• Editorial Board +
• For Contributors +
• Journal Search +
Journal Search Engine
ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.1 pp.118-128
DOI : https://doi.org/10.7232/iems.2017.16.1.118

# The Admissible Multiperiod Mean Variance Portfolio Selection Problem with Cardinality Constraints

Peng Zhang*, Bing Li
School of Management, Wuhan University of Science and Technology, Wuhan, China
Corresponding author : zhangpeng300478@aliyun.com
May 9, 2016 November 4, 2016

## ABSTRACT

Uncertain factors in finical markets make the prediction of future returns and risk of asset much difficult. In this paper, a model,assuming the admissible errors on expected returns and risks of assets, assisted in the multiperiod mean variance portfolio selection problem is built. The model considers transaction costs, upper bound on borrowing risk-free asset constraints, cardinality constraints and threshold constraints. Cardinality constraints limit the number of assets to be held in an efficient portfolio. At the same time, threshold constraints limit the amount of capital to be invested in each stock and prevent very small investments in any stock. Because of these limitations, the proposed model is a mix integer dynamic optimization problem with path dependence. The forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally, to evaluate the model, our result of a meaning example is compared to the terminal wealth under different constraints.

## 1.INTRODUCTION

The portfolio selection problem decides the choice of limited capital to a number of potential assets according to a profitable investment strategy. The pioneering work of the portfolio selection problem is the concept of efficient set developed by Markowitz (1952). In his seminal work based on modern portfolio theory, Markowitz viewed portfolio selection as a mean-variance optimization problem with regard to two criteria: to maximize the expected return of a portfolio, and to minimize the variance of return. More formally, a desirable portfolio is defined to be a tradeoff between risk and expected return. It combines probability theory with optimization techniques to model the investment behavior under uncertainty. Mean variance model makes a basic assumption that the trend of asset markets in future can be simulated by current and before asset data. The mean and covariance are the most important indicators in portfolio selection problem; thus, the assumption means that is, the mean and covariance of assets in future is similar to the past one. Although we have to say that it is very hard to ensure the assumption is always correct in the real ever-changing asset markets, the assumption is much common in lots of works about this problem (see Yu et al., 2010).

The portfolio selection model based on fuzzy probabilities has proposed by Tanaka et al. (2000). The mean vector and covariance matrix in the model designed by Markowitz are respectively replaced by the fuzzy weighted average vector and covariance matrix. It can be regarded as a natural extension of the model given by Markowitz because of the close relationship between probability theory and fuzzy probability. Its approach permits the incorporation of expert knowledge by means of a possibility grade, to reflect the similarity between the future state of asset markets and the state of previous periods. By using fuzzy approaches, the knowledge of the experts and the subjective opinions of the investors can be better integrated into a portfolio selection model. Zhang and Nie (2004), Zhang et al. (2006), and Zhang and Wang (2008) discussed the admissible efficient portfolio selection using the assumption that the expected return and risk of assets have admissible errors to reflect the uncertainty in real investment actions. Then, an analytic derivation of admissible efficient frontier is given, when short sales were not allowed on all risky assets. Dubois and Prade (1988) defined an interval-valued expectation of fuzzy numbers which is set as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. Carlsson and Fullér (2001) employed possibility distributions by introducing lower and upper possibilistic mean values of fuzzy numbers. Huang (2008) proposed mean risk curve portfolio selection models. Li et al. (2010) proposed mean- varianceskewness fuzzy portfolio which can be solved by a genetic algorithm and fuzzy simulation. Carlsson et al. (2002) introduced a possibilistic approach to select portfolios with highest utility score under the assumption that the returns of assets are trapezoidal fuzzy numbers.

Many modifications are proposed to improve the accuracy of the Markowitz’s mean variance portfolio selection model, for example, limiting the number of assets to be held in an efficient portfolio (cardinality constraints) or prescribes lower and upper bounds on the fraction of the capital invested in each asset (threshold constraints). These improvements are based on real-world practices; however, it is clearly not desirable to administer a portfolio made up of a large number of assets because of transaction costs, complexity of management, or the policies of asset management companies. The model (often called cardinality constrained Markowitz model), and its variations have been fairly intensively studied in the last decade. Some researchers proposed exact solution methods (ie., Bienstock, 1996; Bertsimas and Shioda, 2009; Li et al., 2006; Shaw et al., 2008; Murray and Shek, 2012); Cesarone et al., 2013; Cui et al., 2013; Sun et al., 2013; Le Thi et al., 2009, 2014). Since exact solution methods were able to solve only a fraction of practically useful LAM models, many heuristic algorithms had also been proposed (ie., Fernández and Gómez, 2007; Ruiz-Torrubiano and Suarez, 2010; Anagnostopoulos and Mamanis, 2011; Woodside-Oriakhi et al., 2011; Deng et al., 2012). In these studies, it appears that the computational complexity of the solution given by the LAM (Limited Asset Markowitz) model is much greater than the one spent by the classical Markowitz model. Indeed, the standard Markowitz model is a convex quadratic programming problem, while the LAM model is a mixed integer quadratic programming problem which is a NP-hard problem. Clearly, the computational complex of LAM model is much improved.

Although the case of a long-term investment is of greater importance in practice, much less has been done in that area. Although it is heavily discussed in the recent literatures (see e.g., Li and Ng, 2000; Zhu et al., 2004; Brandt et al., 2006; Gülpınar and Rustem, 2007; Çelikyurt and Özekici, 2007; Calafiore, 2008; Yan et al., 2009, 2012; Yu et al., 2010, 2012; Wu and Li, 2012; Li and Li, 2012; Zhang et al., 2012, 2014; Liu et al., 2012, 2013; Zhang and Zhang, 2014; Bodnar et al., 2015). To the best of our knowledge, a closed-form solution is not available in the general case up to now. Closed-form solutions are only presented with the assumption of independence by Li and Ng (2000), Zhu et al. (2004), Yan et al. (2009, 2012), Yu et al. (2010, 2012), Wu and Li (2012), Li and Li (2012). For more general models, the solution is frequently determined by a numerical procedure (see e.g., van Binsbergen and Brandt, 2007; Mansini et al., 2007; Gülpınar and Rustem, 2007; Zhang et al., 2012, 2014; Liu et al., 2012, 2013; Zhang and Zhang, 2014; Köksalan and Şakar, 2014).

Since the return of asset is in a fuzzy uncertain economic environment and varies all the time, its expected return and risk cannot accurately be predicted. Based on this fact, it is reasonable to assume that the expected return and risk have admissible errors. This paper deals with the portfolio selection problem based on this assumption that the expected return and risk of asset have admissible errors to show the uncertainty in real investments. The contribution of this work is as follows. We propose a new admissible multiperiod mean variance portfolio selection with upper bound on borrowing riskfree asset constraints, transaction costs, threshold constraints and cardinality constraints. Cardinality constraints limit the number of assets held in an efficient portfolio. Threshold constraints limit the amount of capital to be invested in each stock; moreover, very small investments in any stock are prevented according to its strategy. Then, a novel forward dynamic programming method is employed to find the solution. Finally, we give an important example to illustrate the idea of the model. More importantly, the results compared to other methods of the example demonstrate the effectiveness of the designed algorithm.

This paper is organized as follows. In Section 2, we present the admissible efficient multiperiod portfolio model and define the upper and lower admissible efficient multiperiod portfolio frontiers by the spreads of the portfolio expected returns and risks from the upper and lower bounds of admissible errors. Transaction costs, upper bound on borrowing constraints, the threshold constraints and cardinality constraints are formulated into the multiperiod portfolio. A new admissible multiperiod portfolio selection model is constructed. A forward dynamic programming method is proposed to solve it in Section 3. In Section 4, A numerical example is given to illustrate our proposed effective means and approaches, and the terminal wealth under different constraints are compared in Section 4, Finally, some conclusions are given in Section 5.

## 2.THE ADMISSIBLE MULTIPERIOD PORTFOLIO SELECTION MODEL

In this paper, we take a very interest assumption that there are n risky assets and one risk-free asset in the financial market. An investor allocates his initial wealth W1 among n + 1 assets at the beginning of time period 1, and obtains the terminal wealth at the end of time period T. The total process of this investment last from the beginning of period 1 to the end of period T. The investor can modification the choice of the n risky assets at the beginning of each period during the process of investment. Suppose that the returns of portfolios among different periods are independent of each other. In order to describe conveniently, we use the following notations:

• xit the investment proportion of risky asset i at period t;

• xi0 the initial investment proportion of risky asset i at period 1;

• xt the portfolio at period t, where xt = (x1t, x2t , … , xnt)’;

• xft the investment proportion of risk-free asset at period t, where $x f t = 1 − ∑ i = 1 n x i t$;

• xbft upper bound on borrowing risk-free asset at period t, where xftxbft, xbft ≤ 0;

• Rit the random return of risky asset i at period t;

• rit the expected return of Rit, where rt = (r1t, r2t , …, rnt)’;

• σijt the covariance of Rit and Rjt;

• rpt the expected return rate of the portfolio xt at period t;

• rNt the net return rate of the portfolio xt at period t;

• uit the upper bound constraints of xit;

• lit the lower bound constraints of xit;

• Wt the holding wealth at the beginning of period t;

• rft the borrowing/lending rate of the risk-free asset at period t;

• rbt the borrowing rate of the risk-free asset at period t;

• rlt the lending rate of the risk-free asset at period t;

• cit the unit transaction cost of risky asset i at period t;

• K the desired number of assets in the portfolio at period t.

where prime (’) denotes matrix transposition and all non-primed vectors are column vectors.

Let the borrowing rate of the risk-free asset at period t be larger than the lending rate of the risk- free asset, ie. rbtrlt. If the borrowing is allowed on the risk free asset, then the expected return associated with portfolio xt are given as follows:(1)

$r p t = ∑ i = 1 n r i t x i t + r f t ( 1 − ∑ i = 1 n x i t ) , t = 1 , ⋯ , T$
(1)

where

$r f t = { r l t , 1 − ∑ i = 1 n x i t ≥ 0 r b t , 1 − ∑ i = 1 n x i t ≤ 0 . 1 − ∑ i = 1 n x i t ≥ 0 ,$

means that lending of the risk-free asset is allowed; otherwise , it represents the borrowing from the riskfree asset .

The variance of the portfolio xt can be expressed as(2)

$V t ( x t ) = ∑ i = 1 n ∑ j = 1 n x i t σ i j t x j t$
(2)

Since the economic environment is uncertain and rit, I = 1, … , n varies in the process of investment, the future states of n risky assets returns cannot be predicted accurately. We extend the admissible average returns and covariances of singleperiod portfolio selection in Zhang and Nie (2004), Zhang et al. (2006), and Zhang and Wang (2008) to multiperiod portfolio selection. Thus, the admissible average returns and covariances at period t are, respectively, defined as(3)

$r i t ¯ = r i t + ϕ i t , ϕ i l t ≤ ϕ i t ≤ ϕ i h t , i = 1 , ⋯ , n , t = 1 , ⋯ , T$
(3)

and(4)

$σ i j t ¯ = σ i j t + ε i j t , ε i j l t ≤ ε i j t ≤ ε i j h t , i = 1 , ⋯ , n , t = 1 , ⋯ , T$
(4)

where $ϕ i t$ denotes the admissible error for rit; φil t expresses the lower bound of $ϕ i t$ is the upper bound of φ it; $ε i j t$ defines the admissible error for $σ i j t ; ε i j l t$ indicates the lower bound of $ε i j t ; ε i j h t$ shows the upper bound of $ε i j t$. $ϕ i j t , ϕ i h t , ε i j l t and ε i j h t$ can be estimated by combining the forecasted information of the assets return with the opinion of experts opinion. Correspondingly, the intervals $[ r i t + ϕ i l t , r i t + ϕ i h t ]$ and $[ σ i j t + ε i j l t , σ i j t + ε i j h t ]$ are determined. $ϕ i t$ and $ε i j t$ can be selected by an investor based on his attitude to return and risk.

The admissible average return vector can be defined by(5)

$r t ¯ = r t + ϕ t , ϕ l t ≤ ϕ t ≤ ϕ h t , t = 1 , ⋯ , T$
(5)

where $r t = ( r 1 t , r 2 t , ⋯ , r n t ) ′ , ϕ t = ( ϕ 1 t , ϕ 2 t , ⋯ , ϕ n t ) ′ , ϕ l t = ( ϕ 1 l t , ϕ 2 l t , ⋯ , ϕ n l t ) ′ , and ϕ h t = ( ϕ 1 h t , ϕ 2 h t , ⋯ , ϕ n h )$

Similarly, the admissible covariance matrix can be denoted by(6)

$V t ¯ = V t + ε t , ε l t ≤ ε t ≤ ε h t , t = 1 , ⋯ , T$
(6)

where $V t = ( s i j t ) n × n , e t = ( e i j t ) n × n , e 1 t = ( e i j 1 t ) n × n$ and $e h t = ( e i j h t ) n × n$

The admissible expected value and variance of the return associated with portfolio xt are respectively given by(7)

$r p t ¯ = ( r t + ϕ t ) ′ x t + r f t ( 1 − e ′ x t ) , t = 1 , ⋯ , T$
(7)

and(8)

$V t ( x t ) ¯ = x t ′ ( V t + ε t ) x t$
(8)

where e = (1, 1, …, 1).

For any xt ≥ 0, it follows that

$( r t + ϕ l t ) ′ x t + r f t ( 1 − e ′ x t ) ≤ r p t ¯ ≤ ( r t + ϕ h t ) ′ x t + r ( 1 − e ′ x t )$

and

$x t ′ ( V t + ε l t ) x t ≤ V t ( x t ) ¯ ≤ x t ′ ( V t + ε h t ) x t$

We assume in the sequel that the transaction costs at period t is a V shape function of difference between the tth period portfolio xt = (x1t, x2t , ⋯ , xnt) and the t−1 th period portfolio xt−1 = (x1t−1 ,x2t−1 , ⋯ , xnt−1). That’s to say, the transaction cost for asset i at period t can be expressed by(9)

$C i t = c i t | x i t − x i t − 1 |$
(9)

Hence, the total transaction costs of the portfolio xt = (x1t, x2t , … , xnt) at period t can be represented as(10)

$C t = ∑ i = 1 n c i t | x i t − x i t − 1 | , t = 1 , ⋯ , T$
(10)

Thus, the admissible net return rate of the portfolio xt at period t can be denoted as(11)

$r N t ¯ = ( r t + ϕ t ) ′ x t + r f t ( 1 − e ′ x t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | = ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 |$
(11)

Then, the admissible holding wealth at the beginning of the period t can be written as(12)

$W t + 1 = W t ( 1 + r N t ¯ ) = W t ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) , t = 1 , ⋯ , T$
(12)

To formulate cardinality constraints into the multiperiod portfolio model, zero-one decision variables are added as:(13)

$z i t = { 1 if any of asset i of period t ( i = 1 , ⋯ , n ; t = 1 , ⋯ , T ) 0 otherwise$
(13)

where $∑ i = 1 n z i t ≤ K$.

The threshold constraints of multiperiod portfolio selection can be expressed as(14)

$l i t ≤ x i t ≤ u i t$
(14)

where lit and uit are respectively the lower and upper bounds constraints of xit.

A rational investor wishes to not only maximize admissible net return but also minimize the admissible variance of the rate of return on a portfolio. Thus, it is very important to get a tradeoff of these two objectives. Let (1−θ) and θ be the preference coefficients associated with criteria $r N t ¯$ and $V t ( x t ) ¯$ respectively. Then the investor attempts to maximize(15)

$F t ( r N t ¯ , V t ( x t ) ¯ ) = ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) − θ ( ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j t ) x j t )$
(15)

where $ϕ i t$ denotes the admissible error for rit, $ϕ i t$ denotes the admissible error for $σ i j t$. Here the parameter θ can be interpreted as the risk aversion factor of a investor. The greater the factor w is, the more risk aversion the investor has. In this paper, we assume that the investor is of risk aversion, i.e., 0 ≤ θ ≤ 1.

Thus, we construct the following admissible multiperiod portfolio selection model with cardinality con straints:

$max ∑ t = 1 T ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) − θ ( ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j t ) x j t ) )$

$s . t . { W t + 1 = ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) W t ( a ) 1 − ∑ i = 1 n x i t ≥ x f t b ( b ) ∑ i = 1 n z i t ≤ K , z i t ∈ { 0 , 1 } ( c ) l i t z i t ≤ x i t ≤ u i t z i t , i = 1 , ⋯ , n , t = 1 , ⋯ , T ( d )$
(16)

where constraint (a) denotes the wealth accumulation constraint; constraint (b) indicates the upper bound on borrowing risk-free asset constraints at period t; constraint (c) represents the desired number of assets in the portfolio must not exceed the given value K; constraint (d) states threshold constraints of xit.

If $( ϕ t , ε t ) = ( ϕ h t , ε l t ) ,$ then (16) can be rewritten as:

$max ∑ i = 1 T ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i h t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) W t ) − θ ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j l t ) x j t$

$s . t . { W t + 1 = ( 1 + ∑ i = 1 n ( r i t + ϕ i h t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) W t 1 − ∑ i = 1 n x i t ≥ x f t b ∑ i = 1 n z i t ≤ K , z i t ∈ { 0 , 1 } l i t z i t ≤ x i t ≤ u i t z i t , i = 1 , ⋯ , n , t = 1 , ⋯ , T$
(17)

If $( ϕ t , ε t ) = ( ϕ h t , ε l t )$, then (16) can be rewritten as:

$max ∑ i = 1 T ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i l t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) − θ ( ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j h t ) x j t ) )$

$s . t . { W t + 1 = ( 1 + ∑ i = 1 n ( r i t + ϕ i l t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t | x i t − x i t − 1 | ) W t 1 − ∑ i = 1 n x i t ≥ x f t b ∑ i = 1 n z i t ≤ K , z i t ∈ { 0 , 1 } l i t z i t ≤ x i t ≤ u i t z i t , i = 1 , ⋯ , n , t = 1 , ⋯ , T$
(18)

The Model (17) means that the investor optimistically estimates the return and risk. The Model (18) means that the investor pessimistically estimates the return and risk. The Model (16) covers the scenario when the investor makes his portfolio selection neither too optimistically nor too pessimistically.

Definition 1. The optimal solution of (16), $x t ( r t + ϕ t , σ t + ε t )$ is called an admissible efficient portfolio. The optimal solution of (17), $x t ( r t + ϕ h t , σ t + ε l t )$ is called an upper admissible efficient portfolio. The optimal solution of (18), $x t ( r t + ϕ l t , σ t + ε h t )$ is called a lower admissible efficient portfolio.

For each admissible error couple (φt, εt) given by the investor, an admissible efficient frontier can be obtained by (16). If φt = 0 and εt = 0, then the admissible efficient frontier is the classical efficient frontier. It is obvious that the admissible portfolio selection model shown in (16) is an extensions of the conventional multiperiod portfolio selection models.

## 3.SOLUTION ALGORITHM

In this section we formulate a numerical solution to (16) based on an essential assumptions. Our results about (16) will require the following assumptions to be satisfied.

Assumption 1. (i) rt +φt≠kl, for any kR, (ii) σt+εt is a semi-positive definite matrix.

Assumption 1 (i) is essential. Assumption 1 (ii) is easily satisfied by a proper selection of εt.

Let $y i t = | x i t − x i ( t − 1 ) |$. Then the Model (16) can be turned into as follows.

$max ∑ i = 1 T ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t y i t ) − θ ( ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j t ) x j t ) )$

$s . t . { W t + 1 = ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t y i t ) W t 1 − ∑ i = 1 n x i t ≥ x f t b y i t ≥ x i t − x i ( t − 1 ) y i t ≥ − ( x i t − x i ( t − 1 ) ) ∑ i = 1 n z i t ≤ K , z i t ∈ { 0 , 1 } l i t z i t ≤ x i t ≤ u i t z i t , i = 1 , ⋯ , n , t = 1 , ⋯ , T$
(19)

where the Model (19) is the admissible multiperiod portfolio selection.

In this section, the forward dynamic programming method is proposed to solve the Model (19).

The sub-problem of period t of the Model (19) can be transformed into(20)

$ma ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t y i t ) − θ ( ∑ i = 1 n ∑ j = 1 n x i t ( σ i j t + ε i j t ) x j t ) )$

$s . t . { W t + 1 = ( 1 + ∑ i = 1 n ( r i t + ϕ i t ) x i t + r f t ( 1 − ∑ i = 1 n x i t ) − ∑ i = 1 n c i t y i t ) W t 1 − ∑ i = 1 n x i t ≥ x f t b y i t ≥ x i t − x i ( t − 1 ) y i t ≥ − ( x i t − x i ( t − 1 ) ) ∑ i = 1 n z i t ≤ K , z i t ∈ { 0 , 1 } l i t z i t ≤ x i t ≤ u i t z i t , i = 1 , ⋯ , n$
(20)

In the following section, we provide the detailed procedure of the forward dynamic programming method for finding optimal solutions of the Model (19). The procedure of the algorithm can be showed as follows:

Algorithm The forward dynamic programming method:

Step1. When t = 1, W1 and x0 = (x10, …, xn0)’ have been given, the sub-problem of period 1 of the Model (19) can be transformed into

$max ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i 1 + ϕ i 1 ) x i 1 + r f 1 ( 1 − ∑ i = 1 n x i 1 ) − ∑ i = 1 n c i 1 y i 1 ) − θ ( ∑ i = 1 n ∑ j = 1 n x i 1 ( σ i j 1 + ε i j 1 ) x j 1 )$

$s . t . { W 2 = ( 1 + ∑ i = 1 n ( r i 1 + ϕ i 1 ) x i 1 + r f 1 ( 1 − ∑ i = 1 n x i 1 ) − ∑ i = 1 n c i 1 y i 1 ) W 1 1 − ∑ i = 1 n x i 1 ≥ x f 1 b y i 1 ≥ x i 1 − x i 0 y i 1 ≥ − ( x i 1 − x i 0 ) ∑ i = 1 n z i 1 ≤ K , z i 1 ∈ { 0 , 1 } l i 1 z i 1 ≤ x i 1 ≤ u i 1 z i 1 , i = 1 , ⋯ , n$
(21)

If the matrix $( σ i j 1 + ε i j 1 ) n × n$ is a semi-positive definite matrix, the Model (21) is a cardinality constrained convex quadratic programming problem. The optimal solution of the Model (21), $x 1 * = ( x 11 * , ⋯ . x n 1 * )$ can be obtained by branch-and-bound implementation with pivoting algorithms (Bertsimas and Shioda, 2009). At the same time,

$( 1 − θ ) ( 1 + ∑ i = 1 n ( r i 1 + ϕ i 1 ) x i 1 * + r f 1 ( 1 − ∑ i = 1 n x i 1 * ) − ∑ i = 1 n c i 1 y i 1 * ) and - θ ( ∑ i = 1 n ∑ j = 1 n x i 1 * ( σ i j 1 + ε i j 1 ) x j 1 * )$

$W 2 ∗$ can be obtained, respectively.

Step2. When $t = m ( m ≥ 1 and m ∈ Z + ) , W m + 1 *$ and $x m * = ( x 1 m * , ⋯ . x n m * )$ have been obtained, the sub-problem of period m of the Model (19) can be transformed into

$max ( 1 − θ ) ( 1 + ∑ i = 1 n ( r i ( m + 1 ) + ϕ i 1 ) x i ( m + 1 ) + r f 1 ( 1 − ∑ i = 1 n x i ( m + 1 ) ) − ∑ i = 1 n c i ( m + 1 ) y i ( m + 1 ) ) − θ ( ∑ i = 1 n ∑ j = 1 n x i ( m + 1 ) ( σ i j ( m + 1 ) + ε i j ( m + 1 ) ) x j ( m + 1 ) )$

$s . t . { W ( m + 2 ) = ( 1 + ∑ i = 1 n ( r i ( m + 1 ) + ϕ i ( m + 1 ) ) x i ( m + 1 ) + r f ( m + 1 ) ( 1 − ∑ i = 1 n x i ( m + 1 ) ) − ∑ i = 1 n c i ( m + 1 ) y i ( m + 1 ) ) W ( m + 1 ) ∗ 1 − ∑ i = 1 n x i ( m + 1 ) ≥ x f ( m + 1 ) b y i ( m + 1 ) ≥ x i ( m + 1 ) − x ( m + 1 ) * y i ( m + 1 ) ≥ − ( x i ( m + 1 ) − x ( m + 1 ) * ) ∑ i = 1 n z i ( m + 1 ) ≤ K , z i ( m + 1 ) ∈ { 0 , 1 } l i ( m + 1 ) z i ( m + 1 ) ≤ x i ( m + 1 ) ≤ u i ( m + 1 ) z i ( m + 1 ) , i = 1 , ⋯ , n$
(22)

If the matrix $( σ i j ( m + 1 ) + ε i j ( m + 1 ) ) n × n$ is a semi-positive definite matrix, the Model (22) is a cardinality constrained convex quadratic programming problem. The optimal solution of the Model (22), $x m + 1 * = ( x m + 1 * , ⋯ . x n m + 1 * )$ can be obtained by branch-and-bound implementation with pivoting algorithms (Bertsimas and Shioda, 2009). At the same time,

$( 1 − θ ) ( 1 + ∑ i = 1 n ( r i ( m + 1 ) + ϕ i 1 ) x i ( m + 1 ) * + r f 1 ( 1 − ∑ i = 1 n x i ( m + 1 ) * ) − ∑ i = 1 n c i ( m + 1 ) y i * ) - θ ( ∑ i = 1 n ∑ j = 1 n x i ( m + 1 ) * ( σ i j ( m + 1 ) + ε i j ( m + 1 ) ) x j ( m + 1 ) * )$

and $W m + 2 *$ can be obtained, respectively.

Step3. If t = T, then the maximization of the terminal utility

$( 1 − θ ) ( 1 + ∑ i = 1 n ( r i T + ϕ i T ) x i T * + r f T ( 1 − ∑ i = 1 n x i T * ) − ∑ i = 1 n c i T y i T * ) - θ ( ∑ i = 1 n ∑ j = 1 n x i T * ( σ i j T + ε i j T ) x j T * ) and W T + 1 * can be obtained ,$

and $W T + 2 *$ can be obtained, respectively. Otherwise t = m+1, then turn Step 2.

At period t, the global optimal solutions of the Model (21) and Model (22), which are the sub-problem of the Model (19) . can be obtained by branch-and-bound implementation with pivoting algorithm (Bertsimas and Shioda (2009)). So the global optimal solution of the Model (19) can also be known by the forward dynamic programming method. As a result, the global optimal solution of Model (16) can also be obtained.

The optimal solutions of (17) can be obtained as $( ϕ t , ε t ) = ( ϕ h t , ε l t )$. The optimal solutions of (18) can be obtained as $( ϕ t , ε t ) = ( ϕ l t , ε h t )$. Correspondingly, both the upper and lower admissible optimal solutions can also be indicated by the forward dynamic programming method.

## 4.NUMERICAL EXAMPLE

In this section, a numerical example is given to express the idea of the proposed model. Assume that an investor chooses twenty stocks from Shanghai Stock Exchange for his investment. The stocks codes are respectively S1 (600036), S2 (600002), S3 (600060), S4 (600362), S5 (600519), S6 (601111), S7 (601318), S8 (600900), S9 (600887), S10 (600690), S11 (6000970), S12 (600000), S13 (600009), S14 (600019), S15 (600029), S16 (600104), S17 (600315), S18 (600518), S19 (600570), S20 (600880). He intends to make five periods of investment with initial wealth W1 = 1 and his wealth can be adjusted at the beginning of each period. It is assumed that the returns, risk and turnover rates of the twenty stocks at each period are represented as trapezoidal fuzzy numbers. We collect historical data of them from April 2006 to March 2015 and set every three months as a period to handle the historical data.

The transaction costs of assets of the two periods take the same value cit = 0.003 (i = 1, …, 20; t = 1, …, 5), the desired number of assets in the portfolio K = 0, …, 9 at period t, t = 1, …, 5, upper bound on borrowing riskfree asset xbft = −0.5, preference coefficient θ = 0, 0.1,…1, the borrowing rate of the risk-free asset rbt = 0.017, the lending rate of the risk-free asset rlt = 0.009, t = 1, …, 5, the lower lit = 0.05 and upper bound constraints uit = 0.2 (i = 1, …, 20; t = 1, …, 5).

In this example, x0, $ϕ h t , ϕ l t and ε i j t$ are given by

• x0 =  (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)’,

• $ϕ h t$ (0.025, 0.015, 0.01, 0.015, 0.025, 0.015, 0.045, 0.015, 0.045, 0.015, 0.025, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.015, 0.01, 0.015)’,

• $ϕ l t = ϕ h t$ φ ht,

• $ε i j t$ =  0, I = 1, …, 20, j = 1, …, 20.

By using the forward dynamic programming method to solve the Model (16), Model (17) and Model (18), the corresponding results can be obtained as follows.

If $ϕ i t$ = 0, $ε i j t$ = 0, θ = 0.5, K = 6, the optimal solution of the admissible multiperiod portfolio selection (Model (16)) is shown in the Table 1.

When $ϕ i t$ =0, $ε i j t$ = 0, θ = 0.5, K = 6, the optimal investment strategy at period 1 is x11 = 0.2, x41 = 0.2, x51 = 0.2, x61 = 0.2, x71 = 0.1, x101 = 0.2, xf1 = −0.2 and the rest of variables equal to zero, which means investor should allocate his initial wealth on asset 1, asset 4, asset 5, asset 6, asset 7 , asset 10, risk-free asset and otherwise assets by the proportions of 20%, 20%, 30%, 20%, 20% , 20%, −20% and the rest of variables equal to zero, respectively. In Table 1, the optimal investment strategies at period 2, period 3, period 4 and period 5 can also be obtained. In this case, the available terminal wealth is 1.9236.

If $ϕ i t = 0 , ε i j t = 0$, θ = 0.5, K = 8, the optimal solution of Model (16) will be obtained as the Table 2. The available terminal wealth is 2.0611.

To display the influence of K on the optimal solution of multiperiod, its value is respectively set as 6 and 8, and the Model (16) for portfolio decision-making will be used afterwards. After using forward dynamic programming method, the corresponding optimal investment strategies can be obtained as shown in Table 1 and Table 2. According to the results shown in Table 1 and Table 2, it can be seen that most of assets of the optimal solutions of K = 6 and K = 8 are the same. There are six assets of the same values in period 1, i.e. asset 1, asset 4, asset 5, asset 6, asset 7, asset 10.

If $ϕ h t$ = (0.025, 0.015, 0.01, 0.015, 0.025, 0.015, 0.015, 0.015, 0.015, 0.015, 0.025, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.015, 0.01, 0.015)’, $ε i j t$ = 0, θ = 0.5, K = 6, the optimal solution of he upper admissible multiperiod portfolio selection (Model (17)) will be obtained as the Table 3. When θ = 0.5, K = 6, the available terminal wealth is 2.0946. If $ϕ l t = − ϕ h t , ε i j t = 0$, θ = 0.5, K = 6, the optimal solution of the lower admissible multiperiod portfolio selection (Model (18)) will be obtained as the Table 4. Its available terminal wealth is 1.8107.

Table 1, Table 3 and Table 4show thatwhenθ = 0.5, K = 6, the optimal solutions of the Model (16), the Model (17) and the Model (18) are not the same, and the available terminal wealthof the three models are different. The terminal wealth of the upper admissible multiperiod portfolio selection (Model (17)) is the largest, and the terminal wealth of the lower admissible multiperiod portfolio selection (Model (18)) is the smallest.

When θ = 0.5, K = 0, 1, …, 9, the optimal terminal wealth of the Model (16), the Model (17) and the Model (18) can respectively be obtained as shown in Table 5. W6 is the terminal wealth of the admissible mutiperiod portfolio selection when $ϕ i t$ = 0, $ε i j t$ = 0. UW6 is the terminal wealth of the optimistically admissible mutiperiod portfolio selection (the Model (17)). LW6 is the terminal wealth of the pessimistically admissible mutiperiod portfolio selection (the Model (18)).

Where W6 is the terminal wealth of the admissible mutiperiod portfolio selection which $ϕ i t$ = 0, $ε i j t$ = 0. UW6 is the terminal wealth of the optimistically admissible mutiperiod portfolio selection (the Model (17)). LW6 is the terminal wealth of the pessimistically admissible mutiperiod portfolio selection (the Model (18)).

In the used data sets, the same optimal solutions at K = 8 is suitable for K ≥ 9. The experiments in this paper correspond to the values of K is the interval [0, 9]. It can be seen that, elaborated in Table 5, the terminal wealth also becomes larger, when the preset the desired number of assets in the portfolio become larger, this reflects the influence of the desired number of assets on portfolio selection.

In order to address the relationship between the K and the terminal wealth of the Model (16), the Model (17) and the Model (18), Figure 1 based on the data in Table 5 is pointed as follow: K-W6 is the admissible terminal wealth of the Model (16) with different K; K-UW6 is the upper admissible terminal wealth of the Model (17) with different K; K-LW6 is the lower admissible terminal wealth of the Model (18) with different K,

In Figure 1, with the same value of Kthe terminal wealth of the upper admissible multiperiod portfolio selection is larger than the terminal wealth of the lower admissible multipriod portfolio selection. For the terminal wealth under the conditions of $ϕ i l t ≤ ϕ i t ≤ ϕ i h t$, the admissible multiperiod portfolio selection is at middle position between upper and lower admissible multiperiod portfolio selection.

When K = 8, θ = 0, 0.1, …, 0.9, 1, the optimal terminal wealth of the Model (16), the Model (17) and the Model (18) can be respectively obtained as shown in Table 6. W6 is the terminal wealth of the admissible mutiperiod portfolio selection which $ϕ i t$ = 0, $ε i j t$ = 0. UW6 is the terminal wealth of the upper admissible mutiperiod portfolio selection (the Model (17)). LW6 is the terminal wealth of the lower admissible mutiperiod portfolio selection (the Model (18)).

where W6 is the terminal wealth of the admissible mutiperiod portfolio selection which $ϕ i t$ = 0, $ε i j t$ = 0. UW6 is the terminal wealth of the upper admissible mutiperiod portfolio selection (the Model (17)). LW6 is the terminal wealth of the lower admissible mutiperiod portfolio selection (the Model (18)).

From Table 6, the Figure 2, which reflect the relationship between the preference coefficients θ and the terminal wealth of the Model (16), the Model (17) and the Mod- el (18), can be obtained as follows. PC is preference coefficients; PC-W6 is the admissible terminal wealth of the Model (16) with different θ; PC-UW6 is the upper admissible terminal wealth of the Model (17) with different θ ; PC-LW6 is the lower admissible terminal wealth of the Model (18) with different θ .

In Figure 2, it can be seen that, if the θ has the same value same, the terminal wealth of the upper admissible multiperiod portfolio selection is above the terminal wealth of the lower admissible multipriod portfolio selection. For $ϕ i l t ≤ ϕ i t ≤ ϕ i h t , ε i j l t ≤ ε i j t ≤ ε i j h t$, the terminal wealth of the admissible multiperiod portfolio selection is the middle of the terminal wealth of upper and lower admissible multiperiod portfolio selection. The terminal wealth becomes smaller, when preference coefficient θ become larger, which gives the influence of preference coefficient θ on portfolio selection.

## 5.CONCLUSIONS

In this paper, we have discussed the admissible multiperiod portfolio selection problem with transaction cost, borrowing constraints, threshold constraints and cardinality constraints. In the proposed model, the cardinality constraints are used to control the number of assets held in an efficient portfolio. Because of the transaction cost and cardinality constraints, the multiperiod portfolio selection is a mix integer dynamic optimization problem with path dependence. The forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally, an example using real data from the Shanghai Stock Exchange is given to illustrate the behavior of the proposed model and the designed algorithm.

## ACKNOWLEDGMENTS

This research was supported by the National Natural Science Foundation of China (no. 71271161).

## Figure

The relationship between the K and the terminal wealth of the Model (16-18).

The relationship between the θ and the terminal wealth of the Model (16-18).

## Table

The optimal solution of Model (16) when φit = 0, εijt = 0, θ = 0.5, K = 6

The optimal solution when φit = 0, εijt = 0, θ = 0.5, K = 8

The optimal solution of Model (17) when θ = 0.5, K = 6

The optimal solution of Model (18) when θ = 0.5, K = 6

The optimal terminal wealth of the Model (16-18) when θ = 0.5, K = 1, , 9

The optimal terminal wealth of the Model (16-18) when K = 8 θ= 0, 0.1, , 0.9, 1

## REFERENCES

1. Anagnostopoulos K P , Mamanis G (2011) The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms , Expert Systems with Applications, Vol.38 ; pp.14208-14217
2. Bertsimas D , Shioda R (2009) Algorithms for cardinality-constrained quadratic optimization , Computational Optimization and Applications, Vol.43 ; pp.1-22
3. Bienstock D (1996) Computational study of a family of mixed-integer quadratic programming problems , Mathematical Programming, Vol.74 ; pp.121-140
4. Bodnar T , Parolya N , Schmid W (2015) A closedform solution of the multi-period portfolio choice problem for a quadratic utility function , Annals of Operations Research, Vol.229 (1) ; pp.121- 158
5. Brandt M , Santa-Clara P (2006) Dynamic portfolio selection by augmenting the asset space , The Journal of Finance, Vol.61 ; pp.2187-2217
6. Calafiore G C (2008) Multi-period portfolio optimization with linear control policies , Automatica, Vol.44 (10) ; pp.2463-2473
7. Carlsson C , Fullér R (2001) On possibilistic mean value and variance of fuzzy numbers , Fuzzy Sets and Systems, Vol.122 ; pp.315-326
8. Carlsson C , Fullér R , Majlender P (2002) A possibilistic approach to selecting portfolios with highest utility score , Fuzzy Sets and Systems, Vol.131 ; pp.13-21
9. Cesarone F , Scozzari A , Tardella F (2013) A new method for mean-variance portfolio optimization with cardinality constraints , Annals of OperationsResearch, Vol.205 ; pp.213-234
10. Cui X T , Zheng X J , Zhu S S , Sun X L (2013) Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems , Journal of Global Optimization, Vol.56 ; pp.1409-1423
11. Çlikyurt U , Öekici S (2007) Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach , European Journal of Operational Research, Vol.179 (1) ; pp.186-202
12. Deng G F , Lin W T , Lo C C (2012) Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization , Expert Systems with Applications, Vol.39 ; pp.4558-4566
14. Fernández A , Gómez S (2007) Portfolio selection using neural networks , Computers & Operations Research, Vol.34 ; pp.1177-1191
15. Gülp?nar N , Rustem B (2007) Worst-case robust decisions for multi-period mean-variance portfolio optimization , European Journal of Operational Research, Vol.183 (3) ; pp.981-1000
16. Huang X (2008) Risk Curve and Fuzzy Portfolio Selection , Computers and Mathematics with Applications, Vol.55 ; pp.1102-1112
17. Köksalan M , ?akar C T (2014) An interactive approach to stochastic programming-based portfolio optimization , To Appear in Annals of Operations Research,
18. Le Thi H A , Moeini M , Dinh T P (2009) Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA , Computational Management Science, Vol.6 ; pp.459-475
19. Le Thi H A , Moeini M (2014) Long-short portfo- lio optimization under cardinality constraints by difference of convex functions algorithm , Journal of Optimization Theory and Applications, Vol.161 ; pp.199-224
20. Li C J , Li Z F (2012) Multi-period portfolio optimization for asset-liability management with bankrupt control , Applied Mathematics and Computation, Vol.218 ; pp.11196-11208
21. Li D , Ng W L (2000) Optimal dynamic portfolio selection: Multiperiod mean-variance formulation , Mathematical Finance, Vol.10 (3) ; pp.387-406
22. Li D , Sun X , Wang J (2006) Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection , Mathematical Finance, Vol.16 ; pp.83-101
23. Li X , Qin Z , Kar S (2010) Mean-variance-skewness model for portfolio selection with fuzzy returns , European Journal of operational Research, Vol.202 ; pp.239-247
24. Liu Y J , Zhang W G , Xu W J (2012) Fuzzy multi-period portfolio selection optimization models using multiple criteria , Automatica, Vol.48 ; pp.3042-3053
25. Liu Y J , Zhang W G , Zhang P (2013) A multiperiod portfolio selection optimization model by using interval analysis , Economic Modelling, Vol.33 ; pp.113- 119
26. Mansini R , Ogryczak W , Speranza M G (2007) Conditional value at risk and related linear programming models for portfolio optimization , Annals of Operations Research, Vol.152 ; pp.227-256
27. Markowitz H M (1952) Portfolio selection , Journal of Finance, Vol.7 ; pp.77-91
28. Murray W , Shek H (2012) A local relaxation method for the cardinality constrained portfolio optimization problem , Computational Optimization and Applications, Vol.53 ; pp.681-709
29. Ruiz-Torrubiano R , Suarez A (2010) Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains , IEEE Computational Intelligence Magazine, Vol.5 ; pp.92-107
30. Shaw D X , Liu M S , Kopman L (2008) Lagrangian relaxation procedure for cardinality-constrained portfolio optimization , Optimization Methods & Software, Vol.23 ; pp.411-420
31. Sun X L , Zheng X J , Li D (2013) Recent advances in mathematical programming with semicontinuous variables and cardinality constraint , Journal of the Operations Research Society of China, Vol.1 ; pp.55-77
32. Tanaka H , Guo P , Türksen I B (2000) Portfolio selection based on fuzzy probabilities and possibility distributions , Fuzzy Sets and Systems, Vol.111 ; pp.387-397
33. van Binsbergen J H , Brandt M (2007) Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function? , Computational Economics, Vol.29 ; pp.355-367
34. Woodside-Oriakhi M , Lucas C , Beasley J E (2011) Heuristic algorithms for the cardinality constrained efficient frontier , European Journal of Operational Research, Vol.213 ; pp.538-550
35. Wu H L , Li Z F (2012) Multi-period meanvariance portfolio selection with regime switching and a stochastic cash flow , Insurance: Mathematics and Economics, Vol.50 ; pp.371-384
36. Yan W , Li S R (2009) A class of multi-period semi-variance portfolio selection with a four-factor futures price model , Journal of Applied Mathematics and Computing, Vol.29 ; pp.19-34
37. Yan W , Miao R , Li S R (2007) Multi-period semi-variance portfolio selection: Model and numerical solution , Applied Mathematics and Computation, Vol.194 ; pp.128-134
38. Yu M , Takahashi S , Inoue H , Wang S Y (2010) Dynamic portfolio optimization with risk control for absolute deviation model , European Journal of Operational Research, Vol.201 (2) ; pp.349-364
39. Yu M , Wang S Y (2012) Dynamic optimal portfolio with maximum absolute deviation model , Journal of Global Optimization, Vol.53 ; pp.363-380
40. Zhang W G , Nie Z K (2004) On admissible efficient portfolio selection problem , Applied Mathematics and Computation, Vol.159 ; pp.357-371
41. Zhang W G , Liu W A , Wang Y L (2006) On admissible efficient portfolio selection problem: Models and algorithms , Applied Mathematics and Computation, Vol.176 ; pp.208-218
42. Zhang W G , Wang Y L (2008) An analytic derivation of admissible efficient frontier with borrowing , European Journal of Operational Research, Vol.184 ; pp.229-243
43. Zhang W G , Liu Y J , Xu W J (2012) A possibilistic mean-semivariance-entropy model for multiperiod portfolio selection with transaction costs , European Journal of Operational Research, Vol.222 ; pp.41-349
44. Zhang W G , Liu Y J , Xu W J (2014) A new fuzzy programming approach for multi-period portfolio Optimization with return demand and risk control , Fuzzy Sets and Systems, Vol.246 ; pp.107-126
45. Zhang P , Zhang W G (2014) Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints , Fuzzy Sets and Systems, Vol.255 ; pp.74-91
46. Zhu S S , Li D , Wang S Y (2004) Risk control over bankruptcy in dynamic portfolio selection: ageneralized mean-variance formulation , IEEE Transactions on Automatic Control, Vol.49 (3) ; pp.447-457