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1 Portfolo Optmzaton Usng Mult-Obectve Partcle Swarm Optmzaton Vrya Ymyng, Natonal Insttute of Development Admnstraton, Thaland Ohm Sornl, Natonal Insttute of Development Admnstraton, Thaland The Asan Conference on Technology, Informaton & Socety 2015 Offcal Conference Proceedngs Abstract Portfolo optmzaton s an mportant problem n fnance. Its goal s to dscover an effcent fronter whch yelds hghest expected return on each level of portfolo varance. The problem has multple obectves, and ts search space s large. Multobectve partcle swarm optmzaton s a mult-obectve optmzaton method, developed from partcle swarm optmzaton, by ncorporatng non-domnated sortng and crowdng dstance. Ths research proposes a portfolo optmzaton technque based on mult-obectve partcle swarm optmzaton. Two obectves used n the research are maxmzaton of return and mnmzaton of portfolo rsk. The technque s evaluated usng daly stock total return ndex gross dvdends from Stock Exchange of Thaland between 2006 and The technque s deployed n unknown tradng perods, and the results are compared wth standard market benchmarks. The results show that the proposed technque performs well n comparsons wth the market benchmarks. Keywords: Portfolo optmzaton, Mult-obectve partcle swarm optmzaton, Markowtz s model, portfolo management afor The Internatonal Academc Forum

2 Introducton Investors throughout the world are nterested n portfolo management. The man focuses for ths problem are on expected return and rsk management. Portfolo theory, frst ntroduced by (Markowtz, 1952, 1959), s appled to portfolo allocaton to ad securty selecton and asset allocaton to gan the hghest expected return whle havng an acceptable rsk level. Later on, ths theory was developed nto others theores such as the captal market theory. There are many constrants that a fund manager has to consder before makng decsons on nvestment allocaton, such as those defned by the nvestment commttee and by the securtes and exchange commsson, such as the maxmum and the mnmum weghts of shares, the portfolo rsk, and the acceptable value at rsk. Besdes, there are other factors that the fund manager should consder such as lqudty and dvdend yeld (Clarke et al., 2002). Because the search space of portfolo optmzaton s large and not sutable for the Brute force method whle the populaton random samplng yelds nconsstent solutons. A better approach s needed to obtan accurate and sutable solutons quckly. Mult-obectve partcle swarm optmzaton (MOPSO) s developed from partcle swarm optmzaton (PSO), ntroduced by Eberhart & Kennedy (1995), and based on the herd s behavor or swarm ntellgence. A flock of brds or a swarm seeks for food by communcatng wth one another to assemble where they fnd good food. Along the way, f better food sources are dscovered, they communcate back and fly to the best sources together. Later, Moore & Chapman (1999) appled PSO to search for mult-obectve solutons, and there currently are numerous researches on applyng PSO to varous problems. At the same tme, Raquel & Naval Jr. (2005) presented MOPSO whch employs non-domnant sortng and crowdng dstance methods from Non-Domnated Sortng Genetc Algorthm-II (NSGA-II), created by Deb et al. (2000) and mutaton by Coello et al. (2002, 2004). MOPSO has the same prncple as PSO whch males t sutable to fnd the best search space spot n a short tme. PSO uses real-valued encodng and vector-based calculaton and thus lends tself well to real-valued problems (Coello et al., 2002). Moreover, Mshra et al. (2009) compared the results of MOPSO and those of NSGA-II on a portfolo optmzaton problem wthout nvestment constrants. The results show the superorty of MOPSO over NSGA-II. Ths research presents a portfolo optmzaton technque usng MOPSO wth nvestment constrants. In the rest of the paper, Secton 2 presents the proposed technque. In Secton 3, the technque s evaluated usng actual stock prces from the stock exchange of Thaland, and the results are presented. Secton 4 provdes concludng remarks.

3 Proposed Technque PSO s a populaton-based search algorthm, smulatng the socal behavor of brds wthn a flock. It s found to be very effectve n a wde varety of applcatons and able to produce good results at a very low computatonal cost. PSO reles on two mechansms: parent representaton and fne tunng of the parameters. A partcle s a member (ndvdual) of the swarm. Each partcle represents a potental soluton to the problem beng solved. The poston of a partcle s determned by the soluton t currently represents. PSO uses an operator that sets the velocty of a partcle to a partcular drecton. The drecton s defned by both the partcle s greatest success (personal best or pbest) and the best partcle of the entre swarm (global best or gbest). If the drecton of the personal best s smlar to the drecton of the global best, the angle of potental drectons wll be small, whereas a larger angle wll provde a larger range of exploraton. Partcles are flown through the search space. Changes to the postons of partcles wthn the search space are based on the socal-psychologcal tendency of ndvduals to emulate the success of other ndvduals. The soluton set of a problem wth multple obectves does not consst of a sngle soluton. Instead, n mult-obectve optmzaton, we am to fnd a set of dfferent solutons,.e., the Pareto optmal set. In MOPSO, a swarm s frst ntalzed. A set of leaders s also ntalzed wth the non-domnated partcles from the swarm. The set of leaders s usually stored, and qualty measures are calculated for all the leaders. At each generaton, a partcle s flown. The partcle s evaluated, and ts correspondng pbest s updated. A new partcle replaces ts pbest partcle usually when ths partcle s domnated or f both are ncomparable (.e., they are both non-domnated wth respect to each other). After the partcles are updated, the set of leaders s updated. Fnally, the qualty measure of the set of leaders s recalculated. Ths process s repeated for a certan number of teratons. Portfolo Optmzaton Usng MOPSO The MOPSO process s shown n Fgure 1. Frst, a number of partcles are defned. Too few partcles wll not yeld nclusve soluton whle too many partcles wll slow down the MOPSO process. From experments, we fnd that the most sutable number s 200 partcles, whch s then set as the number of partcle vectors. Elements of a vector are varables of a soluton,.e., portfolo weghts. The ntal values of weghts are randomly set, and the sum of all weghts w s equal to 1. n w = = 1 1 where w s the nvestment weght of securty, and n s the number of elements n a vector.

4 Fgure 1: The MOPSO algorthm Each vector s checked for any volaton of the constrants. Obectve values for each vector are then calculated. Two obectve functons used n ths study are: Obectve 1: Maxmzng the expected return: Maxmze E( rp ) = = w r 1 n where E(r p ) s the expected rate of return of portfolo p w s the nvestment weght of securty n portfolo p r s the expected rate of return of securty. Obectve 2: Mnmzng the portfolo rsk 2 m 2 2 m Mnmze p = w + = 1 = 1 = 1 m 2 w w The covarance of securtes and ( ) can be calculated as: 1 = m k n = 1 ( r k E[ r ])( r k E[ r ]) where 2 p m s the portfolo varance s the covarance of securtes and s the number of days

5 2 s the varance of securty r k s the daly return of securty on day k. When s equal to, can be calculated as: becomes 2 (the varance of securty ). The daly return r k close prcek number of shares k r k = 1 + total ( close prce k number of shares ), 1, k 1 ± ( adust prce adusted shares ) dvdend yeld k total dvdend yeld k dvdend per share number of shares, k 1 = ( close prce number of shares 1) ± ( adust prce adusted, k 1, k shares ) where close prce,k number of shares,k dvdend per share adust prce adusted share s the closng prce of securty on day k s the number of outstandng shares on day k s the cash dvdend per share of securty s the prce after adustment (by the corporate) s the number of shares after adustment. Non-domnant partcles yeld values on the Pareto front whch are the best solutons of a mult-obectve problem. Non-domnant sortng s performed to fnd nondomnant partcles by comparng each partcle to other partcles wth respect to each obectve. If a partcle s worse than any partcle n an obectve, that partcle s domnated and elmnated. When non-domnant results are obtaned, they are stored n the Pareto set n a sorted order, and crowdng dstances between two consecutve partcles are calculated from the populaton. The crowdng dstance of partcle can be calculated as follows: d = n = 1 f f, + 1,max f f, 1,mn where n s the number of obectves, and s the partcle order. Once the crowdng dstances for all partcles are obtaned, they are sorted from maxmum to mnmum before selectng the values to be the goal selecton. From experments, we fnd that the selecton should be performed n steps. If the sze of the Pareto set s less than 5, all partcles are selected whle f there are more than 5 partcles, select the top 30 partcles. Then, the gbest values are determned. Each partcle value s pbest to calculate the velocty n order to fnd ts new poston x accordng to the equaton below: x, ( t + 1) = x, ( t) + v, ( t + 1) v, ( t + 1) = ωv, ( t) + c1r1, ( t)[ pbest, ( t) x, ( t)] + c2r2, [ gbest ( t) x, ( t)] where x (t) s the -th partcle s poston at teraton t wth respect to obectve v, (t) s the velocty of partcle at teraton t wth respect to obectve c 1, c 2 are constant veloctes where c +c 4 1 2

6 r 1, r 2 are random values for speed adustment where r 1, r 2 U(0,1 ) ω s the nerta weght where 1 ω > ( c + c ) 2 1 In our research, we adopt a commonly used values of c 1 = and c 2 = (Van den Bergh, 2006), and r 1 and r 2 are random values between 0 and 1, and they are ndependent from each other. The number of teratons s set at 3,000. However, we fnd that after 1,500 teratons, the Pareto front generally s unchanged. The nerta weght ω helps reducng the velocty of a partcle to control severe movements. Its value s vared accordng to (Corazza & Komlov, 2009) as follows: wmax wmn ω = w max ( ) teraton teraton where w max and w mn are the maxmum and mnmum allowable securty weghts. max 2 1 Portfolo Optmzaton Constrants Two constrants are mposed on weghts of securtes n a portfolo. Frst, to not overemphasze on a partcular stock, each stock must account for no more than 10% of the total portfolo. In addton, the proporton of an ndustry must not exceed 40% of the total portfolo. Mutaton Operaton wth Constrants After updatng velocty, the mutaton operaton s performed. Our mutaton operaton s a modfcaton from the orgnal operaton by Coello et al. (2002) whch mutates only one varable of a partcle vector. Snce a portfolo optmzaton problem consders many constrants, mutatng only one varable decreases the effects of mutaton. Our modfed mutaton makes changes to every value n a vector to expand the search space, as shown n Fgure 2. for unt = 1 to number_of_partcles mutaton_rate = 0.5; f (1-teraton/max_teraton)^(5/mutaton_rate) > rand for element = 1 to vector_sze mutaton_range = (weght_max weght_mn) * (1- teraton/max_teraton)^(5/mutaton_rate); UB = partcle(unt, element) + mutaton_range; LB = partcle(unt, element) mutaton_range; partcle(unt, element) = (rand(1)*(ub-lb))+lb; end for end f end for Fgure 2: The mutaton operaton

7 Each weght n a partcle vector s verfed f there s any volaton of the constrants. If any, adustments are made to lmt the weght values accordng to the constrants. The ndustry proportons are checked, and adustments are performed to lmt those proportons by equally updatng the weghts that are n the same drecton of the dfference. Then, the fnal adustment s made n order for the total sum of weghts n a portfolo to be 1(or 100%). After adustments, the process of fndng non-domnant partcles as descrbed earler s repeated untl reachng the specfed number of teratons. Expermental Evaluatons Accordng to the captal market theory by Markowtz (1952), rsk can be dvded nto two types: systematc rsk and unsystematc rsk. Systematc rsk cannot be elmnated because t s the stock market rsk whle unsystematc rsk can be elmnated through dversfcaton. Ths s because stocks and shares n dfferent ndustres have dfferent returns dependng on the busness cycles. For ths reason, we select 5 stocks wth hghest captals from each ndustry group,.e., Agro and Food (AGRO), Consumer Products (CONSUMP), Fnancals (FINCIAL), Industrals (INDUS), Property and Constructon (PROPCON), Resources (RESOURC), Servces (SERVICE), and Technology (TECH) ndustres, as follows: AGRO ndustry conssts of CPF, MINT, TUF, TF and KSL CONSUMP ndustry conssts of SUC, ICC, MODERN, TR and SITHAI FINCIAL ndustry conssts of SCB, KBANK, BBL, KTB, BAY and TPC INDUS ndustry conssts of TPC, STANLY, TCCC, VNT and SSI PROPCON ndustry conssts of SCC, CPN, SCCC, LH and PS RESOURC ndustry conssts of PTT, PTTEP, GLOW, TOP and RATCH SERVICE ndustry conssts of CPALL, AOT, BDMS, MAKRO and BIGC TECH ndustry conssts of ADVANC, INTUCH, TRUE, DELTA and JAS. The proposed method s evaluated usng four sets of data whch span 4 dfferent perods of tme whch are: set 1 ( ), set 2 ( ), set 3 ( ), and set 4 ( ). Results To fnd an optmal portfolo, fnancers typcally apply a proporton varaton calculaton to create an effcent fronter lne. The Monte Carlo method s a popularly used one. It randomzes the varables accordng to the constrants for a portfolo. Once a random portfolo s obtaned, portfolo rsk and expected return are calculated. The portfolo wth the hghest return at the same rsk level wll be on the effcent fronter. Effcent Fronters Generated by MOPSO and the Monte Carlo Method Results of portfolo optmzaton by MOPSO and the Monte Carlo method are shown n Fgures 3, 4, 5 and 6. We can see that MOPSO yelds better effcent fronters than does the Monte Carlo method.

8 Fgure 3: Effcent fronters by MOPSO Fgure 4: Effcent fronters by MOPSO and Monte Carlo methods and Monte Carlo methods Fgure 5: Effcent fronters by MOPSO Fgure 6: Effcent fronters by MOPSO and Monte Carlo methods and Monte Carlo methods Tradng Results In order to evaluate the tradng performance of the proposed method, three types of portfolos by MOPSO are selected, and ther returns are calculated whch consst of: 1. Portfolo wth the hghest expected return, 2. Portfolo wth the lowest portfolo rsk, and 3. Portfolo wth the mnmum coeffcent of varaton. We compare the returns of the 3 types of portfolos generated by MOPSO wth the performance of SET, SET50, SET100 and SETHD ndces. These ndces are the market representatves and used as the standard comparatve ndces for nvestment.

9 Portfolo Type Investment Perod Table 1: Total returns of MOPSO and benchmark ndces Total Return Mnmum Coeff. of Varaton Maxmum Return Mnmum Rsk SET SET50 SET100 SET HD % 15.65% 18.27% 3.69% 3.74% 3.23% NA % 59.70% 34.49% 40.53% 35.94% 37.69% 26.14% % 3.55% 0.04% -3.63% -3.53% -4.07% -8.57% % 19.18% 12.25% 19.12% 16.98% 18.18% 8.91% The results are shown n Table 1. The results show that the portfolo wth the hghest expected return outperforms all other portfolo types and all ndces n every year. The portfolo wth the mnmum coeffcent of varaton performs better than all ndces n almost every year. Only n 2014 that t generates the return whch are 3.68%, 1.54%, and 2.74% less than SET, SET50, and SET100 ndces, respectvely. The portfolo wth the lowest rsk performs better than all ndces n 2011 and 2013 whle t performs worse than the ndces (except for SETHD) n 2012 and Overall, we can see that the proposed technque generate portfolos that perform well n comparsons wth standard nvestment ndces. Concluson Portfolo optmzaton ams to dscover an effcent fronter whch shows hghest expected return on each level of portfolo varance. Due to large varatons of varables and constrants, manual portfolo optmzaton s neffcent. Mult-obectve partcle swarm optmzaton s an optmzaton technque whch s sutable for solvng numerc optmzaton and yelds hgh qualty results. It s used n ths research to construct effcent fronters for portfolo optmzaton. The proposed method s evaluated usng daly stock total return ndex gross dvdends from Stock Exchange of Thaland between years 2006 and Its performance n actual tradng s compared wth the total returns net dvdend from SET, SET50, SET100 and SETHD, wdely used nvestment performance ndces. The results ndcate that the returns from the proposed method are better than the standard ndces n most nvestment perods.

10 References Clarke, R., De Slva, H., & Thorley, H. (2002). Portfolo constrants and the fundamental law of actve management. Fnancal Analysts Journal. 58(3), Coello, C. A., Toscano, P.G., & Salazar, L.M. (2004). Handlng multple obectves wth partcleswarm optmzaton. IEEE Transactons on Evolutonary Computaton, 8(3), Coello, C. A., Lamont, G. B., & Van Veldhuzen, D.A. (2002). Evolutonary Algorthms for Solvng Mult-Obectve Problems. Boston : Kluwer. Corazza, M., & Komlov, N. (2009). Pratcle Swarm Intellgence: an alternatve approach n Portfolo Optmzaton. Facolta d Economa, Unversta CA Foscar Veneza. Deb, K., Agrawal, S., Pratab, A., & Meyarvan, T. (2000). A Fast and Fast and Eltst Multobectve Genetc Algorthm: NSGA-II. Proc. Parallel Problem Solvng From Nature VI Conf, Eberhart, R. C., & Kennedy, J. (1995). Partcle Swarm Optmzaton. IEEE Internatonal Conference on Neural Networks, Goldberg, D. E. (1989). GenetcAlgorthm n Search, Optmzaton, and Machne Learnng. Addson Wesley. Markowtz, H.M. (1952). Portfolo Selecton. The Journal of Fnance, 7 (1), Markowtz, H.M. (1959). Portfolo Selecton. Effcent Dversfcaton of Investments. NewYork: John Wley. Mshra S. K., Panda G., & Meher S. (2009). Mult-obectve partcle swarm optmzaton approach to portfolo optmzaton. In Nature & Bologcally Inspred Computng, NaBIC 2009, World Congress, Moore, J., & Chapman, R. (1999). Applcaton of Partcle Swarm to Multobectve Optmzaton. Department of Computer Scence and Software Engneerng, Auburn Unversty. Raquel, C.R., & Naval Jr, P.C. (2005). An Effectve Use of Crowdng Dstance n Multobectve Partcle Swarm Optmzaton. In Proceedngs of the Genetc and Evolutonary Computaton Conference (GECCO-2005), Van den Bergh, F. (2006). An analyss of partcle swarm optmzers. Doctoral dssertaton, Unversty of Pretora. Contact emal: vryay@sso.go.th