A HEURISTIC APPROACH TO THE INDEX TRACKING PROBLEM: A CASE STUDY OF THE TEHRAN EXCHANGE PRICE INDEX

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1 Asan Academy of Management Journal, Vol. 18, No. 1, 19 34, 2013 A HEURISTIC APPROACH TO THE INDEX TRACKING PROBLEM: A CASE STUDY OF THE TEHRAN EXCHANGE PRICE INDEX Mohsen Varse 1*, Naser Shams 2, Behnam Fahmna 3 and Abbas Yazdanpanah 4 1,2 Department of Industral Engneerng, Amrkabr Unversty of Technology, Tehran, Iran 3 School of Management, Unversty of South Australa, Adelade, Australa 4 Department of Computer & Informaton Technology Engneerng, Amrkabr Unversty of Technology, Tehran, Iran *Correspondng author: mohsen.varse@unsa.edu.au ABSTRACT Index trackng, the most popular form of passve fund management, s a portfolo selecton problem n whch the return of one of the stock market ndexes s reproduced by creatng a trackng portfolo consstng of a subset of the stocks ncluded n the ndex. Index trackng has been known as an NP-Hard problem, and sophstcated approaches have been proposed n the lterature to solve ths problem. Ths paper presents an easyto-mplement heurstc soluton to ths complex problem. The proposed approach was mplemented to develop a trackng portfolo of 438 stocks lsted n the Tehran Exchange Prce Index. The numercal results ndcate that the approach s able to dentfy qualty solutons wthn reasonable model runtme. Keywords: ndex trackng, portfolo selecton, fund management, heurstc approach INTRODUCTION Fund management concerns the nvestgaton of stocks and securtes of those companes whose stocks are represented n stock markets worldwde (Beasley, Meade, & Chang, 2003). There are two common strateges n fund management. The frst s the actve strategy, whch s used when the goal s to make nvestments n stocks that are expected to outperform other stocks n the market or outperform the average outcomes of the stock market (Alexander & Dmtru, 2005). In ths strategy, performance depends on the fund managers' experence and judgment. The other strategy s called passve strategy, whch s used by less experenced fund managers who are reluctant to take great rsks. In passve strategy, the prmary focus s placed on the long-term performance of the market nstead of the short-term, temporary extra return achevement (Alexander & Dmtru, 2005). Asan Academy of Management and Penerbt Unverst Sans Malaysa, 2013

2 Mohsen Varse et al. Emprcal analyss n recent years has revealed that actvely managed funds cannot exceed ther comparatve ndex, and ths has resulted n more attenton beng pad to passvely managed funds n whch managers can accrue profts wthout takng excessve rsks. Passve management of funds, partcularly ndex trackng, has ganed popularty n the U.S., Europe, Australa and Eastern Asa (European Asset Management Assocaton, 2001; Marnger, 2008). The most common model of passve fund management s called the ndex fund or the ndex trackng portfolo, whch takes two forms: full replcaton and partal replcaton (Marnger, 2008). In full replcaton, the nvestor purchases all of the stocks avalable n the ndex, whereas n partal replcaton, only a subset of stocks are purchased (Shapcott, 1992). Ths paper uses partal replcaton for solvng the ndex-trackng problem, n whch choosng a subset of stocks leads to a combnatoral optmsaton problem. We have developed an heurstc approach to solvng ths complex problem. Here, the problems are choosng the best subset of stocks to be ncluded n the ndex trackng portfolo and calculatng the optmal weght for each stock n the trackng portfolo. We tackle the two problems n a dsaggregated approach. Although some complcated solutons have been proposed n the lterature to solve the ndex trackng problem, ths paper proposes a smple heurstc approach for selectng the optmal subset of stocks, and the concept of pseudo nverse n advanced algebra s used to determne the optmal weght of each stock n the trackng portfolo. LITERATURE REVIEW The ndex trackng problem (ITP) s a portfolo optmsaton problem, a popular research area n the feld of management scence and operatons research (MS/OR). Despte the mportance of the ITP, only a few publshed artcles address practcal solutons to ths problem. In fact, there were only 15 publshed artcles that focused on solvng the ITP before 2003 (Beasley, Meade, & Chang, 2003). Several approaches and data collectons have been used to solve ths problem, the comparson and analyss of whch are beyond the scope of ths paper. Here, we only refer to the most recent and relevant artcles n ths area. By establshng the standard mean-varance model for portfolo optmsaton, Markowtz created an ndex trackng portfolo n whch the optmal weght of each stock s determned by solvng Markowtz's model wth ts well-recognsed objectve functon and constrants (Dergs & Nckel, 2003; Markowtz, 1952). Hodges (1976) used Markowtz's model and for the frst tme presented a comparson between the ndex tradeoff curve and the ndex trackng portfolo tradeoff curve. Later on, Roll (1992) used Markowtz's model; however, the 20

3 A Heurstc Approach to the Index Trackng Problem theoretcal defcency prevented ths model from beng wdely used durng the past two decades (Beasley et al., 2003). After Markowtz, several approaches have been used for modellng the ITP. Rudd (1980) created a sngle factor model for the ITP of the S&P 500 wth an heurstc soluton for handlng the problem constrants. He ncluded the transacton costs of purchasng and sellng stocks n the objectve functon. Corell and Marcellno (2006) presented mult-factor models for solvng the ITP. Connor and Leland (1995) consdered the cash management problem n ther model for buldng a trackng portfolo and ncluded the transacton costs as a fxed percentage of the money nvested. Buckley and Korn (1998) modfed Connor's model and consdered transacton costs as the man model constrant. Rudolf, Wolter and Zmmermann (1999) developed four dfferent forms of lnear functons for ndex trackng models. They changed the trackng error mnmsaton model to a lnear model but could only solve the problem for a small set of data. Frno and Gallagher (2001) studed the mpact of seasonal factors on trackng errors and found that the trackng error s hgher n January because of the market fluctuatons at the begnnng of the year. Because of the complexty of the ITP, tradtonal methods cannot solve the problem optmally; therefore, heurstc and meta-heurstc approaches have recently been adopted to dentfy the optmal solutons to the problem (Beasley et al., 2003; Dergs & Nckel, 2003; Krnk, Mttnk, & Paterln, 2009; Marnger & Oyewum, 2007; Mao, 2007; Ruz-Torrubano & Suárez, 2009). Beasley et al. (2003) used an evolutonary heurstc approach for dentfyng the best stocks to be placed n the portfolo. Dergs and Nckel (2003) adopted a smulated annealng meta-heurstc to desgn a decson support system to be used by fund managers. Okay and Akman (2003) used a constrant aggregaton method for the frst tme to solve the ITP; ths method was ntally developed by Beasley et al. (2003). They demonstrated that ther approach yelds smlar computatonal results wthn a shorter model runtme. Oh, Km and Mn (2005) used a Genetc Algorthm (GA) approach to follow the South Korean stock market ndex. Dose and Cncott (2005) developed a clusterng model to buld a portfolo to track the S&P 500 ndex. In ths model, the stocks are grouped based on a dstance measure between two stocks, and only one sngle stock s chosen as a representatve of that group. By solvng a quadratc problem, the optmal weght for each s stock s calculated. Krnk et al. (2009) suggested a dfferental, evoluton-based meta-heurstc and compared the results wth those of other meta-heurstc algorthms such as GA, smulated annealng and partcle swarm optmsaton. It was shown that ths approach s more effcent n solvng more complcated problems. Marnger and Oyewum (2007) used an dentcal approach to re-create the Dow Jones ndex 21

4 Mohsen Varse et al. performance and demonstrated several advantages n terms of the number of parameters nvolved and also ts ease of mplementaton. Prmbs and Sung (2008) used stochastc recedng horzon control for solvng an ITP. They developed a predctve control model and formulated the ITP as a stochastc lnear quadratc control problem. Carakgoz and Beasley (2009) used a lnear regresson to solve the problem n whch both the ITP and mproved ndexaton problem are transformed nto a mxed ntegrated lnear program that can be solved by standard lnear solvers such as IBM ILOG CPLEX Optmzaton Studo. Barro and Canestrell (2009) studed a dynamc ITP that consders the mnmum number of stocks n an ndex portfolo n terms of transacton costs. They proposed a mult-stage trackng error model that was solved usng a stochastc plannng technque based on a progressve hedgng algorthm. Ruz-Torrubano and Suárez (2009) ntroduced a hybrd strategy based on the combnaton of an evolutonary algorthm and quadratc programmng. They used the formulaton of Beasley et al. (2003) and demonstrated that the computatonal tmes can be reduced usng ther hybrd approach. In ths paper, we propose an heurstc approach for solvng a complex ITP. Our heurstc approach helps the nvestor determne both the optmal subset of stocks for the trackng portfolo and the optmal weght of each stock. To nvestgate the effectveness of the soluton proposed n ths paper, the approach s mplemented to develop the desred trackng portfolo from 438 stocks presented n the TEPI. ITP FORMULATION We assume havng N stocks n a gven ndex durng tme perod T. The objectve s to develop an ndex trackng portfolo wth K stocks (n whch K<<N) to track the ndex from the tme of T to the future tme of L. The answers to the followng queres are sought n a typcal ITP: (1) whch stocks must be ncluded n the trackng portfolo.e., the optmal K stocks from the set of N stocks; and (2) what proporton of the nvested fund should be allocated to each of the K stocks. A basc approach for modellng an ITP s an hstorcal approach that assumes that the past drects the future. Lke many prevous works, such as those of Beasley et al. (2003) and Marnger and Oyewum (2007), we nvestgated the valdty of ths assumpton by dvdng the data set nto n-sample data and out-of-sample data. Therefore, the problem formulaton presented n ths secton s smlar to the formulatons of Beasley et al. (2003), Marnger and Oyewum (2007), Ruz- Torrubano and Suárez (2009) and Okay and Akman (2003). 22

5 A Heurstc Approach to the Index Trackng Problem Mathematcal Modellng We use the followng parameters for the ITP formulaton. N : Total number of stocks n the concerned stock market ndex K : Desred number of stocks n the trackng portfolo e : Mnmum proporton of nvestment n each stock u x : Maxmum proporton of nvestment n each stock T : Tme perod (0, 1, 2,, T) n whch the tme unt can be dsaggregated to days or weeks S t, : Value of one unt of stock ( = 1, 2,, N) at tme t (t = 0,, T ) r t, I : Sngle perod contnuous tme return of stock at tme t (derved from Equaton 1). ( / ) rt, = Loge S t, S t 1, (1) B o : Amount of nvestment to create the ndex trackng portfolo (.e., the avalable budget) n : Number of stocks (I = 1,..., N) n the ndex trackng portfolo (decson varable) b : Stock selecton bnary varable; b = 1 f stock s selected to be ncluded n the trackng portfolo, b = 0 otherwse (decson varable) P : Value of ndex trackng portfolo n tme t (derved from Equaton 2): t N n St, = = Pt 1 (2) r, : Sngle perod contnuous tme return of the trackng portfolo at tme t t P (derved from Equaton 3): rt, p= Loge ( Pt / Pt 1) (3) I t : Value of the ndex at tme t r, : Sngle perod contnuous tme return of the concerned ndex at tme t t I (derved from Equaton 4): r = Log I / I ( ) ti, e t t 1 (4) 23

6 Mohsen Varse et al. Usng the above parameters, Equaton 5 formulates the objectve functon for the proposed ITP. 2 ( / ) 1/2 t = 1 T ( ) = ( tp, ti, ) Mn TE Mn r r T Subject to: (5) N b 1 K = 1 N (6) = l x b ( ns / B ) x b = 1.N (7) u t, o N ns T, Bo = 1 N (8) = 1 [ 0,1] b = 1.N (9) n = 0, 1, 2,3,, N = 1.N (10) Equaton 5 represents the objectve functon of the proposed ITP. The mean square error rato s used to accommodate the theoretcal weakness n usng varance measure. In addton, the objectve functon does not ntend to acheve excess return of the concerned ndex, but to track t as closely as possble. Equaton 6 s the man model constrant that ensures there are K stocks avalable n the trackng portfolo. Equaton 7 ensures that f stock s not avalable n the ndex portfolo (b = 0), then n wll be equal to zero; and f stock s avalable n the ndex portfolo (b = 1), then the mnmum and maxmum nvestment lmts on each stock must be consdered n the soluton approach. The complete use of the avalable budget s mposed by Equaton 8. Beasley et al. (2003), Ruz- Torrubano and Suárez (2009) and Krnk et al. (2009) argue that t s not necessary to consder ths constrant n an ITP because a small nvestment n some stocks or budget reducton can smply rectfy small devatons for the complete satsfacton of ths constrant. Equatons 9 and 10 enforce restrctons on the nteger varables. Modfed Mathematcal Modellng usng Contnuous Decson Varables There are two nteger varables n the proposed formulaton: (1) b s a bnary varable ndcatng whether stock s selected n the trackng portfolo, and (2) n s an nteger varable standng for the number of stocks n the portfolo. Obvously, n s a dscrete number because the number of purchased stocks cannot be a decmal or a ratonal number. To tackle the complexty of solvng a 24

7 A Heurstc Approach to the Index Trackng Problem dscrete model, we convert n to a contnuous form as s also proposed n Dergs and Nckel (2003), Frno, Gallagher and Oetomo (2005), and Ruz-Torrubano and Suárez (2009). We defne x as a contnuous varable to show the proporton of nvestment n stock derved from Equaton 11. x = n S t, t, B 0 = 1.N (11) Usng ths approach, n can fluctuate dynamcally over tme, although the proporton of the nvestment n stock remans constant. Thus, r t,p can be reformulated as follows: r N = r x tp, t, = 1 (12) Wth ths modfcaton, the formulaton of the objectve functon s changed to the followng: 1 N T Mn ( TE ) = Mn r x r t = 1 T = 1 t, ti, 2 1/2 (13) If stock exsts n the trackng portfolo, then b = 1 and constrants are effectve: L U x x x (14) N = 1 N x = 1 b 1 K If stock does not exst n the trackng portfolo, then b = 0 and thus x = 0. x R b 0,1 25 (15) (16) (17) [ ] = 1.N (18) In ths formulaton, Constrant 14 s replaced wth Constrant 7, ndcatng the floor and the celng proportons of the amount of money nvested n stock.

8 Mohsen Varse et al. Constrant 15 s replaced wth Constrant 8, ensurng the complete use of the avalable budget. Constrant 16 s smlar to Constrant 6, whch s the major constrant of the ITP, restrctng the number of stocks n the trackng portfolo. Short sellng s allowed n our model, and therefore Equaton 17 ndcates that x can also obtan negatve numbers. A HEURISTIC SOLUTION APPROACH Defnng contnuous varables n prevous secton can sgnfcantly smplfy the development of a soluton approach for the proposed ITP. However, a bnary varable b remans n the model formulaton. The problem of selectng the stocks to be ncluded n the trackng portfolo can sgnfcantly ncrease the problem dmenson, whch n many cases results n an extremely large problem search space (Rudolf et al., 1999). Ths s why heurstc and meta-heurstc methods have been adopted n recent years. In general, two questons must be answered to buld a trackng portfolo: (1) whch K stocks of N stocks should be selected from the concerned ndex for ncluson n the trackng portfolo, and (2) what s the optmal weght of each selected stock n the trackng portfolo. Assumng that the answer to the frst queston s known to be S comprsed of K stocks and also assumng that all model constrants are relaxed, the problem can be solved usng Equaton 19. ( ) 2 1/2 1 T t = 1 t, ti, (19) T S Mn TE = Mn r x r For better llustraton of ths formulaton, we defne the followng matrxes to convert Equaton 19 nto a matrx form. A T N s the return matrx of N stocks n the tme horzon T, and X N 1 s the decson varable matrx of the optmal weghts of stocks n the trackng portfolo. I T 1 s also defned as a matrx for the return of ndex n tme horzon T. Therefore, the matrx form of Equaton 19 s presented n Equaton ( ) ( ) = ( ) Mn TE Mn A X I (20) 26

9 A Heurstc Approach to the Index Trackng Problem For Equaton 20 to be solved, Equaton 21 must be true because TE s mnmsed (.e., equal to zero) only f the return of the ndex-trackng portfolo (AX) equals the return of the ndex (I). AX = I (21) Although matrx A s a non-square matrx, Equaton 21 can be calculated usng the pseudo nverse technque from the context of advanced lnear algebra (Hefferon, 2008). Hence, matrx X can be calculated from Equaton 22. = T ( ) 1 T X A A A I (22) In summary, by relaxng all the model constrants, the optmal objectve functon can be calculated from Equaton 22. However, the feasblty of the soluton s not guaranteed f model constrants are taken nto consderaton. In addton, the second queston of the ITP has nevertheless remaned unanswered. In any ITP, a tme seres of data (.e., ndex tme seres) s tracked by the use of another tme seres of data (portfolo tme seres). Smlarty between the concept of ''the correlaton between two tme seres of data'' and the ITP leads us to development of an heurstc approach n whch the selected stock to be ncluded n the portfolo s the one wth the strongest postve correlaton wth the ndex. In the proposed approach, the correlaton between the tme seres of the value of N stocks (ncluded n the ndex) and the tme seres of the value of the ndex s calculated. The frst K + L stocks wth the hghest postve correlaton are selected to form the reduced search space, and the remanng N-K-L stocks are gnored. L s an nteger between 0 and 10 amng to make the search space a lttle wder for achevng the optmal trackng portfolo (t wll be demonstrated n the next secton why the value of L should be n ths range). Usng ths approach wll help consderably n reducng the sze of the search space, makng the problem easer to solve optmally. Hence, the proposed approach presented n ths secton can be summarsed n the followng four steps: 1. Select K + L stocks out of N stocks that have the hghest postve correlaton wth the ndex. Ths approach reduces the problem search space n whch the number of canddates n the trackng portfolo s K out of K + L ( K + L K ) nstead of K out of N ( N K ). 2. From Equaton 22, calculate the optmal weght of the stocks n every selected portfolo. 27

10 Mohsen Varse et al. 3. From Equaton 20, calculate the TE value for each trackng portfolo. 4. Select the portfolo wth mnmum TE to represent the ndex. Although the complex ITP can easly be handled usng ths approach, there are nevertheless uncertantes n terms of the feasblty of the generated solutons because of the gnored model constrants. To deal wth ths, we defne a constrant volaton measure representng the devatons aganst the major model constrants. If the value of the constrant volaton measure s large and sgnfcant, then our heurstc approach must be adopted to take the model constrants nto consderaton. If the constrant volaton measure s not substantal, we gnore the consderaton of model constrants because the proposed approach has prevously resulted n a feasble soluton. The detaled dscusson of the mplementaton of ths model on the TEPI s presented n the next secton. MODEL IMPLIMENTATION To evaluate the performance of the proposed heurstc approach n ths paper, we mplemented our heurstc method to develop the trackng portfolo n a real-lfe case study. The most demandng stock market n Iran s the Tehran Stock Exchange (TSE), the prce ndex of whch s referred to as the TEPI (also known as TEPIX). The TSE s made up of 438 stocks provdng a representaton of the country's economy throughout the year. For our data set, we decded to use the daly stock prces of the TEPI n 2003 because n that year the country experenced the most stable economy of the past decade and n 2003 there was much less mssng data, enablng us to create a qualty data set. Smlar to the methods of Marnger and Oyewum (2007), the average of the exstng prces of the adjacent days was used for the mssng data. Havng 205 values for each stock n TEPI, we clustered the tme perod [0, 103] for the n-sample data set and the tme perod [103, 205] for the out-of-sample data set. The proposed model was run on a SONY VAIO laptop wth a 2.1 GHz dual-core processor and 2GB of RAM. In addton, we used x l = 0.01 and x u = 1 as the mnmum and maxmum proportons of nvestment n each stock, respectvely. Table 1 shows the acheved numercal results for the frst TEPI trackng portfolo for K = 5 and K = 10. As prevously mentoned, the man ssue n the presented heurstc approach s to dentfy whether the generated trackng portfolo s feasble. For ths, we measured the percentage of constrant volaton for both floor and celng constrants as well as the budget constrant (.e., the rato of the volated portfolos to all possble portfolos n the problem search space). 28

11 A Heurstc Approach to the Index Trackng Problem Columns 5 and 6 n Table 1 (constrant volaton) demonstrate that these values are qute small n scale, whch valdates the feasblty of the solutons found. To evaluate the effectveness of the generated search space obtaned from our heurstc approach, the mean and standard devatons of trackng error (TE) from all possble portfolos n the search space are calculated n columns 7 and 8, respectvely. The low values of ''standard devaton of TE'' and the proxmty of the values of ''mean of TE'' and ther equvalent n column 2 (.e., the mnmum TE) not only demonstrate the effectveness of ths approach n generatng a practcable search space but can also valdate the fundamental concept behnd the proposed approach n ths paper (.e., the correlaton-based selecton for the trackng portfolo refer to prevous secton for more nformaton). Table 1 Computatonal results for the TEPI for K = 5 and K = 10 Trackng error (TE) and Tme (Second) Constrant volaton Search space effectveness K Standard Mean Floor and Out-of Insample devaton of Budget Tme of TE Celng -sample TE e-05 e-04 00e-03 97e-04 4e e-05 e-04 54e e-04 4e The promsng results demonstrate that the approach could develop an effectve trackng portfolo n terms of both trackng error and model runtme. The TE values n Table 1 show only a small dfference between n-sample data and outof-sample data. Moreover, the results for K = 10 are slghtly better than those for K = 5, whch ndcates that TE decreases wth an ncrease n the number of stocks n the trackng portfolo. The acheved numercal results ndcate that the ndex-trackng problem wth an hstorcal look-back approach (refer to Beasley et al., 2003) can be effectvely addressed usng the concept of the matrx correlaton (whch s naturally qute smlar to the concept of ndex trackng). In ths manner, the complcated ndextrackng problem can be solved easer and faster (refer to column 4 n Table 1 for the runtme report). A feasble trackng portfolo consstng of fve stocks could be produced n about two seconds. Fgure 1 and Fgure 2 llustrate the performance of our trackng portfolo wth K = 5 for n-sample and out-of-sample data, respectvely. The acheved results for K = 10 are shown n Fgure 3 and Fgure 4. 29

12 Mohsen Varse et al Portfolo Return Index Return Retrun Tme lag Fgure 1. In-sample trackng performance for TEPIX wth K = Portfolo Return Index Return Retrun Tme lag Fgure 2. Out-of-sample trackng performance for TEPIX wth K = 5. In the former secton, we mentoned that the value of L should be wthn the range of 0 to 10 as an ndcaton of the sze of the search space. To evaluate ths assumpton, we run our heurstc model for both K = 5 and K = 10 when the value of L vares between 0 and 10. In Fgures 5 and 6, the vertcal axs demonstrates the TE values and the horzontal axs represents K + L, n whch L changes from 0 to 10. Both fgures show an ntal sharp decrease n the value of trackng error (TE) whle K + L ncreases to approxmately 11 (.e., L = 6) and 18 (.e., L = 8), respectvely. The TE value seems to reman almost unchanged after these ponts. Ths ndcates that the suggested movement range for L s well set to be between 0 and 10, resultng n an approprate search space for our heurstc. 30

13 A Heurstc Approach to the Index Trackng Problem 0.02 Portfolo Return Index Return Retrun Tme lag Fgure 3. In-sample trackng performance for TEPIX wth K = Portfolo Return Index Return Retrun Tme lag Fgure 4. Out-of-sample trackng performance for TEPIX wth K =

14 Mohsen Varse et al. Fgure 5. Changes n TE value wth fluctuatons n the value of L (K = 5) TE TE K+L K+L Fgure 6. Changes n TE value wth fluctuatons n the value of L (K = 10) CONCLUSIONS The ndex trackng problem s commonly faced by fund managers who ntend to develop trackng portfolos for followng or out-performng the average stock market performance. The avalable heurstc and meta-heurstc technques proposed for solvng ndex trackng problems are complex n nature and requre 32

15 A Heurstc Approach to the Index Trackng Problem long computatonal tmes. Ths paper presented an easy-to-mplement heurstc approach usng a correlaton-based method to select the approprate stocks for ncluson n the trackng portfolo as well as the concept of pseudo nverse to determne the optmal weght of the selected stocks. The proposed approach was mplemented to develop a trackng portfolo from 438 stocks lsted n the Tehran Exchange Prce Index. The numercal results ndcate that our heurstc approach yelds qualty outcomes wth small trackng error values n both n-sample and out-of-sample data sets wthn reasonable model runtme. Wth ths proven applcaton, the proposed method can easly be mplemented n other stock markets to assst fund managers n dealng wth the complexty of the ndex-trackng problem. REFERENCES Alexander, C., & Dmtru, A. (2005). Indexng, contegraton and equty market regmes. Internatonal Journal of Fnance & Economcs, 10(3), Barro, D., & Canestrell, E. (2009). Trackng error: a multstage portfolo model. Annals of Operatons Research, 165(1), Beasley, J. E., Meade, N., & Chang, T. J. (2003). An evolutonary heurstc for the ndex trackng problem. European Journal of Operatonal Research, 148(3), Buckley, I. R. C., & Korn, R. (1998). Optmal ndex trackng under transacton costs and mpulse control. Internatonal Journal of Theoretcal and Appled Fnance, 1(3), Canakgoz, N. A., & Beasley, J. E. (2009). Mxed-nteger programmng approaches for ndex trackng and enhanced ndexaton. European Journal of Operatonal Research, 196(1), Connor, G., & Leland, H. (1995). Cash management for ndex trackng. Fnancal Analysts Journal, 51(6), Corell, F., & Marcellno, M. (2006). Factor based ndex trackng. Journal of Bankng & Fnance, 30(8), Dergs, U., & Nckel, N.-H. (2003). Meta-heurstc based decson support for portfolo optmzaton wth a case study on trackng error mnmzaton n passve portfolo management. OR Spectrum, 25(3), Dose, C., & Cncott, S. (2005). Clusterng of fnancal tme seres wth applcaton to ndex and enhanced ndex trackng portfolo. Physca A: Statstcal Mechancs and ts Applcatons, 355(1), European Asset Management Assocaton. (2001). Indexaton and nvestment, a collecton of essays (Vol. 1). London, UK: Heronsgate Ltd. Frno, A., & Gallagher, D. R. (2001). Trackng S&P 500 ndex funds. The Journal of Portfolo Management, 28(1), Frno, A., Gallagher, D. R., & Oetomo, T. N. (2005). The ndex trackng strateges of passve and enhanced ndex equty funds. Australan Journal of Management, 30(1),

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