SOLVIG CARDIALITY COSTRAIED PORTFOLIO OPTIMIZATIO PROBLEM BY BIARY PARTICLE SWARM OPTIMIZATIO ALGORITHM Aleš Kresta Klíčová slova: optmalzace portfola, bnární algortmus rojení částc Key words: portfolo optmzaton, bnary partcle swarm optmzaton Abstrakt Př řešení úloh optmalzace portfola se většnou využívají metody matematckého programování. Tyto metody však nemohou být použty, pokud zavedeme omezení počtu držených aktv. K řešení takto defnovaného problému je nutno použít jednu z mnoha heurstckých metod (genetcké algortmy, neuronové sítě nebo algrotmus rojení částc). V tomto příspěvku je využto bnárního algortmu rojení částc a metody kvadratckého programování př hledání efektvní množny řešení př optmalzac portfola. V článku jsou použty dvě množny vstupních dat. První množnu tvoří akce zahrnuté do ndexu Dow Jones Industral Average, druhou pak akce zahrnuté do ndexu Standard & Poor's 500. V závěru příspěvku jsou grafcky srovnány nalezené efektvní množny pro různá omezení počtu držených akcí. Abstract Mathematcal programmng methods domnate n the portfolo optmzaton problems, but they cannot be used f we ntroduce a constrant lmtng the number of dfferent assets ncluded n the portfolo. To solve ths model some of the heurstcs methods (such as genetc algorthm, neural networks and partcle swarm optmzaton algorthm) must be used. In ths paper we utlze bnary partcle swarm optmzaton algorthm and quadratc programmng method to fnd an effcent fronter n portfolo optmzaton problem. Two datasets are utlzed. Frst dataset conssts of the stocks ncorporated n the Dow Jones Industral Average, second dataset contans stocks from the Standard & Poor's 500. The comparson of found effcent fronters for dfferent lmtaton on the number of stock held s made at the close of the paper. Introducton Proper allocaton of the funds s nowadays gettng more and more mportant. Wth the ncreasng amount of the money fund managers admnster, ther responsblty s ncreasng and quanttatve approaches get more attenton than qualtatve. In the feld of the portfolo optmzaton the poneer work was Markowtz mean-varance model []. Assumng that assets returns follow a multvarate normal dstrbuton, we are concerned only n the portfolo expected return and varance. We are thus lookng for the portfolos wth maxmum expected return and mnmal varance. Ths Pareto effcent set of portfolos s called an effcent fronter (EF). Selecton of one optmal portfolo from the effcent fronter then depends only on the rsk atttude. 24
Ths relatvely smple model s easly solvable by the quadratc programmng methods. However for a practcal purpose, there are two major weaknesses: () there s an evdence that assumed multvarate normal dstrbuton of assets returns does not hold [2]; () n the real world, we are nterested n nteger constrant of a number of assets desred to hold, or to lmt the proporton of the assets nto a certan nterval. Lmtng the number of dfferent assets ncluded n the portfolo decreases transacton cost, whch can be very hgh utlzed result of the unconstraned model and buyng a small number of each asset. Also the montorng of news and frm s results s easer and so less costly wth few assets n the portfolo. One approach to deal wth ths could be the use of mean-absolute devaton [3] nstead of varance as a measure of rsk. Problem s then solvable by lnear programmng methods. However, ths approach has been showed to be nsenstve to some extremes. Second approach s to use some heurstc method such as the genetc algorthms, neural networks, smulated annealng or partcle swarm optmzaton [4-0]. These methods have showed good results. The goal of the paper s to exemplfy the use of bnary partcle swarm optmzaton algorthm n fndng the effcent fronter n non-lnear portfolo optmzaton problem. The paper s organzed as follows. Secton s focused on the portfolo optmzaton problem, both general and constraned verson are descrbed. In secton 2 the attenton to partcle swarm optmzaton algorthm as a used optmzaton method s gven. In secton 3 applcaton and ts results on the data from large-cap common stocks actvely traded n the Unted States are presented.. Descrpton of portfolo optmzaton problems Portfolo optmzaton conssts of a portfolo selecton problem n whch we want to fnd the optmum way of nvestng a partcular amount of the money n a gven set of securtes or assets [7]. Whle wthout any nformaton about the nvestor's rsk atttude we are not able to fnd one optmal portfolo, we are lookng for the Pareto effcent fronter. Ths fronter contans all possble portfolos for whch we are not able to lower the rsk wth the same level of the expected return or to heghten the expected return wthout ncreasng the rsk. The model n whch we consder smple portfolo optmzaton wthout any lmtaton on the maxmum number of the dfferent assets we want to hold wll be referred as general portfolo optmzaton problem (Problem P). The model n whch we lmt the number of dfferent assets held wll be referred as cardnalty constraned portfolo optmzaton problem (Problem P2). Problem P - General portfolo optmzaton problem Objectve functon: mn x j x j = j= σ, subject to: x = =, (C) = x E( R ) = E( R ), (C2) x [0,], =,...,. (C3) 25
In ths problem s the number of the avalable assets, x s the proporton of avalable amount nvested n asset, σ s the covarance between assets and j, E R ) s the expected j return of -th asset and E ( R ) s a desred expected portfolo return. The above descrbed mnmzng problem s easly solvable by the nonlnear (quadratc) programmng methods. The effcent fronter s constructed when solvng ths model for dfferent values of E ( R ). As any other model even the unconstraned portfolo optmzaton model has many premses, whch make t very smplfed. On the one hand t allows us to fnd the soluton easly; on the other hand the model could not be utlzed for a real world applcaton. For example, suppose we are lookng for the mnmal rsk portfolo, whch s compounded from the assets ncorporated n the Standard & Poor's 500 ndex. In fact t s not a bg problem to fnd such a portfolo but t would be nearly mpossble to nvest money nto t. It s due to hgh number of assets we must nvest to, whch stands for a hgh transacton costs. These transacton costs are generally dependent on the fact how often we rebalance the portfolo. For smplcty we wll abstract from these costs n the paper. So n the real world we want to lmt the number of assets we nvest n. Introducng K as the desred number of dfferent assets n the portfolo, we can extend the prevous model to the cardnalty constraned case (Problem P2). Problem P2 - Cardnalty constraned portfolo optmzaton problem Objectve functon: mn z = j= x σ x z, j j j ( subject to: = z x =, (C) = = z = K, (C2) x z E( R ) = E( R ), (C3) x [0,], =,...,, (C4) { 0,} z, =,,. (C5) All varables have the same meanng as n the Problem P and z s a bnary varable takng the value or 0 dependng on f -th asset s or s not ncluded n the portfolo. The constraned portfolo optmzaton problem s a mxed-nteger nonlnear (quadratc) programmng problem for whch the computatonally effectve algorthm does not exst [8]. To solve ths problem we can utlze: () algorthms for solvng mxed-nteger nonlnear programs (see [-5]); () some heurstc method such as partcle swarm optmzaton [4], smulated annealng [5, 6, 8], neural networks [7], genetc algorthms [8-0] and others [6-8]. In ths paper a bnary partcle swarm optmzaton s used to fnd the optmal z and the weghts are determned by the more standard quadratc programmng approach. 26
2. Descrpton of bnary partcle swarm optmzaton algorthm The orgnal partcle swarm algorthm was ntroduced and dscussed n [9, 20]. It mtates brds flockng and fsh schoolng as t s searchng n D-dmensonal real numbers space for the best poston. In ths algorthm the certan number of partcles s utlzed, each partcle's poston representng soluton of the problem. Partcles move across the search space partally randomly and partally n the dependence of the personal and global best poston dscovered so far. Objectve functon mtates the space rchness for food. So partcles are clearly determned by ther veloctes and postons: v t + ) = w( t) v ( t) + C ϕ p x ( t) + C ϕ g x ( ), () ( ) ( ) ( 2 2 t x ( t + ) = x ( t) + v ( t + ), (2) where v ( t +) s the velocty of the j-th partcle n the -th dmenson n the teraton t +, w (t) stands for momentum, C and C 2 are constants, ϕ and ϕ 2 are random numbers from nterval <-,>, p s the personal best poston found so far, g s the global best poston found so far, x (t) s the poston of the j-th partcle n the -th dmenson n the teraton t. The poston of the j-th partcle can be found as the sum of ts prevous poston x j (t) and the current velocty ( t +). The whole algorthm s shown n Fg.. After a partcles v ntalzaton, there s the loop, n whch the velocty and the poston are repeatedly updated accordng to () and (2). Swarm ntalzaton Start o Evaluate the performance of each partcle Compute new personal best Compute new global best Change the velocty vector Change postons of partcles umber of teraton acheved Yes End Fg. : Partcle swarm optmzaton algorthm dagram In [2] the partcle swarm algorthm was modfed to operate on a bnary varables and dscrete bnary verson of the partcle swarm algorthm was ntroduced (BPSO). There s n fact just one modfcaton partcle's velocty v now represents the probablty of x takng the value. Snce the probablty must be n the nterval [0,], a logstc transformaton of s used. The poston equaton (2) s then changed as follows: f ( rand() S( v )) then x = else x = 0, (3) < where rand () s a random number from nterval [0,], S( ) stands for the logstc transformaton functon: S( x) =. (4) x + e Equaton () for the velocty remans unchanged, but x (t), p and g are now not real numbers but take values only ether 0 or. v 27
In ths algorthm t s smple to mplement the constrant C2 n Problem P2 we repeat generatng random numbers and f they are smaller than S v ) then we change the correspondng portfolo K. x to untl x = ( s equal to the desred number of assets held n the 3. Applcaton part effcent fronter estmaton In ths secton the effcent fronter of constraned portfolo optmzaton problem defned by Problem P2 s searched. Used data conssts of two datasets, whch correspond to weekly prces between January 2002 and December 2007 for the assets ncluded n ndces Dow Jones Industral Average (DJI, yahoo fnance tcker ^DJI) and Standard & Poor's 500 (S&P, yahoo fnance tcker ^GSPC). Both datasets have been acqured from web server http://yahoo.fnance.com, [22]. For both datasets stocks wth the mssng values were elmnated - datasets thus consst of 30 and 465 stocks respectvely. Expected returns E R ) and covarance matrx σ j were estmated on weekly bass from downloaded tme seres (January 2002 to December 2007), thus we assume that the returns and covarances of assets wll reman unchanged. For each dataset we are lookng for the 500 dfferent optmal portfolos n dependence of the desred expected return E ( R ) and the desred number of assets K. The problem of fndng optmal portfolo s solved usng the BPSO and the quadratc programmng. The BPSO s used to fnd the optmal vector of logcal varables z whle the quadratc programmng s utlzed n the objectve functon to fnd the optmal weghts of the selected assets. The mplementaton of BPSO algorthm s wrtten n MATLAB, utlzng MATLAB s pre-defned functons. ( 3.. Dow Jones Industral Average Snce the DJI dataset contans only 30 assets, t s possble to compare the results of BPSO wth the results obtaned by tryng every possble combnaton of the assets held (brutal force method). Ths comparson s made on Fg. 2. We can see that the results are vsually the same. As showed n [8], effcent fronter of the cardnalty constraned model s a dscontnuous curve wth jumps. Ths means that some values of E ( R ) are not ratonal (snce there exst portfolos wth the same or lesser rsk and hgher return). The tme needed for the optmzaton s showed n Tab.. Tab. : Tme complexty of the brutal force method and BPSO algorthm The desred number of assets held (K) Brutal force BPSO 2 0.220 seconds.78 seconds 3 3.336 seconds 2.080 seconds 4 29.842 seconds 2.72 seconds As we can see for K=2 the BPSO s slower than tryng every possble combnaton of chosen assets. Ths s evoked by the small number of possble combnaton n the brutal force method. For K=3 the BPSO was qucker. For K=4 the BPSO was much qucker, usng only 0% of the tme needed by the brutal force method. 28
Fg. 2: Cardnalty constraned effcent fronter (CCEF) found by brutal force (left) and BPSO (rght) for K=2 (top), K=3 (mddle), K=4 (bottom) 3.2. Standard & Poor's 500 Because of the hgh number of the stocks n the S&P dataset the brutal force method s not utlzed for ts tme complexty. We consder only the BPSO algorthm, results of whch are shown n Fg. 3. For better clarty the fgure s zoomed and thus not the whole effcent fronters are shown. We can see that the effcent fronter for K=3 s not delneated very well. From 500 portfolos found we utlze only 46 to draw effcent fronter, we exclude every portfolo whch s not ratonal (whch has the same or bgger rsk and lesser return compared to other portfolos). For K=2 only 60 and for K=4 about 36 portfolos are used to draw the proper effcent fronter. Ths sgnfes that searchng for effcent fronter n Problem P2 when dealng wth vector of 465 stocks s hard to solve for BPSO algorthm. On the other hand the results are gven n the reasonable tme. Vsually CCEF for K=2 lays below CCEF for K=3, CCEF for K=3 lays below CCEF K=4. Ths s caused by the dversfcaton. Wth the ncreasng number of dfferent assets we nvest n, the rsk s decreasng whle the expected return stays the same. We can see that effcent fronters are very close to each other. Dfference between CCEF for K=2 and K=4 s around 0.% for varances below 0.0025. Ths means 0.% extra expected return f we allow nvestng nto 4 assets nstead of 2. 29
Fg. 3: Cardnalty constraned effcent fronters (CCEF) for S&P wth lmtng the number of dfferent stocks ncluded n the portfolo 4. Concluson Ths paper was focused on the problem of portfolo optmzaton. The cardnalty constraned mean-varance model was utlzed and solved usng the bnary partcle swarm optmzaton algorthm and the quadratc programmng. Two datasets were used. These datasets correspond to the weekly prces between January 2002 and December 2007 for the assets ncluded n the ndces Dow Jones Industral Average for the frst dataset and Standard & Poor's 500 for the second dataset. Datasets consst of 30 and 465 assets respectvely. For each dataset 500 dfferent optmal portfolos were found and the cardnalty constraned effcent fronter was drawn. The results were presented n secton 3. We can conclude that for smaller dataset DJI the bnary partcle swarm optmzaton outperform the brutal force method n terms of the tme complexty. The S&P dataset was a hard nut to crack for the proposed algorthm. But we can say that for ths dataset the algorthm gves reasonable result n reasonable tme. Acknowledgements The research has been elaborated n the framework of the IT4Innovatons Centre of Excellence project, reg. no. CZ..05/..00/02.0070 supported by Operatonal Programme 'Research and Development for Innovatons' funded by Structural Funds of the European Unon and state budget of the Czech Republc. The research was also supported by VŠB-TU Ostrava under the SGS project SP20/7. 30
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[20] KEEDY, J.; EBERHART, R. Partcle swarm optmzaton. In IEEE Internatonal Conference on eural etworks - Conference Proceedngs. 995. [2] KEEDY, J.; EBERHART, R. A dscrete bnary verson of the partcle swarm algorthm. In Proceedngs of the World Multconference on Systemcs, Cybernetcs and Informatcs. 997. p. 404-409. [22] Yahoo! Fnance - Busness Fnance, Stock Market, Quotes, ews [onlne]. 2009 [ct. 2009-09-0]. Dostupné z WWW: <http://fnance.yahoo.com/>. [23] ZMEŠKAL, Z.; TICHÝ, T.; DLUHOŠOVÁ, D. Fnanční modely. Praha: Ekopress, 2004. ISB 80-869-87-4. Classfcaton JEL: C6, G Ing. Aleš Kresta Katedra fnancí Ekonomcká fakulta VŠB-TU Ostrava Sokolská tř. 33, 70 2 Ostrava ales.kresta@vsb.cz 32
Appendx Pseudo code for BPSO algorthm for 500 dfferent R randomly ntalze all partcles evaluate all partcles fnd global best g and for each partcle personal best repeat for each partcle j compute new velocty accordng to Eq. for each dmenson f v < v mn then v = v mn for each dmenson f v > v max then v = v max x = 0 for each dmenson repeat untl = rand(, number of dmenson) f ( rand(0,) S( v )) then x = x = =K < evaluate partcle j end fnd new global best g and for each partcle new personal best untl number of teratons reaches 00 end p p 33