Fuzzy multi-period portfolio selection optimization models using multiple criteria

Automatica 48 (2012) 3042–3053

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Fuzzy multi-period portfolio selection optimization models using multiple criteria✩ Yong-Jun Liu, Wei-Guo Zhang 1 , Wei-Jun Xu School of Business Administration, South China University of Technology, Guangzhou, 510641, PR China

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Article history: Received 9 July 2011 Received in revised form 30 March 2012 Accepted 6 July 2012 Available online 28 September 2012 Keywords: Risk management Fuzzy multi-period portfolio selection Multiple criteria decision-making TOPSIS-compromised programming

abstract To simulate the real transactions in financial market, multiple decision criteria in portfolio selection should be considered to provide investors with additional choices. This paper deals with multi-period portfolio selection problems in fuzzy environment by considering some or all criteria, including return, transaction cost, risk and skewness of portfolio. Two possibilistic portfolio optimization models by using multiple criteria are first presented for the basic multi-period portfolio selection problem. Then, they are naturally extended to dynamic feedback models with closed-loop control policies. A TOPSIScompromised programming approach is designed originally to transform the proposed models into single objective models. After that, a genetic algorithm is devised for obtaining optimal solutions. Furthermore, a numerical example is given to illustrate the advantage of the proposed models and the efficiency of the designed algorithm over the existing approaches. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction On the basis of Markowitz (1952) originally proposed the mean–variance (M–V) model for a single period portfolio selection problem, numerous scholars have extended the single period portfolio selection into multi-period portfolio selections by using different approaches. Mossin (1968) presented optimal multiperiod portfolio selection policies by using dynamic programming approach. Hakansson (1971) analyzed the multi-period mean–variance by means of a general theory of portfolio choice. Li, Chan, and Ng (1998) employed dynamic programming approach to deal with the multi-period safety-first portfolio selection problem. Using the same approach, Li and Ng (2000) considered the mean–variance formulation for the multi-period portfolio selection problem and determined the optimal portfolio selection policy and an analytical expression of the mean–variance efficient frontier. Zhu, Li, and Wang (2004) further extended their model by taking bankruptcy constraint into consideration, and presented a

✩ This research was supported by the National Natural Science Foundation of China (No.70825005, 71171086), GDUPS (2010) and the Program for New Century Excellent Talents in University (No.NCET-10-0401). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Oswaldo Luiz V. Costa under the direction of Editor Berç Rüstem. E-mail addresses: [email protected] (Y.-J. Liu), [email protected], [email protected] (W.-G. Zhang), [email protected] (W.-J. Xu). 1 Tel.: +86 208711421; fax: +86 208711405.

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.08.036

generalized mean–variance model. In addition, Yu, Takahashi, Inoue, and Wang (2010) proposed a dynamic portfolio selection optimization with bankruptcy control for absolute deviation model. In continuous-time version of dynamic portfolio selection, Zhou and Li (2000) investigated the mean–variance portfolio selection problem where short-selling of assets was admissible. Li, Zhou, and Lim (2002) further studied the mean–variance portfolio selection problem with no short-selling constraints. Recently, some researchers have investigated the multi-period portfolio selection problem by using a stochastic market process in order to modulate various parameters of the financial model to make it more realistic; see for example Calafiore (2008, 2009), Çelikyurt and Özekici (2007), Costa and Araujo (2008), Gülpınar and Rustem (2007), Wei and Ye (2007) and Yiu, Liu, Siu, and Ching (2010). In the multi-stage case, the investor decides based on expectations and/or scenarios up to some intermediate times prior to the horizon. These intermediate times correspond to rebalancing or restructuring periods. Sun, Fang, Wu, Lai, and Xu (2011) proposed a novel variant of particle swarm optimization, called drift particle swarm optimization, and applied it to solve the multi-stage portfolio optimization problem. The previous studies on multi-period portfolio selections have been dominated by using the variance or absolute deviation of the return of portfolio as risk measure and ignoring the transaction costs which are entailed by buying or selling assets to rebalance the existing portfolio during the whole investment horizon. However, in real financial market, transaction costs play a crucial role in transaction. If ignoring the transaction costs, investors may fail to obtain the efficient portfolio. Bertsimas and Pachamanova (2008) incorporated transaction costs into consideration to study the multi-period portfolio selection problem.

Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

As above the literatures mentioned, they were proposed on the basis of probability theory. They often characterized a financial asset as a random variable. Though probability theory is one of the main techniques used for analyzing uncertainty in finance, the financial market is also affected by many nonprobabilistic factors such as vagueness and ambiguity as presented by Lacagnina and Pecorella (2006). It is well known that the financial market is extremely complex. The returns of assets are usually affected by many factors including economic, social, political and people’s psychological factors as proposed by Huang (2011). Especially, the influences of people’s psychological factors cannot be neglected when they evaluate the returns of assets. Huang (2008) summarized the influences on the returns of assets as the following three main types. The first one is the influence of general economic that mainly contains the influence of the fiscal policy, monetary policy, inflation, and such economic events as international monetary devaluations, political events as election in a country or a province, social events as upheavals in a country, etc. The second one is the influence of industry that influences an industry to prosper or suffer in the long run or during the expected near-term economic environment. Such industry factors include import or export quotas or taxes, excess supply or shortage of a resource, or government-imposed regulations on an industry. The third one comes from the performances of concerned company. Good or bad performances of the company will affect the relative asset return. In addition, financial markets are usually very sensitive. An accident or a hard-to-verify message may affect asset prices or returns greatly. Östermark (1996) presented that the portfolio selection problem should consider all necessary aspects of modern business management (e.g., forecasting, inventory management, accounting information, risk analysis, taxation, economic friction, computer support, etc.). Notice that all the factors mentioned above are affected by human’s subjective intention. Thus, in these cases, it is impossible for us to predict the probability distributions of the returns of risky assets. In this context, decision makers are usually provided with information which is characterized by vague linguistic descriptions such as high risk, low profit, high interest rate as proposed by Sheen (2005). With the widely use of fuzzy set theory in Zadeh (1965), people have realized that they could use fuzzy set theory to handle the vagueness or ambiguity in financial market. In fact, the fuzzy portfolio selection problem was researched from 1990s. Numerous researchers have studied fuzzy portfolio selection problems (see for example Carlsson, Fullér, & Majlender, 2001; Fang, Lai, & Wang, 2006; Giove, Funari, & Nardelli, 2006; Inuiguchi & Tanino, 2000; León, Liern, & Vercher, 2002; Sadjadi, Seyedhosseini, & Hassanlou, 2011; Watada, 1997; Zhang, Liu, & Xu, 2012; Zhang, Xiao, & Xu, 2010; Zhang, Zhang, & Xu, 2010). It can be seen that fuzzy set theory is a powerful tool used to describe an uncertain environment with vagueness, ambiguity or some other type of fuzziness, which are always involved in not only the financial markets but also the behavior of the financial managers’ decisions. Wang and Zhu (2002) pointed out that using fuzzy approaches, quantitative analysis, qualitative analysis, the experts’ knowledge and the investors’ subjective opinions could be better integrated in a portfolio selection model to catch variations of stock markets more efficiently. So it is reasonable to construct portfolio selection model by using fuzzy set theory in real investment world. Though great progress has been made in the previous studies, there are still some problems in the existing portfolio studies as follows. Most of the existing multi-period portfolio selection models have focused on only two fundamental factors, i.e., expected return and risk of portfolio. However, in practical investment, only using expected return and risk as decision criteria cannot capture all the relative information for a portfolio decision. To provide investors with additional choice, more criteria should

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be integrated into portfolio selection models. To our knowledge, studies that have considered multiple decision making criteria are few for the multi-period portfolio selection problem. Therefore, the purpose of this paper is to investigate the fuzzy multi-period portfolio optimization problem by using multiple criteria. Four fuzzy multi-period portfolio selection optimization models in open-loop or in closed-loop, which consist of some or all criteria, including return, transaction cost and skewness of portfolio, are proposed to provide investors with additional choices. In the proposed models, the return is characterized by the possibilistic mean value and the risk is measured by possibilistic variance. The skewness is quantified by the third order moment about the possibilistic mean value of a return distribution. To solve the proposed models, we first present a TOPSIS-compromised programming approach to convert them into single objective programming models. Then, we design a genetic algorithm with penalty term to solve them. The remainder of this paper is organized as follows. For the better understanding of the paper, we will introduce some basic definitions of fuzzy numbers in Section 2. In Section 3, we give the details of the modeling process. In Section 4, we first design a TOPSIS-compromised programming approach to convert the proposed models into single objective models. Then, we design a genetic algorithm to solve them. In Section 5, a numerical example is given to illustrate the ideas of the proposed models. In addition, we also provide approach comparison to express the effectiveness of our designed algorithm for solving our models. Finally, we conclude the paper in Section 6. 2. Preliminaries Let us first review some basic concepts about fuzzy number, which we need in the following section. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by  F (R). A γ -level set of a fuzzy number A is denoted by [A]γ = {x ∈ R|µA (x) ≥ γ } if γ > 0 and [A]γ = cl{x ∈ R|µA (x) ≥ γ } (the closure of the support of A) if γ = 0. It is well known that if A is a fuzzy number then [A]γ is a compact subset of R for all γ ∈ [0, 1]. The above-mentioned definition can be found in Dubois and Prade (1980). In the following, we will introduce the notions of possibilistic mean value, variance and covariance of fuzzy numbers introduced in Carlsson and Fullér (2001), and Saeidifar and Pasha (2009). Definition 1. Let A ∈  F (R) be a fuzzy number with [A]γ = [a(γ ), a(γ )] for all γ ∈ [0, 1]. The possibilistic mean value of A is defined as E(A) =

1



γ (a(γ ) + a(γ )) dγ ,

(1)

0

it follows that E(A) is the nearest weighted point to A ∈  F (R) which is unique. Theorem 1. Let A, B ∈  F (R) be two fuzzy numbers and let λ, µ ∈ R be real numbers. Then, we have E(λA + µB) = λE(A) + µE(B). Definition 2. Let A ∈  F (R) be a fuzzy number with [A]γ = [a(γ ), a(γ )] for all γ ∈ [0, 1]. Then, the variance of A is defined as Var(A) =

1



γ [(a(γ ) − E(A))2 + (a(γ ) − E(A))2 ] dγ . 0

(2)

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Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

Definition 3. For any two given fuzzy numbers A with [A]γ = [a(γ ), a(γ )] for all γ ∈ [0, 1] and B with [B]γ = [b(γ ), b(γ )] for all γ ∈ [0, 1], the possibilistic covariance of A and B is defined as Cov(A, B) =

1



γ (a(γ ) − E(A))(b(γ ) − E(B)) dγ 0 1



γ (a(γ ) − E(A))(b(γ ) − E(B)) dγ .

+

(3)

0

Theorem 2. Let A and B be any two fuzzy numbers, and let λ, µ ∈ R. Then

+ 2|λµ|Cov(φ(λ)A, φ(µ)B), 1, 0, −1,

x>0 x=0 x<0



(4)

Theorem 3. . Let A1 , A2 , . . . , An be n fuzzy numbers, and let λ1 , λ2 , . . . , λn be n real numbers. Then Var



 λi Ai

n

=



+2

otherwise.

0, −1,

n 

|λi λj |Cov(φ(λi )Ai , φ(λj )Aj ),

∀γ ∈ [0, 1].

β −α 6

.

(7)

From Definition 2, the possibilistic variance of A can be represented as Var(A) =



b−a 2

+

α+β

2

6

+

(β + α)2 + (β − α)2 72

.

(8)

From Definition 4, the possibilistic skewness of A can be computed as PS(A) = M3 (A) =

+

x>0 x=0 x<0

19(β 3 − α 3 ) 1080

(αβ − βα 2 ) 72

+

(b − a)(β 2 − α 2 ) 24

.

(9)

(5) 3. The formulation of multi-period portfolio selection optimization

is a sign function of x ∈ R.

Saeidifar and Pasha (2009) introduced the notion of the possibilistic skewness of a fuzzy number as follows. Definition 4. Let A ∈  F (R) be a fuzzy number with [A]γ = [a(γ ), a(γ )] for all γ ∈ [0, 1]. Then the possibilistic skewness (PS) of fuzzy number A is defined as follows: M3 (A) PS(A) = √ 3 , Var(A)

2

+

2

i
1,

a+b

λ2i Var(φ(λi )Ai )

i=1

i=1

where φ(x) =

if b ≤ x ≤ b + β,

By Definition 1, the possibilistic mean value of A can be expressed as

is a sign function of x ∈ R.

From Theorem 2 above, we can easily deduce the following theorem.

n

if a − α ≤ x ≤ a, if x ∈ [a, b],

[A]γ = [a − (1 − γ )α, b + (1 − γ )β],

E(A) =

Var(λA + µB) = λ2 Var(φ(λ)A) + µ2 Var(φ(µ)B)



 a−x   1 − α ,  1, µA (x) = x−b   , 1−   β  0,

The γ -level set of A can be computed as

Furthermore, the following conclusion is shown in Zhang and Wang (2007).

where φ(x) =

tolerance interval [a, b], left width α > 0 and right width β > 0 if its membership function takes the following form

(6)

1

where M3 (A) = 0 γ [(a(γ ) − E(A))3 + (a(γ ) − E(A))3 ] dγ is the third order moment about the possibilistic mean value, E(A), of A. The possibilistic skewness, PS(A), of fuzzy number A shows the weight of fuzzy number at the left or right sides. Note that the defined PS(A) has the following rules: 1. If PS(A) > 0, then it means that A is skewed to the right, that is, it has a long right tail and a very shorter left tail; 2. If A is a fuzzy number with the symmetric membership function, then we have PS(A) = 0, that is, its right tail and left tail are equal; 3. If PS(A) < 0, then it means that A is skewed to the left, that is, it has a long left tail and a very shorter right tail. To reduce the computation complexity, in this paper, we use M3 (A) as the possibilistic skewness of A. Denoted it as PS(A) = M3 (A). For convenience of description, we call the possibilistic skewness of a fuzzy variable as skewness in the following sections. From the definitions above, we can obtain the following formulas for a trapezoidal fuzzy number A = (a, b, α, β) with

In this section, we discuss the possibilistic return, risk and skewness of portfolio for the multi-period portfolio selection problem. Then, to express investors’ preferences more flexible, we formulate four different multiple criteria models for the multiperiod portfolio selection problem in fuzzy environment (see the following four cases in Sections 3.2 and 3.3). Two of them are basic (open-loop) optimization models. The others are dynamic feedback models with closed-loop control policies, which are the corresponding extended models of the basic optimization models. Assume that there are n risky assets in a financial market for trading, and the returns of assets are denoted as trapezoidal fuzzy numbers. An investor hopes to allocate his/her initial wealth W1 among the n risky assets at the beginning of period 1, and obtain the terminal wealth at the end of period T . The investor’s wealth can be reallocated at every beginning of the following T − 1 consecutive time periods. For notational convenience, we first introduce the following notations: ri,t : the return of risky asset i at period t, where ri,t = (ai,t , bi,t , αi,t , βi,t ); r (t ): the minimum given return level of the portfolio at period t; xi,t : the crisp form investment proportion of risky asset i at period t; x(t ): the vector of the crisp form portfolio at period t, where x(t ) = (x1,t , x2,t , . . . , xn,t )′ ; ci,t : the unit transaction cost for risky asset i at period t. RN ,t : the net return of the portfolio at period t; Wt : the expected value of the wealth at the beginning of period t; ui,t : the upper bound constraint of xi,t , i = 1, 2, . . . , n; t = 1, 2, . . . , T .

Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

3.1. The criteria for the multi-period portfolio selection problem

According to Definition 4, the cumulative skewness of the T period investment can be expressed as

As we know, in practical investment, different investors with different preferences may result from investment strategies. In order to achieve greater flexibility in portfolio selection, it is necessary to consider multiple criteria for expressing investors’ preferences. In the following subsections, we will introduce the criteria, including return, transaction cost, risk and skewness of portfolio. We will quantify return by the possibilistic mean value, risk by possibilistic variance and skewness by the third order possibilistic moment about the fuzzy return of the asset. Assume that the whole investment process is self-financing, that is, the investor does not invest the additional capital during the portfolio selection. From the assumption in the previous section, ri,t = (ai,t , bi,t , αi,t , βi,t ) (i = 1, 2, . . . , n; t = 1, 2, . . . , T ) are trapezoidal fuzzy numbers. Derived from Definition 1, the possibilistic mean value of the portfolio x(t ) = (x1,t , x2,t , . . . , xn,t )′ at period t can be expressed as

 E

n 

 =

xi,t ri,t

n   ai , t + b i , t

i=1

+

2

i=1

βi,t − αi,t



6

E(RN ,t ) = E

T  1  19

24 t =1  45

 xi,t ri,t − Ct

,

(11)

for t = 1, 2, . . . , T . Then, the expected value of the wealth at the beginning of period t + 1 can be expressed as Wt +1 = Wt E(RN ,t ),

(12)

for t = 1, 2, . . . , T . From (12), we have Wt +1 = W1 j=1 E(RN ,j ). Thus, the expected value of the terminal wealth at the end of period T is

t

E(RN ,t ).

(13)

−

By Definition 2 and Theorem 2, the cumulative risk of over T period investment can be represented as V ( x) =





Var

n 

t =1

+

t =1

=

xi,t ri,t

x2i,t Var(ri,t ) + 2

i=1

   

 +

n 

i =1



n

   T  n    t =1

i =1

xi,t βi,t

 −

i=1

n 

3  xi,t αi,t



i=1

n 

i=1

xi,t βi,t

 n 

n 

2  xi,t αi,t 

i =1

 xi,t (bi,t − ai,t )

(15)

3.2. The basic multi-period portfolio optimization models Similar to most multi-period portfolio optimization models, we first discuss the basic multi-period portfolio approach. The basic approach takes the viewpoint of a decision maker that at time t = 0 wants to compute and freeze the whole sequence of optimal portfolio x(1), x(2), . . . , x(T ). Assume that the transaction cost Ct is a V-shaped function of differences between the tth period portfolio x(t ) = (x1,t , x2,t , . . . , xn,t )′ and the t − 1th period portfolio x(t − 1) = (x1,t −1 , x2,t −1 , . . . , xn,t −1 )′ . Then, the transaction cost of the portfolio x(t ) = (x1,t , x2,t , . . . , xn,t )′ at period t can be expressed as Ct =

n 

ci,t |xi,t − xi,t −1 |,

x i ,t

(16)

Thus, we get E(RN ,t ) =

n   ai,t + bi,t

−

2

n 

+

βi,t − αi,t 6

 xi,t

ci,t |xi,t − xi,t −1 |,

(17)

i=1



WT +1 = W1

xi,t xi,t Cov(ri,t , rj,t )

i
bi,t − ai,t 2

+

βi,t + αi,t

2

n 

i =1

2

+

βi,t − αi,t 6

 xi,t

 ci,t |xi,t − xi,t −1 | .

(18)

i=1

Since, in real world, investors with different preferences may result from different investment strategies. In order to achieve greater flexibility, we propose two basic multi-period portfolio optimization models, which consist of criteria including return, transaction cost, risk, and skewness of portfolio.

6

2  n 2     xi,t (βi,t + αi,t ) + xi,t (βi,t − αi,t )   i=1

72

 T n    ai,t + bi,t t =1

− 

t = 1, 2, . . . , T .

i=1

for t = 1, 2, . . . , T . Hence, the expected value of the terminal wealth at the end of period T is



T

=

3

i=1

i =1

 n  



i=1



n 

i=1

i =1

t =1

T 

3



  2 n n    xi,t αi,t xi,t βi,t



T

WT +1 = W1

1

+

xi,t ri,t

i=1



=



(10)

i=1



PS

 n 

t =1

xi,t ,

Let Ct be the transaction cost of the portfolio x(t ) = (x1,t , x2,t , . . . , xn,t )′ at period t. Then, the possibilistic mean value of the net return of the portfolio x(t ) = (x1,t , x2,t , . . . , xn,t )′ at period t can be computed as n 

PS(x) =



T 

 2  2  n n    ×  xi,t βi,t − xi,t αi,t  .  i=1 i=1

t = 1, 2, . . . , T .



3045

   

. (14)

Case 1. The model (P1 ) consists of two objectives, namely, the maximization of the expected value of the terminal wealth and the minimization of the cumulative risk of T period investment which

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Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

    n T n     βi,t − αi,t ai , t + b i , t   + xi,t − ci,t |xi,t − xi,t −1 | max WT +1 = W1   2 6  i=1 t =1 i=1    n 2  n 2               xi,t (βi,t + αi,t ) + xi,t (βi,t − αi,t )    2  T  n      bi,t − ai,t βi,t + αi,t  i=1 i=1  min V ( x ) = + + x  i,t    i=1  2 6 72   t =1            n n  ai,t + bi,t  (P1 ) βi,t − αi,t  s.t . + x i ,t − ci,t |xi,t − xi,t −1 | ≥ r (t )   2 6   i=1 i=1  n     xi,t = 1     i=1      n  n     βi,t − αi,t ai , t + b i , t   + xi,t − ci,t |xi,t − xi,t −1 |  Wt +1 = Wt  2 6  i=1 i=1 0 ≤ xi,t ≤ ui,t , i = 1, 2, . . . , n; t = 1, 2, . . . , T

(a)

(19)

(b) (c) (d)

Box I.

is measured by the sum of the possibilistic variance of the portfolio return at each period. The details of the model (P1 ) are shown in Box I, where constraint (19)(a) represents the portfolio return must achieve or exceed the given minimum return constraint at each period; constraint (19)(b) indicates the proportion at period t sum to one; constraint (19)(c) denotes the wealth accumulation constraint; constraint (19)(d) states the lower and upper bound constraints of xi,t . For notational simplicity, we denote the feasible region of the model (P1 ) as x ∈ D. Case 2. The model (P2 ) includes all the criteria of the model (P1 ). In addition, this model adds an objective to maximize the cumulative skewness of the portfolio over all investment horizon as given in Box II. 3.3. The multi-period portfolio optimization models with linear recourse The models (P1 ) and (P2 ) are open-loop optimization problems. Although formulations of (P1 ) and (P2 ) are crisp and easily obtained, the influence of the historical information about returns of assets on portfolio decision-making is not reflected in the proportion adjustments in the current decision stage. In many cases, the process of adjustment on portfolio is of closed-loop nature, in which the adjustment decision depends on the historical information. To demonstrate the whole process of adjustment, we extend the two open-loop optimization models into corresponding dynamic feedback optimization models by using the closed-loop approach. 3.3.1. Dynamic feedback policy of portfolio To demonstrate the closed-loop nature of the multi-period portfolio decision problem, in this subsection, the dynamic feedback control policy will be constructed for the whole adjustment sequence. We assume that the adjustment policies of portfolios are affine functions about their one-period backwards return deviations, where the coefficients of these functions become the decision variables of the problem. That is to say, the adjustment amount of the portfolio at period t depends on the return deviation of the portfolio at period t − 1. Under this hypothesis, the dynamic feedback adjustment policies can be expressed as the following causal functions:

△ x(0) = △x(0), △ x(t ) = △x(t ) + Θ (t − 1)[R(t − 1) − R(t − 1)], t = 1, 2, . . . , T ,

(20) (21)

where △x(0) = (△x1,0 , △x2,0 , . . . , △xn,0 )′ represents the ‘‘hereand-now’’ decision variable; △ x(t ) = (△ x1,t , △ x2,t , . . . , △ xn,t )′ is the dynamic feedback adjustment proportion of the portfolio at period t; △x(t ) = (△x1,t , △x2,t , . . . , △xn,t )′ , t = 1, 2, . . . , T , are the nominal portfolio adjustments; Θ (t ) = (θij (t ))n×n denotes the market relation matrix at the period t; R(t ) = (r1,t , r2,t , . . . , rn,t )′ is the vector of the return of the portfolio at period t , R(t ) = (r 1,t , r 2,t , . . . , r n,t )′ represents the vector of the predetermined return of the portfolio at period t. Remark 1. Notice that the control policy in Eq. (21) is an affine function about one-period backwards return deviation from expectation [R(t − 1) − R(t − 1)]. Similar to the interpretation in Calafiore (2009), the control policy in Eq. (21) can be expressed as follows: the nominal adjustment proportion of the portfolio, △x(t ), performs at period t when the market performed as expected at period t − 1. However, if the market did not perform as expected at period t − 1, we should adjust △x(t ) with a term proportion to the market deviations from expectation [R(t − 1) − R(t − 1)] at period t − 1. The coefficients of the adjustment are the elements in the market relation matrix Θ (t − 1) at period t − 1. The element θij (t − 1) in row i and column j of Θ (t − 1) denotes the sensitivity of the control action in asset i, △ xi,t , with respect to deviations from expectation of the return of asset j at period t. Based on the discussion above, the iterative dynamic equation of the investment proportion of the portfolio at period t + 1 can be expressed by

 x(t + 1) =  x(t ) + △ x(t + 1)  = x(t ) + △x(t + 1) + Θ (t )R(t ),

(22)

where R(t ) = (r1,t − r 1,t , r2,t − r 2,t , . . . , rn,t − r n,t ) is the vector of the fuzzy deviation of the portfolio at period t. By Eqs. (21) and (22), we can obtain the general formula of the portfolio at period t as follows: ′

 x( t ) =  x(0) +

t  [Θ (k − 1)R(k − 1) + △x(k)],

(23)

k=1

where  x(0) = x(0) is the investment proportion of the portfolio at the beginning of period 1. Since the returns of assets at each period are characterized by fuzzy variables, the investment proportions of assets at each period

Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

3047

    n T n     β ai,t + bi,t i,t − αi,t   + xi,t − ci,t |xi,t − xi,t −1 | max WT +1 = W1   2 6  i=1 t =1 i=1    n 2  n 2             2    x (β + α ) + x (β − α )    i ,t i ,t i,t i ,t i,t i,t T   n     b − a β + α i , t i , t i , t i , t i = 1 i = 1   min V (x) = xi,t + +    i=1  2 6 72    t =1              max PS(x)      3  3  n  n       19 (P2 )  2   2  − xi,t αi,t xi,t βi,t T  n n n n         i = 1 i = 1 1 1   +  xi,t αi,t xi,t βi,t − xi,t βi,t xi,t αi,t  =   24 t =1  45 3    i = 1 i = 1 i = 1 i = 1                           2 2   n n n           + x ( b − a ) x β − x α i,t i,t i,t i,t i,t i,t i,t      i =1 i =1 i=1         s.t . x ∈ D Box II.

are also fuzzy variables. By Definition 1, the crisp form investment proportion of the portfolio at period t can be represented by x(t ) = E [ x(t − 1) + △ x(t )]

E(RN ,t ) =

= E [ x(t − 1)] + Θ (t − 1)E [R(t − 1)] + △x(t ) = x(0) +

t  [Θ (k − 1)E (R(k − 1)) + △x(k)],

(24)

k=1

  t n   = x i ,0 + [θij (k − 1)E (rj,k−1 − r j,k−1 )] + △xi,k . k=1

j =1

Remark 2. Notice that with assumption Θ (t ) ̸= 0 (t = 1, 2, . . . , T − 1), the dynamic equation Eq. (24) indicates an affine feedback control policy. The investment proportion of the portfolio at period t , x(t ), depends on the sequence of past nominal proportion adjustments △x(t ) (t = 1, 2, . . . , T ) and on the sequence of past market reaction matrixes Θ (t ) (t = 1, 2, . . . , T − 1). So, using the subsequent adjustment approach above, the historical information can be better used for decision making. In this dynamic feedback optimization setting, the constraint of self-financing is represented by E

 n 

 △ x i ,t

= 0,

i=1

That is, for all t = 1, 2, . . . , T , we have

i=1

n 

ci,t |E(△ xi,t )|

n  i=1

  n      ci,t  [θij (t − 1)E(rj,t −1 − r j,t −1 )] + △xi,t  ,  j =1 

t = 1, 2, . . . , T .

2

i=1

βi,t − αi,t



6

+

xi,t −

n 

ci,t |E(△ xi,t )|

i =1

βi,t − αi,t

 xi,t

6

   n  aj,t −1 + bj,t −1  θij (t − 1) − ci,t   j =1 2     βj,t −1 − αj,t −1  − r j ,t − 1 +△xi,t  +  6

(26)

for t = 1, 2, . . . , T . Consequently, the expected value of the wealth at the beginning of period t + 1 can be expressed as

 Wt +1 = Wt

n   ai,t + bi,t

2

i=1

−

n 

+

βi,t − αi,t 6

 xi,t

 ci,t |E(△ xi,t )| ,

(27)

i =1

 T n    ai,t + bi,t t =1

(25)

j=1

i=1

=

=

 n  ai,t + bi,t

WT +1 = W1

The expected transaction costs of the portfolio at period t can be expressed as Ct =

2

+

for t = 1, 2, . . . , T . From (27), the expected value of the terminal wealth at the end of period T is

t = 1, 2, . . . , T .

  n n   [θij (t − 1)E(rj,t −1 − r j,t −1 )] + △xi,t = 0.

n   ai , t + b i , t i=1

for all t = 1, 2, . . . , T . Thus, the crisp form investment proportion of risky asset i at period t can be expressed by xi,t

Thus, the possibilistic mean value of the net return of the portfolio at period t can be computed as

−

n 

i =1

2

+

βi,t − αi,t 6

 xi,t

 ci,t |E(△ xi,t )| .

(28)

i=1

The formulas of the cumulative risk and the cumulative skewness of the T period investment are the same as Eqs. (14) and (15), respectively. 3.3.2. Explicit dynamic feedback portfolio selection formulations In contrast to the open-loop optimization models in the previous section, two dynamic feedback portfolio models will be

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Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

formulated by using the closed-loop optimization approach. In the proposed two dynamic feedback models with linear recourse, we consider the influence of the return deviation of historical data on the portfolio decision making (see the following two cases). Case 3. Based on the same decision criteria as the model (P1 ), the explicit model (P3 ) for multi-period portfolio selection in is shown in Box III, where xi,t )  E (△  closed-loop 

=

n

j =1

xi,t = xi,0 +

 − r j,k−1

aj,t −1 +bj,t −1

θij (t − 1) t

2

k=1

  n j =1

+

βj,t −1 −αj,t −1

θij (k − 1)

6



− r j ,t − 1

aj,k−1 +bj,k−1 2

+

+ △xi,t ;

βj,k−1 −αj,k−1 6



+ △xi,k ; 1′ = (1, 1, . . . , 1)′ is a vector of ones of

n-dimension; constraint (29)(a) represents the portfolio return must achieve or exceed the given minimum return level at each period; constraint (29)(b) denotes the self-financing constraint; constraint (29)(c) shows the relationship among elements in the market relation matrix at period t; constraint (29)(d) states the lower and upper bound constraints of xi,t . For notational simplicity, we denote constraints (29)(a)–(d) as x ∈ D0 . Case 4. Based on the same decision criteria as the model (P2 ), the explicit model (P4 ) for multi-period portfolio selection in closedloop is given in Box IV. Notice that the above two dynamic feedback models (P3 ) and (P4 ) can be, respectively, reduced to the open-loop models (P1 ) and (P2 ) by zeroing the market relation matrix Θ (t ) for all t. 4. Algorithm analysis Note that the proposed four models in Section 3 are multiobjective programming models. Since the incommensurability among objectives, it is not easy to directly solve them by using traditional algorithms. In order to remove the effects of the incommensurability of their objectives, in the following subsection, a TOPSIS-compromised programming approach will be presented to transform them into single objective models, respectively. Then, a genetic algorithm with penalty term will be presented to solve them. 4.1. TOPSIS-compromised programming technique for the proposed models

solutions nearest to the ideal point, which is composed of the optimum values of each objective function. However, in most practical situations, decision-maker may concern about the deviations from both the ideal point and the anti-ideal point, namely, the worst values of each objective function. In other words, decision-maker may like to have a decision which not only makes as much profit as possible, but also avoids as much risk as possible. Recently, there has been a continuing effort in designing new approaches for solving the multi-objective programming problem. Abo-Sinna and Amer (2005) extended the concept of TOPSIS for multiple objective decision making problems to obtain a compromise solution for large-scale multi-objective nonlinear programming problems. The major problem with this approach is the complexity of computation with the increasing of the number of variables. Yang (2000) presented a minimax reference point (MRP) approach for multi-objective optimization. But this approach only concerns about the distance of each objective function from its targeted value or goal. Sakawa, Kato, and Nishizaki (2003) proposed an interactive fuzzy satisficing (IFS) method for a multi-objective linear programming problems with random variable coefficients in objective functions and/or constraints. In contrast to the MRP approach, the IFS method only considers the deviation of each objective function from its negative point. To combine the information of the deviations from both ideal point and negative point, in the following section, we will present a novel TOPSIS-compromised programming approach for solving the proposed four models. To illustrate the advantage of the proposed TOPSIS-compromised programming approach, we will add a comparison analysis with the existing approaches in the next section. First, let us introduce the basic procedure of the designed approach. Here, we use models (P1 ) and (P2 ) to demonstrate the fundamental procedures of our designed approach. Similar to Abo-Sinna and Amer (2005) and Zeleny (1982), the fundamental procedure of the designed TOPSIS-compromised programming approach can be summarized as follows: Step 1: Calculate the ideal and anti-ideal solutions of each objective under the given constraints. For the proposed models (P1 ) and (P2 ), we can see that the objective of the cumulative risk of portfolio, V (x), is an attribute of the type of ‘‘less is better’’, namely, it is a type of cost objective function. Then, its ideal and anti-ideal solutions can be obtained by solving the following mathematical programming problems, respectively, V + (x) = min V (x),

Since the proposed four models are all multi-objective programming models, a common technique for solutions is to transform them into corresponding single objective programming models. In traditional multiple objective decision-making, many researchers have proposed some approaches such as goal programming, fuzzy goal programming and compromise programming. In Charnes and Cooper (1968) originally proposed the goal programming approach. The goal programming approach minimizes the weighted sum of absolute deviation to goal values subjectively given by the decision-maker. Its main shortcoming lies in the incommensurability between objectives of different measures. Different metrics of objectives may lead to incommensurable goal values and deviations from them. Summing up these incommensurable deviations is simply unreasonable. The fuzzy goal programming approach was first proposed by Zimmermann (1978). In the fuzzy goal programming approach, an aspiration level for each objective and the admissible violation constants for each goal should be given. However, these aspiration levels and admissible violation constants are subjectively determined by the decision-maker. In 1982, Zeleny developed a novel methodology, ‘‘compromise programming’’, to compare the performance of alternatives in a multiobjective decision problem. Its basic idea is to select those efficient

x∈D

V − (x) = max V (x). x∈D

However, the objectives of the expected value of the terminal wealth and the cumulative skewness are all attributes of the type of ‘‘more is better’’, that is, they are all the types of ‘‘benefit functions’’. So the ideal solutions of WT +1 and PS(x) can be obtained by solving the following single objective problems, respectively, WT++1 = max WT +1 , x∈D

PS+ (x) = max Skw(x). x∈D

The anti-ideal solutions of WT +1 and PS(x) can be obtained by solving the following mathematical programming problems, respectively, WT−+1 = min WT +1 , x∈D

PS− (x) = min Skw(x). x∈D

Then, the ideal and anti-ideal points of each model can be obtained. Here, we denote (WT++1 , V + (x)) and (WT++1 , V + (x), PS+ (x)) as the ideal points of (P1 ) and (P2 ), respectively. And their anti-ideal points are denoted as (WT−+1 , V − (x)) and (WT−+1 , V − (x), PS− (x)), respectively. Step 2: Use the obtained ideal and anti-ideal solutions to construct normalized positive and negative deviations for each objective, respectively. For the objective of the expected value of the terminal

Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

3049

    n T n     βi,t − αi,t ai,t + bi,t   + x i ,t − ci,t |E(△ xi,t )| max WT +1 = W1   2 6   i=1 t =1 i =1   n 2  n 2               xi,t (βi,t + αi,t ) + xi,t (βi,t − αi,t )    2  T  n      b i , t − ai , t βi,t + αi,t  i =1 i=1   min V ( x ) = + + x i ,t    i=1  2 6 72   t =1          (P3 )   n   βi,t − αi,t ai,t + bi,t    + xi,t − ci,t |E (△ xi,t )| ≥ r (t ) s.t .   2 6  i =1          n n    aj,t −1 + bj,t −1 βj,t −1 − αj,t −1   + △ x i ,t = 0 θij (t − 1) + − r j ,t − 1   2 6   i =1 j =1   ′   1 Θ (t − 1) = 0 0 ≤ xi,t ≤ ui,t , i = 1, 2, . . . , n; t = 1, 2, . . . , T

(29) (a)

(b) (c) (d)

Box III.

    n T n     βi,t − αi,t ai,t + bi,t    + xi,t − ci,t |E (△ xi,t )| max WT +1 = W1   2 6  i=1 t =1 i=1     n 2  n 2              xi,t (βi,t + αi,t ) + xi,t (βi,t − αi,t )    2  T  n       βi,t + αi,t bi,t − ai,t i=1 i=1   min V ( x ) = + + x  i,t    i=1  2 6 72   t =1                n 3  n 3           2 x β − x α 19 i , t i , t i , t i , t T  n n  (P4 )    i =1 i=1 1 1   +  xi,t αi,t xi,t βi,t max PS(x) =   24 t =1  45 3   i=1 i=1                    2     2  2      n n n n n             x β − x β x α + x ( b − a ) − x α  i , t i , t i , t i , t i , t i , t i , t i , t i , t i , t i , t      i=1 i =1 i=1 i=1 i=1         s.t . x ∈ D0 Box IV.

wealth WT +1 , its normalized positive and negative deviations are, respectively, defined as d1 (x) = +

WT +1 −

WT−+1

WT++1

,

−

d1 =

WT++1 WT++1

− W T +1 . − WT−+1

V − (x) − V (x) V − (x)

,

d− 2 (x) =

V ( x) − V + ( x) V − ( x)

−

V + (x)

PS(x) − PS− (x) PS (x) +

,

d− 3 (x) =

PS+ (x) − PS(x) PS+ (x) − PS− (x)

.

Step 3: Formulate the Lp -metric and Dp -metric for the objectives of



p

p

 1p

,

 1p

,

where weights θ1 and θ2 are the relative weights of WT +1 and V (x), and p = 1, 2 . . . , ∞. For model (P2 ), the Lp -metric of its objectives is defined as p

p

p

+ + p p p L2p = λ1 [d+ 1 (x)] + λ2 [d2 (x)] + λ3 [d3 (x)]



 1p

p

p

p

− − p p p D2p = λ1 [d− 1 (x)] + λ2 [d2 (x)] + λ3 [d3 (x)]



the given problem by aggregating and weighting the normalized positive and negative deviations, respectively. For model (P1 ), the Lp -metric of its objectives is defined as + p p L1p = θ1 [d+ 1 (x)] + θ2 [d2 (x)]

p

,

and the Dp -metric of its objectives is defined as

.

Similarly, for the objective of the cumulative skewness over T period investment PS(x), we have d+ 3 ( x) =

p

− p p D1p = θ1 [d− 1 (x)] + θ2 [d2 (x)]



For the objective of the cumulative risk over T period investment V (x), its normalized positive and negative deviations are, respectively, calculated by d+ 2 ( x) =

and the Dp -metric of its objectives is defined as

 1p

,

where weights λ1 , λ2 and λ3 are the relative weights of WT +1 , V (x) and PS(x), and p = 1, 2 . . . , ∞. Notice that the Lp -metrics above the proposed models measure the weighted sum of the positive deviations from their ideal points, and the Dp -metrics above measure the weighted sum of the negative deviations from their negative points. By varying the value of p, we can obtain different deviation measures for each model such that the greater value assigned to p the more importance given to the biggest individual positive and negative deviations. In other words, both Lp -metric and Dp -metric of each model decrease as the parameter p increases, and greater

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Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

emphasis is given to the largest positive and negative deviations in forming the total deviations. Specifically, p = 1 implies that an equal importance is assigned to each deviation, while p = 2 indicates that these deviations are weighted proportionately with the largest deviation having the largest weight. Finally, for p = ∞, the corresponding deviation is dominated by the largest regret distance between the individual ones. Step 4: Use the simple weighted average method to deal with both Lp -metric and Dp -metric, and then the TOPSIS-compromised problem can be obtained. For model (P1 ), we can obtain its corresponding TOPSIS-compromised problem in the following (P1′ ):

  p 1   max f1 (x) = τ θ1 [d+ (x)]p + θ2p [d+ (x)]p p  1 2  1  p − p p p − (1 − τ ) θ1p [d− (P1′ ) 1 (x)] + θ2 [d2 (x)]   s.t . x ∈ D,   θ1 + θ2 = 1, θj ∈ [0, 1], j = 1, 2. Similarly, for model (P2 ), we have

  p  max f ( x ) = τ λp1 [d+  2 1 (x)]      1p    p + p + p p + λ2 [d2 (x)] + λ3 [d3 (x)] (P2′ )    1  p − p − p p p p  − (1 − τ ) λp1 [d−  1 (x)] + λ2 [d2 (x)] + λ3 [d3 (x)]   s.t . x ∈ D,  λ1 + λ2 + λ3 = 1, λj ∈ [0, 1], j = 1, 2, 3. where τ ∈ (0, 1) can be considered as the preference coefficient of Lp -metric. The greater τ is, the more preference for Lp -metric the investor is. By using the same approach above, the models (P3 ) and P4 can also be transformed into two corresponding single objective optimization models. Denote them as (P3′ ) and (P4′ ), respectively. 4.2. Genetic algorithm Notice that the four transformed models are all complex nonlinear programming problems, if using the traditional algorithm may fail to obtain the optimal solutions. In order to avoid getting stuck at a local optimal solution, we will design a genetic algorithm to solve them. Genetic algorithm is a type of stochastic searching technique based on the mechanism of genetic and natural selection, which was originally proposed by Holland (1975). In the original genetic algorithm all parents are replaced by their offsprings to form a new generation, which is called generational replacement. However, this replacement process may lead to offsprings less than their parents because the genetic algorithm is blind. So it may cause some fitter chromosomes to be lost from the evolutionary process. To overcome the above-mentioned short-coming of the genetic algorithm, some improved genetic algorithms have been proposed such as Chootinan and Chen (2006), Gen and Cheng (1997) and Nanakorn and Meesomklin (2001). In this section, we design a genetic algorithm with penalty term to solve our models. Without loss of generality, let us take the model (P1 ) for example. Here, we first introduce its representation, initialization, evaluation function, selection, crossover and mutation. Representation and initialization. In this algorithm, we encode a randomly generated solution x = (x1,1 , . . . , xn,1 ; . . . ; x1,T , . . . , xn,T ) into chromosome by real-valued representation C = (c1,1 , . . . , cn,1 ; . . . ; c1,T , . . . , cn,T ), where the genes ci,t (i = 1, 2, . . . , n; t = 1, 2, . . . , T ) are restricted in the corresponding variable bounds. Repeat this operation pop− size times, then pop− size chromosomes, C1 , C2 , . . . , Cpop− size , can be obtained.

Evaluation function with penalty term. Since the randomly generated pop− size solutions may not satisfy the feasible region. Inspired by Lin and Liu (2008), we construct the following evaluation function with penalty term ev al(x) = exp(ω(f (x) − Kp(x))), where ω is a positive constant; K is a sufficiently large positive number; f (x) denotes the value of the objective function; p(x) represents a penalty term, and it is defined as

 0, if x ∈ feasible region,    T T  p(x) = max{gt (x), 0}2 + max{|ht (x)|, 0}2 ,   t = 1 t = 1  otherwise.

Here, gt (x) =

n  ai,t +bi,t i=1

2

+

βi,t −αi,t 6





xi,t − ci,t |xi,t − xi,t −1 | −

r (t ) and ht (x) = i=1 xi,t − 1 denote the inequality and equality constraints of (P1′ ), respectively.

n

Selection process. The selection process is based on the proportional selection. Similar to Gen and Cheng (1997), to prevent the populations from degenerating during iteration, both parents and their immediate offsprings are candidates for the new generation. We compare the parents with their offsprings by evaluating their fitness values to select the fitter ones to store in a mating pool. After the comparison operation, we can obtain pop− size chromosomes with higher fitness values. Then, we perform the proportional selection operation. The chromosome k is selected for a new population by the following probability Ps (xk ): P s ( xk ) =

ev al(xk ) ,  ev al(xk )

pop− size

k = 1, 2, . . . , pop− size.

k=1

In this process, the chromosomes of the current population with higher fitness values have higher chance as the parents to reproduce the offsprings. Crossover operation. The crossover process is based on arithmetic crossover. We first denote the crossover probability of the genetic algorithm as Pc . The crossover operation performs as follows. Generate a random number υ from interval (0, 1) and the chromosome Ck (k = 1, 2, . . . , pop− size) is selected as a parent provided that υ < Pc . Repeat this process pop− size times and Pc · pop− size chromosomes are expected to be selected to perform the crossover operation. The crossover operation on C1 and C2 will generate two offsprings C1′ and C2′ as follows: C1′ = υ C1 + (1 − υ)C2 ,

C2′ = (1 − υ)C1 + υ C2 .

Mutation operation. The mutation process is based on direction mutation operation. We denote Pm as the mutation probability of the genetic algorithm. Similar to the crossover process, the chromosome Ck is selected as a parent to perform the mutation operation if κ < Pm , where κ is a random number in [0, 1]. For any selected parent C , the gene ci,t of C = (c1,1 , . . . , cn,1 ; . . . , ci,t , . . . ; c1,T , . . . , cn,T ) is selected for mutation as follows: ci,t ′ = ci,t + κ y(ui,t − li,t ),

i = 1, 2, . . . , n; t = 1, 2, . . . , T ,

where y is a random number in interval [−0.5, 0.5]; li,t and ui,t are the lower and upper bounds of ci,t , respectively. After repeating the above operation pop− size times, Pm · pop− size chromosomes are expected to be selected to perform the mutation operation. The procedure of the designed genetic algorithm can be summarized as follows: Step1. Input parameters pop− size, Pc and Pm ; Step2. Initialize the randomly generated pop− size chromosomes;

Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053 Table 1 The asset return possibility distributions at each period. t

Asset i

ai,t

bi,t

αi,t

βi,t

t =1

Asset 1 Asset 2 Asset 3 Asset 4

1.0346 1.0388 1.0241 1.0159

1.1118 1.0972 1.0838 1.0852

0.2800 0.3036 0.2057 0.3541

0.7781 0.3860 0.3573 0.3737

t =2

Asset 1 Asset 2 Asset 3 Asset 4

1.0313 1.0390 1.0260 1.0170

1.1008 1.1072 1.0853 1.0886

0.2804 0.3126 0.2018 0.3556

0.7901 0.3891 0.3892 0.3786

t =3

Asset 1 Asset 2 Asset 3 Asset 4

1.0350 1.0340 1.0256 1.0169

1.1218 1.0989 1.0856 1.0882

0.2805 0.3136 0.2157 0.3541

0.7881 0.3876 0.3673 0.3937

Step3. Calculate the evaluation function values for all chromosomes; Step4. Perform the selection operation by proportional selection; Step5. Update the chromosomes by crossover and mutation operations; Step6. Repeat the third to fifth steps for given number of cycles; Step7. Report the best chromosome as the optimal solution. 5. Numerical example In this section, a numerical example based on real world data from Chinese Stock Exchange Market will be given to illustrate the ideas of our proposed models and the advantage of our designed algorithm over the IFS and the MRP approaches for solving our models. Assume that an investor wants to choose 4 risky assets for his investment, and he could reallocate his wealth at the beginning of each period. He intends to make a three-period investment with initial wealth W1 = 10 000. Original data come from the weekly closing prices of the 4 risky assets from Jan. 2001 to Jan. 2010. We set every three years as an observation to handle these historical data. Using the simple estimation method in Zhang, Xiao et al. (2010), the trapezoidal possibility distributions of the returns of assets can be obtained as shown in Table 1. Suppose that the transaction costs of assets at each period are identical, i.e., ci,t = 0.003 for all i = 1, 2, 3, 4; t = 1, 2, 3. Let the minimum expected return of the portfolio at the tth period r (t ) = 1.085 (t = 1, 2, 3), and the investment proportion xi,t ∈ [0, 0.5]. The values of p and τ are set to 1 and 0.5, respectively. To solve the proposed models with the designed genetic algorithm, we let the population size pop− size be 30, the crossover probability Pc be 0.8, the mutation probability Pm be 0.01 and the maximum iteration number be 1000. Now we first use the basic open-loop optimization models in Section 3.2 to reallocate the investor’s assets. By using Step 1 of Section 4.1, the ideal and anti-ideal solutions of each objective can be obtained. Using Steps 2–4, the transformed models (P1′ ) and (P2′ ) of problems (P1 ) and (P2 ) can be obtained, respectively. For solving the transformed models, we set θ1 = θ2 = 0.5 and λ1 = λ2 = λ3 = 31 . After running the designed genetic algorithm with 1000 generations, the corresponding compromised portfolio strategies of the two transformed models are listed in types I and II of Table 2, respectively. Case j For the sake of description, we denote W4 as the expected value of the terminal wealth of Case j for j = 1, 2, 3, 4. From Table 2, we can find that if the investor only pays attention to the expected value of the terminal wealth and the cumulative risk of the three period investment, his/her optimal investment strategies are listed in type I. As shown in type I, assets 1, 2 and 3 are selected for investment and asset 4 is excluded over the whole investment horizon. The corresponding expected value of the terminal wealth

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is W4Case 1 = 13 689.94. However, if the investor prefers to use the expected value of the terminal wealth, cumulative risk and skewness of portfolio as decision making criteria, he/she should adjust his/her wealth by the strategies shown in type II. From type II, we can find out that the investor should allocate most of his/her wealth to assets 1 and 3 over the whole investment periods. The expected value of the terminal wealth is W4Case 2 = 13 982.26. On the contrary, for solving models (P3 ) and (P4 ), we set the following parameters in the two dynamic feedback optimization models. Let the predetermined returns of portfolios at periods 1 and 2 be R(1) = (1.0732, 1.0680, 1.0792, 1.0538)′ and R(2) = (1.0661, 1.0731, 1.0869, 1.0566)′ , respectively. The initial investment proportion is set as x(0) = (0.25, 0.25, 0.25, 0.25)′ . Using the designed algorithm mentioned above, the optimal policy parameters of the two models can be obtained as follows. For model (P3 ), the nominal proportion adjustments of the three period investment are △x(1) = (0.1672, −0.1672, 0.25, −0.25)′ , △x(2) = (0.0002, −0.0001, 0.0084, −0.0085)′ and △x(3) = (0.0663, −0.0654, −0.0007, −0.0002)′ , respectively. And the corresponding market relation matrices are

 −0.0019  0.0005 Θ (1) =  −0.0985 0.0999  −0.007 −0.0022 Θ (2) =  0.0074 0.0018

−0.0003 0.0001 −0.0163 0.0165

0 0 0 0

0 0 , 0 0

−0.0019 −0.0012 0.0016 0.0015

0 0 0 0

0 0 . 0 0





For model (P4 ), the nominal proportion adjustments of the three period investment are △x(1) = (0.25, −0.25, 0.25, −0.25)′ , △x(2) = (−0.0323, 0.0286, 0.0272, −0.0235)′ and △x(3) = (0.011, 0.0002, −0.0235, 0.0124)′ , respectively. The obtained market relation matrices are



0.3776 −0.3346 Θ (1) =  −0.3186 0.2756

0.0707 −0.060 −0.057 0.0463

 −0.1259 −0.0035 Θ (2) =  0.2715 −0.1421

−0.0229 0.0098 0.0390 −0.0259

0 0 0 0 0 0 0 0



0 0 , 0 0



0 0 . 0 0

Then, the corresponding compromised portfolio strategies of models (P3 ) and (P4 ) can be obtained, as shown in types III and IV of Table 2, respectively. From type III, we can find that the portfolio strategy at period 1 of the model (P3 ) is x(1) = (0.4172, 0.0828, 0.5, 0)′ . This means that the investor should allocate 41.72%, 8.28% and 50% of his/her initial wealth in assets 1, 2 and 3, respectively. The investment strategy of period 2 is x(2) = (0.4173, 0.0827, 0.5, 0)′ , which means the investor should invest 41.73%, 8.27% and 50% of the wealth at the end of period 1 in assets 1 and 3, respectively. The portfolio at period 3 is x(3) = (0.4829, 0.0171, 0.5, 0)′ . This indicates that the investor will invest 48.29%, 1.71% and 50% of the wealth at the end of period 2 in assets 1, 2 and 3, respectively. The corresponding expected value of the terminal wealth is W4Case 3 = 13 833.06. However, if using the dynamic feedback model (P4 ) for portfolio decision-making, the corresponding investment strategies are shown in type IV. From type IV, we can find that the investor should averagely allocate his/her wealth between assets 1 and 3 over the whole investment horizon. At the end of period 3, the expected value of the terminal wealth is W4Case 4 = 13 994.01. From Table 2, we find that the dynamic feedback model performs better than the corresponding open-loop one based on

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Y.-J. Liu et al. / Automatica 48 (2012) 3042–3053

Table 2 The investment strategies of the four models with our designed approach. Case j

Type

Case j

t

Asset 1

Asset 2

Asset 3

Asset 4

W4

I

Case 1

t =1 t =2 t =3

0.3422 0.3420 0.4818

0.1578 0.1580 0.0182

0.5000 0.5000 0.5000

0.0000 0.0000 0.0000

13 689.94

II

Case 2

t =1 t =2 t =3

0.5000 0.5000 0.5000

0.2688 0.0000 0.0000

0.2312 0.5000 0.5000

0.0000 0.0000 0.0000

13 982.26

III

Case 3

t =1 t =2 t =3

0.4172 0.4173 0.4829

0.0828 0.0827 0.0171

0.5000 0.5000 0.5000

0.0000 0.0000 0.0000

13 833.06

IV

Case 4

t =1 t =2 t =3

0.5000 0.5000 0.5000

0.0000 0.0000 0.0000

0.5000 0.5000 0.5000

0.0000 0.0000 0.0000

13 994.01

Table 3 The comparative investment strategies of the models (P1 ) and (P2 ) obtained by the MRP and the IFS approaches. Case j

Case j

t

Asset 1

Asset 2

Asset 3

Asset 4

W4

Case 1

t =1 t =2 t =3

0.4802 0.1748 0.3329

0.0198 0.3253 0.1674

0.5000 0.4999 0.4997

0.0000 0.0000 0.0000

13 504.54

Case 2

t =1 t =2 t =3

0.4031 0.2736 0.3122

0.0969 0.2264 0.1878

0.5000 0.5000 0.5000

0.0000 0.0000 0.0000

13 514.25

Case 1

t =1 t =2 t =3

0.2718 0.4736 0.2748

0.2282 0.0267 0.2398

0.5000 0.4997 0.4850

0.0000 0.0000 0.0004

13 496.00

Case 2

t =1 t =2 t =3

0.2486 0.2486 0.4746

0.2514 0.2518 0.0254

0.5000 0.4996 0.5000

0.0000 0.0000 0.0000

13 515.04

IFS:

MRP :

the same decision criteria. As can be seen, the expected value of the terminal wealth order of Case 1 and Case 3 is W4Case 1 < W4Case 3 and the resulting order of Case 2 and Case 4 is W4Case 1 < W2Case 4 . In addition, after comparing the expected values of the terminal wealth, shown in type I and type II (or type III and type IV), we can find that the model (P2 ) (or (P4 )) with skewness performs better than the model (P1 ) (or (P3 )), which has no skewness criterion. So, we can conclude that the criterion of skewness pays an important role in portfolio selection, especially when the distribution of the return on the asset is asymmetric. In order to illustrate the effectiveness of the designed algorithm, we also use the MRP approach and the IFS method to solve the models (P1 ) and (P2 ). The comparative results are shown in Table 3. From Table 3, it can be seen that if using the IFS method to solve the proposed two models, the expected values of the terminal wealth are 13 504.54 and 13 514.25, respectively. If using the MRP approach to solve them, the expected values of their final wealth are 13 496.00 and 13 515.04, respectively. After comparison, we can find that our designed algorithm performs better than the MRP approach and the IFS method for solving the proposed models. As can be seen, our designed algorithm earns much higher terminal wealth than the aforementioned two approaches. 6. Conclusions In this paper, we investigate the multi-period portfolio selection problem with fuzzy returns, and propose four models to express investors with different preference for decision making criteria which consider some or all criteria including return, transaction cost, risk and skewness of portfolio. Since the proposed models are multi-objective nonlinear programming models, a TOPSIScompromised programming approach is presented originally to

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Yongjun Liu received his M.S. degree in Applied Mathematics from Guangxi University, Nanning, China in 2009. He has been working on a Ph.D. Degree in Department of Decision Science, School of Business Administration, South China University of Technology. His research interest includes mathematical finance, finance engineer and investment decision. Currently, he has particular interests in portfolio selection and risk management.

Weiguo Zhang received the B.E. degree from Ningxia University in 1983, M.E. degree from Sichuan University in 1995 and Ph.D. degree from Xi’an Jiaotong University in 2003, respectively. He has been a professor since 1998. His research interest has been in the areas of decision analysis, information processing, artificial intelligences, uncertainty reasoning, design and analysis of algorithms and their application to portfolio managements. His articles have been published in European Journal of Operational Research, Information Sciences, Information Processing Letters, Insurance: Mathematics and Economics, Mathematical and Computer Modelling, and others.

Weijun Xu received his B.S. and M.S. degrees in Applied Mathematics and Financial Mathematics from Ningxia University of Department of Applied Mathematics in 1999 and 2002, respectively. He also worked for his Ph.D. degree from 2002 to 2005 in the School of Management at Xi’an Jiaotong University, China. He is currently a professor in South China University of Technology. His research interest has been in the areas of decision theory and methodology and financial engineering.