Heuristic optimization of complex constrained portfolio sets with short sales

Transcription

1 Heurstc optmzaton of complex constraned portfolo sets wth short sales G A Vjayalakshm Pa Dept of Math. & Computer Applns. PSG College of Technology Combatore, INDIA pagav@mca.psgtech.ac.n Therry Mchel Tactcal Asset Allocaton and Overlay Lombard Oder Darrer Hentsch Pars, FRANCE Therry.Mchel@lodh.com Machne Intellgence Research day, Inda Workshop, Nagpur, Jan 24, 2009

2 Portfolo Optmzaton Outlne Computatonal Intellgence based soluton methods K-means cluster analyss for portfolo optmzaton Evoluton Strategy wth Hall of Fame desgn and mplementaton Dfferental Evoluton (rand/1/bn)- desgn and mplementaton Expermental Studes and Results Conclusons 2

3 Portfolo Optmzaton A fnancal portfolo s a basket of tradable assets such as bonds, stocks, securtes etc The problem of portfolo optmzaton ncorporates the twn objectves of maxmzng return and/or mnmzng rsk Obtan the optmal weghts Trace the effcent fronter whch s a rsk-return return tradeoff curve 3

4 Effcent fronter Tokyo Stock Exchange Bombay Stock Exchange Nkke225 dataset BSE200 dataset (March 2002-March 2007) (July 2001-July 2006) Expected portfolo annual return(%) Annualzed rsk(%) Expected portfolo annual return(%) Annualzed rsk (%) 4

5 Mathematcal formulaton mnmze N N WWjj (1 ) 1j 1 W Subject to N 1 0 W W 1 [0,1] 1, 1,2,... N (basc constrants) Markowtz Mean-Varance model 5

6 Markowtz (1952) framework assumed a perfect market For such markets, the mathematcal formulaton reduced to a quadratc optmzaton problem and tradtonal methods of soluton could be drectly appled In realty, portfolo selecton problems are dffcult to solve especally when market frctons, nvestor preferences are consdered The mathematcal model calls for augmentng the objectve functon wth several constrants, renderng the optmzaton problem complex enough for drect solvng by analytcal methods. 6

7 Incluson of cardnalty constrant Emprcal fndngs have revealed that nvestors prefer to nvest n rather a lmted number of assets,, consderng the fact that transacton costs, portfolo management fees and nformaton gatherng costs are dependent on the number of assets that are ncluded for nvestment. To acheve ths the cardnalty constrant calls for the ncluson of only K assets out of a unverse of N assets, for a pre-specfed value of K chosen by the nvestor. 7

8 Mathematcal formulaton turns complex! mn N N WWjj (1 ) 1j 1 W Subject to N 1 0 W W 1 [0,1] 1, 1,2,... N (basc constrants) N 1 Z K where Z 1 0, f W 0 otherwse (cardnalty constrants) 8

9 Computatonal Intellgence based soluton methods A brute force approach callng for a choce from dfferent alternate solutons to the problem N CK combnatoral exploson for large values of N [Farrell (1997)] groupng avalable assets nto asset classes based on the ndustry, sze, geographcal aspects etc and makng a selecton from each of these classes nferor solutons largely due to gnorng ng the correlatons between the assets 9

10 Sngle agent local search algorthms such as Smulated Annealng and Threshold Acceptng (TA) (Chang et al., 2000; Wnker 2001) perls of gettng stuck n local optma Mult-agent methods such as Ant systems, Genetc algorthms and Evolutonary algorthms or hybrd search methods (Marnger, 2005) near optmal or mxed results 10

11 [Fernandez and Gomez (2007)] Hopfeld neural networks for the soluton of the problem no one heurstc approach outperformed others n all knds of nvestment polces Pa and Mchel (2007, 2009) k-means cluster analyss to tackle cardnalty constrant 11

12 k-means cluster analyss for portfolo optmzaton - a bref revew Cluster analyss could be vewed as an exploratory data analyss tool whch can classfy objects nto clusters dependng on the maxmum degree of assocaton shared by objects belongng to the same cluster. k-means clusterng unlke other clusterng technques addresses the need when exactly K dfferent clusters wth the greatest possble dstncton are requred for a pre-specfed value of K. 12

13 Observatons k-means clusterng though prmarly adopted to elmnate the cardnalty constrant yelds a smplfed mathematcal model whch s amenable for drect solvng by tradtonal optmzaton methods such as Quadratc Programmng, besdes heurstc strateges The relablty [ Tola, 2005] of k-means clustered portfolo sets turn out to be better than those obtaned by the Markowtz and RMT fltered counterparts 13

14 For large K, the effcent fronters traced by the k-means clustered portfolo sets were found to be n close proxmty to the exact or deal effcent fronter. Heurstc approaches are senstve to the number of desgn varables n the problem. k-means clusterng promotes dmensonalty reducton whch could be put to ts best use by the approaches concerned, thereby leadng to faster convergence. 14

15 An nvestable unverse I j j j { a 1 K, a C } j defned as a set of K assets a each of whch s randomly or preferentally chosen from cluster C, 1 K s extracted 15

16 Mathematcal formulaton turns complex! mn N N WWjj (1 ) 1j 1 W Subject to N W 1 1 [0,1] (basc constrants) N 1 Z K where Z 1 0, f W 0 otherwse (cardnalty constrants) 16

17 , Incluson of other constrants a a W 3 K ; b, 1,2,... N 3 b 1 K (Short sales) For each class of assets j j W 0 1 j j j j (class constrants) 17

18 Heurstc Portfolo Optmzaton process Mn( rsk) and Max(return) K-means cluster analyss Subject to Basc constrants Cardnalty constrant Short sales Class constrants Optmal weghts Heurstc Optmzaton technques Weght standardzaton algorthms Mn( rsk) and Max(return) Subject to Basc constrants Short sales Class constrants Expected portfolo annual return(%) Annualzed rsk(%) Effcent fronter 18

19 Heurstc Portfolo Optmzaton strateges Evoluton Strategy wth Hall of Fame Dfferental Evoluton (rand/1/bn) Quadratc Programmng 19

20 Evoluton Strategy wth Hall of Fame: Desgn and mplementaton Parent Populaton Arthmetc varable pont cross over Real number unform mutaton (Andrzej Osyczka, 2002) New populaton s : u Offsprng Populaton Hall of Fame 20

21 Each chromosome n the populaton comprses of K genes representatve of the portfolo weghts that need to be optmzed. Durng each generaton the chromosomes undergo a weght standardzaton The weght standardzaton ensures that each chromosome n each populaton represents a feasble soluton, before they compete among themselves and the one n the Hall of Fame to determne the best soluton. The objectve functon of the portfolo optmzaton problem yelds the ftness value that s used to rank chromosomes That chromosome reportng the mnmum ftness value s declared as the best ft chromosome 21

22 Dfferental Evoluton (rand/1/bn): Desgn and mplementaton Parent Populaton Mutaton operator Tral vector Populaton Determnstc selecton Offsprng Populaton Bnary Cross over operator New populaton 22

23 Expermental Studes and Results Portfolo optmzaton problem defnton over BSE200 data set cardnalty constrant K=20 short sellng bounds (-0.15, +1.15) N=200 assets n the BE200 data set Class constrants: Banks: 0.01 w t t 0.3 Technology: 0.01 w s s 0.4 Ol and Gas: 0.01 wr r

24 Table 1: Composton of assets n the Investable unverses (K =20) of BSE200 data set, satsfyng the class constrants Number of Assets n the chosen asset class and ther respectve class constrants Banks Technology Ol and Gas Investable unverse 1 Investable unverse 2 Investable unverse

25 Table 2 Choce of parameters defnng the Evoluton Strategy wth Hall of Fame durng the expermental studes Parameter Values set Chromosome length (number of genes) K = 20 Populaton sze 100 Generatons 2000 Parent, chld chromosomes rato n a generaton (1:2) Choce of values for the Rsk averson 18 ponts parameter λ to graph the effcent fronter 25

26 Table 3 Choce of parameters defnng Dfferental Evoluton (rand/1/bn) durng the expermental studes Parameter Values set Chromosome length (number of genes) K = 20 Populaton sze 200 Generatons 1000 Scalng factor β 0.5 Probablty of recombnaton 0.87 Choce of values for the Rsk averson parameter λ to graph the effcent fronter 18 ponts 26

27 Experment 1 Effcent fronters for Short sellng and Class constraned Investable unverses of BSE200 data set (K=20) usng Quadratc Programmng 27

28 Experment 2 Effcent fronters traced by Evoluton Strategy wth Hall of Fame for varous runs on an Investable unverse of the BSE200 (July 2001-July 2006) data set 28

29 Experment 3 Effcent fronters traced by Dfferental Evoluton (rand/1/bn) for varous runs on an Investable unverse of the BSE200 (July 2001-July 2006) data set 29

30 Experment 4 Effcent fronters traced by the Evoluton Strategy wth Hall of Fame, Dfferental Evoluton (rand/1/bn) and Quadratc Programmng for a specfc nvestable unverse of BSE200 (July 2001-July 2006) data set 30

31 Data Envelopment Analyss (DEA) of optmal portfolos Data Envelopment Analyss (DEA) s a non-parametrc, determnstc methodology to asses the relatve effcences and performances of a collecton of comparable enttes called Decson Makng Unts (DMUs) whch transform nputs to outputs. DEA determnes the effcency scores of each DMU relatve to the others and makes use of lnear programmng to compute the effcency scores. To obtan the effcency scores of the optmal portfolos each rsk return couple s chosen to represent a DMU. The effcent fronters graphed n Experment 4 were the source nputs to the DEA. 31

32 Summary of the DEA of constraned optmal portfolos obtaned by ES HoF and DE(rand/1/bn) for the BSE200 data set Portfolo optmzaton method Evoluton Strategy wth Hall of Fame Dfferental Evoluton (rand/1/bn) Number of DMUs (n) consdered by the DEA Average effcency score (µ) Standard devaton of the effcency scores (σ) Coeffcent of Varaton (%)

33 Conclusons k-means clusterng promotes dmensonalty reducton whch could be exploted by the heurstc approaches n reducng the desgn varables and consequently leadng to faster convergence the smplfed problem model obtaned after k-means clusterng s amenable for soluton by both tradtonal and heurstc methods Both the heurstc methods yelded consstent results rrespectve of the nvestable unverses 33

34 Conclusons (Contd..) A Data Envelopment Analyses of the optmal portfolos obtaned by the two competng heurstc approaches revealed that the optmal portfolos obtaned by DE(rand/1/bn) approach were robust n comparson to those obtaned by ES HoF Statstcal nferences drawn wth regard to the techncal effcences ences of the optmal portfolos, obtaned by ES HoF and DE(rand/1/bn) concluded that there was a sgnfcant dfference n the means of the effcency scores of the two methods and hence these are dstnctvely tvely dfferent n behavor. 34

35 References Chang T J, N Meade, J B Beasley and Y M Sharaha,, Heurstcs for cardnalty constraned portfolo optmzaton, Computers and Operatons Research,, 27: , 1302, Marnger Detmar, Portfolo management wth heurstc optmzaton, Sprnger, Pa Vjayalakshm G A and Therry Mchel, On measurng the relablty of k-means k clustered fnancal portfolo sets for cardnalty constraned portfolo optmzaton, n Mathematcal and Computatonal Models, (eds.) R Nadarajan,, R Antha and C Porkod,, pp , 323, Narosa Publshng House, Pa Vjayalakshm G A and Therry Mchel, Evolutonary optmzaton of constraned k-means clustered assets for dversfcaton n small portfolos, IEEE Trans. On Evoluton Computaton,, 2009 (to appear) 35

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