Optimal Portfolio Construction (A Case Study of LQ45 Index in Indonesia Stock Exchange)

Transcription

1 Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): Optmal Portfolo Constructon (A Case Study of LQ45 Index n Indonesa Stock Exchange) Dhea Ayu Pratw 1, Irn Yunta 1 Telkom Unversty, School of Economcs and Busness, Bandung, Jawa Barat, Indonesa Abstract: The purpose of ths paper s to test whether sngle ndex model or constant correlaton model of portfolo selecton offers better nvestment alternatves to nvestors n IDX. Sample taken for ths research s stocks consstently oned n LQ45 ndex for the tme perod of February 010 to January 015. After obtanng the optmal portfolos, the performance of each portfolo s evaluated and analyzed n terms of ther expected return and rsk. The measurement of that performance uses the rsk adusted methods: Sharpe, Treynor, and Jensen Index. By usng the sngle ndex model, t s observed that only sx out of sample stocks are allowed to be ncluded n the optmal portfolo. Meanwhle the optmal portfolo constructed by usng the constant correlaton model conssts of eght stocks. The fnal results then show that the optmal portfolo constructed by usng the sngle ndex model has a better performance. The three ndexes Sharpe, Treynor, and Jensen gve the same performance rankng for both portfolos, t means that both portfolos have been well-dversfed. Keywords: optmal portfolo, Sngle Index Model, Constant Correlaton Model, Indonesa Stock Exchange. 1. Introducton The theory of nvestment says that the decson-makng n nvestment s not only based on the expected rate of return but also the rsk level. Ether return or rsk are two nseparable thngs snce the consderaton of an nvestment s both s trade off. The return and the rsk have a postve relatonshp: the bgger the rsk whch has to be borne, the bgger the return whch has to be compensated [1]. The rsk-return trade off becomes a dlemma n decson-makng of nvestment for the nvestors. In most cases, the rsk mght be able to be reduced by combnng some sngle securtes nto a portfolo. Ths effect of reducng rsks by ncludng a large number of nvestments or securtes n a portfolo s called dversfcaton []. Ths dversfcaton s mportant, snce studes ndcate that dversfcaton can help nsulate one s nvestment aganst market and management rsks wthout much sacrfcng the desred level of return [3]. The part of rsk whch can be elmnated by constructng the well-dversfed portfolo s the unsystematc rsk. Ths rsk s unque for a company, so the bad thng occurs n a company can be offset by the good thng occurs n another company [1]. It s nvestors ob to fnd the optmal combnaton of stocks for ther portfolos that can mnmze the rsks and maxmze the returns. However, an nvestor s faced wth a choce from among an enormous number of stocks. When one consders the number of possble stocks and the varous possble proportons n whch each can be held, the decson process seems overwhelmng. s accepted (both assumptons wll be dscussed n the next part). Generally, both models wll be able to determne whether a stock can be ncluded n the optmal portfolo or not by usng a unque rankng crtera. The stock wll be sorted based on the performance measured by usng a rato of excess return aganst the rsk, so f a stock s ncluded nto an optmal portfolo, any hgher ranked stock must also be ncluded nto the optmal portfolo. After obtanng the optmal portfolos, t s mportant to measure the performance of each portfolo. The Sharpe, Treynor, and Jensen ndex are the measures of portfolo performance whch have nserted the return or the rsk factor. The Sharpe and Treynor ndex are ratos of compensaton aganst a rsk. The dfference s that the ndcator of the rsk of Sharpe ndex s a total rsk, meanwhle the Treynor ndex s a systematc rsk. The Jensen ndex s an ndex whch shows the dfference between the level of actual return ganed by the portfolo and the level of expected return f the portfolo s n lne wth the captal market [6].. Obectve of the Study The man obectve of ths research s to construct the optmal portfolo whch contans the stock of ndex LQ45 usng the sngle ndex model and constant correlaton model as well as ts performance analyss by usng the rsk adusted performance approach: Sharpe, Treynor, and Jensen. The LQ45 Index s a captalzaton-weghted ndex of the 45 most heavly traded stocks on the Indonesa Stock Exchange. In a seres of publcatons and books, Elton, Gruber, and Padberg [4]-[5] propose smple technques used to select optmal portfolo. These models allow the development of a system for computng the composton of optmum portfolos that s so smple t can often be performed wthout the use of computer. That model can be used f the assumpton of the sngle ndex model or the constant correlaton model regardng the covarance structure return among ts securtes 3. Research Method Volume 4 Issue 6, June The type of ths research s descrptve quanttatve research. The sample of research s stocks part of LQ45 ndex selected by usng the purposve samplng technque. The crteron of stock selecton s the stock whch s consstently oned n LQ45 ndex durng February 010 January 015. The data are from secondary data, such as the adusted Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY

2 Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): weekly closng stock prce, IHSG as the market ndex, and SBI as the proxy of rsk free rate for the observaton perod durng February 010-January Theoretcal Foundaton and Methodology 4.1 Sngle Index Model Sngle ndex model has an assumpton that one reason securty returns mght be correlated s because of a common response to market changes. Ths case comes up from an observaton that when the market goes up (as measured by any of the wdely avalable stock market ndexes), most stocks tend to ncrease n prce, and when the market goes down, most stocks tend to decrease n prce [4]. Thus, the return on a stock and the return on stock market ndex can be commonly stated as a relatonshp: Explanaton: R = return on a stock or securty R Rm e (1) = beta s the coeffcent whch measures the change of R caused by the change of R m R = rate of return on the market ndex m = the expected value of the component of securty s return that s ndependent of the market s performance. e = resdual error, whch s the random varable wth zero expected value or E(e ) = 0. If the assumpton of sngle ndex model s met as the descrpton of the covarance structure among securtes, the method of optmal portfolo selecton whch wll be explaned can be used. ote that all procedures used to calculate the optmal portfolo wll only consder the short sales dsallowed stuaton. 4. Constructng the Optmal Portfolo Based on the Sngle Index Model The calculaton to determne the optmal portfolo should be eased f t s only based on a number whch can determne whether securtes can be ncluded nto the optmal portfolo or not. The number s the excess return to beta rato [1]. The excess return to beta rato (ERB) shows the relatonshp between return and rsk. Ths ERB measures the relatve excess aganst a unt of rsk whch can be dversfed measured by beta. E( R) Rf ERB () Explanaton: R = rsk-free asset return f The optmal portfolo wll contan securtes wth hgh ERB values. Thus, the cut off pont s needed to determne the lmtaton of ERB values whch are consdered as hgh. Ths cut off pont can be determned by usng these followng steps: 1. Sortng the securtes based on the hghest ERB values to the lowest ERB values. Volume 4 Issue 6, June Calculatng the value of A and B for each securtes [ E( R ) R ]. A f e Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY and B (3) Where e s the varance from the resdual error of securtes whch s also a unque rsk or unsystematc rsk. 3. Calculatng the value of C, that s the value of C for securtes, calculated based on the followng formula: C 1 m 1 A m 1 B e (4) where: a. The cut-off pont (C*) s the value of C where the last ERB value s stll greater than the value of C. b. The securtes ncluded nto the optmal portfolo are those whch posses the bgger ERB value or same wth the ERB value n C* pont. To determne the amount of the proporton of each selected securtes n the optmal portfolo, the formula s: w Wth the value of : ( *) ERB C e Explanaton: w = proporton of securty 1 e = varance from resdual error of securty Expected return based on the sngle ndex model can be calculated by usng ths formula: (5) E( R ). E( R ) (6) p p p m Meanwhle, the varance or the rsk of portfolo can be calculated by usng ths formula: p p m w. e (7) Constant Correlaton Model The constant correlaton model assumes the correlaton between all pars of securtes s the same. The procedures assumng a constant correlaton coeffcent exactly parallel those presented for the case of sngle ndex model [1]. Thus the correlaton coeffcent value s the average of correlaton coeffcent value among those securtes. If the assumpton of constant correlaton model s met as the descrpton of the covarance structure among securtes, the procedures of optmal portfolo determnaton whch wll be explaned can be used. ote that all procedures used to calculate the optmal portfolo wll only consder the short sales dsallowed stuaton.

3 Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): Constructng the Optmal Portfolo Based on the Constant Correlaton Model The procedures of determnng the optmal portfolo by usng ths model are the same wth the procedures by usng the sngle ndex model, but the selectons of securtes whch are ncluded nto the portfolo are based on the excess return to standard devaton value (ERS). ERS can be calculated by usng ths formula: ( E( R) Rf ) ERS (8) Ether sngle ndex model or constant correlaton model dvde the excess return value wth the rsk. However, n the constant correlaton model, the standard devaton ( ) substtutes the beta as ts rsk ndcator. Furthermore, the value of C s used to determne the cut-off pont calculated based on followng formula: E( R) Rf C 1 (9) where s a correlaton coeffcent assumed constant for all securtes. To meet ts assumpton that the correlaton coeffcents among stocks are constant, the used value s the average value of correlaton coeffcent value ( ) among stocks, as followng: where the number of whch are calculated as followng: o nn ( 1) Volume 4 Issue 6, June ext, all stocks or securtes whch have a hgher excess return to standard devaton than C* are ncluded nto the optmal portfolo. The amount of the optmum nvestment for each securtes s calculated as followng: w (10) Where, Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY 1 1 (1 E( R) R f C* The expected return of portfolo s the weghted average of expected returns from each sngle securtes n the portfolo [1]. The expected return of portfolo s calculated as followng: n E( R ) w. E( R ) (11) P 1 Then, the rsk of ths portfolo can be calculated by usng the followng equaton: n n n p w w. w (1) Explanaton: = varance return of securty = covarance between securtes and return 5. Results and Dscusson 5.1 Sngle Index Model The calculaton results are presented to determne the optmal portfolo by usng sngle ndex model as followng: Table 1: The Constructon of Optmal Portfolo (Sngle Index Model) wth R f = % and E(R m )= 0.307% Issuers Code E(R) e ERB A B A 1 B C 1 1 JSMR TLKM KLBF UVR ASII GGRM BBI BBCA BBRI LPKR IDF BMRI SMGR LSIP ITP PGAS AALI UTR BDM PTBA ITMG ADRO

4 Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): Table 1 s a stock lst sorted based on the ERB value. From the table, t can be seen that there are three stocks whch have a negatve ERB value so those three stocks are removed from the optmal portfolo canddate. It s due to those stocks have the smaller return compared to the SBI return (rsk free rate). However, those stocks are rsky assets. The ratonal nvestors certanly wll not decde to nvest on the rsky assets f they cannot get the compensaton from the rsk. From the table, t s also known that there are 1 stocks whch have more than one beta value so t s called aggressve stocks, whch are a stock possessng a hgh rsk. Ths stock wll experence the hgher ncrease from the market ncrease and the sharper decrease f the market decreases. Then, nne stocks have less than one beta value. Ths stock s called defensve stock. Ths stock has smaller senstvty aganst the market. It s due to the stock prce mght move to the same drecton wth the market, however, the amount s not same (smaller). PTBA has one beta value, as bg as the beta of market so ths stock s neutral. Furthermore, the results of cut-off pont (C*) smulaton requre that only stocks wth ERB C value whch wll be ncluded nto the optmal portfolo. From 19 samples, there are only sx stocks whch have bgger ERB values than C. Thus, the sx stocks become the last canddate ncluded nto the optmal portfolo. The value of C* s known as 0.4. Then, the amount of nvestment allocaton percentage on the selected sx stocks s as followng: Fgure 1: Allocaton of Investment Funds (%) Table : Return, Rsk, and Portfolo Beta E(Rp) % Rsk % p % From the fgure 1 above, t can be seen that JSMR, KLBF and TLKM get the bggest allocaton of nvestment funds whlst GGRM gets the smallest porton whch s 0.6%. Meanwhle, table shows the calculaton results of the rsk and the return of portfolo. Based on the table, ths optmal portfolo can gve the level of portfolo return by % weekly calculated, wth the rsk stated n the standard devaton by.8437%. The amount of the portfolo rsk s smaller than the ndvdual stock rsk (the ndvdual rsk s n table 3). It proves that the dversfcaton of portfolo can decrease the portfolo rsk, by remanng to mantan the rate of portfolo return. o 5. Constant Correlaton Model The calculaton results are presented to determne the optmal portfolo by usng the constant correlaton model (table 3) as followng: Table 3: The Establshment of Optmal Portfolo (Constant Correlaton Model) wth 0.36 Issuers Code E(R) Excess Return ERS 1 1 KLBF JSMR ASII TLKM UVR BBI BBCA BBRI BMRI GGRM IDF LPKR SMGR ITP PGAS LSIP UTR AALI BDM PTBA ITMG ADRO Volume 4 Issue 6, June Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY E(R ) R f C

5 Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): From stocks, there three stocks whch have negatve ERS values so those three stocks are removed from the optmal portfolo canddate. It s due to those stocks have smaller return compared to SBI return (rsk free rate), although those stocks are rsky assets. From the cut-off pont smulaton, t s known that from 19 stock canddates, there are only eght stocks whch have more ERS value than C. Thus, those eght stocks become the last canddate whch wll be ncluded nto the optmal portfolo. The C* value s known as After the stocks are selected, the next step s to calculate the percentage or the proporton of nvestment funds for each stock. ndcator shows the same performance rankng for both portfolos, t means that both portfolos have been welldversfed. Based on the Sharpe ndex, the optmal portfolo constructed by sngle ndex model wll be able to gve the compensaton of portfolo return aganst each total rsk by 0.18%. Meanwhle, the Treynor ndex for the compensaton of portfolo return aganst systematc rsk (beta) s %. Seen from the Jensen ndex, the portfolo establshed by usng sngle ndex model has a bgger return from the expected return f the portfolo s n the captal market lne of 0.488%. 6. Concluson From the research above, there are some man ponts of the concluson. Frst, the optmal portfolo constructed by usng sngle ndex model contans sx combnatons of stock, whlst the optmal portfolo by usng constant correlaton model contans eght stocks. JSMR and KLBF are the stocks wth the bggest contrbuton n the portfolo snce each of them gets the bggest porton of funds (around 9%-3%). Fgure : Allocaton of Investment Funds (%) From the fgure above, t can be seen that KLBF has the bggest percentage of allocaton of nvestment funds whch s 3.4%, then t s followed by JSMR wth the allocaton of funds by 9.84%. Whle BBRI gets the smallest percentage of allocaton by 3.67%. Table 4: Return, Rsk, and Portfolo Beta E(Rp) % Total Rsk % p 0.750% Second, the amount of portfolo rsk s smaller compared to each ndvdual rsk. It ndcates that the unsystematc rsk has been successful to be reduced by establshng the welldversfed portfolo. It occurs due to the decrease of return happened n a stock or some stocks n the portfolo have been able to be covered by the ncrease of another stock n the portfolo. Last, the value of three portfolo performance ndcator ndexes: Sharpe, Treynor, and Jensen, consstently show that the portfolo constructed by usng sngle ndex model has a better performance compared to the portfolo constructed by usng constant correlaton model. So, the portfolo constructed by sngle ndex model automatcally becomes a recommendaton from the wrter for the nvestors to nvest. Besdes, the three performance ndcator shows the same performance rankng for both portfolos, t means that both portfolos have been well-dversfed. The amount of portfolo rsk s smaller than the rsk of each stock. It means that the unsystematc rsk of stock has been able to be reduced by establshng the well-dversfed portfolo. It occurs due to the decrease of return happened n a stock or some stocks n the portfolo have been able to be covered by the ncrease of another stock n the portfolo. 5.3 The Comparson of Portfolo Performance Table 5: The Comparson of Optmal Portfolo Performance Sngle Index Model Constant Correlaton Model Sharpe (%) Treynor (%) Jensen (%) The results of three portfolo performance ndcator ndexes show that the optmal portfolo constructed by sngle ndex model has a hgher rankng or a hgher performance value compared to the portfolo constructed by usng constant correlaton model. Besdes, due to the three performances References Volume 4 Issue 6, June [1] J. Hartono, Teor Portofolo dan Analss Investas (8 th ed), BPFE-YOGYAKARTA, Yogyakarta, 013. [] A. J. Keown, S. Trman, J. D. Martn, Fnancal Management (11 th ed), Pearson Prentce Hall, Boston, 011. [3] Y. Maheshwar, Investment Management, PHI Learnng Prvate, ew Delh, 008. [4] E. J. Elton, M. J Gruber, Modern Portofolo Theory and Investment Analyss (3 rd ed.), John Wlley & Sons, ew York, [5] E. J. Elton, M. J Gruber, M. W Padberg, Smple Crtera for Optmal Portofolo Selecton: Tracng Out The Effcent Fronter, The Journal of Fnance, 33(1), pp [6] E. Tandelln, Portofolo Investas (1 st ed). KAISIUS, Yogyakarta, 010. Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY

6 Author Profle Internatonal Journal of Scence and Research (IJSR) ISS (Onlne): Index Coperncus Value (013): 6.14 Impact Factor (013): Dhea Ayu Pratw born on July, Receved Bachelor n Busness Management of Telecommuncaton and Informatcs (015) from Telkom Unversty, Indonesa. Irn Yunta receved Bachelor n Industral Engneerng (004) from Gaah Mada Unversty, and Master of Management n Busness Management from Padaaran Unversty (008). She s now a permanent lecturer n Telkom Unversty, Indonesa. Volume 4 Issue 6, June Paper ID: SUB Lcensed Under Creatve Commons Attrbuton CC BY