Analyss of sectoral erformance n Borsa Istanbul: a game theoretc aroach Nesrn Ozkan Bursa Orhangaz Unversty, Turkey Deartment of Bankng and Fnance Keywords Game Theory, mnmax, sectoral ortfolo, lnear rogrammng, Borsa Istanbul. Abstract Game theory has been the subject of many dversfed felds of study n lterature. Wthn game theory framework, ortfolo otmzaton s one of the alcatons used n fnance. The man goal of the study s to create otmal ortfolos n Borsa Istanbul by usng a game theoretc aroach and to analyze relatve erformances of sectoral ortfolos. In the study, ortfolo otmzaton s set u as a zero sum-u game and converted to a lnear otmzaton model. The stock market and nvestors are dentfed as layers (oonents) n the model. Players have nvested n the ortfolos accordng to the mnmum rsk wth maxmum return crtera. The monthly data for the erod between 2009 and 2014 s obtaned from Borsa Istanbul. The data set ncludes the stock return of 229 comanes and also ncludes returns of Borsa Istanbul four sector ndexes: fscal, ndustral, servce and technology. Relatve erformances of sectoral ortfolos are evaluated by Share Performance Index and Varaton Coeffcent. The man fndng s that the model can be used n ortfolo otmzaton. Technology-based ortfolo attans the hghest return wth lowest ortfolo concentraton. Moreover, the relatve erformance of technology ortfolo s hgher than the other three sectors. Introducton In fnancal markets, the rsk and uncertanty are unversal matters. All nvestment decsons are shaed by rsk and uncertanty. The rsk comes from the obscurty of future. Inward and outward both economcal and oltcal challenges affect the markets. Rsk bascally nherts the nature of fnancal markets. As t s nevtable as a matter, the nvestors have to take t nto account n decson-makng. The nvestng decsons are shaed by choosng from a vast array of fnancal nstruments. Investors ether nvest n a sngle fnancal nstrument or create a dversfed ortfolo that nvolves many assets. Whle nvestng more than one asset, the nvestor seeks to otmze the ortfolo comoston. In ortfolo nvestment, the nvestor has to take two crucal decsons. One s to decde whch fnancal nstrument s ncluded n the ortfolo, and another s the weght of them. (Prgent, 2007). For that urose, many models have been roosed n nvestment theory. The models dffer n the ersectve how they aroach the otmzaton roblem, and measure the rsk. Some models accet the mnmum varance as a measure of rsk whle some other uses sem varance, beta coeffcent, varaton coeffcent or maxmum loss. Modern ortfolo theory was ut forth by Harry Markowtz (1952) by hs ublshng Portfolo Selecton. In hs study, Markowtz develoed a quadratc rogrammng model. The model s based on Mean-Varance Model, andthe rsk s dentfed as varance. It defnes the nvestors as rsk averse, keen to otmze the ortfolo by maxmzng the exected return for a gven level of rsk. In case the nvestor takes on hgher rsk, the exected return must be hgher. The rsk averson of nvestors determnes how much rsk to take on. Hence, all the nvestors hold ther ortfolos as subject to ther rsk erceton. Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 61
In Mean-Varance Model, the roblem aears n workng wth large asset unverse.in case, the asset unverse s large, the estmaton of covarances become qute dffcult. In order to overcome the dffculty n calculaton, Wllam Share (1963) develoed Sngle Index Model and then followng other ndex models. The model measures the nterrelaton between the mean return of market and stocks nstead of correlatons among the stocks. The measure of relaton s defned as Beta Coeffcent by Share, and t can be exressed by smle lnear regresson. Wllam Share (1971) remarked that f the ortfolo analyss roblem was arorate to aly lnear rogrammng technques, the success of alcaton would be substantally enhanced. Lnear rogrammng has found a wde usage n nvestment decson models n the lterature. Martn R.Young (1998) was the frst aled Mnmax Theorem to the ortfolo selecton roblem. Young s model solve the ortfolo selecton roblem by lnear rogrammng. The assumtons of the model are based on game theory. Young s ortfolo selecton model, exressed Mnmax Model ams to mnmze the maxmum loss for a gven level of return. The rsk s dentfed as the mnmum return contrary to the varance models. In the model, the ortfolo maxmzes the mnmum return or, alternatvely, mnmzes the maxmum loss. Mnmax Theorem was ntroduced by John von Neumann n 1928. The theorem s bascally used n decson makng under uncertanty. Game theory was then acceted as a dsclne n 1944 wth ''The Theory of Games and Economc Behavor'' by John von Neumann and Oscar Morgenstern. Game theory formulates the strateges of oonents mathematcally, n the decson-makng rocess. In game theory, games are layed under rsk and uncertanty. From ths ont of vew, fnancal markets and games are smlar n all ther asects. The ortfolo otmzaton roblem can be bult u two-erson zero-sum games. In the game, the loss of one layer equals the gan of the other and so the sum of the game s zero. The market and the nvestor mght be dentfed as oonents. Each behaves ratonally and chooses the best strategy for ther own. The game fundamentally s a sngle- layer game that s layed aganst Nature. (Shubk, 1989).The market reresents the Nature wth all ts characterstcs. (Fredman, 1997). It has numerous strateges that are affected by varous economc, oltcal and socal turmol s lke the volatlty of exchange rates and nterest rates. In consderaton of the comlexty of the factors, we cannot solely determne the effect of each factor on the overall market. The assumton s vald for the stock exchange market as well. The change n asset values also reflect the economcal, oltcal and socal changes on the market, butanone of them s solely resonsble for ths change. In the game, the oonents choose ther strateges n order to maxmze the exected mnmum returns (Maxmn crteron) or to mnmze the maxmum exected loss (Mnmax crteron).in other words, the market seek to mantan the mnmum loss and the nvestor as an oonent, choose the best alternatve among the worst outcome wth mnmum rsk. Hence, the model resents a qute conservatve aroach. In lterature, Paahrstodoulou and Dotzauer (2004) comare the Mean-Varance model (MV), Mean Absolute Devaton Model (MAD) and Mnmax model (MM)n Stockholm Stock Exchange between the erods 1997 and 2000. They note that Maxmn Model s most robust accordng to other models. Ca et al. (2004) comare the Mnmax wth Markowtz's quadratc rogrammng models n terms of rsk n Hong Kong Stock Exchange. They fnd that the results of both models has smlar erformance. Nevertheless, Mnmax s not senstve to the data. Faras et al. (2007) comare MV Model, MM Model and MAD Model n Brazlan Stock Market (BOVESPA). They ont out that Mnmax Model s sueror to others. Hassan et al. (2012) roduce the asset allocaton by usng Maxmn Model between the crss years 1997 and 2001 n Malaysa. Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 62
They remark that the model rovde consstent result n crss erod. Schaar Schmdt and Schanbacher (2014) analyze the effectveness of Mnmax Model n ortfolo allocaton between the erod 1990 and 2010. The analyss reveal that the ortfolo weghts are stable over tme n Mnmax Model and model erforms better n terms of rsk aroach. The analyss n ths aer can contrbute to the lterature as follows: The study ams to examne an alternatve aroach for game theoretc models. In order to measure the effectveness of the model, the analyss focuses on four sectoral ortfolos. We can dvde the analyss nto three hases. In the frst hase, the ortfolos are roduced based on Mnmax aroach by whch the weghts for sectoral ortfolos s obtaned. In the second hase, the ortfolo return s calculated accordng to the average return of stocks durng the analyss erod. Followng that, the rsk of ortfolo s comuted by the covarances between the selected stocks. In the thrd hase, the relatve erformances of the four ortfolos are evaluated by usng Share Performance Index and Correlaton Coeffcent. At the end of the analyss, the erformance of ortfolos s comared wth ndces values. We note that the model mght rovde a useful and effcent aroach for the nvestment decson and an alternatve model for ortfolo selecton. Data and Methodology The am of research to obtan the stock allocaton of ortfolo generated va Mnmax aroach n Borsa Istanbul to assess the relatve erformance of four sectoral ortfolos and to determne the sueror ortfolo to all other. Data The data conssts of 229 stock rces and sectoral ndces values. The data set nvolves four comlatons of data. Each data comlaton contans the stocks tradng n man sectors of Borsa Istanbul. The man sectors used n the analyss are fscal, ndustral, servce and technology. Besdes the ortfolos, the sectoral ndces are comuted. All the data are obtaned from Borsa Istanbul. Transacton costs and taxes are gnored n the study. The analyss s run between the erod January 2009 and December 2014. Monthly closng rces are used to assess the change n stock values. The ntermttent stocks are excluded from analyss. The rsk-free rate s taken from the webste of the Central Bank of the Reublc of Turkey. The rate s comuted as the average of comounded government bond yelds as observed wthn the analyss erod. The ndces values are used as a measure n comarson wth ortfolo returns. All smulatons are run va Mcrosoft Excel Software. Methodology The ayoff matrx s generated accordng to the returns of stock. The years are dslayed on the matrx rows (market strateges), and the stocks are on columns (nvestor strateges).each cell n the matrx reresents a ayoff value.the stocks can be held n the ortfolo for a mnmum erod of 1 month. Intally, the log returns of stocks are calculated as followng: Rt Rt 1 RM t R t1 where RM Is the monthly return of the stock t t R s the stock rce at the tme(month) t R s the stock rce at the tme(month) t-1 t 1 Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 63
The ayoff matrxes are constructed for 12 months. The matrxes were bult for the four sectoral ortfolos searately. It s almost mossble to know whch strategy the market wll choose. However, we mght assgn robabltes, say S,to each stock. In a zero-sum u game, the outcome of the game can be acheved by lnear rogrammng. The model s used only to decde whch stock s ncluded the ortfolo and what s the weght of the stock n the ortfolo. The model s transformed to lnear rogrammng. The technology ndces are comosed of 12 stocks and the formulaton for technology sector s exressed as follows: 12 a S V, j 1,2,3,4,5 1 j where s the strategy of the nvestor j s the strategy of themarket aj s the ay off for an nvestor wth strategy when the market strategy sj S s the robablty of strategy (nvestment share of strategy ) V s the value of the game The exected returns are calculated for each month n the same way. The robabltes sum u to one. 12 S 1 1 Both sdes of equatons are dvded byv, as defned by equaton: S P V After the transformaton, the equaton s descrbed as 12 ap1 1 j The am of nvestor s maxmzng the gan (V). We can aroach the am n another sde by mnmzng 1/V. In ths sense, the objectve functon s as follows: 12 1 MnZ Mn P V 1 All robabltes have to be ostve as well. P 0 When all smulatons were solved wth lnear rogrammng model, the stock allocaton for sectoral ortfolos was acqured. Stock Code Comany Name Portfolo Allocaton 1 ALCTL Alcatel Lucent Teletas 0.002 2 ANELT Anel Telecom 0.156 3 ARENA Arena Comuter 0.145 4 ARMDA Armada Comuter 0.224 5 ASELS Aselsan 0.128 6 DGATE Datagate Comuter 0.113 7 ESCOM Escort Technology 0.009 8 INDES Indeks Comuter 0.095 9 KAREL Karel Electronc 0.035 10 LOGO Logo Software 0.052 11 NETAS Netas Telecom 0.042 Table 1: Portfolo allocaton for technology sector Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 64
In the second hase, we calculated rsk and return of ortfolos. The ortfolo return wth hstorcal data wll be calculated as follows: R N w. R 1 where R s the average return of the ortfolo w s the weght of stock n theortfolo R s the average return of stock The rsk of the ortfolo s comuted wth varance. The square root of varance gves the standard devaton of the ortfolo. Standard devaton denotes the rsk assocated wth an asset or ortfolo. N N N 2 2 2 w w wjcovj 1 1 j1 where s the varance of the ortfolo 2 2 s the varance of stock Cov s the covarance between stock and stockj j The varance of each stocks s not solely a measure for overall ortfolo rsk, the covarance matrx was bult for each sectoral ortfolos n the analyss. Technology Sector Standard Devaton 0.0769 Portfolo Return 0.0305 Number of Stocks n Portfolo 11 Fscal Sector Standard Devaton 0.0728 Portfolo Return 0.0277 Number of Stocks n Portfolo 28 Servce Sector Standard Devaton 0.0773 Portfolo Return 0.0239 Number of Stocks n Portfolo 24 Industral Sector Standard Devaton 0.0764 Portfolo Return 0.0274 Number of Stocks n Portfolo 25 Table 3: Rsks and returns of sectoral ortfolos Table 2: Rsk and return of sectoral ortfolos From Table 3, we can deduce that the return of technology-based ortfolo rovded the hghest return to the nvestor n the erod 2009 and 2014. Snce the rsk s crtcally mortant for the decson maker, esecally for the nvestor wth rskaverse characterstcs, the rsks were also comared ortfolos. The fscal-based ortfolo carres the lowest rsk wth the standard devaton of 0, 0728.The dfference between the hghest (servce sector) and lowest values (fscal sector) s roughly 6% n ortfolos. In general, the rsks are qute close between the sectors. Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 65
On the other hand, the concentraton of ortfolos was evaluated relatvely. Deste the stock unverse was consderably large for ndustral sector, otmal ortfolo comoston was acheved by 25 stocks. The fscal-based ortfolo has the greatest number of stocks, and the technology-based ortfolo contans 11 stocks wth the mnmum number among all. That means the otmal ortfolo comoston was acheved wth less stock than other ortfolos durng that erod. Technology Fscal Servce Industral Portfolo Indce Portfolo Indce Portfolo Indce Portfolo Indce Return 0.0305 0.0342 0.0277 0.0207 0.0239 0.0181 0.0274 0.0241 Rsk 0.0769 0.0897 0.0728 0.0911 0.0773 0.0558 0.0764 0.0638 Table 4: Comarson of sectoral ortfolos wth ndce value Table 3: Comarson between sectoral ortfolos and ndces Indces are commonly used as a benchmark n the analyss. Hence, ortfolo returns and rsks are comared wth relatve ndces n Table 4.In comarson, three ortfolos excet technology attaned hgher return than the ndce. Snce each ortfolo has dfferent return and rsk, we need a common measure to examne the relatve erformances of ortfolos. In the thrd hase, the relatve erformances of four ortfolos are evaluated by Share Performance Index and Correlaton Coeffcent. In the analyss, both are used as benchmark. In ortfolo management, the erformance of a ortfolo s a crucal matter. As the erformance resents the success of ortfolo, t hels the nvestor to decde whether the ortfolo wll be held for future. Wllam Share (1994) develoed an ndex to assess the erformance of a ortfolo. The erformance ndex s bult as the excess return of the ortfolo over the rsk-free rate, dvded by the ortfolo rsk. The ndex s a rsk- adjusted measure of returns. Share used three varables n assessng the erformance of ortfolo: the return of a rsk-free asset, the average return of an asset and the standard devaton of return. It s commonly used n rsk/return measures. Share erformance ndex s formulated as follows: R Rf SharePerformanceIndex Where R s the average return of the ortfolo R f s the rsk-free rate s the standard devaton of the ortfolo The hgher the ndex, the better the erformance of the ortfolo. Varaton Coeffcent s a rato of the average return to the standard devaton. It s a useful tool to the nvestor n the evaluaton of alternatves wth dfferent exected return. VaratonCoeffcent R In general, the lower the varaton coeffcent, the better the erformance of a ortfolo. Results and Consderatons The sectoral stocks n Borsa Istanbul are used n our analyss. The reason for usng a sectoral analyss s to gude the nvestor about whch sector brngs gan more than others. Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 66
For a better understandng of the erformance results, we can exress the dfferences between the return of sectoral ndces wth ortfolo returns n ercentage terms. The am of the comarson of ndces s to comrehend what f the nvestor refers nvestng the ndces nstead of holdng ortfolo. If the ndces were chosen for nvestment, the nvestor recorded loss for three sectors: ndustral, fscal and servce. The sectoral ndex that rovdes gans the nvestor more than generated ortfolos only the technology one. If the nvestment was made to the technology ndces n the erod 2009 and 2014, the nvestor s gan was aroxmately 12% hgher than ortfolo return. On the other hand, the gan of technology ndex accomanes wth hgh volatlty. Also, an outstandng nference mght be done for the servce sector. The servce ndces was brought almost 41% loss for the nvestor. As a sequence, we can deduce that the gan of ndces excet technology ndces s lower than generated ortfolos. Sectoral Portfolos Varaton Coeffcent Share Performance Index Technology Sector 2.519 0.237 Fscal Sector 2.624 0.212 Industral Sector 2.785 0.198 Servce Sector 3.231 0.150 Table 4: Performance of sectoral Table ortfolos 4: Performances of sectoral ortfolos Table 4reresents the relatve erformances of generated sector-based ortfolos. The ortfolos are ranked from the lowest varaton coeffcent to the hghest. Accordng to the results, technology-based ortfolo outerforms wth the lowest varaton coeffcent and the hghest Share Performance Index. The sueror ortfolo s generated by the stocks tradng n technology sector. In recent years, the technology sector has been growng steadly n Turkey as well as n the World, and shares of technology comanes offer hgh returns to the nvestors n stock market. As a consequence, the results sustan that the method s effcent n ortfolo otmzaton that allocates the convenent stocks for ortfolo dversfcaton. For future studes, the game theoretc model can be examned wth other ortfolo otmzaton models n the am of a benchmark. Furthermore, an analyss mght be conducted wth sub- Sectors, and that wll rovde noteworthy results. Conclusons In ths study, we am to buld u a comaratve analyss of sect oral ortfolos between the years 2009 to 2014 n Borsa Istanbul and to gude the nvestors for future nvestment. Also, we attan to buld ortfolos by a game theoretc model. The analyss s run n three stes. At frst, we search for otmal ortfolo allocaton n fscal, ndustral, servce and technology sectors accordng to the Mnmax aroach n decson-makng. We then construct ortfolos wth the mnmum rsk and maxmum return. The game theoretc model acheves to buld u otmal ortfolos. The return and rsks of generated ortfolos are comared wth each other, and also wth the ndce. At the thrd ste, the relatve erformance of sectors s comared wth Share Performance Index and Varaton Coeffcent. Snce the analyss s run on hstorcal data, the aroach has a determnstc structure lke most nvestment analyss models. In the ortfolo, the erformances of the stocks, as well as Internatonal Conference on Restructurng of the Global Economy, 22-23 June 2015, Rome, Italy 67
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