IMPROVED PORTFOLIO OPTIMIZATION MODEL WITH TRANSACTION COST AND MINIMAL TRANSACTION LOTS

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1 Vol.7 No.4 (200) p73-78 Joural of Maageme Scece & Sascal Decso IMPROVED PORTFOLIO OPTIMIZATION MODEL WITH TRANSACTION COST AND MINIMAL TRANSACTION LOTS TIANXIANG YAO AND ZAIWU GONG College of Ecoomcs & Maageme Najg Uversy of Iformao Scece ad Techology Najg Cha ABSTRACT. Ths paper frs chages he bary objecve model o sgle objecve model by adopg lear weghed mehod. Whe sudyg he mmal rasaco los hs paper sudes he egral cosra ad he dffere mmal rasaco los. Whe sudyg he rasaco cos hs paper sudes he dffere rasaco cos rao. The paper he sudes he suao wh ew vesme ad he sof cosra. Fally he paper esablshes he porfolo opmzao model wh rasaco cos ad mmal rasaco lo ad coducs emprcal aalyss o he real daa of he Shagha sock marke. Keywords: Porfolo; Trasaco Cos; Mmal Trasaco Los. Iroduco. Markowz developed hs heory of porfolo allocao uder uceray 952. He pu forward ha he ules of porfolo s he fuco of expeced reur raes ad varace [3-4]. Usually hgh reur raes are accompaed by hgh varace. Whe varace s fxed vesors pursue reur raes as hgh as possble. Whe reur raes are fxed vesors pursue varace as low as possble. Raoal vesors maxmze her expeced ules by selecg effecve porfolo. I Markowz s mea-varace model covarace mus be calculaed of rsk asses ad he calculao s very dffcul. Sharp s capal asse prce model dvde rsk o sysem rsk ad o-sysem rsk. The model regards capal reur as rsk compesao. Sharp s sgle expoeal model decreases he calculao work. Koo ad Yamazak [2] pu forward he absolue varace ca be ulzed o measure rsk ad aalyzed he Tokyo sock marke. Markowz s classc mea-varace model eglecs some mpora facors he vesme pracce such as he lmao of mmal rasaco los ad rasaco expese. I rece years some porfolo opmzao models pu forward by Mas ad Graza cosder he mmal rasaco los [56] ad he vesme formg process s close o pracce suao. The exsed resuls dcae he mea-varace model wh mmal rasaco los ad rasaco expese s a egral programmg model wh src cosra. Whe he model oly cosders mmal rasaco los ad does o cosder rasaco expese he model s a NP-hard problem []. Whe cosder mmal rasaco los Mas argued wheher he soluo ca be foud or o depeded rsk fuco bu he adoped he mea-absolue error model pu forward by Kooad Yamazak. Because Mas dd o adop Markowz s mea-varace model covarace bewee dffere asses were o cosdered ad here exsed esmao rsk. Ths paper cosder dffere facors he sock marke such as he lmao of

2 74 Joural of Maageme Scece & Sascal Decso Dec.200 vesme value he mmal rasaco los rasaco expese cos ad share allome. Fally hs paper esablshed a mproved porfolo model wh mmal rasaco los ad rasaco cos. 2. Markowz s Mea-Varace Model. Markowz s porfolo heores ca be expressed as he followg quadrac programmg. s. m 2 σ x = j= xxσ j j xr = r () = r s he expeced where s he umber of asses x s he proorao rao of h asse 2 reur rae of h asse r s he expeced reur rae of porfolo σ s he varace of porfolo ad σ s he covarace of he reur rae of h asse ad j h asse. The double objecve programmg problem of mmzg he varace ad maxmzg he reveue ca be rasformed o he followg parameers programmg. max ( λ) xr λ xxσ j j j= s.. x = (2) x 0 = 2... Where λ s rsk averso coeffce. The more λ s he more rsk averso s. 0 λ. 3. The Porfolo Model wh Mmal Trasaco Los ad Trasaco Cos. I he praccal rasaco process he mmal rasaco los usually exs. Le he mmal rasaco los C of h sock ca be expressed as c = Np (3) Nb whe k + k N Ns whe k + < k Where N s he mmal rasaco los of h sock N b s he mmal umber of h sock for he buyer N s s he mmal umber of h sock for he seller ad p s he prce of h sock. Whe cosder he mmal rasaco los he vesme weghs of he model are adjused as x kc / I (4) Where k s he u of h sock I s he upper lmao of he vesme. Whe we adjus he porfolo he rasaco cos wll occur. Take h sock for

3 IMPROVED PORTFOLIO OPTIMIZATION MODEL 75 example. Le x ad x + be he vesme proporo of h perod ad + h perod. TC + s he rasaco cos of h sock of + h perod so TC = d x x (5) Where d + s rasaco cos rae of h sock of + h perod. Assume ha buyg sock ad sellg sock have dffere rasaco cos raes. Le m b be rasaco cos rae of buyer ad le m s be rasaco cos rae of seller so mb whe k + k d + (6) ms whe k + < k The oal rasaco cos of he porfolo a + h perod ca be expressed as TC = d x x (7) Uder he lmao codo of mmal rasaco los whe we adjus he porfolo he oal rasaco cos of he h sock ca be expressed as TC = d k k c (8) Where d + s he rasaco cos rae of h sock of + h perod. k ad k + are he umbers of he vesme respecvely of h perod ad + h perod. c + s he mmal rasaco lo. The oal rasaco cos of he porfolo ca be expressed as TC = d k k c (9) If we requre he oal vesme s more ha 95% of he upper lm of he vesme so he porfolo opmzao model wh mmal rasaco los ad rasaco cos ca be expressed as d. k k. c max ( λ) x r λ x xj σ = j + j= I+ s.. x + x + = kc / I+ c = Np x 0 = k N I k c I

4 76 Joural of Maageme Scece & Sascal Decso Dec.200 Where N whe k k N N whe k < k m whe k + k d + m whe k + < k m s he rasaco cos rae of he buyer. b he seller. u of h sock of + h perod. b + s + b s m s s he rasaco cos rae of k + s he vesme k s he vesme u of h sock of h perod. I + s he upper lm of he vesme of + h perod. c + s he mmal rasaco lo of h sock. share of h sock. p s he prce of h sock. N s he mmal rasaco r + s he expeced reur rae of of h sock of + h perod. σ j s he covarace of reur rae bewee h sock ad j h sock of h perod. 4. Posve Aalyss. I our posve aalyss we selec 8 socks Shagha sock marke. The sock codes are ad Ulze Dazhhu sofware o dowload he hsory rasaco daa of every moh from Jeuary 200 o March The reur raes are calculaed accordg o he followg formula. (0 + a) P /0 P + D r = (0) P Where r s he reur rae of he sock h moh. a dcaes every 0 shares are alloed a shares h moh. P s he close prce of he las rasaco day h moh. P s he close prce of he las rasaco day h moh. dvded h moh. The covarace marx of reur raes are show Table. TABLE. The covarace marx of reur raes D s he Assume ha he average reur rae of he hsory mohs be he reur rae + perod ad he average close prce of he hsory mohs be he close prce + perod. Le k = 5 ( = 2...8). Le λ = 0.5. Adop he close prce of he frs rasaco day Aprl Assume ha we buy 5 u sock for every sock. Namely we buy 500 shares

5 IMPROVED PORTFOLIO OPTIMIZATION MODEL 77 for every sock. Le he upper lm of he vesme be RMB6000. The moh reur raes ad prces of 8 socks are show Table 2. TABLE 2. The moh reur raes ad prces of 8 socks Codes Reur rae prce Le d N + 00 whe k + 5 whe k + < whe k whe k + < 5 () (2) Solve he programmg wh lgo0.0 ad oba dffere reur raes ad varace correspodg o dffere λ Table 3. TABLE 3. Reur raes ad varace correspodg o dffere λ λ (kk2...k8) Reur raes varace 0 ( ) ( ) ( ) ( ) ( ) From Table 3 we ca see ha whe he reur rae creases varace creases. Because we cosder he mmal rasaco lo whe λ = 0 ad λ = 0.2 he reur raes ad varace are equal. 4. Coclusos. Ths paper adoped he lear weghed mehod o rasform he double objecve programmg o sgle objecve programmg. Cosder he mmal rasaco los ad rasco expese hs paper esablshed he comprehesve o-lear programmg. Fally hs paper adoped he real daa he Shagha sock marke o make posve aalyss. Ackowledgme. Ths paper s suppored by Humaes ad Socal Sceces Foudao of Msry of Educao of Cha (09YJC63029) ad Phlosophcal ad Socal Scece Foudao of Hgher Educao of Jagsu Provce of Cha (09SJD630059). REFERENCES [] H. Kellerer R. Mas ad M.G. Speraza (2000) O selecg a porfolo wh fxed coss ad mmum los Aals of Operaos Research vol.99 o.3 pp

6 78 Joural of Maageme Scece & Sascal Decso Dec.200 [2] H. Koo ad H. Yamazak (99) Mea-varace devao porfolo opmzao model ad s applcaos o Tokyo sock marke Maageme Scece vol.37 o.5 pp [3] H. Markowz Porfolo Seleco (952) Joural of Facevol.3 o.7 pp [4] H. Markowz (959) Porfolo Seleco: Effce Dversfcao of Ivesme New York: Wley. [5] R. Mas ad M.G. Speraza (999) Heursc algorhms for he porfolo seleco problem wh mmum rasaco los Europea Joural of Operaoal Researchvol.4 o.4 pp [6] R. Mas (997) Mxed Ieger Lear Programmg Models for Facal Problems: Aalyss Algorhms ad Compuaoal Resuls Ph.D. Thess Uversy of Bergamo.