On portfolio management with value at risk and uncertain returns via an artificial neural network scheme

Journal Pre-proofs On portfolio management with value at risk and uncertain returns via an artificial neural network scheme Sahar Mohammadi, Alireza Nazemi PII: DOI: Reference:

S1389-0417(19)30496-6 https://doi.org/10.1016/j.cogsys.2019.09.024 COGSYS 903

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Cognitive Systems Research

Received Date: Revised Date: Accepted Date:

12 April 2019 14 August 2019 22 September 2019

Please cite this article as: Mohammadi, S., Nazemi, A., On portfolio management with value at risk and uncertain returns via an artificial neural network scheme, Cognitive Systems Research (2019), doi: https://doi.org/10.1016/ j.cogsys.2019.09.024

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On portfolio management with value at risk and uncertain returns via an artificial neural network scheme Sahar Mohammadi1,∗, Alireza Nazemi1,†

1

Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-

316, Tel-Fax No:0098 23-32300235, Shahrood, Iran.

ABSTRACT

This paper focuses on the computation issue of portfolio optimization with

scenario-based Value-at-Risk. The main idea is to replace the portfolio selection models with linear programming problems. According to the convex optimization theory and some concepts of ordinary differential equations, a neural network model for solving linear programming problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns. Several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

Keywords: Uncertain variables, portfolio selection, Value at risk, crisp equivalent programming, neural network, stability, convergent.

1. Introduction In 1952 Markowitz in [1] invented the mean-variance analysis which provides an easy and intelligent solution of the optimal portfolio selection problem. Following this approach the weights of the optimal portfolio, i.e. the parts of the investors wealth invested into the selected assets, are obtained by minimizing the variance of the portfolio for a given level of the expected return. Depending on the chosen level of the expected return, different optimal portfolios are obtained. All these portfolios lie in a set in the mean-variance space, the socalled efficient frontier, and they possess the property that a larger value of the expected ∗ [email protected] † [email protected]

(Corresponding Author)

1

return corresponds to a larger value of the risk. Hence, it is impossible to increase the profit of the portfolio without increasing its risk. Merton in [2] studied this point in detail. He showed that the dependence between the expected return and the risk is non-linear and derived the equation of the efficient frontier which is a parabola in the mean-variance space. Usually, the variance of the portfolio is chosen as a measure of the portfolio risk. This method of constructing portfolio weights has one serious drawback, namely, the variance is not always an appropriate measure of the risk since it takes into account a two-sided risk. This means that high profits might increase the risk calculation if the variance is used as a measure of risk. Better risk measures are based on the probability or the value of losses. In other words, it is desirable to have measures which depend only on the positive values of the loss function or negative values of the asset return and are known as downside risk measures (see [3]). Recent developments in risk theory suggest that quantile-based measures are well suited functions to compute the risk. The simplest and most popular measure is the value-atrisk (VaR), which is recommended as a standard tool for banking supervision by the Basel Committee and it is equal to the upper percentile of the loss distribution. The concept of the VaR was first introduced by Baumol in 1963. The popularity of this risk measure is mostly related to its simple and easy understandable representation of high losses. The VaR is nowadays widely used by fund managers. Moreover, the recent regulations of the Basel Committee may lead to an increase of the application of the VaR as the relevant measure of risk. Alexander and Baptista in [5] proposed to use the VaR as a risk proxy instead of the variance in the Markowitzs theory for constructing an optimal portfolio. They presented an explicit solution of the minimum VaR problem for portfolio selection under the assumption that the asset returns have a multivariate normal distribution and showed that the minimum VaR portfolio lies above the global minimum variance (GMV) portfolio on the mean-variance efficient frontier. The VaR has been used as a risk-measure method in stochastic portfolio-selection models (PSMs). In [6], Jorion defined the VaR as the worst expected loss over a given horizon under normal market conditions at a given level of confidence, and he observed that the VaR can be used as a risk measure in portfolio selection; furthermore, he also introduced the calculation of the VaR in stochastic models. Focusing on a decentralized portfolio management system, which is widespread among financial institutions, Garcia [7] employed the VaR as a riskmeasure method and a risk-control tool. The VaR can be used in a wide variety of situations. To address the robust portfolio-selection problem, where only partial information is available on the exit time distribution and the conditional distribution of the portfolio return, Huang [8] extended the worst-case VaR approach and formulated the corresponding problems as semidefinite programs. He presented some numerical results by the use of real market data 2

to demonstrate the practicability of using the VaR in portfolio selection. The security returns in conventional models are always determined by the precise historical data. However, such precise data are not always available, and nowadays, with the development of stock markets, it is hard to forecast security returns with stochastic values. Therefore, to handle such imprecise uncertainty, it is more reasonable to treat security returns as variables with imprecise distributions, i.e., uncertain variables. To construct uncertain PSMs (FPSMs), various risk-measure techniques have been used, such as mean variance, mean semivariance and mean entropy. Watada [9] introduced uncertain set theory to PSMs. He extended Markowitzs mean-variance idea to the uncertain environment. Based on the concept of the semivariance of a uncertain variable, Huang [10] proposed two uncertain mean-semivariance PSMs and presented a uncertain-simulation based genetic algorithm (GA) for the solution. Huang [11] also constructed mean-entropy models for uncertain portfolio selection in which the entropy is used as a measure of risk: The smaller the entropy, the safer the portfolio will be. The models in [4] and [10] are solved by the GA. The aforementioned models find the optimal solutions by minimizing the variance or entropy and, thus, maximizing the stability of the portfolio. However, these evaluation methods do not focus on the risk of future loss, which is significant to investors. Over the years, neural networks for optimization and their engineering applications have been widely investigated [14]-[32]. These neural networks are essentially governed by a set of dynamic systems characterized by an energy function, which is the combination of the objective function and constraints of the original optimization problem, and three common techniques, such as penalty functions, Lagrange functions and primal and dual functions. In addition, neural networks for solving optimization problems are hardware-implementable; that is, the neural networks can be implemented by using integrated circuits. In many scientific applications, real-time online solutions of the PSMs problems are often desired. Various numerical procedures and traditional algorithms have been presented over decades for solving PSM problems. For example one can see ([40]-[45]). Since the computing time required for solving PSM greatly depends on the dimension and the structure of the problem, the conventional numerical methods are usually less effective in real-time applications. One promising approach to handle on-line applications is to employ recurrent neural networks based on circuit implementation. Nevertheless, PSMs with VaR have not yet been well established by neural network schemes. In this paper, we focus on neural network approach to the portfolio selection problems with uncertain returns and VaR as a risk-measure and we must solve it. Our neural network will be aimed to solve an equivalent optimality system whose solutions are candidates of original problem. With motivation from the above discussions and following the properties of uncertain variables, we first transform the portfolio selection problem which the securities are assumed to be uncertain variables 3

into linear programming (LP) problems. According to the Karush-Kuhn-Tacker (KKT) optimality conditions [13], a neural network model can be constructed. It is shown that the limit equilibrium points sequence of the proposed neural network can approximately converge to an optimal solution of linear programming problem. Simulation results on two numerical examples of the portfolio selection problem show the effectiveness and performance of the neural network model. The remainder of this paper is organized as follows. In the next section, the preliminaries relevant to uncertain variables are introduced. In Section 3, the VaR model and deterministic equivalents with various reformulations are presented. In section 4, a neural network is derived. The convergence of the proposed modelling framework is proved in Section 5. Simulation results on two numerical examples of portfolio selection problem with normal uncertain variables and rectangular uncertain variables is given in Section 6 to demonstrate the performance of the model. Finally, Section 7 concludes this paper.

2. Preliminaries In this section, we present some preliminaries for uncertain risk analysis within the frame work of uncertain theory ([18,19]). Uncertain theory can be traced back to the earlier work in [33]-[37]. Definition 2.1 Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event. The set function M is called an uncertain measure if it satisfies the following four axioms: (i): (Normality) M{Γ} = 1; (ii): (Monotonicity) M{A} ≤ M{B} whenever A ⊂ B; (iii): (Self-Duality) M{Λ} + M{Λc } = 1 for every event Λ; (iv): (Countable Subadditivity) For every countable sequence of events {Λi }, we have M ∑M i=1 M{Λi }.

{∪ i

} Λi ≤

The triplet (Γ, L, M) is called an uncertainty space. Definition 2.2 An uncertain variable is defined as a measurable function from an uncertain space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ | ξ(γ) ∈ B}, is an event. 4

(1)

Definition 2.3 The uncertainty distribution Φ : R → [0, 1] of a uncertain variable ξ is defined by Φ(t) = M{ξ ≤ t}.

(2)

Definition 2.4 The uncertain variables ξ1 , ξ2 , ..., ξn are said to be independent if n {∩ } M {ξi ∈ Bi } = min M{ξi ∈ Bi }, 1≤i≤n

i=1

(3)

for any Borel sets B1 , B2 , ..., Bn of real numbers. Theorem 2.5 The uncertain variables ξ1 , ξ2 , ..., ξn are independent if and only if n {∪ } M {ξi ∈ Bi } = max M{ξi ∈ Bi }, 1≤i≤n

i=1

(4)

Theorem 2.6 Let ξ1 , ξ2 , ..., ξn be independent uncertain variables with continuous uncertainty distributions Φ1 , Φ2 , ..., Φn , respectively, and Ψ the uncertainty distribution of the −1 −1 sum k1 ξ1 + k2 ξ2 + ... + kn ξn . If Φ−1 1 (α), Φ2 (α), ..., Φn (α) are unique for each α ∈ (0, 1),

we have Ψ−1 (α) =

n ∑

Ψ−1 i (α) =

i=1

n ∑

ki Φ−1 i (α),

0 < α < 1.

(5)

i=1

Definition 2.7 Let ξ be an uncertain variable. Then the expected value of ξ is defined by

∫

∞

E[ξ] =

∫ M{ξ ≥ r}dr −

0

0

−∞

M{ξ ≤ r}dr,

(6)

provided that at least one of the two integrals is finite. Theorem 2.8 Let ξ be an uncertain variable whose uncertainty distribution is Ψ. If its expected value exists, then

∫

1

E[ξ] =

Φ−1 (α)dα.

(7)

0

Theorem 2.9 Let ξ1 and ξ2 be independent uncertain variables with finite expected values. Then for any real numbers a1 and a2 , we have E[a1 ξ1 + a2 ξ2 ] = a1 E[ξ1 ] + a2 E[ξ2 ].

(8)

Definition 2.10 Let ξ be an uncertain variable with finite expected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2 ]. The standard deviation of ξ is defined as √ δ = V [ξ]. 5

3. VaR models and crisp equivalents The accurate definition of uncertain VaR is the starting point of uncertain VaR-based uncertain risk analysis. In order to describe the risk of uncertain variables, we revisit the pessimistic value of uncertain variable and precisely define the following uncertain VaR which paves the foundation of uncertain risk analysis. Definition 3.1 Let ξ be a uncertain variable, and β ∈ (0, 1] be a confidence level. Then the Value-at-Risk of ξ, is the function VaR : (0, 1] → R defined by VaR(β) = sup{x | M{ξ ≥ x} ≥ β}.

(9)

Theorem 3.2 For a risk confidence level β ∈ (0, 1], we have VaR(β) = Φ−1 (1 − β),

(10)

where Φ−1 (1 − β) denotes the generalized inverse function of Φ(β). In Markowitz models, security returns were regarded as random variables. As discussed in introduction, there exist situations that security returns may be uncertain variable parameters. In this situation, we can use uncertain variables to describe the security returns. Let xi denotes the investment proportion in the ith security, ξi represents uncertain return of the ith security, i = 1, 2, ..., n , respectively. We quantify investment return by the expected value of a portfolio, and risk by the value at risk. An optimal portfolio can then be one with maximum expected return for the given VaR level. In fact, when the investors preset VaR level that they feel satisfactory and want to maximize the expected return for this given level of VaR, the optimization model becomes

max E[

n ∑

xi ξi ]

i=1

  VaR(β) ≤ S   ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n

(11)

where S is the maximum risk level that investor can tolerate. This is known as an optimistic model. Another optimal portfolio may be one with minimal VaR for the given expected return level. Let the investors preset expected return level that they feel satisfactory and want to minimize the VaR for the given level of expected return. In this case, the portfolio selection model can be stated as 6

min VaR(β)  ∑ n  E[ xi ξi ] ≥ R     ni=1 s.t. ∑ xi = 1     i=1  xi ≥ 0 , i = 1, 2, . . . , n

(12)

where R denotes the minimum expected investment return that the investors can accept. This is known as an unfortunately model. The traditional solution methods are required to convert the objective function and the constraints to their respective deterministic equivalents. As we know, this process is usually hard to perform and only is successful for some special cases. Let us consider the following forms of the uncertain return rates.

3.1. Linear uncertain variables The crisp equivalents of the problems (12) and (11) in term that security returns may be linear uncertain variable parameters are studied. Lemma 3.3 Consider a linear uncertain variable ξ = (a, b) where a < b. a) The expected value of ξi is obtained as a+b 2

E[ξ] =

(13)

b) The uncertainty distribution of the linear uncertain variable ξi is    0 r≤a    r−a Φ(r) = a≤r≤b  b−a    1 r≥b

(14)

c) for 0 < β ≤ 1, VaR(β) = βa + (1 − β)b

(15) 2

Proof. The proof for each cases is clear and thus we omit it.

Suppose that the return rate ξi of the ith security is a linear uncertain variable, i.e. ∑n ξi = (ai , bi ), i = 1, 2, ..., n, where ai < bi . Then i=1 xi ξi is a linear uncertain variable as ∑n ∑n ( i=1 xi ai , i=1 xi bi ), with E[

n ∑ i=1

xi ξi ] =

(

∑n i=1

xi ai + 2

∑n

7

i=1

xi bi )

∑n =

i=1

xi (bi + ai ) . 2

The mean-VaR model (11) for 0 < β ≤ 1 can be reduced to the following crisp LP problem max

n ∑ i=1

xi (

ai + bi ) 2

∑ n   i=1 xi (ai β + (1 − β)bi ) ≤ S  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n

(16)

The mean-VaR model (12) for 0 < β ≤ 1 can be also reduced to the following crisp LP problem

min

n ∑

xi (ai β + (1 − β)bi )

i=1

∑ n a +b   i=1 xi ( i 2 i ) ≥ R  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n

(17)

3.2. Zigzag uncertain variables The crisp equivalents of the problems (12) and (11) in term that security returns may be zigzag uncertain variable parameters are studied. Lemma 3.4 Consider a zigzag uncertain variable ξ = (a, b, c) where a < b < c. a) The expected value of ξi is obtained as E[ξ] =

a + 2b + c 4

(18)

b) The uncertainty distribution of the zigzag uncertain variable ξi is    0     r−a    2(b a) Φ(r) = r + − c − 2b     2(c − b)     1 c) VaR(β) =

r≤a a≤r≤b (19) b≤r≤c r≥c

  (2β − 1)a + (1 − β)2b

0<β≤

 2bβ + (1 − 2β)c

1 2

(20)

≤β<1

Proof. The proof for each cases is clear and thus we omit it. 8

1 2

2

Suppose that the return rate ξi of the ith security is a zigzag uncertain variable, i.e. ∑n ξi = (ai , bi , ci ), i = 1, 2, ..., n, where ai < bi < ci . Then i=1 xi ξi is a zigzag uncertain ∑n ∑n ∑n variable as ( i=1 xi ai , i=1 xi bi , i=1 xi ci ), with E[

n ∑

xi ξi ] =

i=1

∑n ∑n ∑n ∑n ( i=1 xi ai + 2 i=1 xi bi + i=1 xi ci ) xi (ai + 2bi + ci ) = i=1 . 4 4

The mean-VaR model (11) for 0 < β ≤

1 2

can be reduced to the following crisp LP

problem max

n ∑

xi (

i=1

ai + 2bi + ci ) 4

∑ n   i=1 xi ((2β − 1)ai + (1 − β)2bi ) ≤ S  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n and for

1 2

(21)

≤β<1 n ∑

max

xi (

i=1

ai + 2bi + ci ) 4

∑ n   i=1 xi (2bi β + (1 − 2β)ci ) ≤ S  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n The mean-VaR model (12) for 0 < β ≤

1 2

(22)

can be also reduced to the following crisp LP

problem min

n ∑

xi ((2β − 1)ai + (1 − β)2bi )

i=1

∑ n ai +2bi +ci  )≥R  i=1 xi ( 4  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n and for

1 2

(23)

≤β<1 min

n ∑

xi (2bi β + (1 − 2β)ci )

i=1

∑ n ai +2bi +ci  )≥R  i=1 xi ( 4  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n 9

(24)

3.3. Trapezoidal uncertain variables The crisp equivalents of the problems (12) and (11) in term that security returns may be rectangular Trapezoidal variable parameters are studied. Lemma 3.5 Consider a trapezoidal uncertain variable ξ = (a, b, c, d) where a < b < c < d. a) The expected value of ξi is obtained as E[ξ] =

a+b+c+d 4

(25)

b) The uncertainty distribution of the triangular uncertain variable ξi is    r≤a 0     r−a   a≤r≤b     2(b − a) Φ(r) =

c) VaR(β) =

1

2      r + d − 2c    2(d − c)     1

b≤r≤c

(26)

c≤r≤d r≥d

  (2β − 1)a + (1 − β)2b

0<β≤

 2βc + (1 − 2β)d

1 2

1 2

(27)

≤β<1

Proof. The proof for each cases is clear and thus we omit it.

2

Suppose that the return rate ξi of the ith security is a trapezoidal uncertain variable, i.e. ∑n ξi = (ai , bi , ci , di ), where ai < bi ≤ ci < di , i = 1, 2, ..., n. Then i=1 xi ξi is a trapezoidal ∑n ∑n ∑n ∑n uncertain variable as ( i=1 xi ai , i=1 xi bi , i=1 xi ci , i=1 xi di ) with ∑n ∑n ∑n ∑n ∑n n ∑ ( i=1 xi ai + i=1 xi bi + i=1 xi ci + i=1 xi di ) xi (ai + bi + ci + di ) E[ xi ξi ] = = i=1 , 4 4 i=1 The mean-VaR model (11) for 0 < β ≤

1 2

can be then converted to the following crisp LP

problems

max

n ∑ i=1

xi (

ai + bi + ci + di ) 4

∑ n   i=1 xi ((2β − 1)ai + 2bi (1 − β)) ≤ S  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n 10

(28)

and for

1 2

≤β<1 n ∑

max

xi (

i=1

ai + bi + ci + di ) 4

∑ n   i=1 xi (2βci + (1 − 2β)di ) ≤ S  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n The mean-VaR model (12) for 0 < β ≤

1 2

(29)

can be also converted to the following crisp

LP problem min

n ∑

xi ((2β − 1)ai + 2bi (1 − β))

i=1

∑ n ai +bi +ci +di  )≥R  i=1 xi ( 4  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n and for

1 2

(30)

≤β<1 min

n ∑

xi (2βci + (1 − 2β)di )

i=1

∑ n ai +bi +ci +di  )≥R  i=1 xi ( 4  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n

(31)

3.4. Normal uncertain variables The crisp equivalents of the problems (12) and (11) in term that security returns may be normal uncertain variable parameters are studied. Lemma 3.6 Consider a normal uncertain variable ξ = N (e, σ). a) The expected value of ξi is obtained as E[ξ] = e

(32)

b) The uncertainty distribution of the normal uncertain variable ξi is ( ( ))−1 π(e − r) √ Φ(r) = 1 + Exp x ∈ R, σ > 0 3σ c)

√ VaR(β) = e +

3σ ln π

11

(

1−β β

(33)

) (34)

Proof. The proof for each cases is clear and thus we omit it.

2

Suppose that the return rate ξi of the ith security is normally distributed with parameters ei and σi > 0, i = 1, 2, ..., n. i.e. ξi ∼ N (ei , σi ). Then we have E[

n ∑

xi ξi ] =

i=1

n ∑

xi ei .

i=1

The mean-VaR model (11) for 0 < β ≤ 1 can be then reduced to the following LP problem max

n ∑

xi ei

i=1

( ( )) √  ∑n 1−β 3σi  x e + ln ≤S  i=1 i i π β  ∑ n s.t. x =1  i=1 i    xi ≥ 0 , i = 1, 2, . . . , n

(35)

The mean-variance model (12) for 0 < β ≤ 1 can be also reduced to the following LP problem min

n ∑

( xi

√

ei +

i=1

3σi ln π

(

∑ n   i=1 xi ei ≥ R  ∑ n xi = 1 s.t.  i=1    xi ≥ 0 , i = 1, 2, . . . , n

1−β β

))

(36)

From the above analysis, it is seen that the portfolio optimization problems with uncertain returns and VaR as the risk measurement can be converted to the LP problems. Thus we consider a general form of the LP problems given by

min cT x

(37)

s.t. Ax − b = 0,

(38)

Bx − d ≤ 0,

(39)

B ) = m + l (0 ≤ m, l < n). A Throughout this paper, we assume that problem (37)-(39) has a unique optimal solution. where A ∈ Rl×n , b ∈ Rl , B ∈ Rm×n , d ∈ Rm , x ∈ Rn and rank(

In the next section, we will try to propose a high performance neural network model for solving LP problem (37)-(39). 12

4

The neural network model

In this section, we give the proposed neural network model in [23] for solving LP problem (37)-(39). Let x(.), u(.) and v(.) be some time dependent variables. The aim is to construct a continuous-time dynamical system that will settle down to the KKT point of the problem (37)-(39). We propose a neural network as dy = τ ϕ(y), dt y(t0 ) = y0 = (xT0 , uT0 , v0T )T ∈ Rn+m+l , τ > 0, where

( )  − c + B T (u + Bx − d)+ + AT v   + . ϕ(y) =  (u + Bx − d) − u   Ax − b

(40) (41)



(42)

An indication on how the neural network (40)-(41) can be implemented on hardware is provided in Figure 1. The stability and convergence properties of the model (40) and (41) are similar ones in [23]. Therefore we only state some results and omit here their proofs. Theorem 4.1 Let y ∗ = (x∗ T , u∗ T , v ∗ T )T be the equilibrium point of the neural network (40) and (41). Then x∗ is a KKT point of the problem (37)-(39). On the other hand, if x∗ ∈ Rn is an optimal solution of problem (37)-(39), then there exist u∗ ∈ Rm and v ∗ ∈ Rl such that y ∗ = (x∗ T , u∗ T , v ∗ T )T is an equilibrium point of the network (40) and (41). Lemma 4.2 The Jacobian matrix ∇ϕ(y) of the mapping ϕ defined in (42) is a negative semidefinite matrix. Lemma 4.3 The function ∥(u + Bx − d)+ ∥2 of (u + Bx − d)+ defined in (42) is convex and continuously differentiable on Rn × Rm . Theorem 4.4 The neural network in (40) and (41) is stable in the sense of Lyapunov. Lemma 4.5 (i) For any initial point y(t0 ) = (x(t0 )T , u(t0 )T , v(t0 )T )T , there exists a unique continuous solution y(t) = (x(t)T , u(t)T , v(t)T )T for system (40) and (41). (ii) Let y(t) = (x(t)T , u(t)T , v(t)T )T be the state trajectory of (40) and (41) with the initial point y(t0 ) = (x(t0 )T , u(t0 )T , v(t0 )T )T . If u(t0 ) ≥ 0, then u(t) ≥ 0. Theorem 4.6 The state trajectory of the neural network (40) and (41) converges to an equilibrium point for any initial point y0 = (x(t0 )T , u(t0 )T , v(t0 )T )T ∈ Rn+m+l . In particular, neural network (40) and (41) with any initial point y0 = (x(t0 )T , u(t0 )T , v(t0 )T )T ∈ Rn+m+l is globally asymptotically stable when D∗ has unique equilibrium point, where D∗ is the optimal point set of (37)–(39) and its dual. 13

Figure 1: A simplified block diagram for the neural network (40)-(41).

14

Theorem 4.7 The convergence rate of the neural network in (40) and (41) increases as τ increases.

5. Illustrative examples In order to demonstrate the effectiveness of the proposed neural network (40) and (41), in this section, we test several illustrative examples. The simulation is conducted on Matlab 7, the ordinary differential equation solver engaged is ode45s. A question that appears in all examples is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (11) or one formed on (12). When we want to compare the VaR of different assets with different expected returns, we need to use the relative measure. We propose the coefficient of variation (C.V): C.Vi =

V aRpi . E(Rpi )

Example 1: Assume that there are 10 securities and the returns of these securities are all linear uncertain variables ξi = (ai , bi ), i = 1, ..., 10, and are shown in Table 1. Table 1: Securities are all linear uncertain variables security

return

security

return

1

(1 , 1.2)

6

(0.5 , 1.5)

2

(0.8 , 1)

7

(0.1 , 0.6)

3

(2.1 , 2.3)

8

(0.3 , 0.7)

4

(0.9 , 1.4)

9

(1.7 , 2.4)

5

(2 , 3.3)

10

(0.4 , 0.9)

Suppose that the risk is not allowed to exceed 0.6, β = 0.1 and the minimum expected return that investor can accept is 2.5; then the LP models (11) and (12) are respectively as follows: max

1.1x1 + 0.9x2 + 2.2x3 + 1.15x4 + 2.65x5 + x6 + 0.35x7 + 0.5x8 + 2.05x9 + 0.65x10

(43)     1.18x1 + 0.98x2 + 2.28x3 + 1.35x4 + 3.17x5 + 1.4x6 + 0.55x7 + 0.66x8 + 2.33x9 + 0.85x10 ≤ 0.6 s.t.

  

x1 + x2 + · · · + x10 = 1 xi ≥ 0 , i = 1, 2, . . . , 10

(44) 15

and

min

1.18x1 + 0.98x2 + 2.28x3 + 1.35x4 + 3.17x5 + 1.4x6 + 0.55x7 + 0.66x8 + 2.33x9 + 0.85x10

(45)     1.1x1 + 0.9x2 + 2.2x3 + 1.15x4 + 2.65x5 + x6 + 0.35x7 + 0.5x8 + 2.05x9 + 0.65x10 ≥ 2.5 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (46)

The optimal solution of (43) and (44) is x∗ = (0, 0, 0, 0, 0, 0, 0.55, 0.45, 0, 0)T and the value of objective function is 0.42. This means that in order to gain maximum expected return with the VaR not greater than 0.6, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 0.42. We apply the proposed neural network in (40) and (41) to solve (43) and (44). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 2 displays the transient behavior of (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of the model (45) and (46) is x∗ = (0, 0, 0.33, 0, 0.67, 0, 0, 0, 0, 0)T , and the value of objective function is 2.88. This means that in order to minimize the VaR with the expected value greater than 2.5, the investor should assign his capital according to x∗ . The corresponding minimum VaR is 2.88. Figure 3 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (45) and (46). The question that appears here is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (43) and (44) or one formed on (45) and (46)? We have

(C.V )model(43,44) =

0.6 = 1.43, 0.42

(C.V )model(45,46) =

2.88 = 1.15. 2.5

We can consequently say that the portfolio formed by model (43) and (44) is a riskier investment than portfolio formed on the basis of model (45) and (46). Thus, the portfolio formed based on model (45) and (46) is more suitable for investors. 16

Figure 2: Transient behavior of x1 , x2 , . . . , x10 of LP model (43) and (44).

Figure 3: Transient behavior of x1 , x2 , . . . , x10 of LP model (45) and (46).

Example 2: Assume that there are 10 securities and the returns of these securities are all zigzag uncertain variables ξi = (ai , bi , ci ), i = 1, ..., 10, and are shown in Table 2. 17

Table 2: Securities are all zigzag uncertain variables security

return

security

return

1

(-0.2 , 2.1 , 2.5)

6

(-0.2 , 2.5 , 3)

2

(-0.1 , 1.9 , 3)

7

(-0.2 , 3 , 3.5)

3

(-0.4 , 3 , 4)

8

(-0.4 , 2.5 , 4)

4

(-0.1 , 2 , 2.5)

9

(-0.3 , 2.8 , 3.2)

5

(-0.6 , 3 , 4)

10

(-0.3 , 2 , 2.5)

Suppose that the risk is not allowed to exceed 3.2, β = 0.2 and the minimum expected return that investor can accept is 2; then the LP models (11) and (12) are respectively as follows:

max

1.625x1 + 1.675x2 + 2.4x3 + 1.6x4 + 2.35x5 + 1.95x6 + 2.325x7 + 2.15x8 + 2.125x9 + 1.55x10

(47)     3.48x1 + 3.1x2 + 5.04x3 + 3.26x4 + 5.16x5 + 4.12x6 + 4.92x7 + 4.24x8 + 4.66x9 + 3.38x10 ≤ 3.2 s.t.

  

x1 + x2 + · · · + x10 = 1 xi ≥ 0 , i = 1, 2, . . . , 10

(48) and

min

3.48x1 + 3.1x2 + 5.04x3 + 3.26x4 + 5.16x5 + 4.12x6 + 4.92x7 + 4.24x8 + 4.66x9 + 3.38x10

(49)     1.625x1 + 1.675x2 + 2.4x3 + 1.6x4 + 2.35x5 + 1.95x6 + 2.325x7 + 2.15x8 + 2.125x9 + 1.55x10 ≥ 2 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (50) The optimal solution of (47) and (48) is x∗ = (0, 0.91, 0, 0, 0, 0, 0, 0.09, 0, 0)T and the value of objective function is 1.72. This means that in order to gain maximum expected return with the risk not greater than 3.2, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 1.72. We apply the proposed neural network in (40) and (41) to solve (47) and (48). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 4 displays the transient behavior of 18

(x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of model (49) and (50) is x∗ = (0, 0.32, 0, 0, 0, 0, 0, 0.68, 0, 0)T , and the value of objective function is 3.86. This means that in order to minimize the VaR with the expected value greater than 2, the investor should assign his capital according to x∗ . The corresponding minimum VaR is 3.86. Figure 5 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (49) and (50).

The question that appears here is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (47) and (48) or one formed on (49) and (50)? We have

(C.V )model(47,48) =

3.2 = 1.86, 1.72

(C.V )model(49,50) =

3.86 = 1.93. 2

Figure 4: Transient behavior of x1 , x2 , . . . , x10 of LP model (47) and (48).

19

Figure 5: Transient behavior of x1 , x2 , . . . , x10 of LP model (49) and (50).

We can consequently say that the portfolio formed by model (49) and (50) is a riskier investment than portfolio formed on the basis of model (47) and (48). Thus, the portfolio formed based on model (47) and (48) is more suitable for investors. Example 3: Assume that there are 10 securities and the returns of these securities are all zigzag uncertain variables ξi = (ai , bi , ci ), i = 1, ..., 10, and are shown in Table 3.

Table 3: Securities are all zigzag uncertain variables security

return

security

return

1

(-0.2 , 2.1 , 2.5)

6

(-0.2 , 2.5 , 3)

2

(-0.1 , 1.9 , 3)

7

(-0.2 , 3 , 3.5)

3

(-0.4 , 3 , 4)

8

(-0.4 , 2.5 , 4)

4

(-0.1 , 2 , 2.5)

9

(-0.3 , 2.8 , 3.2)

5

(-0.6 , 3 , 4)

10

(-0.3 , 2 , 2.5)

Suppose that the risk is not allowed to exceed 3.2, β = 0.6 and the minimum expected return that investor can accept is 2; then the LP models (11) and (12) are respectively as follows: 20

max

1.625x1 + 1.675x2 + 2.4x3 + 1.6x4 + 2.35x5 + 1.95x6 + 2.325x7 + 2.15x8 + 2.125x9 + 1.55x10

(51)     2.02x1 + 1.68x2 + 2.8x3 + 1.9x4 + 2.8x5 + 2.4x6 + 2.9x7 + 2.2x8 + 2.72x9 + 1.9x10 ≤ 3.2 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (52) and

min

2.02x1 + 1.68x2 + 2.8x3 + 1.9x4 + 2.8x5 + 2.4x6 + 2.9x7 + 2.2x8 + 2.72x9 + 1.9x10

(53)     1.625x1 + 1.675x2 + 2.4x3 + 1.6x4 + 2.35x5 + 1.95x6 + 2.325x7 + 2.15x8 + 2.125x9 + 1.55x10 ≥ 2 s.t.

  

x1 + x2 + · · · + x10 = 1 xi ≥ 0 , i = 1, 2, . . . , 10

(54) The optimal solution of (51) and (52) is x∗ = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0)T and the value of objective function is 2.4. This means that in order to gain maximum expected return with the VaR not greater than 3.2, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 2.4. We apply the proposed neural network in (40) and (41) to solve (51) and (52). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 6 displays the transient behavior of (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of model (53) and (54) is x∗ = (0, 0.32, 0, 0, 0, 0, 0, 0.68, 0, 0)T , and the value of objective function is 2.03. This means that in order to minimize the VaR with the expected value greater than 2, the investor should assign his capital according to x∗ . The corresponding minimum risk is 2.03. Figure 7 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (53) and (54). The question that appears here is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (51) and (52) or one formed on (53) and (54)? We have (C.V )model(51,52) =

3.2 = 1.33, 2.4

(C.V )model(53,54) = 21

2.03 = 1.02. 2

Figure 6: Transient behavior of x1 , x2 , . . . , x10 of LP model (51) and (52).

Figure 7: Transient behavior of x1 , x2 , . . . , x10 of LP model (53) and (54).

We can consequently say that the portfolio formed by model (51) and (52) is a riskier investment than portfolio formed on the basis of model (53) and (54). Thus, the portfolio formed based on model (53) and (54) is more suitable for investors. 22

Example 4: Assume that there are 10 securities and the returns of these securities are all trapezoidal uncertain variables ξi = (ai , bi , ci , di ), i = 1, ..., 10, and are shown in Table 4. Table 4: Securities are all trapezoidal uncertain variables security

return

security

return

1

(0.2 , 0.4 , 1 , 1.1)

6

(-0.5 , 1.7 , 3 , 3.7)

2

(0.5 , 0.7 , 1.3 , 1.5)

7

(2 , 2.5 , 3.1 , 4)

3

(0.2 , 0.8 , 2 , 2.2)

8

(-2 , 0.2 , 1 , 1.6)

4

(0.1 , 0.3 , 0.5 , 0.9)

9

(-0.2 , 3 , 3.5 , 4)

5

(-0.7 , 0.7 , 3 , 3.2)

10

(0.1 , 0.9 , 2 , 2.2)

Suppose that the risk is not allowed to exceed 0.7, β = 0.3 and the minimum expected return that investor can accept is 1.5; then the LP models (11) and (12) are respectively as follows:

max

0.675x1 + x2 + 1.3x3 + 0.45x4 + 1.55x5 + 1.975x6 + 2.9x7 + 0.2x8 + 2.575x9 + 1.3x10

(55)     0.48x1 + 0.78x2 + 1.04x3 + 0.38x4 + 1.26x5 + 2.58x6 + 2.7x7 + 1.08x8 + 4.28x9 + 1.22x10 ≤ 0.7 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (56) and

min

0.48x1 + 0.78x2 + 1.04x3 + 0.38x4 + 1.26x5 + 2.58x6 + 2.7x7 + 1.08x8 + 4.28x9 + 1.22x10

(57)     0.675x1 + x2 + 1.3x3 + 0.45x4 + 1.55x5 + 1.975x6 + 2.9x7 + 0.2x8 + 2.575x9 + 1.3x10 ≥ 1.5 s.t.

  

x1 + x2 + · · · + x10 = 1 xi ≥ 0 , i = 1, 2, . . . , 10

(58) The optimal solution of (55) and (56) is x∗ = (0.72, 0, 0, 0, 0.28, 0, 0, 0, 0, 0)T and the value of objective function is 0.92. This means that in order to gain maximum expected return with the risk not greater than 0.7, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 0.92. 23

We apply the proposed neural network in (40) and (41) to solve (55) and (56). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 8 displays the transient behavior of (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of model (57) and (58) is x∗ = (0.06, 0, 0, 0, 0.94, 0, 0, 0, 0, 0)T , and the value of objective function is 1.21. This means that in order to minimize the risk with the expected value greater than 1.5, the investor should assign his capital according to x∗ . The corresponding minimum risk is 1.21. Figure 9 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (57) and (58). The question that appears here is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (55) and (56) or one formed on (57) and (58)? We have

(C.V )model(55,56) =

0.7 = 0.76, 0.92

(C.V )model(57,58) =

1.21 = 0.81. 1.5

Figure 8: Transient behavior of x1 , x2 , . . . , x10 of LP model (55) and (56).

24

Figure 9: Transient behavior of x1 , x2 , . . . , x10 of LP model (57) and (58).

We can consequently say that the portfolio formed by model (57) and (58) is a riskier investment than portfolio formed on the basis of model (55) and (56). Thus, the portfolio formed based on model (55) and (56) is more suitable for investors. Example 5: Assume that there are 10 securities and the returns of these securities are all trapezoidal uncertain variables ξi = (ai , bi , ci , di ), i = 1, ..., 10, and are shown in Table 5.

Table 5: Securities are all trapezoidal uncertain variables security

return

security

return

1

(0.2 , 0.4 , 1 , 1.1)

6

(-0.5 , 1.7 , 3 , 3.7)

2

(0.5 , 0.7 , 1.3 , 1.5)

7

(2 , 2.5 , 3.1 , 4)

3

(0.2 , 0.8 , 2 , 2.2)

8

(-2 , 0.2 , 1 , 1.6)

4

(0.1 , 0.3 , 0.5 , 0.9)

9

(-0.2 , 3 , 3.5 , 4)

5

(-0.7 , 0.7 , 3 , 3.2)

10

(0.1 , 0.9 , 2 , 2.2)

Suppose that the risk is not allowed to exceed 0.7, β = 0.7 and the minimum expected return that investor can accept is 1.5; then the LP models (11) and (12) are respectively as follows: 25

max

0.675x1 + x2 + 1.3x3 + 0.45x4 + 1.55x5 + 1.975x6 + 2.9x7 + 0.2x8 + 2.575x9 + 1.3x10

(59)     0.96x1 + 1.22x2 + 1.92x3 + 0.34x4 + 2.92x5 + 2.72x6 + 2.74x7 + 0.76x8 + 3.3x9 + 1.92x10 ≤ 0.7 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (60) and

min

0.96x1 + 1.22x2 + 1.92x3 + 0.34x4 + 2.92x5 + 2.72x6 + 2.74x7 + 0.76x8 + 3.3x9 + 1.92x10

(61)     0.675x1 + x2 + 1.3x3 + 0.45x4 + 1.55x5 + 1.975x6 + 2.9x7 + 0.2x8 + 2.575x9 + 1.3x10 ≥ 1.5 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (62) The optimal solution of (59) and (60) is x∗ = (0, 0, 0, 0.85, 0, 0, 0.15, 0, 0, 0)T and the value of objective function is 0.82. This means that in order to gain maximum expected return with the risk not greater than 0.7, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 0.82. We apply the proposed neural network in (40) and (41) to solve (59) and (60). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 10 displays the transient behavior of (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of model (61) and (62) is x∗ = (0, 0, 0, 0.57, 0, 0, 0.43, 0, 0, 0)T , and the value of objective function is 1.37. This means that in order to minimize the risk with the expected value greater than 1.5, the investor should assign his capital according to x∗ . The corresponding minimum risk is 1.37. Figure 11 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (61) and (62). The question that appears here is: which of the above portfolios are more suitable for the investors The portfolio formed based on the model (59) and (60) or one formed on (61) 26

and (62) We have (C.V )model(59,60) =

0.7 = 0.85, 0.82

(C.V )model(61,62) =

1.37 = 0.91. 1.5

Figure 10: Transient behavior of x1 , x2 , . . . , x10 of LP model (59) and (60).

Figure 11: Transient behavior of x1 , x2 , . . . , x10 of LP model (61) and (62).

We can consequently say that the portfolio formed by model (61) and (62) is a riskier 27

investment than portfolio formed on the basis of model (59) and (60). Thus, the portfolio formed based on model (59) and (60) is more suitable for investors. Example 6: Assume that there are 10 securities and the returns of these securities are all normal uncertain variables ξi = N (ei , σi ), i = 1, ..., 10, and are shown in Table 6. Table 6: Securities are all normal uncertain variables security

return

security

return

1

(0.8 , 1.3)

6

(1.1 , 1.8)

2

(0.9 , 1.6)

7

(0.8 , 1.7)

3

(0.8 , 1.5)

8

(0.7 , 1.5)

4

(0.9 , 1.4)

9

(0.8 , 1.6)

5

(1 , 1.8)

10

(0.7 , 0.8)

Suppose that the risk is not allowed to exceed 3.7, β = 0.01 and the minimum expected return that investor can accept is 0.9; then the LP models (11) and (12) are respectively as follows:

max

0.8x1 + 0.9x2 + 0.8x3 + 0.9x4 + x5 + 1.1x6 + 0.8x7 + 0.7x8 + 0.8x9 + 0.7x10 (63)     4.09x1 + 4.94x2 + 4.59x3 + 4.438x4 + 5.549x5 + 5.695x6 + 5.1x7 + 4.49x8 + 4.84x9 + 2.72x10 ≤ 3.7 s.t. x1 + x2 + · · · + x10 = 1    xi ≥ 0 , i = 1, 2, . . . , 10 (64)

and

min

4.09x1 + 4.94x2 + 4.59x3 + 4.438x4 + 5.549x5 + 5.695x6 + 5.1x7 + 4.49x8 + 4.84x9 + 2.72x10

(65)     0.8x1 + 0.9x2 + 0.8x3 + 0.9x4 + x5 + 1.1x6 + 0.8x7 + 0.7x8 + 0.8x9 + 0.7x10 ≥ 0.9 s.t.

  

x1 + x2 + · · · + x10 = 1 xi ≥ 0 , i = 1, 2, . . . , 10

(66) The optimal solution of (63) and (64) is x∗ = (0, 0, 0, 0, 0, 0.33, 0, 0, 0, 0.67)T and the value of objective function is 0.37. This means that in order to gain maximum expected return with the risk not greater than 1.4, the investor should assign his capital according to the optimal solution x∗ . The corresponding maximum expected return is 0.37. 28

We apply the proposed neural network in (40) and (41) to solve (63) and (64). Simulation results show the trajectory of (40) and (41) with any initial point is always convergent to y ∗ = (x∗ T , u∗ T )T . For example, Figure 12 displays the transient behavior of (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t), x7 (t), x8 (t), x9 (t), x10 (t))T based on (40) and (41) with a random initial point. The optimal solution of model (65) and (66) is x∗ = (0, 0, 0, 0, 0, 0.5, 0, 0, 0, 0.5)T , and the value of objective function is 3.39. This means that in order to minimize the risk with the expected value greater than 1.3, the investor should assign his capital according to x∗ . The corresponding minimum risk is 3.39. Figure 13 shows that the trajectories of the proposed neural network in (40) and (41) with a random initial point converge to the optimal solution of the problem (65) and (66). The question that appears here is: which of the above portfolios are more suitable for the investors? The portfolio formed based on the model (63) and (64) or one formed on (65) and (66)? We have

(C.V )model(63,64) =

1.4 = 3.78, 0.37

(C.V )model(65,66) =

3.39 = 2.61. 1.3

Figure 12: Transient behavior of x1 , x2 , . . . , x10 of LP model (65) and (66).

29

Figure 13: Transient behavior of x1 , x2 , . . . , x10 of LP model (63) and (64).

We can consequently say that the portfolio formed by model (65) and (66) is a riskier investment than portfolio formed on the basis of model (63) and (64). Thus, the portfolio formed based on model (63) and (64) is more suitable for investors. To end this section, we answer two natural questions: what are the practical and computational advantages of the network (40) and (41), compared to existing generally available algorithms for LP problems? Are there advantages of our network compared to the existing ones? To answer these, we summarize what we have observed from numerical experiments and theoretical results as below. • Compared with traditional numerical optimization algorithms, the neural network approach has several potential advantages. First, the structure of a neural network can be implemented effectively using very large scale integration and optical technologies. Second, neural networks can solve many optimization problems with time-varying parameters. Third, the dynamical techniques and the numerical ODE techniques can be applied directly to the continuous-time neural network for solving constrained optimization problems effectively. • Compared with some existing gradient neural networks in [15], the neural network (40) and (41) can be implemented without a penalty parameter. Thus this model is wholly suitable to be implemented in hardware. • Changing initial points does not have much effect for our neural network model, whereas it does for some existing models. One can easily achieve the divergence of the solution trajectory of some test problems [23], while it does not affect anything by our neural network model. The reason is that our model is globally convergent to the optimal solution of the 30

problem. • An attractive feature of the method described in this paper is that the starting point need not be in the feasible region, in contrast to some methods such as the classical simplex scheme. • In order to state computational complexity of the proposed neural network (40) and (41) compared with some other neural networks, one can see section Introduction in [23].

6. Conclusion In this paper, we have proposed a high-performance neural network model for solving the portfolio selection problem with uncertain returns and value at risk as a risk measurement. Based on some concepts of linear programming theory and ordinary differential equations, we strictly prove the asymptotic stability of the proposed network. From any initial point, the trajectory of this network converges to an optimal solution of the original programming problem. The structure of the proposed network is reliable and efficient. The other advantages of the proposed neural network are that it can be implemented without a penalty parameter and can be convergent to an exact solution of the problem. It should be also noted that Theorem 6.4 guarantees that the stated model in (40) and (41) converges globally to the unique optimal solution. The results obtained are highly valuable in both theory and practice for solving the portfolio selection with uncertain returns problems in economics and financial mathematics. Compliance with Ethical Standard Funding This study was not funded by any grant. Conflict of interest The authors declare that they have no conflict of interest.

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