Portfolio rebalancing model with transaction costs based on fuzzy decision theory

European Journal of Operational Research 175 (2006) 879–893

www.elsevier.com/locate/ejor

Decision Support

Portfolio rebalancing model with transaction costsbased on fuzzy decision theory

Yong Fang a, K.K. Lai b,c,*, Shou-Yang Wang a

a Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, Chinab Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

c College of Business Administration, Hunan University, Changsha, China

Received 3 December 2002; accepted 27 May 2005Available online 16 August 2005

Abstract

The fuzzy set is one of the powerful tools used to describe an uncertain environment. As well as quantifying anypotential return and risk, portfolio liquidity is taken into account and a linear programming model for portfolio rebal-ancing with transaction costs is proposed. The level of return that an investor might aspire to, the risk and the liquidityof portfolio are vague in an uncertain financial environment. Considering them as fuzzy numbers, we propose a port-folio rebalancing model with transaction costs based on fuzzy decision theory. An example is given to illustrate thebehavior of the proposed model using real data from the Shanghai Stock Exchange.� 2005 Elsevier B.V. All rights reserved.

Keywords: Portfolio rebalancing; Fuzzy set; Fuzzy decision; Transaction costs

1. Introduction

In 1952, Markowitz [14] published his pioneering work which laid the foundation of modern portfolioanalysis. Markowitz�s model has served as a basis for the development of modern financial theory over thepast five decades. However, contrary to its theoretical reputation, it is not used extensively to constructlarge-scale portfolios. One of the most important reasons for this is the computational difficulty associatedwith solving a large-scale quadratic programming problem with a dense covariance matrix. Konno andYamazaki [9] used the absolute deviation risk function to replace the risk function in Markowitz�s model

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2005.05.020

* Corresponding author. Address: Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue,Kowloon, Hong Kong. Tel.: +852 27888563; fax: +852 27888560.

E-mail address: [email protected] (K.K. Lai).

mailto:[email protected]

880 Y. Fang et al. / European Journal of Operational Research 175 (2006) 879–893

and formulated a mean absolute deviation portfolio optimization model. It turns out that the mean abso-lute deviation model maintains the favorable properties of Markowitz�s model and removes most of theprincipal difficulties in solving Markowitz�s model. Simaan [21] provided a thorough comparison ofthe mean variance model and the mean absolute deviation model. Furthermore, Speranza [22] used thesemi-absolute deviation to measure the risk and formulated a portfolio selection model.

Transaction cost is one of the main concerns for portfolio managers. Arnott and Wagner [2] found thatignoring transaction costs would result in an inefficient portfolio. Yoshimoto�s empirical analysis [24] alsodrew the same conclusion. Mao [13], Jacob [8], Patel and Subhmanyam [18] and Morton and Pliska [15]studied portfolio optimization with fixed transaction costs. Pogue [19], Chen et al. [5], and Yoshimoto[24] studied portfolio optimization with changeable transaction costs. Mulvey and Vladimirou [16] andDantzig and Infanger [6] incorporated transaction costs into the multi-period portfolio selection model.Li et al. [11] gave a linear programming algorithm to solve a general mean variance model for portfolioselection with transaction costs. Due to changes of situation in financial markets and investors� preferencestowards risk, most of the applications of portfolio optimization involve a revision of an existing portfolio,i.e., portfolio rebalancing.

Usually, expected return and risk are two fundamental factors which investors consider. In some cases,investors may consider other factors such as liquidity. Liquidity has been measured as the degree of prob-ability involved in the conversion of an investment into cash without any significant loss in value. Arenaset al. [1] took into account three criteria (return, risk and liquidity) and used a fuzzy goal programmingapproach to solve the portfolio selection problem.

In 1970, Bellman and Zadeh [3] proposed fuzzy decision theory. Ramaswamy [20] presented a bond port-folio selection model using fuzzy decision theory. A similar approach for portfolio selection using fuzzydecision theory was proposed by Leon et al. [10]. Using the fuzzy decision principle, Ostermark [17] pro-posed a dynamic portfolio management model by fuzzifying the objective and the constraints. Watada[23] presented another type of portfolio selection model based on the fuzzy decision principle. The modelis directly related to the mean–variance model, where the goal rate (or the satisfaction degree) for an ex-pected return and the corresponding risk are described by logistic membership functions.

In this study, considering liquidity, we propose a linear programming model for portfolio rebalancingwith transaction costs. Furthermore, based on fuzzy decision theory, a portfolio rebalancing model withtransaction costs is proposed.

This paper is organized as follows. In Section 2, we introduce briefly fuzzy decision theory and the max-imization decision. In Section 3, as well as quantifying any potential return and risk, liquidity of a portfoliois taken into account and a linear programming model for portfolio rebalancing with transaction costs isproposed. In Section 4, the investor�s vague aspiration levels are considered as fuzzy numbers. Based on thefuzzy decision theory, a portfolio rebalancing model with transaction costs is proposed. In Section 5, anexample is given to illustrate the behavior of the proposed model using real data from the Shanghai StockExchange. A few concluding remarks are given in Section 6.

2. Fuzzy decision and maximization decision

Due to incomplete knowledge and information, it is not enough to use precise mathematics to model acomplex system. In order to represent vagueness in everyday life, Zadeh [25] introduced the concept offuzzy sets in 1965. Based on this concept, Bellman and Zadeh [3] presented fuzzy decision theory. They de-fined decision-making in a fuzzy environment with a decision set which unifies a fuzzy objective and a fuzzyconstraint.

Suppose that fuzzy sets are defined on a set of alternatives, X. Let the fuzzy set for the fuzzy objective beidentified as G, the fuzzy set for the constraint as C, and let decision set D be a unifying factor between G

Y. Fang et al. / European Journal of Operational Research 175 (2006) 879–893 881

and C. Then, decision set D can be defined as a intersection set between fuzzy objective G and fuzzy con-straint C, D = G \ C. Decision set D is a fuzzy set named a fuzzy decision. The corresponding membershipfunction of the decision set D is given by

lDðxÞ ¼ minðlGðxÞ; lCðxÞÞ; 8x 2 X .

More generally, if there are m fuzzy goals Gi (i = 1, . . . , m) and n fuzzy constraints Cj (j = 1, . . . , n), then thefuzzy decision is defined by the following fuzzy set:

D ¼ fG1 \ G2 \ � � � \ Gmg \ fC1 \ C2 \ � � � \ Cng.

Its membership function is characterized in the following:

lDðxÞ ¼ minðlG1ðxÞ; lG2

ðxÞ; . . . ; lGmðxÞ; lC1

ðxÞ; lC2ðxÞ; . . . ; lCn

ðxÞÞ; 8x 2 X .

The above concepts illustrate that there is no difference between the objective and constraint in a fuzzyenvironment.

Bellman and Zadeh [3] proposed a maximization decision. The maximization decision is defined by thefollowing non-fuzzy subset:

D� ¼ fx� 2 X jx� ¼ argmaxflDðxÞg ¼ argmaxfminðlGðxÞ; lCðxÞÞgg;
where m fuzzy objectives and n fuzzy constraints are given, the maximization decision can be denoted asfollows:
D� ¼ fx� 2 X jx� ¼ argmaxflDðxÞg¼ argmaxfminðlG1

ðxÞ; lG2ðxÞ; . . . ; lGm

ðxÞ; lC1ðxÞ; lC2

ðxÞ; . . . ;lCnðxÞÞgg.

The maximization decision can be considered to be an optimal decision.

3. Linear programming model for portfolio rebalancing with transaction costs

Suppose an investor allocates his/her wealth among n securities offering random rates of return. Theinvestor starts with an existing portfolio and decides how to reallocate assets.

The expected rate of return ri of security i without transaction costs is given by

ri ¼1

T

XT

t¼1

rit; i ¼ 1; 2; . . . ; n;

where rit can be determined by historical or forecast data.Let xþ ¼ ðxþ1 ; xþ2 ; . . . ; xþn Þ and x� ¼ ðx�1 ; x�2 ; . . . ; x�n Þ, where xþi is the proportion of the security i,

i = 1, 2, . . . , n bought by the investor, x�i is the proportion of the security i, i = 1, 2, . . . , n sold by the inves-tor. Then the transaction costs of the security i, i = 1, 2, . . . , n can be expressed as

Ciðxþi ; x�i Þ ¼ pðxþi þ x�i Þ; i ¼ 1; 2; . . . ; n;

where p is the rate of transaction costs for the securities. So the total transaction costs can be expressed as

Cðxþ; x�Þ ¼Xn

i¼1

pðxþi þ x�i Þ.

We assume that the investor does not invest the additional capital during the portfolio rebalancing pro-cess. Thus, we have


Xn

i¼1

ðx0i þ xþi � x�i Þ þ

Xn

i¼1

pðxþi þ x�i Þ ¼ 1;

where x0i is the proportion of the security i, i = 1, 2, . . . , n owned by the investor before portfolio

rebalancing.The expected net return on the portfolio after paying transaction costs is given by
Xn
i¼1

riðx0i þ xþi � x�i Þ �

Xn

i¼1

pðxþi þ x�i Þ.

The semi-absolute deviation of return on the portfolio x = (x1, x2, . . . , xn) below the expected returnover the past period t, t = 1, 2, . . . , T can be represented as

wtðxÞ ¼ min 0;Xn

i¼1

ðrit � riÞxi

( )�� ¼ j

Pni¼1ðrit � riÞxij þ

Pni¼1ðri � ritÞxi

2;

where xi ¼ x0i þ xþi � x�i .

So the expected semi-absolute deviation of the return of the portfolio x = (x1, x2, . . . , xn) below the ex-pected return can be represented as

wðxÞ ¼ 1

T

XT

t¼1

wtðxÞ ¼XT

t¼1

jPn

i¼1ðrit � riÞxij þPn

i¼1ðri � ritÞxi

2T;

where xi ¼ x0i þ xþi � x�i .

We use w(x) to measure the portfolio risk.Liquidity has been measured as the degree of probability of being able to convert an investment into cash

without any significant loss in value. The turnover rate of a security is the proportion of turnover volume totradable volume of the security, and is a factor which may reflect the securities liquidity. Generally, inves-tors prefer greater liquidity, especially since in a bull market for securities, returns on securities with highliquidity tend to increase with time. Here, we use the turnover rates of securities to measure their liquidity.It is known that turnover rates of securities in the future cannot be accurately predicted in a securities mar-ket. Possibility theory has been proposed by Zadeh [26] and advanced by Dubois and Prade [7] where fuzzyvariables are associated with possibility distribution. In this study, we assume that the turnover rates ofsecurities are modelled by possibility distributions rather than by probability distributions. That is, theturnover rates on the securities will be represented by fuzzy numbers. In many cases, it might be easierto estimate the possibility distributions of turnover rates on securities, rather than the corresponding prob-ability distributions. In this study, we regard trapezoidal possibility distribution as the possibility distribu-tion of the turnover rates on the securities.

A fuzzy number A is called trapezoidal with tolerance interval [a, b], left width a and right width b if itsmembership function takes the following form:

AðtÞ ¼

1� a�ta if a� a 6 t 6 a;

1 if a 6 t 6 b;

1� t�bb if a 6 t 6 bþ b;

0 otherwise

8>>><>>>:

and we denote A = (a, b,a,b). It can easily be shown that

½A�c ¼ ½a� ð1� cÞa; bþ ð1� cÞb�; 8c 2 ½0; 1�;
where [A]c denotes the c-level set of A.


Let [A]c = [a1(c), a2(c)] and [B]c = [b1(c), b2(c)] be fuzzy numbers and let k 2 R be a real number. Usingthe extension principle we can verify the following rules for addition and scalar multiplication of fuzzynumbers:

½Aþ B�c ¼ ½a1ðcÞ þ b1ðcÞ; a2ðcÞ þ b2ðcÞ�;½kA�c ¼ k½A�c.

Carlsson and Fuller [4] introduced the notation of crisp possibilistic mean value and crisp possibilisticvariance of continuous possibility distributions, which are consistent with the extension principle. The crisppossibilistic mean value of A is

EðAÞ ¼Z 1

0

cða1ðcÞ þ a2ðcÞÞdc.

It is clear that if A = (a, b,a,b) is a trapezoidal fuzzy number, then

EðAÞ ¼Z 1

0

c½a� ð1� cÞaþ bþ ð1� cÞb�dc ¼ aþ b2þ b� a

6.

Denote the turnover rate of the security j by the trapezoidal fuzzy number lj ¼ ðlaj; lbj; aj; bjÞ. Then theturnover rate of the portfolio x = (x1, x2, . . . , xn) is

Pnj¼1ljxj.

By the definition, the crisp possibilistic mean value of the turnover rate of security j is represented asfollows:

EðljÞ ¼Z 1

0

c½laj � ð1� cÞaj þ lbj þ ð1� cÞbj�dc ¼ laj þ lbj

2þ

bj � aj

6.

Therefore, the crisp possibilistic mean value of the turnover rate of the portfolio x = (x1, x2, . . . , xn) canbe represented as

EðlðxÞÞ ¼ EXn

j¼1

ljxj

!¼Xn

j¼1

laj þ lbj

2þ

bj � aj

6

� �xj.

In the study, we use the crisp possibilistic mean value of the turnover rate to measure the portfolio liquidity.Assume the investor wants to maximize return and minimize risk after paying transaction costs. At the

same time, he/she requires that the portfolio liquidity is not less than a given constant by rebalancing the exist-ing portfolio. Based on the above discussions, the portfolio rebalancing problem is formulated as follows:

ðP1Þ maxXn

i¼1

riðx0i þ xþi � x�i Þ �

Xn

i¼1

pðxþi þ x�i Þ

minXT

t¼1

jPn

i¼1ðrit � riÞxij þPn

i¼1ðri � ritÞxi

2T

s:t:Xn

j¼1

laj þ lbj

2þ

bj � aj

6

� �xj P l;

Xn

i¼1


Xn

i¼1


xi ¼ x0i þ xþi � x�i ; i ¼ 1; 2; . . . ; n;

0 6 xþi 6 ui; i ¼ 1; 2; . . . ; n;0 6 x�i 6 x0

i ; i ¼ 1; 2; . . . ; n;

where l is a given constant by the investor.


Eliminating the absolute function of the second objective function, the above problem can be trans-formed into the following problem:

ðP2Þ maxXn

j¼1

rjðx0j þ xþj � x�j Þ �

Xn

j¼1

pðxþj þ x�j Þ

min1

T

XT

t¼1

yt

s:t:Xn

j¼1

laj þ lbj

2þ

bj � aj

6

� �xj P l;

yt þXn

i¼1

ðrit � riÞxi P 0; t ¼ 1; 2; . . . ; T ;

Xn

i¼1


Xn

i¼1


xi ¼ x0i þ xþi � x�i ; i ¼ 1; 2; . . . ; n;

0 6 xþi 6 ui; i ¼ 1; 2; . . . ; n;

0 6 x�i 6 x0i ; i ¼ 1; 2; . . . ; n;

yt P 0; t ¼ 1; 2; . . . ; T ;

where l is a given constant by the investor.The above problem is a bi-objective linear programming problem. One can use several multiple objective

linear programming algorithms to solve it efficiently.

4. Portfolio rebalancing model based on fuzzy decision theory

Since investment is generally influenced by disturbances to social and economical circumstances, an opti-mization approach is not always the best. In some cases, a satisfaction approach is much better than anoptimization one. An investor always has aspiration levels for expected return and risk. In the real worldof financial management, the expert�s knowledge and experience are very important in decision-making.Based on an experts� knowledge, the investor may decide his/her aspiration levels for expected portfolioreturn and risk. Watada [23] employed a logistic function, i.e., a non-linear S shape membership function,to express aspiration levels of an investor�s expected return rate and risk. The S shape membership functionis given by

f ðxÞ ¼ 1

1þ expð�axÞ .

The function has a similar shape to the tangent hyperbolic function employed by H. Leberling, but it ismore easily handled than the tangent hyperbolic function. Therefore, it is more appropriate to considerthe logistic function to denote a vague goal level, which an investor considers. According to maximizationprinciple and using variance to measure the portfolio risk, Watada [23] proposed a fuzzy portfolio selectionmodel. The model extended the Markowitz�s mean–variance model to the fuzzy case.

In the portfolio rebalancing model proposed in Section 3, the two objectives (return and risk) and theconstraint on the liquidity of the portfolio are considered. Since the expected return, the risk and the liquid-ity are vague and uncertain, we use the non-linear S shape membership functions proposed by Watada [23]


to express the aspiration levels of expected return, risk and liquidity of the portfolio. Using the semi-abso-lute deviation risk function [22] to measure the portfolio risk, we propose a fuzzy portfolio rebalancingmodel based on Bellman–Zadeh�s maximization principle.

The membership functions of the goals for expected return, risk and liquidity are given as follows:

(a) Membership function of the goal for expected portfolio return

lrðxÞ ¼1

1þ expð�arðEðrðxÞÞ � rMÞÞ; ð1Þ

where rM is the mid-point where the membership function value is 0.5 and ar can be given by theinvestor based on his/her own degree of satisfaction for the expected return, rM represents the middleaspiration level for the portfolio return. Fig. 1 shows the membership function of the goal for ex-pected return.

(b) Membership function of the goal for portfolio risk

lwðxÞ ¼1

1þ expðawðwðxÞ � wMÞÞ; ð2Þ

where wM is the mid-point where the membership function value is 0.5 and aw can be given by theinvestor based on his/her own degree of satisfaction regarding the level of risk. wM represents themiddle aspiration level for the portfolio risk. Fig. 2 shows the membership function of the goal forrisk.

(c) Membership function for the goal for portfolio liquidity

llðxÞ ¼1

1þ expð�alðEðlðxÞÞ � lMÞÞ; ð3Þ

where lM is the mid-point where the membership function value is 0.5 and al can be given by the inves-tor based on his/her own degree of satisfaction regarding the liquidity. lM represents the middle aspi-ration level for the portfolio liquidity. Fig. 3 shows the membership function of the goal for liquidity.

Remark. ar, aw and al determine the shapes of membership functions lr(x), lw(x) and llðxÞ and llðxÞrespectively, where ar > 0, aw > 0 and al > 0. The larger parameters ar, aw and al become, the less theirvagueness becomes.

Fig. 1. Membership function of the goal for expected return.

Fig. 2. Membership function of the goal for risk.

Fig. 3. Membership function for the goal for liquidity.


According to Bellman and Zadeh�s maximization principle, we can define

g ¼ minflrðxÞ; lwðxÞ; llðxÞg.
The fuzzy portfolio rebalancing problem can be formulated as follows:
ðP3Þ max g

s.t. lrðxÞP g;

lwðxÞP g;

llðxÞP g;Xn

i¼1


Xn

i¼1


xi ¼ x0i þ xþi � x�i ; i ¼ 1; 2; . . . ; n;

0 6 xþi 6 ui; i ¼ 1; 2; . . . ; n;

0 6 x�i 6 x0i ; i ¼ 1; 2; . . . ; n;

0 6 g 6 1.


By (1)–(3), the fuzzy portfolio rebalancing problem can be rewritten as follows:

ðP4Þ max g

s.t. gþ expð�arðEðrðxÞÞ � rMÞÞg 6 1;

gþ expðawðwðxÞ � wMÞÞg 6 1;

gþ expð�alðEðlðxÞÞ � lMÞÞg 6 1;Xn

i¼1


Xn

i¼1


xi ¼ x0i þ xþi � x�i ; i ¼ 1; 2; . . . ; n;

0 6 xþi 6 ui; i ¼ 1; 2; . . . ; n;

0 6 x�i 6 x0i ; i ¼ 1; 2; . . . ; n;

0 6 g 6 1;

where ar, aw and al are parameter which can be given by the investor based on his/her own degree of sat-isfaction regarding the three factors.

Substituting h ¼ log g1�g then g ¼ 1

1þexpð�hÞ. The S shape membership function is monotonously increasing,

so maximizing g makes h maximize. Therefore, the above problem may be transformed to an equivalentproblem as follows:

ðP5Þ max h

s:t: ar

Xn

i¼1

rixi �Xn

i¼1

pðxþi þ x�i Þ !

� h P arrM;

hþ aw

T

XT

t¼1

yt 6 awwM;

al

Xn

j¼1

laj þ lbj

2þ

bj � aj

6

� �xj � h P allM;

yt þXn

i¼1

ðrit � riÞxi P 0; t ¼ 1; 2; . . . ; T ;

Xn

i¼1


Xn

i¼1


xi ¼ x0i þ xþi � x�i ; i ¼ 1; 2; . . . ; n;

0 6 xþi 6 ui; i ¼ 1; 2; . . . ; n;

0 6 x�i 6 x0i ; i ¼ 1; 2; . . . ; n;

yt P 0; t ¼ 1; 2; . . . ; T ;h P 0;

where ar, aw and al are parameters which can be given by the investor based on his/her own degree of sat-isfaction regarding the three factors.

Problem (P5) is also a standard linear programming problem. One can use several linear programmingalgorithms to solve it efficiently, for example, the simplex method.

Remark. The non-linear S shape membership functions of the three factors may change shape according tothe parameters ar, aw and al. By selecting the values of these parameters, the aspiration levels of the threefactors may be described accurately. On the other hand, different parameter values may reflect different


investors� aspiration levels. Therefore, it is convenient for different investors to formulate investmentstrategies using the proposed portfolio rebalancing model.

5. An example

In this section, we give a numerical example to illustrate the proposed portfolio rebalancing model. As-sume that an investor chooses 30 different stocks from the Shanghai Stock Exchange for his/her investment.The exchange codes of the 30 stocks are given in Table 1.

The rate of transaction costs for stocks is 0.0055 in the two securities markets on the Chinese mainland.Assume that the investor already owns an existing portfolio and he/she will not invest additional capitalduring the portfolio rebalancing process. The exchange codes of stocks in the existing portfolio and the pro-portions of the stocks are listed in Table 2.

The financial market situation changes, meaning that the investor needs to change his/her investmentstrategy. In our example, we assume that the upper bound of the proportions of Stock j owned by the inves-tor is 1. Now we use the fuzzy portfolio rebalancing model proposed in this study to reallocate the investor�sassets. First, we collect historical data on the 30 stocks from January, 1999 to January, 2002. The data aredownloaded from the web-site www.stockstar.com. Then we use one month as a period to obtain the his-torical rates of returns for 36 periods. With these historical data, the expected rates of returns of the stocksare listed in Table 3.

Table 1The exchange codes of 30 stocks

Stock 1 2 3 4 5 6 7 8 9 10Code 600058 600061 600062 600070 600071 600073 600079 600081 600085 600091

Stock 11 12 13 14 15 16 17 18 19 20Code 600094 600098 6000100 600619 600621 600624 600630 600636 600637 600827

Stock 21 22 23 24 25 26 27 28 29 30Code 600831 600834 600843 600847 600850 600853 600857 600860 600864 600874

Table 2The exchange codes and the proportions of stocks in the existing portfolio

Code 600058 600061 600062 600070 600071 600073 600079Proportions 0.05 0.08 0.05 0.35 0.1 0.12 0.25

Table 3The expected return rates of the stocks

Stock 1 2 3 4 5 6 7 8 9 10r 0.0376 0.0314 0.0326 0.0226 0.0285 0.0495 0.0220 0.0478 0.0271 0.0344

Stock 11 12 13 14 15 16 17 18 19 20r 0.0225 0.0293 0.0411 0.0214 0.0225 0.0285 0.0241 0.0274 0.0295 0.0318

Stock 21 22 23 24 25 26 27 28 29 30r 0.0463 0.0391 0.0244 0.0326 0.0366 0.0229 0.0361 0.0274 0.0312 0.0321

http://www.stockstar.com


Since we assume that the future turnover rates of the securities are trapezoidal fuzzy numbers, we need toestimate the tolerance interval, left width and right width of the fuzzy numbers. In the real world of port-folio management, the investor can obtain values of these parameters by using the Delphi Method [12]based on experts� knowledge. In our example, based on historical data of the securities turnover rates,we adopt the frequency statistic method to estimate these parameters. In the following, we give the estima-tion method for the fuzzy turnover rates for Stock 1 in detail. First, we use historical data (daily turnover

Fig. 4. Frequency of turnover rates.

Table 4The fuzzy turnover rates of the stocks

Stock 1 2 3l (0.009, 0.017, 0.008, 0.018) (0.012, 0.025, 0.01, 0.023) (0.009, 0.017, 0.0082, 0.017)

Stock 4 5 6l (0.010, 0.019, 0.008, 0.018) (0.013, 0.027, 0.010, 0.019) (0.010, 0.025, 0.008, 0.018)

Stock 7 8 9l7 (0.010, 0.020, 0.009, 0.019) (0.008, 0.016, 0.007, 0.015) (0.005, 0.015, 0.0045, 0.017)

Stock 10 11 12l (0.007, 0.017, 0.006, 0.016) (0.011, 0.019, 0.009, 0.018) (0.009, 0.024, 0.008, 0.025)

Stock 13 14 15l (0.007, 0.016, 0.0064, 0.018) (0.013, 0.032, 0.011, 0.030) (0.008, 0.015, 0.0076, 0.017)

Stock 16 17 18l (0.011, 0.026, 0.009, 0.028) (0.012, 0.031, 0.011, 0.030) (0.006, 0.031, 0.005, 0.026)

Stock 19 20 21l (0.011, 0.050, 0.0095, 0.047) (0.008, 0.025, 0.006, 0.026) (0.010, 0.023, 0.008, 0.021)

Stock 22 23 24l (0.011, 0.031, 0.010, 0.030) (0.012, 0.046, 0.010, 0.043) (0.009, 0.026, 0.008, 0.019)

Stock 25 26 27l (0.007, 0.027, 0.0065, 0.025) (0.010, 0.036, 0.009, 0.028) (0.011, 0.029, 0.010, 0.026)

Stock 28 29 30l (0.008, 0.043, 0.007, 0.035) (0.007, 0.035, 0.006, 0.034) (0.006, 0.031, 0.0045, 0.029)


rates from January, 1999 to January, 2002) to calculate the frequency of historical turnover rates. Fig. 4shows the frequency distribution of historical turnover rates for Stock 1. Note that most of the historicalturnover rates fall into the intervals [0.008, 0.010], [0.010, 0.012], [0.012, 0.014], [0.014, 0.016] and[0.016, 0.018]. We regard the mid-points of the intervals [0.008, 0.010] and [0.016, 0.018] as the left andthe right endpoint of the tolerance interval respectively. Therefore the tolerance interval of the fuzzy turn-over rate is [0.009, 0.017]. By observing all the historical data, we use 0.001 and 0.035 as the minimum pos-sible value and the maximum possible value of uncertain turnover rates in the future respectively. Thus theleft width is 0.008 and the right width is 0.018. The fuzzy turnover rate of Stock l is (0.009, 0.017,0.008, 0.018). Using a similar method, we obtain the fuzzy turnover rates of all 30 stocks. These are listedin Table 4.

In the following, we present two kinds of computational results arrived at using to conservative andaggressive techniques respectively. First, we assume that the investor has a conservative and pessimisticmind. Then the values of rM, wM and lM which are given by the investor are small. They are as follows:

TableMemb

g

0.8110.8060.785

TablePortfo

StockPurchaSell ra

StockPurchaSell ra

StockPurchaSell ra

rM ¼ 0.029; wM ¼ 0.030; lM ¼ 0.0225.

Considering the three factors (return, risk and liquidity) as fuzzy numbers with a non-linear membershipfunction, we obtain a portfolio rebalancing strategy by solving (P5). In the example, we give three differentvalues of parameters ar, aw and al. The corresponding computational results are listed in Tables 5–8.

Next, we assume that the investor has an aggressive and optimistic mind. Then the values of rM, wM andlM which are given by the investor are big. They are as follows:

rM ¼ 0.032; wM ¼ 0.034; lM ¼ 0.026.

Considering the three factors (return, risk and liquidity) as fuzzy numbers with a non-linear membershipfunction, we obtain a portfolio rebalancing strategy by solving (P5). In the example, we give three differentvalues of parameters ar, aw and al. The corresponding computational results are listed in Tables 9–12.

Since it is possible that the non-linear S shape membership function changes its shape according to thevalues of the parameters, the non-linear membership function can reflect the investor�s mind accurately andsuitably. From the above results, we find that we can obtain the different portfolio rebalancing strategies by

5ership grade g, obtained risk, obtained expected return and obtained liquidity when rM = 0.029, wM = 0.030 and lM = 0.0225

h ar aw al Obtained risk Obtained expected return Obtained liquidity

1.454 600 800 600 0.0282 0.0314 0.03041.425 500 1000 500 0.0286 0.0319 0.03031.295 400 1200 400 0.0289 0.0322 0.0302

6lio rebalancing ratio when rM = 0.029, wM = 0.030, lM = 0.0225, ar = 600, aw = 800 and al = 600

1 2 3 4 5 6 7 8 9 10se ratio 0.000 0.000 0.092 0.000 0.000 0.000 0.000 0.294 0.000 0.021

tio 0.016 0.000 0.000 0.035 0.100 0.051 0.250 0.000 0.000 0.000

11 12 13 14 15 16 17 18 19 20se ratio 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

tio 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

21 22 23 24 25 26 27 28 29 30se ratio 0.100 0.044 0.000 0.000 0.000 0.000 0.000 0.000 0.038 0.177

tio 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 7Portfolio rebalancing ratio when rM = 0.029, wM = 0.030, lM = 0.0225, ar = 500, aw = 1000 and al = 500

Stock 1 2 3 4 5 6 7 8 9 10Purchase ratio 0.000 0.000 0.086 0.000 0.000 0.000 0.000 0.291 0.000 0.009Sell ratio 0.014 0.000 0.000 0.035 0.100 0.040 0.250 0.000 0.000 0.000







Table 9Membership grade g, obtained risk, obtained expected return and obtained liquidity when rM = 0.032, wM = 0.034 and lM = 0.026

g h ar aw al Obtained risk Obtained expected return Obtained liquidity

0.849 1.726 600 800 600 0.0318 0.0349 0.02950.836 1.630 500 1000 500 0.0324 0.0353 0.02930.802 1.396 400 1200 400 0.0328 0.0355 0.0295






solving (P5) in which the different values of the parameters (ar, aw and al) are given. Through choosing thevalues of the parameters ar, aw and al according to the investor�s frame of mind, the investor may achieve afavorite portfolio rebalancing strategy.










6. Conclusion

In addition to the more usual factors of expected return and risk, portfolio liquidity is considered in theportfolio rebalancing process. The turnover rates of securities are used to measure their liquidity. Consid-ering all three factors, a linear programming model for portfolio rebalancing with transaction costs is pro-posed. An investor�s aspiration levels for the expected return: risk and liquidity are vague in an uncertainenvironment. The vague aspiration levels are considered to be fuzzy numbers with a non-linear S shapemembership function. Furthermore, based on fuzzy decision theory, a fuzzy portfolio rebalancing modelwith transaction costs is proposed. An example is given to illustrate the behavior of the proposed fuzzyportfolio rebalancing model using real data from the Shanghai Stock Exchange. The computation resultsshow that the portfolio rebalancing model with a non-linear S shape membership function can generate afavorite portfolio rebalancing strategy according to the investor�s degree of satisfaction.

Acknowledgements

Supported by the National Natural Science Foundation of China under Grant No. 70221001 and CityUniversity of Hong Kong under Grant No. 7100289. The authors are grateful to the two referees for theirvaluable comments and suggestions.

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Portfolio rebalancing model with transaction costs based on fuzzy decision theory