Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: an Application in İstanbul Stock Exchange

EGE AKADEMİK BAKIŞ / EGE ACADEMIC REVIEW Cilt: 11 • Özel Sayı • 2011 ss. 9-17 Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: An Application in İstanbul Stock Exchange

Portföy Optimizasyonunda Ortalama-Varyans-Çarpıklık-BasıklıkYaklaşımı: İMKB Uygulaması

Burcu ARACIOĞLU1, Fatma DEMİRCAN2, Haluk SOYUER3

ABSTRACT ÖZET

Portfolio optimization, the construction of the best combination of Belli kısıtlar altında yatırımcıların temel beklentilerini karşılayacak investment instruments that will meet the investors’ basic expectati- en iyi yatırım araçları karmasının oluşturulması olan portföy opti- ons under certain limitations, has an important place in the finance mizasyonu. finans dünyasında önemli bir yere sahiptir. Portföy op- world. In the portfolio optimization, the Mean Variance model of Mar- timizasyonunda, oluşturulan portföyler için getiri ve risk arasında kowitz (1952) that expresses a tradeoff between return and risk for a bir dengelemeyi ifade eden Markowitz’in (1952) Ortalama Varyans set of portfolios, has played a critical role and affected other studies modeli, bu alanda kritik bir role sahiptir ve yapılan diğer çalışmaları in this area. da etkilemiştir. In the Mean Variance model, only the covariances between securi- Markowitz’in Ortalama-Varyans modelinde, portföyün riski belirle- ties are considered in determining the risk of portfolios. The model nirken sadece menkul kıymet getirilerinin kovaryans değerleri dik- is based on the assumptions that investors have a quadratic utility kate alınmaktadır. Bu model, yatırımcıların kuadratik fayda fonksi- function and the return of the securities is distributed normally. Va- yonuna sahip olduğu ve hisse senedi getirilerin normal dağıldığı rious studies that investigate the validity of these assumptions find varsayımlarına dayandırılmıştır. Bu varsayımların geçerliliğini ince- evidence against them. Asset returns have significant skewness and leyen çok sayıda çalışmada karşıt bulgulara ulaşılmıştır. Varlık ge- kurtosis. In the light of these findings, it is seen that in recent years tirilerinin anlamlı derecede çarpıklık ve basıklık özelliği gösterdiği researchers use higher order of moments in the portfolio selection saptanmıştır. Bu bulgular ışığında, son yıllarda araştırmacıların port- (Konno et al, 1993; Chunhachinda et al, 1997; Liu et al, 2003; Harvey föy seçiminde yüksek dereceden momentleri kullandıkları görülmek- et al, 2004; Jondeau and Rockinger, 2006; Lai et al, 2006; Jana et al, tedir (Konno et al, 1993; Chunhachinda et al, 1997; Liu et al, 2003; 2007; Maringer and Parpas, 2009; Briec et al, 2007; Taylan and Tatlıdil, Harvey et al, 2004; Jondeau and Rockinger, 2006; Lai et al, 2006; Jana 2010). et al, 2007; Maringer and Parpas, 2009; Briec et al, 2007; Taylan and Tatlıdil, 2010). In this study, in the mean- variance- skewness- kurtosis framework, multiple conflicting and competing portfolio objectives such as Bu çalışmada, ortalama-varyans-çarpıklık ve basıklık modeli çer- maximizing expected return and skewness and minimizing risk and çevesinde, beklenen getiri ve çarpıklığın maksimize edilmesi, varyans kurtosis simultaneously, will be addressed by construction of a poly- ve basıklığın minimize edilmesi gibi birbiri ile çelişen ve aynı anda nomial goal programming (PGP) model. The PGP model will be tested karşılanması gereken portföy amaçları, oluşturulacak polinomal hedef on Istanbul Stock Exchange (ISE) 30 stocks. Previous empirical results programlama yöntemi ile ele alınacaktır. Oluşturulacak PGP modeli, indicate that for all investor preferences and stock indices, the PGP İstanbul Menkul Kıymetler Borsası (İMKB) 30 hisse senetleri üzerin- approach is highly effective in order to solve the multi conflicting de test edilecektir. Daha önce yapılmış olan çeşitli ampirik çalışma portfolio goals in the mean – variance - skewness – kurtosis frame- sonuçları, tüm yatırımcı tercihleri ve hisse senedi endeksleri için, work. In this study, portfolios will be formed in accordance with the ortalama-varyans-çarpıklık-basıklık çerçevesinde çoklu çelişen port- investor preferences over incorporation of higher moments. The ef- föy amaçlarının çözümünde PGP yaklaşımının etkili bir yol olduğunu fects of preferences both on the combination of stocks in the port- işaret etmektedir. Bu çalışmada, yatırımcıların yüksek dereceden folios and descriptive statistics of portfolio returns will be analyzed. momentler ile ilgili tercihlerine göre portföyler oluşturulacaktır. Bu Another aim of this study is to investigate the impacts of the incor- tercihlerin hem portföy içindeki hisse senedi dağılımına, hem de poration of skewness and kurtosis of asset returns into the portfolio portföylerin getirilerinin tanımlayıcı istatistiklerine etkileri incele- optimization on portfolios’ returns descriptive statistics. necektir. Bu çalışmanın bir diğer amacı da, portföy optimizasyonunda hisse senetlerinin getirilerinin çarpıklık ve basıklığının göz önünde bulundurulmasının portföy getirilerinin tanımlayıcı istatistikleri Keywords: Portfolio optimization, mean-variance-skewness- üzerinde yarattığı etkilerin de incelenmesidir. kurtosis approach, Istanbul stock exchange (ISE) 30. Anahtar Kelimeler: Portföy optimizasyonu, ortalama–varyans–çarpıklık–basıklık yaklaşımı, İstanbul menkul kıymetler borsası (İMKB) 30

1. INTRODUCTION ments. The aim in this allocation process is to achieve Investors want to maximize their returns by allo- a desired tradeoff between their risk and return pre- cating their capitals among a set of potential invest- ferences. In other words, investors aim to optimize their portfolios in accordance with their preferences.

1 Assist. Prof., Ege University, Faculty of Economics and Administrative Sciences, Department of Business Administration, [email protected] 2 Res. Ass., Ege University, Faculty of Economics and Administrative Sciences, Department of Business Administration, [email protected] 3 Prof., Ege University, Faculty of Economics and Administrative Sciences, Department of Business Administration, haluk. [email protected] 9 Burcu ARACIOĞLU, Fatma DEMİRCAN, Haluk SOYUER

For long years, portfolio selection and optimiza- roach of PGP and existing literature about this app- tion problem is an attractive topic for investors. Fol- roach are discussed in section 3. Section 4 represents lowing the seminal work of Markowitz, returns of our empirical analysis of the PGP approach. And sec- financial assets are typically described their mean, tion 5 concludes the paper. while risk is described by variance (Maringer and 2. PORTFOLIO OPTIMIZATION IN THE MEAN– Parpas, 2009: 219). Subsequently, an abundant litera- VARIANCE–SKEWNESS-KURTOSIS FRAMEWORK ture emerged, questioning the adequacy of the mean-variance criterion proposed by Markowitz (1952) In financial terms, a portfolio is an appropriate for allocating wealth (Xu et. al., 2007: 2488). This lite- mix or collection of investments held by an institu- rature finds evidence against the model and shows tion or private individuals. The portfolio optimization that asset returns are characterized by significant problem is a well-known difficult problem occurring skewness and kurtosis. As a result of these findings, in the finance world. The problem consists of choo- more recently researchers tended to concern for hig- sing an optimal set of assets in order to minimize the her moments in the portfolio optimization problem risk and maximize the profit of the investment. The and lots of techniques have been developed to solve investor’s objective is to get the maximum possible this problem (Konno et al., 1993; Chunhachinda et al., return on an investment with the minimum possible 1997; Liu et al., 2003; Harvey et al., 2004; Jondeau and risk. This objective is achieved through asset diversifi- Rockinger, 2006; Lai et al., 2006; Jana et al, 2007; Ma- cation (Singh et al., 2010: 75). ringer and Parpas, 2009; Briec et al., 2007; Taylan and The mean-variance framework for portfolio selec- Tatlıdil, 2010). tion, developed by Markowitz (1952), continues to Lai(1991), Chunhachinda, et al. (1997), Prakash et be the most popular method for portfolio construc- al. (2003) and Sun and Yan (2003) applied the poly- tion (Kale, 2009: 439). Since Markowitz’s pioneering nomial goal programming approach to the portfolio work was published, the mean-variance model has selection with skewness. Later, kurtosis is incorpora- revolutionized the way people think about portfolio ted into the portfolio selection by Jondeau and Roc- of assets, and numerous studies on portfolio selecti- kinger (2004). on have been made based on only the first two moments of return distributions (Lai et al, 2006: 1) Most In this study, in the mean- variance- skewness- serious investors use mean-variance optimization to kurtosis framework, portfolio optimization problem form portfolios, in part, because it requires know- will be addressed. In the presence of higher order ledge only of a portfolio’s expected return and vari- moments, portfolio selection contains multiple conf- ance. Yet this convenience comes at some expense, licting and competing portfolio objectives such as because the legitimacy of mean-variance optimizati- maximizing expected return and skewness and mi- on depends on questionable assumptions. Either in- nimizing risk and kurtosis simultaneously. In this vestors have quadratic utility or portfolio returns are framework, portfolio allocation depends on investor normally distributed. Neither of these assumptions preferences for these moments. This multi objective is literally true (Cremers et al., 2003:2; Harvey et al., problem will be solved by using a polynomial goal 2004: 4; Lai et al., 2006:1). Strong empirical evidence programming (PGP) model. suggests that returns are driven by asymmetric and/ The existing literature about portfolio optimiza- or fat-tailed distributions (Jondeau and Rockinger, tion indicates that the PGP approach is highly effec- 2006: 29). The mean – variance model by Markowitz tive in order to solve the multi conflicting portfolio is important in portfolio optimization but this model goals in the mean – variance - skewness – kurtosis should be expanded. framework for all investor preferences and stock in- The classical Markowitz (1952, 1959) model for dices. In this study, the PGP model will be tested on portfolio selection has been studied in the past by a small sample of stocks in ISE and the existence of simplifying it or reformulating it into different mo- an optimal solution will be investigated under diffe- dels. Several practitioners pointed out to the com- rent investor preferences. The effects of preferences putational difficulty of Markowitz model which is both on the combination of stocks in the portfolios associated with solving a large-scale quadratic prog- and descriptive statistics of portfolios’ returns will be ramming (Simimou and Thulasiram, 2010: 481). Seve- analyzed. ral alternative approaches have been developed in In this context, the concepts, portfolio and port- the financial literature to incorporate the individual folio optimization are reviewed in section 2. The app- preferences for higher-order moments into optimal

10 Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: An Application in İstanbul Stock Exchange asset allocation problems (Jurczenko et al., 2005: 3. POLYNOMIAL GOAL PROGRAMMING 2; Taylan and Tatlıdil, 2010: 349). Samuelson (1970) Goal programming (GP) is an important category also showed that the higher moment is relevant to in linear programming. In this idea, instead of trying investors’ decision-making in portfolio selection and, to optimize each objective function, the decision furthermore, almost all investors would prefer a port- maker is asked to specify a goal or target value that folio with a larger third moment if the first and se- realistically is the most desirable value for that functi- cond moments are the same (Liu et al, 2003: 255). In on (Hashemi et al. ,2006: 507). The overall purpose of this framework, portfolio selection with skewness is goal programming is to minimize the deviations bet- determined. But the fourth moment, kurtosis, which ween the achievement of goals and their aspiration is neglected by most researchers, is also important levels (Chang, 2002: 62 – 63). for portfolio selection if return distribution is non- In this study, we deal with PGP. The PGP is a mul- normal, or utility functions are higher than quadratic, ti-objective goal programming technique that allows or higher moments are relevant to the investor’s de- us to incorporate higher order moments in portfolio cision (Lai et al, 2006: 1). In the light of these findings, selection. The PGP model accommodates both intra- it is seen that in recent years researchers use higher level and inter-level preference trade-offs via the spe- order of moments in the portfolio selection. cification of the objective function as a polynomial In this study, following Lai (2006) the PGP will be expression (Deckro and Hebert, 2002: 149). used in order to find solutions to portfolio optimiza- There are numerous studies in the literature in- tion problem that contains multiple conflicting and dicating that portfolio returns are not normally dist- competing portfolio objectives that are maximizing ributed. As a result of the evidence against the nor- expected return and skewness and minimizing risk mality assumption of the Markowitz’s model, higher and kurtosis simultaneously. order moments are started to be considered in the As told by Lai et al (2006)’, PGP was first introdu- portfolio selection problem. ced by Tayi and Leonard to facilitate bank balance Starting from this point, Lai (1991) proposed a sheet management with competing and conflicting multiobjective portfolio selection model to incorpo- objectives (Lai et al. 2006: 2). Along with, Lai (1991), rate the skewness of return distributions. The optimal Chunhachinda et al. (1997), and Prakash et al. (2003) solution of this model is to select a portfolio compo- applied the PGP approach to the portfolio selection nent such that its multiple objectives are optimized. with skewness. All these studies provided evidence That is to maximize the expected rate of return and that incorporating skewness into the portfolio deci- skewness, while minimizing the variance (Chen and sion causes major changes in the optimal portfolio Shia, 2007: 133). Like Lai, Harvey et al (2004), Jurc- (Jondeau and Rockinger, 2006: 30; Lai et al. 2006: 2). zenko et al (2006), Lai et al.(2006), Chen, and Shia In the study of Taylan and Tatlıdil (2010), it is seen that (2007) and Taylan and Tatlıdil (2010) applied the PGP the portfolio optimization is achieved by shortage method to portfolio optimization with fourth mo- function and higher order moments. By construction ment. of a PGP, they tried to analyze multiple competing As Chunhachinda et al. (1997) mentioned, the portfolio allocation objectives such as maximizing important features of polynomial goal programming expected portfolio return and skewness, minimizing include (Chen and Shia, 2007: 131): risk and kurtosis simultaneously and investor’s prefe- - The existence of an optimal solution, rences over incorporation of higher moments (Taylan and Tatlıdil, 2010: 348). - The flexibility in incorporating investor preferences, and To sum up, more recently in local and foreign literature, higher order moments -especially mean- va- - The relative simplicity of the computational re- riance- skewness- kurtosis- based portfolio optimi- quirements (Chen and Shia, 2007: 131). zation has attracted a great deal of attention. In this The advantage of the PGP framework is that it is study, to achieve portfolio optimization in the frame- general enough to accommodate investor desires for work of four moments, the PGP is used. In the follo- higher moments: skewness and kurtosis through pre- wing sections of the study a brief review of the PGP ference parameters. It solves the trade-off among the will be given and it will be followed by the research competing objectives for the return distribution pro- section of the study. perties (Proelss and Schweizer 2009: 1).

11 Burcu ARACIOĞLU, Fatma DEMİRCAN, Haluk SOYUER  n Mean  R(x)  X T R  x R n   T i i Mean R(x) Xi1 R  xi Ri In the application of the PGP model, we compute the first four moments ofi1 asset returns (see Lai et al., n n n 2006): T 2 n2 n n n Variance  V (x)  X VX  T xi  i  2 2 xi x j ij  Variance  V (x)  X VX  xi  x x  T i1  ii1 j1  i ij ij j    i  j Mean  R(x)  X R   xi Ri i 1 i 1 j 1 i1

n n n  n n n n T 3  3 3n n 2 n n2 T 2 2 3 3 2 2 Skewness S(x)  E(X (R  RT ))  3xi si  3 ( xi x j siij  xi x j sijj ) (i  j) Variance  V (x)  X VX  xi  i  xi xSkewnessj ij S(x)  E(X (R  R))  xi si 3 ( xi xj siij  xi x j sijj ) (i  j)   i1  i1 j1  j1  i1 i1 j1 i  j i1 i1 j1 j1

 n n n T 4  4 4n n 3 n T 4 4 4 3  n Kurtosis Kn (x)n E(X (R  Rn))  xi si  4 ( xi x j kiiij  3 3 Kurtosis K(2x)  E(X (R  R2))  xi si 4 ( xi x j kiiij  T 3       Skewness S(x)  E(X (R  R))  x s  3 ( x x s  x x i 1s ) i(1i  i j1) j 1 i1 j1  i i   i j iij  i j ijj (i  j)  i1 n n i1 j1n n n n j1 (i j) x x 3k )  63 x 2 x 2k 2 2  i j ijjjxi x j kijjj ) 6i j xiijji x j kiijj j1 j1 i1 j1 i1 j1  n n n T 4 4 4 3 Kurtosis K(x)  E(X (R  R))  xi si  4 ( xi x j kiiij   n    T i1 i1 j1   Mean  R(x)  X R  xi Ri T T  n n n  R(x)  RX(x)R X R (i j) 3 i21 2 xi x j kijjj )  6 xi x j kiijj T T   n n n V(x) VX(xVX)  X VX j1 i1 jT1 2 2 Variance  V (x)  X VX  xi  i  xi x j ij     i1 i1 j1 i  j T T 3 3 − Sand(x) kurtosis ES(xX)  (fromRE(XR the))(R optimal R)) scores of, R*, V*, S* and (5) (5) P1 P1 where R is the distribution of returns and R is K*, respectively. To obtain the optimal scores, the gi- mean of the return, X =(x ,x ,…,x ) is the transpose ven model,K(x) TP1, E is(X dividedT (R4  R into))4 four subproblems and T T 1 2  n n n K(nx) E(X (Rn R)) of the weightR(x )vector X usedR T to combine 3  the portfolio,3 3  solved2 them individually2 (see Lai et. al.(2006)). Skewness S(x) E(X (R R))  xi si 3(T xi x jTsiij  xi x j sijj ) (i j) xi is the percentage of wealth invested in ithe1 ith risky i1 Xj1I  1X I  1j1 T After calculating the optimal scores of each mo- asset. , V, (andx)  Xis theVX variance - covariance, skew- V S K Xment, 0 weX use0 the PGP model that was proposed by ness-coskewness, and kurtosis – cokurtosis matrices   n n Lain et. al. (2006) to find portfolio allocations for diffe- of R, respectively.S(x)  E(X T (TR  R))3 4 4 4 3 Kurtosis K(x)  E(X (R  R))   xi si  4(rentxi investors’x j kiiij  preferences. The PGP model (P2) is To combine the multiple objectives suchi1 that ma-i1 j(5)1 P1  (i  j) ximizationn of the expectedn n T return4 and skewness of d 1 d 2 d 3 d4  4 3  2 2 d1 1 1 d 2 2 2 d3 3 3 d4  4 K(x) E(X (R R)) Z | Z| | |* | | |  *| | | |  *| | | | * | return xi whilex j kijjj minimization)  6 xi ofx j thekiijj variance and kurto- * R * V * S * K j1 i1 j1 R V S K sis of return,T we use the same multiobjective prog-  X I  1  T * T *  ramming technique with Lai et.al. (2006). The formu- X R Xd RR d1 R X  0 1 lation of the model is given below. T * T X VX  *d  V  X VX  d  V 2 T 2 R(x)  X R  T 3 * (6) P2 T E(X (R3  R))  *d  S E(X (R  R))  d  S 3 (6) P2 V(x)  X TVX 3  P2 T 4 * d1 1 d 2 2 d3 3 d4  4 T E(X (R4  R))  d* 4  K Z | | T | | 3  | |  | E| (X (R  R))  d  K S(x)  E* (X (R  R* )) * * 4 R V S K (5) T P1 P1 T  X I 1   X I 1 K(Tx)  E(X T (R*  R))4 X  0 X R d1  R X  0 T    di 0 i 1,...,4 TX I 1 * di  0 i 1,...,4 (6) X VX  d2  V X  0  The PGP problem solution involves a two- step T 3 * (5) (6) P2 E(X (R  R))  d3  S procedure. First the optimal scores of R*, V*, S* and To combine these objectives into a single objec- K*. Then the optimal scores are substituted into P2,  tive function,T we use a PGP4  approach. * Let d1, d2, d3 and the minimum value of Z can be found for a given E(X (dR R)) d d4 Kd d4 and d4 be the goal1  1variables 2 which2  account3 3  for the 4 set of investor preferences (Lai et. al.,2006:3). Z | * | | * | | * | | * | deviationsX T Iof expectedR1 Vreturn, variance,S skewnessK  T * X X 0R d1  R 12 T  * dXi VX0 i d21,..., V4

 T 3 * (6) P2 E(X (R  R))  d3  S

 T 4 * E(X (R  R))  d4  K X T I  1 X  0

di  0 i 1,...,4 Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: An Application in İstanbul Stock Exchange

4. EXPERIMENTAL ANALYSIS 4.2. Experiment Results 4.1. Data Set In this study our main objective is to show the ef- In this study, Istanbul Stock Exchange (ISE) 30 fects of investors’ preferences both on the combinati- stocks are examined. Our data set contains daily pri- on of stocks in the portfolios and descriptive statistics ces of permanently traded stocks in ISE- 30 index du- of portfolios’ returns in the four moment framework. ring the last five years. Among the permanent stocks In this part, we present all process followed in perfor- in ISE-30 Index, we choose the ones with positive ming the PGP approach. The distribution properties average daily returns for the period January 4, 2010 of the analysed stocks are given in the table below. to December 31,2010. As a result, we obtain 8 stocks In addition to individual distribution properties for implementation. We use logarithmic returns in of asset returns, covariance, coskewness and cokur- our analysis. tosis of asset returns are calculated. Tables 2- 4 show these statistics.

13 Burcu ARACIOĞLU, Fatma DEMİRCAN, Haluk SOYUER

14 Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: An Application in İstanbul Stock Exchange

By dividing the P1 model into four subproblems With the optimal solution of individual objective, we and solved them individually, the optimal scores of solve the P2 with the PGP approach. In the Tables 6-7, four moments are obtained. the first four moment and asset allocations for optimal portfolio with different investors’ preferences are given.

The portfolios formed in accordance with the In the first four portfolio, the first, the second, investor’s preferences over incorporation of higher the third and the fourth moment are optimized. The moments are given above. In order to analyze the portfolio 5, (1,1,0,0) is the Markowitz mean-variance effects of preferences both on the combination of portfolio. Investors higher preference for variance in stocks in the portfolios and descriptive statistics of portfolio 6, resulting in an increase in each moment portfolios’ returns, different levels of preferences investigated. When we consider changing the prefe- are investigated. Investors’ preferences of (1,0,0,0), rence parameters from (1,3,0,0) to (1,1,1,1), it is seen (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,1,0,0), (1,3,0,0), (1,1,1,1), that each moment investigated decreases. The decre- (1,1,3,0), (1,3,0,1), (3,1,1,0), (3,1,3,1) and (1,3,1,3) are ase in the preference for variance by holding expec- included in our experiment. ted return is constant and considering the third and

15 Burcu ARACIOĞLU, Fatma DEMİRCAN, Haluk SOYUER fourth moment in addition to the first two moments have to be answered here is how the portfolio will in portfolio formation, leads to lower expected re- be formed and what the best combination of invest- turn, variance, skewness and kurtosis. Portfolio 8-12 ment instruments in the portfolio will be. represent different combinations of investors’ prefe- In this study, we try to answer these questions in rences for expected returns, variance, skewness and the mean- variance- skewness- kurtosis framework kurtosis. by using a PGP model. In this model, multiple conflic- 5. CONCLUSION ting and competing portfolio objectives such as ma- Investors aim to allocate their capitals among a ximizing expected return and skewness and minimi- set of potential investments to achieve a desired tra- zing risk and kurtosis simultaneously are considered deoff between their risk and return preferences. One in accordance with different investors’ preferences. of the most important preferred investment instru- Our results reveal that the investors’ preferences af- ment is the securities. The important questions that fect both asset allocations of portfolio and descriptive statistics of descriptive statistics of asset returns.

16 Mean–Variance–Skewness–Kurtosis Approach to Portfolio Optimization: An Application in İstanbul Stock Exchange

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