSKB 2,055 13,088 6,37 10
HURGZ
0,014 16,621 1187,2 26 TUPRS
1,975 10,259 5,19 3
ISCTR 0,98 11,91 12,15 21 ULKER
1,239 11,998 9,68 20
ISGYO 0,657 12,186 18,55 23 VESTL
0,174 16,227 93,26 25
KCHOL
2,369 12,95 5,47 6 YKBNK
2,207 13,065 5,92 8
*C.V. represents Coefficient of Variation: Mean/Standard Deviation.
Subsequently, in accordance with the second stage, the aspired levels are calculated
solving each subproblems by using linear and nonlinear programming:
Table 3. The aspired levels of four moments
M* V* S* K*
Objective 3,733 148,86 1,184 0,051
By substituting these aspired levels in (P2), we solve our problem with proposed
algorithm. Certainly, the investor preferences not only change in the process, but also affect
the portfolio selection. In order to verify the sensitivity of the proposed approach to changes
in the investor’s preference
1 2 3 4
( , , , )
   
, twelve different levels of preference are investigated

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53
including also the cases (1,1,0,0), (1,1,1,0), i.e., mean-variance and mean-variance-skewness,
respectively. The optimal variables and objective values which are corresponding to the
different combinations of
1 2 3 4
( , , , )
   
are shown in the following table:
Table 4. Optimal value of objectives and trade-off between the four moments
A B C D E F G H I J K L
1

3 3 3 1 1 1 1 3 2 1 1 1
1

1 1 1 3 1 3 2 1 3 1 1 1
1

1 2 3 1 1 1 3 2 3 0 1 1
1

0 1 1 1 3 3 2 3 1 0 0 1
M 2,957 1,714 1,71 1,728 1,721 1,713 1,713 1,713 1,718 3,277 3,15 1,728
V 148,81
148,72
148,33
149,53
148,85
148,85
149,19
148,85
148,85
148,92
148,83
149,53
S 1,18 0,051 0,053 0,049 0,048 0,048 0,048 0,048 0,053 1,187 -0,08 0,049
K 5,288 0,254 0,254 0,254 0,253 0,253 0,253 0,253 0,254 9,757 3,878 0,254
Obj
1,009 5,013 4,973 5,436 62,48 62,48 16,92 62,06 5,107 1,122 2,156 5,436
Investor determines his/her preferences for objective functions with respect to the
targeted risk per unit return. As indicated in (P2), the smaller objective function, the better
solution is. Thus, investor can select the best portfolio according to the minimal objective
functions. But investor should remember the trade-off between objectives since greater
preference on return may cause greater risk. As reported in Table 4, the mean-variance
efficient portfolio has the highest expected return. This result is consistent is with the notion
that the expected return of mean-variance efficient portfolio must dominate any other
portfolios given the same level of variance. On the other hand, if the investor chooses the
mean-variance efficient Portfolio J, then he/she is exposed to the highest probability of
extreme events. To avoid this case, kurtosis can not be neglected as measure of risk. On the
other hand, the minimum kurtosis is achieved in Portfolio E, F and H, but objective values of
these portfolios are very high. Interestingly, if investor prefers lower preference for variance
in Portfolio L rather than Portfolio D, then the same portfolio including also optimal weights
of the stocks is obtained. The skewness is only negative in the mean-variance-skewness case.
Compared Portfolio B where expected return and variance set equal to those of Portfolio A,
higher preference for skewness leads to lower portfolio skewness but also lower portfolio
kurtosis than Portfolio A. Similarly, we also consider changing the preference parameters o f
Portfolio E from (1,1,1,3) to Portfolio A (3,1,1,0) while holding the values of variance and

Port. Sel. With Higher Mom.: A Poly. Goal Prog. App. To ISE-30 Ind.
54
skewness constant. As preference for expected returns increases, the investor must settle for
higher kurtosis.
As can be seen, the expected return, variance, skewness and kurtosis are conflicting
objectives in portfolio selection problem. That is, as a result of the trade-off between the four
moments, at least one of the other three moment statistics deteriorates. Consequently, there is
strong evidence which shows that the incorporation of the skewness and kurtosis into the
investor’s portfolio decision causes a major change in the construction of optimal portfolio
since different combinations of investor’s preferences on four moments lead to optimal
portfolios with substantially different moment characteristics.
Table 5 presents the optimal weights of stocks which should be included in the
portfolio. Accordingly, the corresponding weight sets of different risk preference level yield
the optimal investment portfolio. For example, for the case of risk preference level (1,1,1,1),
the optimal proportion of 26 different stocks is vector (0,052 0,058 0,000 0,075 0,033 0,065
0,000 0,000 0,079 0,067 0,064 0,023 0,071 0,000 0,000 0,067 0,055 0,055 0,000 0,002 0,062 0,072
0,000 0,017 0,037 0,048). Interestingly, FINBN, PETKM, PTOFS, TCELL and TUPRS are not
included in any efficient portfolio. Although TUPRS has high ranking of coefficient, the
exculison can be the evidence of the importance of higher moments. On the other hand, the
lowest ranking stock HURGZ has dominant components except three cases. DENİZ has the
most dominant components of 29 percent in mean-variance efficient frontier and it dos not get
involded in any model with preference for kurtosis since DENİZ has the highest value of
kurtosis. The least preferred stock is ISGYO with the weigth of 2 percent.

Ekonometri ve İstatistik Sayı:13 (12. Uluslararası Ekonometri, Yöneylem Araştırması,
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55
Table 5. Optimal portfolio’s weights with different preferences of investor’s
A B C D E F G H I J K L
1

3 3 3 1 1 1 1 3 2 1 1 1
2

1 1 1 3 1 3 2 1 3 1 1 1
3

1 2 3 1 1 1 3 2 3 0 1 1
4

0 1 1 1 3 3 2 3 1 0 0 1
AKBNK
1
x
0,06 0,05 0,05 0,052 0,051 0,051 0,051 0,05 0,053 _ _ 0,052
ARCLK
2
x
_ 0,058 0,057 0,058 0,058 0,058 0,058 0,058 0,057 _ _ 0,058
DENİZ
3
x
0,210 _ _ _ _ _ _ _ _ 0,29 0,13 _
DOAS
4
x
0,07 0,07 0,07 0,08 0,08 0,08 0,08 0,08 0,07 _ 0,072 0,08
DOHOL
5
x
_ 0,035 0,035 0,033 0,034 0,034 0,034 0,034 0,035 _ _ 0,033
DYHOL
6
x
_ 0,065 0,07 0,065 0,065 0,065 0,065 0,065 0,065 _ _ 0,065
EREGL
7
x
_ _ _ _ _ _ _ _ _ 0,037 _ _
FINBN
8
x
_ _ _ _ _ _ _ _ _ _ _ _
GARAN
9
x
0,180 0,08 0,08 0,079 0,078 0,078 0,078 0,078 0,080 _ 0,029 0,08
HURGZ
10
x
_ 0,067 0,067 0,067 0,068 0,068 0,068 0,068 0,067 _ _ 0,067
ISCTR
11
x
_ 0,065 0,066 0,064 0,064 0,064 0,065 0,064 0,066 _ _ 0,064
ISGYO
12
x
_ 0,023 0,023 0,023 0,025 0,025 0,025 0,025 0,023 _ _ 0,023
KCHOL
13
x
0,08 0,07 0,070 0,071 0,070 0,070 0,070 0,070 0,070 _ _ 0,071
PETKM
14
x
_ _ _ _ _ _ _ _ _ _ _ _
PTOFS
15
x
_ _ _ _ _ _ _ _ _ _ _ _
SAHOL
16
x
_ 0,067 0,068 0,067 0,066 0,066 0,066 0,066 0,068 _ _ 0,067
SISE
17
x
_ 0,054 0,053 0,055 0,056 0,056 0,056 0,056 0,053 _ _ 0,055
SKBNK
18
x
0,045 0,055 0,054 0,055 0,055 0,055 0,055 0,055 0,054 _ 0,049 0,055
TCELL
19
x
_ _ _ _ _ _ _ _ _ _ _ _
THYAO
20
x
0,17 _ _ 0,002 _ _ _ _ 0,002 0,476 0,307 0,002
TOASO
21
x
0,15 0,06 0,061 0,062 0,061 0,061 0,061 0,061 0,061 0,197 0,417 0,062
TSKB
22
x
_ 0,071 0,071 0,072 0,072 0,072 0,072 0,072 0,071 _ _ 0,072
TUPRS
23
x
_ _ _ _ _ _ _ _ _ _ _ _
ULKER
24
x
0,000 0,016 0,016 0,017 0,018 0,018 0,017 0,018 0,015 _ _ 0,017
VESTL
25
x
_ 0,038 0,038 0,037 0,037 0,037 0,038 0,037 0,038 _ _ 0,037
YKBNK
26
x
0,040 0,049 0,049 0,048 0,047 0,047 0,047 0,047 0,049 _ _ 0,048
5. CONCLUSIONS
This study proposes a Polynomial Goal Programming approach to the mean-variance-
skewness-kurtosis based portfolio optimization model. Through the use of the PGP model, an
investor can construct a portfolio which matches his or her risk preference based on trading
strategies as well as the mean-variance-skewness-kurtosis objectives simultaneously. We
illustrate an example in Turkish Stock Market to test our proposed approach with twenty-six

Port. Sel. With Higher Mom.: A Poly. Goal Prog. App. To ISE-30 Ind.
56
stocks from Istanbul Stock Exchange-30 Index. The empirical results indicate that the
incorporation of the skewness and kurtosis into the investor’s portfolio decision causes a
major change in the construction of optimal portfolio since different combinations of
investor’s preferences on four moments lead to optimal portfolios with substantially different
moment characteristics and this confirms our argument that higher moments can not be
neglected in the portfolio selection.
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Appendix
Table 6. The Variance-Covariance (

ij) Matrix

Port. Sel. With Higher Mom.: A Poly. Goal Prog. App. To ISE-30 Ind.
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Table 7. Skewness-Coskewness (
ijj
s
) Matrix

Ekonometri ve İstatistik Sayı:13 (12. Uluslararası Ekonometri, Yöneylem Araştırması,
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Table 8. The Kurtosis-Cokurtosis (
ijjj
k
) Matrix