The Teth Iteratioal Symposium o Operatios Research ad Its Applicatios (ISORA 2011 Duhuag, Chia, August 28 31, 2011 Copyright 2011 ORSC & APORC, pp. 195 202 Liear Programmig for Portfolio Selectio Based o Fuzzy Decisio-Makig Theory Hog-Wei Liu School of Iformatio, Beijig Wuzi Uiversity, Beijig 101149, Chia Abstract I this paper, portfolio selectio i crisp ad fuzzy cases is studied respectively, ad correspodig model ad algorithms i both case are proposed. I two models, the risk is take as the sum of the absolute deviatio of the risky assets i stead of covariace, the trasactio cost is take as v-shaped fuctio of the differece betwee the existig ad ew portfolio. A efficiet way is give to trasform a optimal problem with o-liear objective fuctio or o-liear costrait ito a liear problem, which alleviate the computatioal difficulty greatly. The ivestor s subjective impact is reflected i the model of the fuzzy decisio-makig eviromet. Compariso ad aalysis of the two models is give via a umerical example which has bee used i Markowitz s paper [2]. Keywords portfolio selectio; fuzzy sets; trasactio cost; liear programmig; optimizatio 1 Itroductio Fluctuatio i stock market is upredictable ad it is radom i ature. This is a difficult task to achieve without plaig ad evaluatig ivestmet alteratives. The portfolio must icorporate what the ivestor believes to be a acceptable balace betwee risk ad reward. Markowitz s mea-variace model of portfolio selectio [1, 2] is oe of the best kow models i fiace ad uaimously recogized to cotribute i the developmet of moder portfolio theory. It explores how risk-averse ivestors ca costruct optimal portfolio assets takig ito cosideratio the trade-off betwee expected returs ad market risk. Portfolio selectio issue cotiuously gaiig a iterest amog scholars[3, 4, 14, 11]. Sice the computatioal difficulty of covariace, Markowitz idea o the mea-variace approach the beig expeded by may researchers such as Sharpe, Mossi, ad Liter. The moder portfolio theory the evolved to Capital Asset Pricig Theory[13] whe risk free rate asset was icluded ito the portfolio ad the evolved to Arbitrage Pricig Theory i which the computatio was largely reduced. Koo&Yamazaki proposed absolutely mea-variace deviatio as risk fuctio from aother perspective to reduce the model ad got efficiet result[3]. Furthermore, the Markowitz model is too basic from practical poit of view ad igores may costraits faced by real-world ivestors: tradig limitatios, size of portfolio, trasactio costs, etc[4, 5, 12, 10]. Ivestmet strategies may be theoretically very profitable before takig ito accout trasactio costs ad taxatio E-mail address: ryuhowell@163.com.
196 The 10th Iteratioal Symposium o Operatios Research ad Its Applicatios issue, but the situatio ca become worse (such as iefficiet portfolio ad completely differet whe these last costraits are icorporated. Ay realistic ivestmet portfolio selectio must support trasactio costs amog other practical limitatios. Several papers dealt with the problem usig both quatitative ad qualitative aalysis methods. Oe of the hot research topics i this area is the use of fuzzy set theory. Fuzzy set theory[6] is a powerful tool used to describe a ucertai eviromet with vagueess, ambiguity or some other type of fuzziess, which appears i may aspects of fiacial markets. Studies by Taaka et. al. [8],[9],[7], Wag et al. [15], Bilbao-Terol et al. [16], Vercher et al. [17], Li & Liu, [18] ad Li & Xu, [19] show that the fuzzy approach also applicable i portfolio selectio. I order to be easily applicatio, this paper maaged to propose a model i which cosiderig trasactio cost for avoidig iefficiet portfolio firstly, usig absolute deviatio istead of variace secodly ad lastly formulatig the oliear programmig to liear programmig. 2 Portfolio selectio model uder crisp case Suppose that a ivestor chooses x i, the proportio ivested i asset i, 1 i for assets. The costraits are x i = 1 ad x i 0,i = 1,2,. The retur R i for the ith asset, 1 i, is a radom variable, with expected retur r i = E(R i. Let R = (R 1,R 2,,R T,x = (x 1,x 2,,x T ad r = (r 1,r 2,,r T. I this paper the trasitio cost for the ith asset c i employs v-shape fuctio, that is c i = k i x i x 0 i, i = 1,2,, (1 where x = (x 0 1,x0 2,,x0 T is a give assets ad k i 0 the trasitio cost for the uit of ith asset. So the total trasitio cost is described ad the total retur is R(x = E [ R i x i ] c i = thus the total risk ca be give as follow: V (x = k i x i x 0 i (2 k i x i x 0 i = E (R i E(R i x i = r i x i k i x i x 0 i (3 d i x i (4 where d i = E (R i E(R i x i. I geeral, ivestors expect maximizig returs ad miimizig risk at the meatime.
Liear Programmig for Portfolio Selectio 197 It ca be formulated mathematically as two-objective Programmig Model max R(x = r i x i k i x i xi 0 mi V (x = d i x i s.t. x i = 1,x i 0,i = 1,, (5 Weighted sum approach to simplify the multi-objective problems, we get the followig parametric programmig max (1 λ( r i x i k i x i xi 0 λ d i x i x i = 1 s.t. x i 0,i = 1,, (6 where λ [0,1] is called risk-aversio factor. The greater value λ is, the more awareess of risk aversio. Theorem 1. x = (x 1,x 2,,x is a optimal solutio of the model (6 if ad oly if there exist (y 1,y 2,,y such that (x 1,x 2,,x ;y 1,y 2,,y is a optimal solutio of the followig programmig: max (1 λ( y i + x i xi 0 0 y i x i + xi 0 0 s.t. x i = 1 x i 0,i = 1,, r i x i k i y i λ d i x i Proof. Suppose that x = (x1,x 2,,x is a optimal solutio of the model (6, let y i = xi x0 i. It is obvious that (x 1,x 2,,x ;y 1,y 2,,y is a feasible solutio of (7. It eed prove that (x1,x 2,,x ;y 1,y 2,,y is a optimal solutio of (7. Let x = (x 1,x 2,,x ;y 1,y 2,,y is ay feasible solutio of (7, the x = (x 1,x 2,,x is a feasible solutio of (6. Sice x = (x1,x 2,,x is a optimal solutio of the model (6, we have (1 λ( r i xi k i xi x0 i λ d i xi (1 λ( (1 λ( r i x i k i x i xi 0 λ d i x i r i x i k i y i λ d i x i (7 (8
198 The 10th Iteratioal Symposium o Operatios Research ad Its Applicatios thus (1 λ( (1 λ( r i xi k i y i λ d i xi r i x i k i y i λ (9 d i x i that is, (x 1,x 2,,x ;y 1,y 2,,y is a optimal solutio of (7. O the cotrary, let (x 1,x 2,,x ;y 1,y 2,,y be a optimal solutio of (7. We prove that x = (x 1,x 2,,x is a optimal solutio of the model (6. Obviously, x = (x 1,x 2,,x is a feasible solutio of the model (6 ad y i x i x0 i. If x = (x 1,x 2,,x is ot optimal solutio of the model (6, there exists a feasible solutio x = (x 1,x 2,,x of (6 such that (1 λ( > (1 λ( r i x i k i x i xi 0 λ d i x i r i xi k i xi x0 i λ d i xi let y i = x i xi 0,i = 1,2,,, the x = (x 1,x 2,,x ;y 1,y 2,,y is a feasible solutio of (7 ad (1 λ( r i x i k i y i λ d i x i = (1 λ( > (1 λ( (1 λ( r i x i k i x i xi 0 λ d i x i r i x i r i xi k i y i k i xi x0 i λ d i xi λ d i xi (10 (11 From the above discussio, we ca see that it ca make the complex portfolio selectio simplify if properly costructig the risk fuctio ad simplifyig the portfolio selectio model. 3 Portfolio selectio model uder fuzzy case I a ivestmet, the kowledge ad experiece of experts are very importat i a ivestor s decisio-makig. Due to complexity ad u-predictio i fiacial markets, it is difficult to give the precise expected value about the risk ad retur, thus it ca be take risk ad retur for grated as two fuzzy objectives. Sice a ivestor ca accept the retur greater tha some level ad accept the risk less tha some level of that, the membership fuctio of two fuzzy objectives µ max ad µ mi ca be give by
Liear Programmig for Portfolio Selectio 199 µ max (x = 0, r i x i r i x i k i x i xi 0 S 0, S 0 S 1 S 0 1, µ mi (x = 0, k i x i x 0 i S 0 r i x i k i x i xi 0 S 1 r i x i k i x i xi 0 S 1 d i x i T 0 T 0 d i x i, T 1 d i x i T 0 T 0 T 1 1, d i x i T 1 where S 0,S 1,T 0,T 1 are give by the ivestor. By itroducig the variable µ ad from the theory of fuzzy set ad fuzzy programmig, we ca costruct the followig programmig max µ s.t. r i x i k i x i xi 0 S 0 µ S 1 S 0 T 0 d i x i µ T 0 T 1 x i = 1 x i 0,i = 1,, Theorem 2. x is a optimal solutio of (14 if ad oly if there exists y such that (x,y is a optimal solutio of the followig programmig max µ s.t. r i x i k i y i S 0 µ S 1 S 0 T 0 d i x i µ T 0 T 1 y i + x i xi 0 0 y i x i + xi 0 0 x i = 1 x i 0,i = 1,, (12 (13 (14 (15
200 The 10th Iteratioal Symposium o Operatios Research ad Its Applicatios Proof. It is similar to the proof the theorem 1. We omit it here. 4 Numerical example ad Coclusio I this sectio, we will give a umerical example to illustrate the proposed portfolio selectio model (7 ad (14. We suppose that the ivestor cosiders the stock portfolio selectio i Markowitz s paper [2] where the data as show i the table 1. Table 1: The retur of America Tabacoo, AT&T, Uited Stats, Geeral Motors, Atchiso&Topeka&Sata Fe, Coca-Cola, Borde, Firestoe ad Sharo Steel(1937-1954 Year # 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9 1937-0.305-0.173-0.318-0.477-0.457-0.065-0.319-0.4 ąą -0.435 1938 0.513 0.098 0.285 0.714 0.107 0.238 0.076 0.336 0.238 1939 0.055 0.2-0.047 0.165-0.424-0.078 0.381-0.093-0.295 1940-0.126 0.03 0.104-0.043-0.189-0.077-0.051-0.09-0.036 1941-0.28-0.183-0.171-0.277 0.637-0.187 0.087-0.194-0.24 1942-0.003 0.067-0.039 0.476 0.865 0.156 0.262 1.113 0.126 1943 0.428 0.3 0.149 0.225 0.313 0.351 0.341 0.58 0.639 1944 0.192 0.103 0.26 0.29 0.637 0.233 0.227 0.473 0.282 1945 0.446 0.216 0.419 0.216 0.373 0.349 0.352 0.229 0.578 1946-0.088-0.046-0.078-0.272-0.037-0.209 0.153-0.126 0.289 1947-0.127-0.071 0.169 0.144 0.026 0.355-0.099 0.009 0.184 1948-0.015 0.056-0.035 0.107 0.153-0.231 0.038 0 0.114 1949 0.305 0.038 0.133 0.321 0.067 0.246 0.273 0.223-0.222 1950-0.096 0.089 0.732 0.305 0.579-0.248 0.091 0.65 0.327 1951 0.016 0.09 0.021 0.195 0.04-0.064 0.054-0.131 0.333 1952 0.128 0.083 0.131 0.39 0.434 0.079 0.109 0.175 0.062 1953-0.01 0.035 0.006-0.072-0.027 0.067 0.21-0.084-0.048 1954 0.154 0.176 0.908 0.715 0.469 0.077 0.112 0.756 0.185 All computatios were carried out o a WINDOWS PC usig the LINDO solver. Accordig to the awareess of the ivestor s risk aversio, we ca get the correspodig ivest strategies by solvig to the model (7. The table 2 shows the obtaied part of the results. Table 2: Part results to model 7 λ (x 1,x 2,,x 9 retur risk 0.0 (0, 0, 0, 0, 1, 0, 0, 0, 0 0.193 0.302 0.3 (0, 0, 0, 1, 0, 0, 0, 0, 0 0.168 0.235 0.5 (0, 0, 0, 0, 0, 0, 1, 0, 0 0.123 0.131 1.0 (0, 1, 0, 0, 0, 0, 0, 0, 0 0.057 0.089 Accordig to the ivestor s aspiratio ad the give value S 0,S 1,T 0,T 1, we ca get the correspodig strategies by solvig to the model (14 as show i table 3. Regardig the expected excess retur ad the trackig error as two objective fuctios, we have proposed a bi-objective programmig model for the idex trackig portfolio selectio problem. Furthermore, ivestors vague aspiratio levels for the excess retur ad
Liear Programmig for Portfolio Selectio 201 Table 3: Part solutios to model 14 S 0 S 1 T 0 T 1 (x 1,x 2,,x 9 µ 0.0878 0.1054 0.502 0.202 (0, 0, 0, 0, 0.4152, 0, 0.5848, 0, 0 1 0.0988 0.20 0.402 0.282 (0, 0, 0, 0.1209, 0.8791, 0, 0, 0, 0 0.90087 the trackig error are cosidered as fuzzy umbers. Based o fuzzy decisio theory, we have proposed a fuzzy idex trackig portfolio selectio model. A example is give to illustrate that the proposed fuzzy idex trackig portfolio selectio model. The computatio results show that the proposed model ca geerate a favorite portfolio strategy accordig to the ivestor s satisfactory degree. Ackowledgemets This work is partly supported by Chiese Natioal Sciece Foudatios (grat No. 10701080, Beijig Natural Sciece Foudatios (grat No. 1092011, Foudatio of Beijig Educatio Commissio (grat No. SM200910037005, Fudig Project for Academic Huma Resources Developmet i Istitutios of Higher Learig Uder the Jurisdictio of Beijig Muicipality (grat No. PHR200906210, Fudig Project for Base Costructio of Scietific Research of Beijig Muicipal Commissio of Educatio(grat No. WYJD200902, Beijig Philosophy ad Social Sciece Plaig Project (grat No. 09BaJG258ad Fudig Project for Sciece ad Techology Program of Beijig Muicipal Commissio of Educatio (grat No. KM200910037002. Refereces [1] Markowitz, H. Portfolio selectio, Joural of Fiace, 1952, 3(7: 77-91 [2] Markowitz, H. Portfolio selectio: Efficiet diversificatio of Ivestmet, New York: Wiley, 1959 [3] Koo,H., Yamazaki, H. Mea-variace deviatio portfolio optimazatio model ad its applicatio to tokyo stock market, Maagemet Sciece, 1991, 37(5: 519-531 [4] Arott, R. D., Wager, W. H. The measuremet ad cotrol of tradig cost, Fiacial Aalysts Joural, 1990, 46(7: 73-80 [5] Brea, M. J. The optimal umber of securities i a risky asset portfolio whe there are fixed costs of trasatio: theory ad some empirical results, Joural of Fiacial Quatitative Aalysis, 1975, 10: 483-496 [6] Bellma, R. ad L. A. Zadeh. Decisio Makig i a Fuzzy Eviromet, Maagemet Sciece, 1970, 17, 141-164 [7] Taaka, H., P. Guo, ad I. B. Turkse. Portfolio Selectio Based o Fuzzy Probabilities ad Possibility Distributios, Fuzzy Sets ad Systems, 2000, 111, 387-397 [8] Taaka, H., Hayashi, I., Watada,J. Possibilistic liear regressio aalysis for fuzzy data, Europea Joural of Operatio Research, 1989, 40: 389-396 [9] Taaka, H., Ishibuchi, H., Yoshikawa, S. Expoetial possiblity regressio aalysis, Fuzzy Sets ad Systems, 1995, 69: 305-318 [10] Yoshimoto, A. The mea-variace approach to portfolio optimazatio subject to trasactio costs, Joural of the operatioal research society of Japa, 1996, 39: 99-117 [11] Parra, M. A., A. B. Terol, ad M. V. R. Uria. A Fuzzy Goal Programmig Approach to Portfolio Selectio, Europea Joural of Operatioal Research, 2001, 133, 287-297
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