Using Harmony Search with Multiple Pitch Adjustment Operators for the Portfolio Selection Problem

Transcription

1 2014 IEEE Congress on Evolutonary Computaton (CEC) July 6-11, 2014, Beng, Chna Usng Harmony Search wth Multple Ptch Adustment Operators for the Portfolo Selecton Problem Nasser R. Sabar and Graham Kendall, Senor Member IEEE algorthms to fnd good qualty solutons n acceptable amount of tme. Heurstc and meta-heurstc algorthms can help nvestors n determnng a portfolo that can satsfy ther partcular demand [4]. Example of heurstcs and meta-heurstc algorthms that have been proposed for PS are: genetc algorthm [4], tabu search [4], smulated annealng [4], partcle swarm optmzaton [7] and evolutonary systems [8], [9], [10], [11], [12]. In ths work, we propose a Harmony Search Algorthm (HSA) for portfolo selecton. HSA s a populaton-based algorthm that mmcs the muscan mprovsaton process n solvng optmzaton problems [13]. In HSA, a new soluton s generated usng a memory procedure, whch s then perturbed usng a ptch adustment operator. The perturbaton step of HSA s analogous to the mutaton operator n genetc algorthms. Thus, as n genetc algorthms, where dfferent mutaton operators are suted to dfferent nstances or dfferent stages of the search, dfferent ptch adustment operators mght be needed to deal wth dfferent nstances or problem landscape changes [14], [15], [16]. Therefore, to enhance the effectveness of HSA, we propose an mproved HSA that utlzes multple ptch adustment operators n such a way that dfferent operators may be used at dfferent ponts of the search. The PS benchmark nstances [17] that have been adopted by other researchers are used to evaluate the performance of the proposed HSA. The results demonstrate the effectveness of the proposed HSA over other algorthms that have been presented n the scentfc lterature. Abstract Portfolo selecton s an mportant problem n the fnancal markets that seeks to dstrbute an amount of money over a set of assets where the goal s to smultaneously maxmze the return and mnmze the rsk. In ths work, we propose a harmony search algorthm (HSA) for ths problem. HSA s a populaton based algorthm that mmcs the muscan mprovsaton process n solvng optmzaton problems. At each teraton, HSA generates a new soluton usng a memory procedure whch consders all exstng solutons and then perturbs them usng a ptch adustment operator. To deal wth dfferent nstances, and also changes n the problem landscape, we propose an mproved HSA that utlzes multple ptch adustment operators. The ratonale behnd ths s that dfferent operators are approprate for dfferent stages of the search and usng multple operators can enhance the effectveness of HSA. To evaluate and valdate the effectveness of the proposed HSA, computatonal experments are carred out usng portfolo selecton benchmark nstances from the scentfc lterature. The results demonstrate that the proposed HSA s capable of producng hgh qualty solutons for most of the tested nstances when compared wth state of the art methods. I. INTRODUCTION Portfolo selecton (PS) s of nterest to researchers and practtoners, due to ts mportance n fnancal engneerng [1], [2], [3]. PS s concerned wth how to nvest a gven amount of money over a set of assets. The man goal s to maxmze the return and mnmze the rsk. Gven a set of assets, nvestors select a subset of those assets to form a sngle portfolo that could smultaneously maxmze ther return and mnmze the rsk. However, amng for hgher returns wll usually result n hgher rsk. Consequently, asset selecton s the most crucal part and determnng the best combnaton of assets s not a trval task. Furthermore, the rsk of a formed set of assets mght be less than an ndvdual asset [4], [5]. Markowtz [1], [2] ntroduced the mean varance PS model that takes nto consderaton the expected return and rsk of the formed portfolo. The Markowtz PS model s treated as a quadratc programmng problem and thus, when the number of assets becomes large, an effcent algorthm that can fnd the optmal soluton wthn a reasonable tme s not known to exst [5], [6]. Researchers have therefore resorted to heurstc and meta-heurstc II. PROBLEM DESCRIPTION Markowtz s mean varance model has been crtczed for consderng unrealstc assumptons that mght not exst n the real world [4], [5], [6]. Thus, some extensons and mprovements have been proposed n the lterature. A notable extenson s the constraned portfolo problem that nvolves cardnalty and boundary constrants that am to reduce the transacton costs and avod small/large holdngs. The cardnalty constrant restrcts the number of assets that can be ncluded n each portfolo. The boundary constrant restrcts the proporton of each asset n the formed portfolo wthn a lower and upper bound. In ths work, we consder the formulaton of the extended model that nvolves cardnalty and boundary constrants [4], [5], [6]: Nasser R. Sabar s wth The Unversty of Nottngham Malaysa Campus, Jalan Broga, Semenyh, Selangor, Malaysa (e-mal: Nasser.Sabar@nottngham.edu.my) Graham Kendall s wth The Unversty of Nottngham, UK and also wth The Unversty of Nottngham Malaysa Campus, Jalan Broga, Semenyh, Selangor, Malaysa (e-mal: Graham.Kendall@nottngham.edu.my) /14/$ IEEE n mnmze λ =1 Subect to =1 n w =1 499 n ww α =1 n + (1 λ ) w μ =1 (1) (2)

2 n s = K (3) = 1 ε s w δ s, = 1,..., n (4) s {0,1}, = 1,..., n (5) In ths work, the PS parameters are also set n ths step; the maxmum cardnalty and boundary constrants where n represents the total number of assets, w represents the proporton of the th asset, α s the connvance between th and th assets, λ s the rsk averson, λ=[0, 1], μ represents the expected return of the th asset, K represents the preferred nvested assets n a portfolo, s s a decson varable representng whether the th asset has been selected or not, and ε and δ respectvely represent the upper and lower bounds. The above equatons are treated as a mxed nteger programmng. There s no known effcent algorthm that can solve these models n reasonable tmes. Portfolo selecton can be consdered as a combnaton of two sub-problems: the problem of selectng the subset of assets to form a portfolo and the problem of decdng the proportons of the selected assets. Therefore, n ths work, we propose an mproved harmony search algorthm for PS. III. THE PROPOSED ALGORITHM The harmony search algorthm (HSA), proposed n [13], s a populaton-based stochastc search algorthm that mtates the muscal mprovsaton process. Smlar to other populaton-based algorthms, HSA operates on a populaton of solutons that s teratvely mproved over a number of generatons. At each generaton, HSA generates a new soluton usng three procedures; harmony memory consderaton, random consderaton and ptch adustment. Then the new soluton wll replace the worse one n the populaton f t s better n terms of qualty [18]. HSA has fve steps llustrated n Fgure 1 and descrbed below. Step 1: Intalze HSA parameters. Ths step s concerned wth settng the man parameters of the HSA, these beng: - Harmony memory sze (HMS), whch represents the populaton sze or the number of solutons to be stored n the harmony memory (HM). - Harmony memory consderaton rate (HMCR). Ths parameter s used durng the soluton generaton process whch decdes whether the components or the decson varables of the new soluton should be selected from the exstng ones n the HM or randomly created. HMCR takes a real value between zero and one. - The ptch adustment rate (PAR) takes a real value between zero and one, and s used to decde whether to adust the components that have been chosen from the HM. - The maxmum number of generatons or mprovsatons (MNI) represents the stoppng condton, based on the number of teratons. Fgure 1: HSA algorthm Step 2: Intalze the harmony memory (HM). HM contans a set of soluton and ts sze s equal to HMS. In ths step, HSA creates a set of the solutons usng ether a random or heurstc method and then adds them to the HM. To deal wth PS, n ths work, each soluton s represented by a two-dmensonal vector and the vector sze s equal to the total number of assets, n. Fgure 2 shows an example of PS soluton representaton, where the frst row of the vector represents the cardnalty whch takes ether 0 or 1, where 1 ndcates that the correspondng asset s selected, whle 0 ndcates non-selecton. The second row of the vector represents the boundary value of the selected asset whch takes a real value wthn the predefned boundary constrants. In ths work, the set of the HM ntal solutons are randomly generated by assgnng for each vector cell (soluton decson varable) of the frst row ether 0 or 1 whle makes sure that the maxmum cardnalty s respected. Next, we assgn for each of the selected asset, a real number wthn the predefned boundary constrants that s represented by the second row of the vector. Next, we calculate the ftness value of the created soluton usng Equaton (1) and add the soluton to the HM. We repeat ths procedure untl the number of generated solutons s equal to HMS. 500

3 The portfolo ndex N The cardnalty The boundary value Fgure 2: An example of the HSA soluton representaton for the PS Step 3: Generates or mprovses a new soluton. Ths step generates (mprovses) a new soluton from scratch accordng to HMCR and the PAR values usng the followng rules: - Memory consderaton rule. Ths rule frst creates an empty soluton (an empty vector) wth sze equal to the total number of assets, n. Then t loops through the soluton decson varables (vector cells) one by one and decdes to ether select the value of the current decson varable from the exstng solutons n HM or randomly set t accordng to the HMCR value. More precsely, ths rule generates, for each decson varable, a random number, r, between zero and one. Then, f r s less than HMCR, select one soluton from HM at random and set the current decson varable of the new soluton same as the correspondng decson varable of the selected HM soluton. Otherwse, the current decson varable s randomly ntalzed. Once a complete soluton s generated, we use Equatons (1-5) to check the constrant volatons (the maxmum cardnalty and boundary constrants) and the ftness value. - Ptch adustment rule (PAR). The ptch adustment rule further adusts the decson varable values that have been selected from the HM solutons accordng to the PAR value. Precsely, ths rule generates, for each decson varable that was selected from the HM solutons, a random number, r, between zero and one. If r s less than PAR, the value of ths decson varable wll be adusted by addng or subtractng a predetermned value from t. In ths work, the ptch adustment rule s responsble for two tasks. Frstly, snce the newly generated soluton s not guaranteed to be feasble because the cardnalty constrant mght be volated. The ptch adustment rule seeks to turn an nfeasble soluton nto a feasble one by randomly selectng one decson varable and then flppng ts value. That s, f the value of the selected decson varable s 0 t wll be changed to 1 and ntalze ts boundary; otherwse t wll be changed to 0. Ths process s repeated untl the nfeasble soluton becomes feasble. Secondly, the ptch adustment rule acts as a local search algorthm that seeks to further mprove the current soluton for a certan number of teratons usng multple adustment operators. We use multple adustment operators due to the fact that dfferent nstances have dfferent characterstcs that may requre dfferent operators to effectvely explore the search space. Thus by usng multple adustment operators we can utlze ther strengths to cope wth the landscape changes that may occur [15]. The proposed multple ptch adustment operators work as follows: take the current soluton as an nput and repeat the followng for a predefne number of teratons (we use 20 non-mprovng teratons fxed based on prelmnary experments), randomly select one decson varable and check ts value. If the value of the selected decson varable s 1, then randomly select one of the followng three operators to adust the boundary of the selected asset wthn the predefned range: A parameterzed Gaussan mutaton, N(0, σ2), where σ=0.5 s the standard dvson. Same as above but σ = 0.3. x = x + F *( x1, x2, ) where x s the current decson varable value, x 1, s the decson varable of the best soluton n the populaton, x 2, s the decson varable of the worst soluton n the populaton and F=0.1 [19]. Next calculate the ftness value of the adusted soluton usng Equaton (1). If the ftness value of the adusted soluton s better than the current one replace the current soluton wth the adusted soluton. Otherwse dscard t and start a new teraton. Step 4: Update HM. Ths step compares the ftness value of the newly generated soluton wth the worse one n HM. The worse soluton n HM wll be replaced by the new one f the new one has a better ftness value. Step 5: The termnaton condton. Ths step decdes whether to termnate HSA or start a new teraton. IV. EXPERIMENTAL SETUP Ths secton frst dscusses the characterstcs of the selected benchmark nstances followed by the parameter settngs of the proposed algorthm. The proposed algorthm s mplemented n Java usng a PC runnng Lnux Ubuntu OS wth 2.2 GHz Quad-Core processor and 2 GB RAM. A. Benchmark Instances The benchmark nstances that are avalable va the OR-lbrary [17] are used to evaluate the performance of the proposed algorthm aganst the state of the art methods. The benchmark has fve dfferent nstances that represent the weekly prces for fve dfferent countres. The man characterstcs of these nstances are presented n Table 1, where n s the total number of the assets, K the maxmum number of assets n a formed portfolo (cardnalty), ε (=1,, n) mnmum lmt of asset s proporton and δ (=1,, n) the maxmum lmt of asset s proporton [17]. 501

4 TABLE 1 THE CHARACTERISTICS OF PS BENCHMARK # Name Country n k ε δ 1- Hang Seng Hong Kong DAX 100 Germany FTSM 100 UK S&P 100 USA Nkke Japan B. Parameter settngs The proposed algorthm has a few parameters that need be set n advance and they were set based on a prelmnary experment. The utlzed parameter values are reported n Table 2. The parameter λ of Equaton (1) was tested usng 51 dfferent values and each value s tested for 1000*n ftness evaluatons (the same as n [4] and [7]). TABLE 2 THE PARAMETER SETTINGS # Name Value 1- Harmony memory sze, HMS Harmony memory consderaton rate, HMCR Ptch adustment rate, PAR Ptch adustment stoppng condton 20 non-mprovng teratons 5- Maxmum number of generatons, MNI 1000*n ftness evaluatons V. RESULTS AND COMPARISONS In ths secton, we frst evaluate the effectveness of usng multple ptch adustment operators wthn HSA. Therefore, we tested four dfferent HSA varants as follows: - MHSA: the proposed HSA that use multple ptch adustment operators (three operators). - HSA1: utlze the frst ptch adustment operator only. - HSA2: utlze the second ptch adustment operator only. - HAS3: utlze the thrd ptch adustment operator only. The results of the four HSA varants (MHSA, HSA1, HSA2 and HSA3) are compared usng a Wlcoxon test wth 0.05 crtcal level (the results of 31 runs). The p-value of MHSA versus HSA1, HSA2 and HSA3 s presented n Table 3, where + ndcates that the MHSA s statstcally better than the compared algorthm (p-value < 0.05), - ndcates that the compared algorthm s better than MHSA (p-value > 0.05) and = ndcates both algorthms have smlar performance (p-value=0.05). As can be seen from Table 3, MHSA s statstcally better than HSA1 and HSA2 on all tested nstances. MHSA s better than HSA3 on 3 out of 5 tested nstances, not statstcally sgnfcant on 1 nstance and performs the same as HSA3 on 1 nstance. These postve results ustfy the use of multple ptch adustment operators wthn HSA n order to deal wth varous nstances as well as the ssue of landscape changes. TABLE 3 THE P-VALUE OF THE COMPARED HSA VARIANTS MHSA vs. HSA1 HSA2 HSA3 # Name p-value p-value p-value 1- Hang Seng + + = 2- DAX FTSM S&P Nkke We now compare MHSA results wth the followng algorthms that have been proposed n the scentfc lterature: - Tabu search algorthm (TS) proposed n [4]. - Smulated annealng (SA) proposed n [4]. - Genetc algorthm (GA) proposed n [4]. - Partcle swarm optmzaton (PSO) proposed n [7]. As n [4] and [7], the average results of MHSA over 31 ndependent runs are compared wth TS, SA, GA and PSO based on the mnmum mean percentage error (MP%) whch s shown n Table 4, where the best obtaned results are hghlghted n bold font. As the table ndcates, the proposed MHSA obtaned the best results for 4 out of 5 tested nstances and beng slghtly nferor on one nstance (S&P 100 nstance). Consderng the average results (last row n Table 4), MHSA produced the best average results compared to other algorthms (GA, SA, TS and PSO). TABLE 4 THE RESULTS OF MHSA COMPARED TO OTHER ALGORITHMS # Name MHSA GA SA TS PSO 1- Hang Seng DAX FTSM S&P Nkke Average In Table 5 we compare the computatonal tme (seconds) of MHSA aganst the compared algorthms, where the best computatonal tme s ndcated n bold. As shown n Table 5, the computatonal tme of MHSA s lower than the other algorthms on all tested nstances. Gven the results presented n Tables 4 and 5, we can conclude that the proposed MHSA s an effectve soluton method for portfolo selecton as t has obtaned good qualty results for all tested nstances wthn a small computatonal tme compared to other algorthms. The results also demonstrate that the use of multple ptch adustment operators does assst HSA n obtanng good results for all tested nstances. TABLE 5 THE COMPUTATION TIME OF MHSA COMPARED TO OTHER ALGORITHMS # Name MHSA GA SA TS PSO 1- Hang Seng DAX FTSM S&P Nkke

5 VI. CONCLUSION Ths work has proposed a harmony search algorthm for the constraned portfolo selecton problem. Harmony search algorthm s a populaton-based algorthm that operates on a populaton of solutons and teratvely mproves them for a predefned number of teratons. Our proposed harmony search algorthm uses two types of ptch adustment procedures. The frst one ams to turn an nfeasble soluton nto a feasble one. Whlst, the second one s a local search algorthm that seeks to mprove the current soluton usng multple ptch adustment operators n order to deal wth dfferent nstance characterstcs as well as landscape changes that mght occur durng the search process. The performance of the proposed algorthm s valdated usng the constraned portfolo selecton problem benchmark nstances. The results demonstrate that the proposed algorthm outperforms other algorthms that have been proposed n the scentfc lterature, on 4 out of 5 tested nstances. The computatonal tme of the proposed algorthm s lower than other algorthms across all nstances. These results ndcate that the proposed algorthm s an effectve soluton method for the constraned portfolo selecton problem. REFERENCES [1] H. Markowtz, "Portfolo selecton*," The ournal of fnance, vol. 7, pp , [2] H. Markowtz, Portfolo selecton: effcent dversfcaton of nvestments: John Wley and Sons, New York, [3] H. Varan, "A portfolo of Nobel laureates: Markowtz, Mller and Sharpe," The Journal of Economc Perspectves, vol. 7, pp , [4] T.-J. Chang, N. Meade, J. E. Beasley, and Y. M. Sharaha, "Heurstcs for cardnalty constraned portfolo optmsaton," Computers & Operatons Research, vol. 27, pp , [5] R. Moral-Escudero, R. Ruz-Torrubano, and A. Suarez, "Selecton of optmal nvestment portfolos wth cardnalty constrants," n IEEE Congress on Evolutonary Computaton, CEC 2006., 2006, pp [6] A. Fernández and S. Gómez, "Portfolo selecton usng neural networks," Computers & Operatons Research, vol. 34, pp , [7] G.-F. Deng, W.-T. Ln, and C.-C. Lo, "Markowtz-based portfolo selecton wth cardnalty constrants usng mproved partcle swarm optmzaton," Expert Systems wth Applcatons, vol. 39, pp , [8] G. Kendall and Y. Su, "Imperfect evolutonary systems," Evolutonary Computaton, IEEE Transactons on, vol. 11, pp , [9] G. Kendall and Y. Su, "A partcle swarm optmsaton approach n the constructon of optmal rsky portfolos," n In Proceedngs of the 23rd IASTED Internatonal Mult-Conference Artfcal Intellgence and Applcatons, 2005, pp [10] G. Kendall and Y. Su, "Learnng wth mperfectons-a mult-agent neural-genetc tradng system wth dfferng levels of socal learnng," n 2004 IEEE Conference on Cybernetcs and Intellgent Systems, 2004, pp [11] G. Kendall, "A mult-agent based smulated stock market-testng on dfferent types of stocks," n The 2003 Congress on Evolutonary Computaton, CEC'03., 2003, pp [12] G. Kendall and Y. Su, "The co-evoluton of tradng strateges n a mult-agent based smulated stock market through the ntegraton of ndvdual learnng and socal learnng," n The 2003 Internatonal Conference on Machne Learnng and Applcatons (ICMLA'03), 2003, pp [13] Z. W. Geem, J. H. Km, and G. Loganathan, "A new heurstc optmzaton algorthm: harmony search," Smulaton, vol. 76, pp , [14] E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, et al., "Hyper-heurstcs: A survey of the state of the art," Journal of the Operatonal Research Socety, vol. 64, pp , [15] N. R. Sabar, M. Ayob, G. Kendall, and Q. Rong, "Grammatcal Evoluton Hyper-Heurstc for Combnatoral Optmzaton Problems," Evolutonary Computaton, IEEE Transactons on, vol. 17, pp , [16] N. R. Sabar, M. Ayob, R. Qu, and G. Kendall, "A graph colorng constructve hyper-heurstc for examnaton tmetablng problems," Appled Intellgence, vol. 37, pp. 1-11, [17] J. E. Beasley, "OR-Lbrary: dstrbutng test problems by electronc mal," Journal of the Operatonal Research Socety, vol. 41, pp , [18] M. Hadwan, M. Ayob, N. R. Sabar, and R. Qu, "A harmony search algorthm for nurse rosterng problems," Informaton Scences, vol. 233, pp , [19] R. Storn and K. Prce, "Dfferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces," Journal of global optmzaton, vol. 11, pp ,