Multi-period portfolio optimization under possibility measures

Economic Modelling 35 (2013) 401–408

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Multi-period portfolio optimization under possibility measures Xili Zhang a, Weiguo Zhang b, Weilin Xiao a,⁎ a b

School of Management, Zhejiang University, Hangzhou, China School of Business Administration, South China University of Technology, Guangzhou, China

a r t i c l e

i n f o

Article history: Accepted 16 July 2013 Keywords: Multi-period portfolio selection Possibility theory Central value Partial swarm optimization (PSO)

a b s t r a c t A single-period portfolio selection theory provides optimal tradeoff between the mean and the variance of the portfolio return for a future period. However, in a real investment process, the investment horizon is usually multi-period and the investor needs to rebalance his position from time to time. Hence it is natural to extend the single-period fuzzy portfolio selection to the multi-period case based on the possibility theory. In this paper, we propose the possibilistic expected value and variance for the terminal wealth with fuzzy forms after T periods by using the central value operator. Classes of multi-period possibilistic mean-variance models are formulated originally under the assumption that the proceeds of risky assets are fuzzy variables. Besides, we apply a particle swarm optimization algorithm to solve the proposed multi-period fuzzy portfolio selection models. A numerical example is given to illustrate the performance of the proposed models and algorithm. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Portfolio selection is seeking the best allocation of wealth among different assets. Numerous studies on portfolio selection are based on the probabilistic mean-variance methodology first proposed by Markovitz (1952), such as, Perold (1984), Pang, (1980), Best (2010) and Maringer and Kellerer (2003). It has also gained widespread acceptance as a practical tool for portfolio optimization. However, it is a single period model which makes a one-off decision at the beginning of the period and holds on until the end of the period, while it was natural to extend Markowitz's work to multi-period portfolio selections, such as Smith (1967), Mossin (1968), Merton (1969), Samuelson (1969), Fama (1970), Hakansson (1971), Elton and Gruber (1974), Francis and Kirzner (1991), Dumas and Luciano (1991), Östermark (1991), Grauer and Hakansson (1993), Pliska (1997), Li and Ng (2000) and Chen (2005). The literatures mentioned often used the probability distribution of asset returns. However, in the real world, the financial market behavior is affected by several non-probabilistic factors such as vagueness and ambiguity (see (Lacagnina and Pecorella, 2006)). The returns of assets are usually affected by many factors including economic, social, political and people's psychological factors as proposed by Huang (2011). Decision-makers are usually provided with information which is characterized by vague linguistic descriptions such as

⁎ Corresponding author. Tel./fax: +86 571 88206867. E-mail addresses: [email protected] (X. Zhang), [email protected] (W. Zhang), [email protected] (W. Xiao). 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.07.023

high risk, low profit, and high interest rate. In these cases, it is impossible for us to get the precise probability distribution we need. Furthermore, even if we know all the historical and current data, it is difficult that we predict the future return as a fixed value. Hence we need to consider that the future return has ambiguousness. There are several approaches dealing with ambiguous situations. On the one hand, some authors characterize uncertain distributions by defining a confidence region of their first two moments, so that the portfolio is robust against such uncertainty, see Pflug and Wozabal (2007) and Wozabal (2012). On the other hand, fuzzy set theory and Possibility theory, proposed by Zadeh (1978) and advanced by Dubois et al. (1988), may help to solve problems in uncertain and imprecise environments. In particular, in the field of portfolio selection, investors are faced with forecasting the performance of the assets they manage. Given the uncertainty inherent in financial markets, analysts are very cautious in expressing their guesses. Hence, there exist a lot of published works in the field of finances, which incorporate the approach of fuzzy set theory. Watada (1997) and León et al. (2002) discussed portfolio selection by using fuzzy decision theory. Inuiguchi and Tanino (2000) introduced a possibilistic programming approach to the portfolio selection problem under the minimax regret criterion. Carlsson et al. (2002) and Zhang et al. (2009a) introduced a possibilistic approach for selecting portfolios with the highest utility value under the assumption that the returns of assets are trapezoidal fuzzy numbers. Zhang et al. (2009b) discussed portfolio selection problem under possibilistic mean-variance utility and presented a SMO algorithm for finding the optimal solution. Zhang et al. (2010) proposed a risk tolerance model for portfolio adjusting problem with transaction costs based on possibilistic

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X. Zhang et al. / Economic Modelling 35 (2013) 401–408

moments. Gupta et al. (2008) applied multi criteria decision making via fuzzy mathematical programming to develop comprehensive models of asset portfolio optimization for the investors pursuing either of the aggressive or conservative strategies. In addition, Bilbao-Terol et al. (2006), Zhang and Wang (2008), Li and Xu (2009), Huang (2010) and Zhang et al. (2011) discussed the portfolio selection problems in a fuzzy uncertain environment by following the ideas of probabilistic mean-variance model. However, the researches mentioned above are on the singleperiod portfolio selection problems in fuzzy environment. There have been few literatures on multi-period fuzzy portfolio selection based on possibility theory. Zhang et al. (2012) presented a meansemivariance-entropy model for multi-period portfolio selection based on possibility theory, in which the risk level is characterized by the sum of the lower possibilistic semivariance of portfolio return in each period. The aim of this paper is to develop a multiperiod mean-variance portfolio selection model with fuzzy returns based on possibility theory. By using the central value operator introduced by Fullér and Majlender (2004) and Fullér et al. (2010a,b, 2011), we formulate the possibilistic expected value and possibilistic variance for the terminal wealth after T periods. A class of multi-period possibilistic mean-variance models is formulated originally. Moreover, an efficient solution is achieved for this class of fuzzy multi-period portfolio selection formulation, which makes the derived investment strategy an easy implementation task. The organization of this paper is as follows. Section 2 introduces some basics of possibility distributions. In Section 3, we develop a class of multi-period portfolio selection models based on possibility theory. The optimization models are converted into crisp forms when the return of risky assets is taken as symmetrical triangular fuzzy variables. An efficient solution is derived in Section 4 to generate the optimal portfolio policy. In Section 5, an example is given to illustrate the behavior of the proposed models and algorithm. This paper concludes in Section 6 with some suggestions for future study.

Furthermore, Ai is called the ith marginal possibility distribution of C and notated Ai = πi(C), where πi denotes the projection operator in Rn on the ith axis, i = 1,…,n. That is the marginal possibility distributions are derived by the projection principle from a joint possibility distribution; the projection of C on the ith axis is Ai for i = 1,…,n. In the sense of subsethood of fuzzy sets the largest joint possibility distribution defines the concept of non-interaction. Fuzzy numbers Ai ∈F ; i ¼ 1; …; n are said to be non-interactive if their joint possibility distribution is given by n o μ C ðx1 ; …; xn Þ ¼ min μ Ai ðxi Þ ; ∀x1 ; …; xn ∈R; i

and the equality [C]γ = [A1]γ × … × [An]γ holds for all γ ∈ [0,1]. In the following, we present the definition of the central value which is given in Fullér and Majlender (2004). Definition 2.1. Let C be a joint possibility distribution in Rn, let g : Rn → R be an integrable function, and let γ ∈ [0,1]. Then, the central value of g on [C]γ is defined by Ψ½C γ ðg Þ ¼ Z

½C 

γ

¼Z ½C γ

(
 n x∈R : μ C ðxÞ≥γ ;
 n cl x∈R : μ C ðxÞ≥γ ;

γ N0; γ ¼ 0;

where cl(C) means the closure of support of C. It is clear that if A∈ F is a fuzzy number then [A]γ is a convex and compact subset
of R for all γ ∈  [0,1], i.e. [A]γ = [AL(γ), AU(γ)], where AL ðγ Þ ¼ inf x∈R : μ A ðxÞ≥γ
 and AU ðγÞ ¼ sup x∈R : μ A ðxÞ≥γ . Let Ai ∈F ; i ¼ 1; …; n be fuzzy numbers, and let C in Rn be a fuzzy set. As discussed in Carlsson et al. (2005) and Fullér and Majlender (2004), fuzzy set C is said to be a joint possibility distribution of fuzzy numbers Ai, i = 1,…,n, if it satisfies the relationship sup μ C ðx1 ; …; xn Þ ¼ μ Ai ðxi Þ; ∀xi ∈R; i ¼ 1; …; n:

x j ∈R; j≠i

½C γ

g ðxÞdx Z

1 dx1 …dxn

 γ 1 Ψ½Aγ ðidÞ ≡ Ψ ½A ¼ Z ½Aγ

In this section, we will briefly recall some basics of possibility distributions which are crucial for our study (see (Carlsson and Fullér, 2011), for details). In general, the arithmetic operations on fuzzy numbers can be approached either by the direct use of the membership function (by the Zadeh extension principle) or by the equivalent use of the γ-cut representation (introduced by Goetschel et al. (1986)). A fuzzy number A is a fuzzy set in R that has a normal, fuzzy convex and continuous membership function of bounded support μ A : R→½0; 1.The family of all fuzzy numbers will be denoted by F . Fuzzy numbers can be considered as possibility distributions. If C is a fuzzy set in Rn then its γ-level set is defined by γ

dx

½C γ

g ðx1 ; …; xn Þdx1 …dxn ;

for all γ ∈ [0,1], where Ψ is called as the central value operator. It is obvious that for any fixed possibility distribution C and γ ∈ [0,1] Ψ½C γ is a linear operator. Especially, if n = 1 and g = x is the identity function (g = id) over R , then for any fuzzy number A∈F with [A]γ = [a1(γ), b1(γ)], γ ∈ [0,1], the central value of the identity function is computed by

2. Preliminaries

½C  ¼

Z

1

Z dx

½Aγ

xdx ¼

a1 ðγÞ þ b1 ðγÞ : 2

ð2:1Þ

Let us denote the projection functions on R2 by πx and πy, i.e., πx(u,v) = u and πy(u,v) = v for all u; v∈R. In what follows, we will show two important properties of the central value operator. Lemma 2.1. If A; B∈ F are non-interactive and g = πx + πy is the addition operator on R2 then    γ  γ Ψ½C γ πx þ πy ¼ Ψ½Aγ ðidÞ þ Ψ½Bγ ðidÞ ¼ Ψ ½A þ Ψ ½B ; for all γ ∈ [0,1]. Lemma 2.2. If A; B∈ F are non-interactive and g = πxπy is the multiplication operator on R2 then    γ  γ Ψ½C γ πx πy ¼ Ψ½Aγ ðidÞΨ½Bγ ðidÞ ¼ Ψ ½A Ψ ½B ; for all γ ∈ [0,1]. Based on Definition 2.1, the measure of dispersion is defined in Carlsson et al. (2005) as follows. Definition 2.2. Let A be a possibility distribution in R, and let γ ∈ [0,1]. Then the measure of dispersion of [A]γ is defined by  2   2 2 R½Aγ ðid; idÞ ¼ Ψ½Aγ id−Ψ½Aγ ðidÞ ¼ Ψ½Aγ id −Ψ½Aγ ðidÞ; for all γ ∈ [0,1].

X. Zhang et al. / Economic Modelling 35 (2013) 401–408

For A∈ F with [A]γ = [a1(γ), b1(γ)], γ ∈ [0,1], the dispersion of [A]γ is specified as  γ 1 R ½A ≡ R½Aγ ðid; idÞ ¼ Z ½Aγ

0

Z dx

½Aγ

B 1 2 x dx−B @Z ½Aγ

12

Z dx

½Aγ

C xdxC A ð2:2Þ

ðb ðγ Þ−a1 ðγ ÞÞ2 : ¼ 1 12

Ψ½C γ ðg Þf ðγ Þdγ:

1 0

R½Aγ ðid; idÞ f ðγ Þdγ:

E f ðAÞ ¼

1 0

a1 ðγ Þ þ b1 ðγ Þ f ðγ Þdγ; 2

ð2:3Þ

and the variance of A with respect to f is obtained as Z Var f ðAÞ ¼

 n n γ and ½Rt  ¼ ∑ ξt;i ðγÞxt;i ; ∑ ψt;i ðγ Þxt;i ; γ∈½0; 1, for xt,i ≥ 0. i¼1

i¼1

Let Wt be the wealth from the beginning of period 1 to the beginning of period t. Since the end-of-period capital position is given by the proceeds from current savings, the wealth of the end-of-period is obtained by W tþ1 ¼ W t Rt :

ð3:2Þ

1 0

t

i¼1

i¼1

the terminal capital over T periods is T

T

W Tþ1 ¼ ∏ Rt ¼ ∏ t¼1

n X

t¼1 i¼1

r t;i xt;i :

ð3:3Þ

n

n

i¼1

i¼1

tain the expressions of the expected value and variance of the terminal wealth WT + 1 in the following theorems. Theorem 3.1. Assume that the proceeds per unit of capital invested in a portfolio Rt ∈ F ; t ¼ 1; …; T are non-interactive with [Rt]γ = [ξt(γ), ψt(γ)], γ ∈ [0,1]. Then the possibilistic expected value of the terminal wealth WT + 1 with respect to f is   E f W Tþ1 ¼

2

ðb1 ðγ Þ−a1 ðγ ÞÞ f ðγ Þdγ: 12

t

By setting ξt ðγÞ ¼ ∑ ξt;i ðγ Þxt;i and ψt ðγ Þ ¼ ∑ ψt;i ðγ Þxt;i , we can ob-

Therefore, for any fuzzy number A∈F with [A]γ = [a1(γ), b1(γ)], γ ∈ [0,1], the expected value of A with respect to f is given by Z

ð3:1Þ

Solving Eq. (3.2) recursively, we have W tþ1 ¼ W 1 ∏ Ri ¼ ∏ Ri . Thus,

Definition 2.4. Let A be a possibility distribution in R. Let γ ∈ [0,1] and f be a weighting function. The measure of variance of A with respect to f is defined as Z

r t;i xt;i ;

Let the investor's initial wealth W1 = 1 without loss of generality. 1 0

Var f ðAÞ ¼

n X i¼1

Definition 2.3. Let C be a joint possibility distribution in Rn, let g : Rn →R be an integrable function, and let f be a weighting function defined by Fullér and Majlender (2004) with ∫10 f(γ)dγ = 1. The expected value of g on C with respect to f is defined by E f ðg; C Þ ¼

in n risky assets at time period t within the planning horizon are denoted by rt = (rt,1,…,rt,n) with [rt,i]γ = [ξt(γ), ψt(γ)], γ ∈ [0,1] being proceeds per unit of capital invested in the ith asset at period t. That is, if we invest an amount xt,i in asset i at the beginning of the period t, we will obtain xt,irt,i at the end of that period. Therefore, the proceeds per unit of capital invested in a portfolio in the tth period can be obtained as Rt ¼

Hence, the measure of the expected value and variance of an integrable function on C are defined in Carlsson et al. (2005) by using the measure of central value and dispersion.

Z

403

Z

∏

0 t¼1

ð2:4Þ

The relationships of central value, expected value and variance of possibility distributions and the corresponding probabilistic values have been explained in Carlsson et al. (2005) and Fullér and Majlender (2004). As pointed out in Fullér and Majlender (2004), if all γ-level sets of C are symmetrical, then the covariance between its marginal distributions A and B becomes zero for any weighting function f, that is,

1 T

ξt ðγÞ þ ψt ðγ Þ f ðγ Þdγ: 2

Proof. Suppose that the proceeds per unit of capital invested in a portfolio Rt ∈F ; t ¼ 1; …; T are non-interactive with [Rt]γ = [ξt(γ), ψt(γ)], γ ∈ [0,1]. From Definition 2.1, Lemma 2.2 and Eq. (3.3), it follows that

T    γ Ψ W Tþ1 ¼ Ψ ∏ ½Rt  ¼Ψ t¼1

3. The possibilistic mean-variance model of multi-period portfolio selection For multi-period portfolio selection problems we consider a planning horizon of T periods in discrete time t = 1,…,T + 1. The investor can allocate his wealth among the n risky assets offering fuzzy returns. The portfolio is allocated at t = 1 the beginning of the first period and thereafter restructured at the beginning of each of the following T − 1 consecutive time period before the investor obtains the reward after the final period or at the beginning of period T + 1. We assume that the investor does not invest the additional capital during the T periods. Suppose that a portfolio at period t is xt = (xt,1,…,xt,n), where xt,i denotes the proportion of investment in risky asset i (i = 1,…,n) at the beginning of period t (t = 1,…,T). The proceeds per unit of capital invested

∏ ½Rt γ

T

∏ πt

t¼1



T T  γ ¼ ∏ Ψ½Rt γ ðπt Þ ¼ ∏ Ψ ½Rt  : t¼1

t¼1

t¼1

Cov f ðA; BÞ ¼ 0; even though A and B may be interactive.

T

By Eq. (2.1), the central value on WT + 1 can be expressed as T ξ ðγ Þ þ ψ ðγ Þ   t Ψ W Tþ1 ¼ ∏ t : 2 t¼1

By Definition 2.3, the proof is complete. Theorem 3.2. Assume that the proceeds per unit of capital invested in a portfolio Rt ∈F ; t ¼ 1; …; T are non-interactive with [Rt]γ = [ξt(γ), ψt(γ)], γ ∈ [0,1]. Then the possibilistic variance value of the terminal wealth WT + 1 with respect to f is  ) ½ψt ðγ Þ−ξt ðγ Þ2 ξ ðγÞ þ ψt ðγÞ 2 f ðγ Þdγ þ t 12 2 0 t¼1  Z 1 T ξ ðγ Þ þ ψt ðγÞ 2 ∏ t f ðγ Þdγ: − 2 0 t¼1

  Var f W Tþ1 ¼

Z

1 T

∏

(

404

X. Zhang et al. / Economic Modelling 35 (2013) 401–408

Proof. Suppose that the proceeds per unit of capital invested in a portfolio Rt ∈F ; t ¼ 1; …; T are non-interactive with [Rt]γ = [ξt(γ), ψt(γ)], γ ∈ [0,1]. From the Definition 2.2, Lemma 2.2 and Eq. (3.3), the dispersion measure of WT + 1 can be explored as   R W Tþ1 ¼ R

T

∏ ½Rt γ

T

T

t¼1

t¼1

∏ πt ; ∏ πt

t¼1

¼Ψ

T

∏ ½Rt γ



2 2 ! 6 −6 ∏ πt 4Ψ



T

t¼1

t¼1

T

∏ ½Rt γ

where  is a required terminal wealth,or equivalently, Z max Z s:t:

32  7 ∏ πt 7 5 : T

T ξ ðγ Þ þ ψ ðγ Þ 1 t t f ðγÞdγ 0∏ 2 t¼1 (  T ½ψt ðγ Þ−ξt ðγÞ2 ξt ðγÞ 1 þ ∏ 0 12 t¼1

) þ ψt ðγ Þ 2 f ðγ Þdγ 2 Z T  ξ ðγÞ þ ψt ðγ Þ 2 0 f ðγÞdγ ≤σ;ð3:8Þ − 1∏ t 2 t¼1

W 1 ¼ 1; W tþ1 ¼ W t Rt ; t ¼ 1; …; T;

t¼1

n X

t¼1

xt;i ¼ 1; t ¼ 1; …; T;

xt;i ≥0; i ¼ 1; …; n; t ¼; …; T;

i¼1

From Lemma 2.1 and Eq. (2.1), it follows that 2



6 6Ψ 4

T

∏ ½Rt γ

T



∏ πt

t¼1

32



2  T T 7   2 7 ¼ ∏ Ψ½R γ ðπt Þ ¼ ∏ Ψ ½Rt γ : 5 t t¼1

t¼1

ð3:4Þ

t¼1

From Definition 2.2, it follows    γ 2 2 γ Ψ½Rt γ π t ¼ R ½Rt  þ Ψ ½Rt  ; t ¼ 1; …; T: Then we have

Ψ

T

∏ ½Rt γ

T

∏ πt

t¼1

2 !

  T 2 ¼ ∏ Ψ½Rt γ πt t¼1

t¼1

i T h  γ 2 γ ¼ ∏ R ½ Rt  þ Ψ ½ R t  : t¼1

ð3:5Þ Rt ¼

From Eqs. (2.1) and (2.2), combining Eqs. (3.4) and (3.5), it follows that i T T h    γ 2 γ 2 γ R W Tþ1 ¼ ∏ R ½Rt  þ Ψ ½Rt  − ∏ Ψ ½Rt  t¼1 t¼1 (  ) T  T ½ψt ðγ Þ−ξt ðγÞ2 ξ ðγÞ þ ψt ðγ Þ 2 ξ ðγÞ þ ψt ðγÞ 2 −∏ t þ t : ¼∏ 12 2 2 t¼1 t¼1

i¼1

xt;i ≥0; i ¼ 1; …; n; t ¼ 1; …; T;

bt;i xt;i ;

i¼1 n

n

i¼1

i¼1

n X

α t;i xt;i ;

i¼1

n X

! βt;i xt;i ; t ¼ 1; …; T;

i¼1

where ∑ α t;i xt;i ¼ ∑ βt;i xt;i . Then the γ-level set of Rt is obtained as " n X

bt;i xt;i −ð1−γÞ

i¼1

n X i¼1

α t;i xt;i ;

n X

bt;i xt;i þ ð1−γÞ

i¼1

n X

# α t;i xt;i ; γ∈½0; 1:

i¼1

According to Theorem 3.1, the central value of [Rt]γ can be expressed as n X

 γ Ψ ½ Rt  ¼

bt;i xt;i :

i¼1

If the weighting function is set to be f(γ) = 2γ giving ∫10 f(γ)dγ = 1, the crisp form of the possibilistic expected value of WT + 1 is obtained as   E f W Tþ1 ¼

Z

1 0

T

2γ ∏

n X

t¼1 i¼1

T

bt;i xt;i dγ ¼ ∏

n X

t¼1 i¼1

bt;i xt;i :

ð3:9Þ

If the weighting function is set to be f(γ) = 1 giving ∫10 f(γ)dγ = 1, the crisp form of the possibilistic expected value of WT + 1 is obtained as

T 1 0∏ t¼1

W 1 ¼ 1; W tþ1 ¼ W t Rt ; t ¼ 1; …; T; n X xt;1 ¼ 1; t ¼ 1; …; T;

n X

r t;i xt;i ¼

i¼1

γ

According to Definition 2.4, the proof of the theorem is complete. Based on Theorems 3.1 and 3.2, analogy to the theory of dynamic mean-variance criterion, the investor seeks an optimal investment strategy x = {x1, …,xT} to minimize the variance of the terminal wealth Varf(WT + 1) which subjects to that the expected terminal wealth Ef(WT + 1) is not small than a preselected level or to maximize the expected value of the terminal wealth Ef(WT + 1) which subjects to that the variance of the terminal wealth Varf(WT + 1) is not greater than a preset risk level. A possibilistic mean-variance formulation for multi-period portfolio selection can be posed as one of the following two forms: (  ) ½ψt ðγÞ−ξt ðγÞ2 ξ ðγ Þ þ ψt ðγ Þ 2 f ðγÞdγ min þ t 12 2 Z T  ξ ðγÞ þ ψt ðγ Þ 2 1 f ðγÞdγ − 0∏ t 2 t¼1 Z T ξ ðγ Þ þ ψ ðγ Þ 1 t t f ðγ Þdγ ≥ε; s:t: 0∏ 2 t¼1

n X

½Rt  ¼

ð3:6Þ

Z

where σ is the preselected risk value. Zhang et al. (2010) presented a simple method to estimate the possibility distributions for the returns of assets. After the membership functions of rt,i, i = 1, …, n, t = 1, …, n are given, we can obtain the crisp form of the multi-period portfolio optimization model, Eqs. (3.7) or (3.8). For defuzzification, we assume that the proceeds per unit of capital invested in the ith asset at period t are a symmetrical triangular fuzzy variable denoted by rt,i = (bt,i,αt,i,βt,i) with αt,i = βt,i(t = 1, …, T, i = 1, …, n). Then the γ-level set [rt,i]γ = [bt,i − (1 − γ)αt,i, bt,i + (1 − γ)αt,i], γ ∈ [0,1]. From the operation properties of fuzzy number, since xt,i ≥ 0(i = 1, …, n, t = 1, …, T), the proceeds per unit of capital invested in a portfolio xt are also a symmetrical triangular fuzzy variable with the following form

  E f W Tþ1 ¼ ð3:7Þ

Z

1

T

∏

n X

0 t¼1 i¼1

T

bt;i xt;i dγ ¼ ∏

n X

t¼1 i¼1

bt;i xt;i ;

which is coincident to Eq. (3.9). According to Theorem 3.2, the dispersion of [Rt]γ is  γ R ½R t  ¼

" #2 n X 1 α t;i xt;i : ð1−γÞ 3 i¼1

ð3:10Þ

X. Zhang et al. / Economic Modelling 35 (2013) 401–408

If T = 3, then

If the weighting function is f(γ) = 2γ, then the possibilistic variance value of WT + 1 can be calculated as

Var f ðW 4 Þ



 Var f W Tþ1 ( " #2 Z 1 n T X 1 2γ ∏ α t;i xt;i þ ð1−γÞ ¼ t¼1 3 0 i¼1 !2 Z 1 n T X dγ− 2γ ∏ bt;i xt;i dγ t¼1

0

Z ¼2

"

i¼1

n 1 X ð1−γ Þ ∏ α t;i xt;i 3 0 t¼1 i¼1 !2 n T X dγ− ∏ bt;i xt;i : T

1

t¼1

n X

¼

!2 ) bt;i xt;i

i¼1

i¼1

!2

n X

2

γ þ

ð3:11Þ

!2 #

i¼1

n X

1

n X

I, |S| = k} and I = {1, …,T}. In the similar way, we have " !2 !2 # n n T X 1 X 2 dγ ð1−γÞ ∏ α t;i xt;i γ þ bt;i xt;i 3 i¼1 0 i¼1 3 2 t¼1 !2 !2 Z 1X T 6X n n  X 7 2k 1 X 2kþ1 6 ∏ α t;i xt;i b j;i x j;i 7 ¼2 dγ 5 γ −γ 4 3 0 k¼0 S∈S t∈S; i¼1 i¼1 T X k¼0

j∉S

2

n 6X 1 1 X 6 ∏ α x 4 ð2k þ 1Þðk þ 1Þ S∈S t∈S; 3 i¼1 t;i t;i k

!2

n X

!2 b j;i x j;i

i¼1

j∉S

α 1;i x1;i

i¼1

i¼1 n X

!2 # b1;i x1;i

i¼1 n X

!2

b2;i x2;i

n X

i¼1

!2

n X

α 2;i x2;i

!2

b1;i x1;i

i¼1

!2

n X

α 3;i x3;i

!2 b1;i x1;i

!2

b3;i x3;i

i¼1 n X

!2 # b2;i x2;i

:

i¼1

Based on the above discussions, when the proceeds per unit of capital invested in the ith asset at period t is a symmetrical triangular fuzzy variable and f(γ) = 1, problem (3.7) can be converted into the following form: 2 min

T X k¼1

X 1 6 6 ∏ 2k þ 1 4

S∈Sk t∈S; j∉S

n T X

s:t: ∏

3

!2 b3;i x3;i

i¼1 n X

i¼1

1

k

α 3;i x3;i

!2

i¼1

S∈Sk t∈S; j∉S

k¼0

n X

!2

i¼1

"

i¼1

þ

n X

α 2;i x2;i

i¼1

þ

T B T C k C Note that ∏ ðst y þ pt Þ ¼ ∑ B @ ∑ ∏ st p j Ay ; where Sk = {S : S

¼

þ

i¼1

!2

n X

1 þ 33

bt;i xt;i

0

Z 2

!2 " !2 !2 !2 n n n n 3 X X X X 1 1 ∏ α t;i xt;i þ α 1;i x1;i α 2;i x2;i b3;i x3;i 3 2 7  3 t¼1 i¼1 53 i¼1 i¼1 i¼1 !2 !2 !2 n n n X X X þ α 1;i x1;i α 3;i x3;i b2;i x2;i

i¼1

t¼1

405

t¼1 i¼1 n X

7 7: 5

n 1X

3

!2 α t;i xt;i

i¼1

n X

!2 b j;i x j;i

i¼1

3 7 7 5 ð3:15Þ

bt;i xt;i ≥;

xt;i ¼ 1; t ¼ 1; …; T;

i¼1

xt;i ≥0; i ¼ 1; …; n; t ¼ 1; …; T:

ð3:12Þ

And problem (3.8) can be converted into the following form: Substituting Eq. (3.12) to Eq. (3.11), we obtain T

2   Var f W Tþ1 ¼

T X k¼1

6X 1 1 6 ∏ ð2k þ 1Þðk þ 1Þ 4S∈S t∈S; 3 k

j∉S

!2

n X

α t;i xt;i

i¼1

n X

!2 b j;i x j;i

i¼1

3 7 7; 5

ð3:13Þ for f(γ) = 2γ. In the similar way, if f(γ) = 1, then the possibilistic variance value of WT + 1 can be calculated as 2 



Var f W Tþ1 ¼

T X k¼1

n X 1 6 1 X 6 ∏ α x 4 2k þ 1 S∈S t∈S; 3 i¼1 t;i t;i k

!2

n X

!2 b j;i x j;i

i¼1

j∉S

3 7 7: 5 ð3:14Þ

In Eqs. (3.13) and (3.14), the only difference is the coefficients in each component for different weighting functions. In order to illustrate the above formula, we take T = 2 and T = 3 in Eq. (3.14) for example. When T = 2, it follows that Var f ðW 3 Þ ¼

n 2 X 1 ∏ α t;i xt;i 2 5  3 t¼1 i¼1

!2 þ þ

1 33 n X i¼1

"

n X

!2 α 1;i x1;i

i¼1

α 2;i x2;i

!2

n X

!2 b2;i x2;i

i¼1 n X i¼1

!2 #

b1;i x1;i

:

max ∏

n X

t¼1 i¼1

s:t:

T X k¼1

n X

bt;i xt;i 2

X 1 6 6 ∏ 2k þ 1 4

S∈Sk t∈S; j∉S

n 1X

3

!2 α t;i xt;i

i¼1

n X i¼1

!2 b j;i x j;i

3 7 7 ≤σ; 5

ð3:16Þ

xt;i ¼ 1; t ¼ 1; …; T;

i¼1

xt;i ≥0; i ¼ 1; …; n; t ¼ 1; …; T:

4. The PSO algorithm In this section, we describe the formulations of a particle swarm optimization (PSO) for the proposed multi-period fuzzy portfolio selection problem. PSO is a kind of evolutionary algorithm first proposed by Eberhart and Kennedy (1995) and Kennedy and Eberhart (1995) for nonlinear functions in 1995. It has been successfully applied to many research fields. Generally, PSO is characterized as simple in concept, easy to implement, and computationally efficient. In a PSO system, the swarm is initialized firstly in a set of randomly generated potential solutions and then performs the search for the optimal solution iteratively. One of the key issues in designing a successful PSO algorithm is the representation step, i.e., finding a suitable mapping between problem solution and PSO particle. According to n−1the equation constraint in problem (3.15), we can write xt;n ¼ 1− ∑ xt;i ðt ¼ 1; …; T Þ. Therefore, i¼1 the PSO population is represented as a two-dimensional array consisting of M particles, each represented as an (n − 1)T-dimensional vector, where n is the number of the assets and T is the planning horizon, i.e., Pm = [x1,1, …,x1,n − 1, …,xT,1, …,xT,n − 1], m = 1, …, M. The

406

X. Zhang et al. / Economic Modelling 35 (2013) 401–408

velocity of particle is respectively defined as Vm = [v1,1, …,v1,n − 1, …, vT,1, …,vT,n − 1], m = 1, …, M. Here we are faced with a non-linear objective function along with a set of inequality constraints. Based on the penalty approach described by Di Pillo and Grippo (1989), we have the following fitness function: 2 F¼

T X k¼1

X 1 6 6 2k þ 1 4S∈S k

"

∏

t∈S; ð4:1Þ j∉S

(

n 1 X α x 3 i¼1 t;i t;i

n T X

þwp  max 0; − ∏

t¼1 i¼1

)

!2

T X n X

bt;i xt;i −

n X

!2 b j;i x j;i

i¼1

3 7 7 5 ð4:1Þ

# n o min 0; xt;i ;

t¼1 i¼1

where wp is a very big number about the weights of penalty which prescribes a high cost to infeasible points. Note that in the feasible space of solutions, the contribution from the penalty is zero. The computational flow of PSO for multi-period fuzzy portfolio selection consists of the following steps:

Step 7. If Tc is less than the predetermined maximum iteration number, go to step 3, else output the Pgb and Fgb. 5. Numerical example In this section, we give an example to help understand the optimization ideas. Assume that the investor chooses 3 assets for investment and 3 periods for the planning horizon in discrete time t = 1, …, 4. The investor's initial wealth W1 = 1. Proceeds of the risky assets at each period are regarded to be symmetrical triangular fuzzy variables and the value of membership functions for the asset returns is given in Table 1. We take problem (3.15) for testing. Let the portfolio at period t be xt = (xt,1,xt,2,xt,3), t = 1, 2, 3. Then, the model reified by the data in Table 1 is in the following form: min

Step 1. Initialize the population in the search space: xt,i(1) in Pm(1) and vt,i(1) in Vm(1), t = 1, …, T, i = 1, …, n − 1, m = 1, …, M, are generated by randomly selecting a value with uniform probability over the optimized parameter search space [0,1]

" !2 !2 !2 !2 n n n n X X X 1 3 X 1 ∏ α t;i xt;i þ α 1;i x1;i α 2;i x2;i b3;i x3;i 189 t¼1 i¼1 45 i¼1 i¼1 i¼1 !2 !2 !2 n n n X X X þ α 1;i x1;i α 3;i x3;i b2;i x2;i i¼1 n X

þ þ

n−1

and [0,0.1] respectively. Set Tc = 1 and xt;n ð1Þ ¼ 1− ∑ xt;i ð1Þ

1 9

i¼1

ðt ¼ 1; …; T Þ. Step 2. According to Eq. (4.1), evaluate the fitness function of each particle in the initial population Fm(1). For each particle, set the inpb pb dividual best Ppb m = [(xt,i )]t = 1,…,T, i = 1,…,n − 1, where xt,i = pb xt,i(1) and Fm = Fm(1), m = 1, …, M. Search for the global gb best value Fgb among Fpb m . Set the particle associated with F as the global best Pgb = [(xgb t,i )]t = 1,…,T, i = 1,…,n − 1. Step 3. Update the time counter Tc = Tc + 1. Step 4. Each particle's velocity is updated according to the following equation: 

þ

!2 α 1;i x1;i

i¼1

!2

α 2;i x2;i

i¼1 n X

þ

!2 α 3;i x3;i

i¼1 3

s:t: ∏ 3 X

3 X

t¼1 i¼1

!2 α 3;i x3;i

i¼1

n X

n X

n X

α 2;i x2;i

i¼1

"

i¼1

!2

!2 b2;i x2;i !2

b1;i x1;i

i¼1 n X

!2 # b1;i x1;i

i¼1

n X i¼1

n X

i¼1 n X

!2 b1;i x1;i

i¼1

n X

!2 b3;i x3;i

i¼1 n X

ð5:1Þ

b3;i x3;i

i¼1 n X

!2 !2 #

b2;i x2;i

i¼1

bt;i xt;i ≥;

xt;i ¼ 1; t ¼ 1; 2; 3;

i¼1

xt;i ≥0; i ¼ 1; 2; 3; t ¼ 1; 2; 3:



pb c1 r 1 xt;i −xt;i ðT c −1Þ

vt;i ðT c Þ ¼ vt;i ðT c −1Þ þ   gb þ c2 r 2 xt;i −xt;i ðT c −1Þ ;

where t = 1, …, T, i = 1, …, n − 1. c1 and c2 are two positive constants called acceleration factors. r1 and r2 are uniformly distributed numbers in [0,1]. Step 5. Based on the updated velocities, each particle changes its position according to the following equation: xt;i ðT c Þ ¼ xt;i ðT c −1Þ þ vt;i ðT c Þ; t ¼ 1; …; T; i ¼ 1; …; n−1; n −1 X xt;i ðT c Þ; t ¼ 1; …; T: xt;n ðT c Þ ¼ 1− i¼1

Step 6. Update the individual best and global best. The particle is evaluated according to its updated position: If Fm(Tc) ≤ Fpb m, pb then update individual best as Ppb m = Pm(Tc) and Fm = Fm(Tc), m =1, …, M. Search for the minimum value Fmin among Fpb m (m =1, …, M) and set the particle associated with Fmin as Pmin. If Fmin b Fgb, then update global best as Pgb = Pmin and Fgb = Fmin.

The proposed PSO approach for multi-period fuzzy portfolio selection problem was implemented using the C+ + language. In our implementation, the maximum iteration number is set as 5000 and other parameters are selected as: number of particles M = 100, the acceleration factors c1 = c2 = 2, and the weight of penalty wp = 10000. Varying the value of , we listed the results in Table 2. It can be seen that when the preset terminal wealth  becomes bigger, the variance becomes larger, which reflects the relationship between risk and proceeds. When  = 1.2, the strategy for period 1 is x1 = (0.0109,0.4247,0.5644), which means that the investor should invest 42.47% and 56.44% of his initial wealth in assets 2 and 3 respectively. The portfolio in period 2 is x2 = (0.9249,0.0669,0.0082), which means asset 1 should be invested with 92.49% of the wealth after the first period. At period 3, x3 = (0.8728,0.1027,0.0245) means that the investor should reduce the proportion of asset 1 to 87.28% and increase the rates of the other two assets slightly. 6. Conclusion Financial markets are usually very sensitive and the influences on the returns of assets include general economic, industry and the

Table 1 The symmetrical triangular possibility distribution of 3 assets in 3 periods. Periods

t=1

Assets

1

2

3

t=2 1

2

3

t=3 1

2

3

bt,i αt,i(βt,i)

1.1426 0.5264

1.0629 0.4662

1.0953 0.3088

1.0587 0.3298

0.8412 0.3923

1.218 0.4865

1.0445 0.4158

1.1855 0.4286

1.1253 0.50

X. Zhang et al. / Economic Modelling 35 (2013) 401–408

407

Table 2 The optimal strategies with different values of . E

1

Assets

1

2

3

1

2

3

1

2

3

Period 1 Period 2 Period 3 Varf(W4)

0.0182 0.1345 0.8886 0.151187

0.3445 0.8640 0.07974

0.6373 0.0015 0.03166

0.0109 0.9249 0.8728 0.202574

0.4247 0.0669 0.1027

0.5644 0.0082 0.0245

0.6033 0.5378 0.0331 0.316679

0.0328 0.0016 0.9610

0.3639 0.4606 0.0059

1.2

performances of concerned company, which are affected by human's subjective intention. Fuzzy models are useful to handle the vagueness and ambiguity of the predictions. The possibilistic mean-variance approach has been extended in this paper to multi-period fuzzy portfolio selection problems. With the central value operator and its properties, we have derived the possibilistic expected value and possibilistic variance value for the terminal wealth after some periods. Under the assumption that the proceeds of risky assets are symmetrical triangular fuzzy variables, a class of multi-period possibilistic mean-variance models and their crisp forms are formulated originally. Moreover, we formulate a particle swarm optimization algorithm for this class of multi-period fuzzy portfolio selection problems, which makes the derived investment strategy an easy implementation task. An example is given to show the whole idea of the proposed models and algorithm. Although the assumption of non-interactivity is unduly restrictive, the determination of multi-period portfolio selection in the presence of possibilistic distributions gives insight into the nature of dynamic portfolio optimization in uncertain environment. And it is also pointed out that the covariance between its marginal distributions becomes zero for any weighting function if all γ-level sets of a fuzzy set are symmetrical. Moreover, the theory of interactive fuzzy numbers is still under development. Some of the future research subjects are to investigate the multi-period portfolio optimization when the returns within and/or between periods are interactive, and to investigate the constrained multi-period portfolio problems with transaction costs based on the possibility theory and their efficient solution methodology. Acknowledgments The authors appreciate greatly the editor and the anonymous referees for their useful comments and suggestions concerning the earlier version of this paper. The authors also thank Shubham Gupta for his carefully reading of our manuscript and his detailed comments and suggestions for further improvement. This research was supported by the National Natural Science Foundation of China (Nos. 71101056; 71171086), the major program of National Social Science Foundation of China (11&ZD156), Natural Science Foundation of Guangdong Province, China (No. S2011040005723), Distinguished Young Talents in Higher Education of Guangdong, China (No. WYM11010), the Fundamental Research Funds for the Central Universities, SCUT (No. 2012ZM0029), the Humanity and Social Science Youth foundation of Ministry of Education of China (No. 13YJC630227), Zhejiang Provincial Natural Science Foundation of China (No. LQ13G010001) and the Fundamental Research Funds for the Central Universities. References Best, M., 2010. Portfolio Optimization. Chapman & Hall/CRC. Bilbao-Terol, A., Perez-Gladish, B., Arenas-Parra, M., Rodriguez-Uria, M., 2006. Fuzzy compromise programming for portfolio selection. Applied Mathematics and Computation 173 (1), 251–264. Carlsson, C., Fullér, R., 2011. Possibility for Decision: A Possibilistic Approach to Real Life Decisions. Springer Berlin Heidelberg, Berlin, Heidelberg. Carlsson, C., Fullér, R., Majlender, P., 2002. A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems 131 (1), 13–21.

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