an application of grey wolf optimization algorithm for fuzzy portfolio selection problem

Available online at www.iiec2016.com

International Conference on Industrial Engineering

(ICIE 2016)

ICIE (2016) 000–000Iran Institute ofIndustrial Engineering Kharazmi University

12thAn application of Gray Wolf Optimization algorithm for fuzzy

portfolio selection problem

Kayvan Salehi*

Department of Industrial Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran* Corresponding author: E-mail address: [email protected]

Abstract

Portfolio selection problem is an important issue in financial engineering which has been solved by a variety of heuristic and non-heuristic techniques. The most studies are concentrating on classical mean-variance model. In this paper, mean-variance model isextended to mean-variance-skewness model. The extended model is generally a high-dimensional nonlinear programmingnecessitating the use of efficient heuristics to find the solution. Hence, this paper presents a heuristic method named Gray WolfOptimization (GWO) method. Finally, several numerical examples are provided to illustrate the modeling idea and performanceand effectiveness of the proposed algorithm is compared against the exact approach (LINGO software) in terms of fitness value andrequired computational time. Results show that the proposed GWO is very promising and achieves quality results for fuzzyportfolio selection in a reasonable time..

Keywords: Portfolio selection; Gray Wolf Optimization; Fuzzy programming

1. Introduction

Portfolio selection deals with the problem of allocating one’s capital to a large number of securities to meet investor’ssatisfaction. The earliest work in this field is due to Markowitz (1952) who introduced the well-known mean-variancemodel considering trade-off between return and risk. The main idea of this model was to characterize the securities ofindividual returns as random variables with normal probability distribution. Lots of efforts have been performed byresearchers in order to expand and solve Markowitz’s model. These attempts, regarding the higher moments such asskewness and kurtosis, have tried to make his model more practical (Samuelson, 1970; Arditti, 1971; Lai, 1991;Konno and Suzuki, 1995; Díaz et al. 2009; Zakamouline & Koekebakker, 2009). According to Konno and Suzuki(1995), maximizing the skewness in a portfolio would result in better return and Samuelson (1970) has indicated thatmost of the investors would prefer a portfolio with larger skewness where mean and variance are the same and thehigher moments such as skewness can be neglected only where there are reasons to trust that the returns aresymmetrically distributed (e.g. normal) or the higher moments are irrelevant to the investors’ decisions.

In the field of models solving, some heuristic methods based on Genetic Algorithm (GA) (e.g. Soleimani et al. 2009;Li et al. 2010), Tabu Search (TS) (e.g. Rolland, 1996), Particle Swarm Optimization (Golmakani and Fazel, 2011; Xu,Chen, and Yang 2007), Simulated Annealing (SA) (Crama and Schyns, 2003), Ant Colony Optimization (ACO)(Maringer, 2001) and Memetic Algorithm (MA) (Maringer and Winker, 2003) have been reported in the literatures forthe portfolio selection problem. In addition to, Chang et al. (2000) proposed three heuristics based on GA, TS, and SAto solve the problem and compared the results. After review the literature, the main contribution of this paper istwofold. First we present mean-variance-skewness model in fuzzy environment. Second we will propose a heuristicnamed as GWO in order to solve our model.

To achieve to this purpose the remainder of this paper is organized as follows. Section 2 defines the moments forfuzzy variables. Section 3 presents the fuzzy mean-variance-skewness model and its variations. A GWO for solving



(ICIE 2016)




Kayvan Salehi*


Abstract



1. Introduction






(ICIE 2016)




Kayvan Salehi*


Abstract



1. Introduction




Author name / IIEC 00 (2016) 000–00

the models is developed in Section 4. Section 5 provides numerical examples to illustrate the effectiveness of proposedalgorithm and the paper ends up with conclusions and summarizes the research in Section 6.

2. Preliminaries

The possibility theory is a branch of mathematics that studies the behavior of fuzzy phenomena which has beenintroduced by Zadeh (1965) via membership function. To measure a fuzzy event, Zadeh (1978) has also proposed theconcept of possibility measure. Although the possibility measure is widely used in portfolio selection problems, it hassome limitations. One of its important limitations is that the measure does not have self-duality property. Using themeasure which is not self-dial may result in the same possibility value for two fuzzy events with different occurringchances. In addition, if the possibility value of a portfolio return being greater than a target value is less than one, thepossibility value of the opposite event, i.e. the portfolio return being less than or equal to the target value, has themaximum value of one. Similarly, if the possibility value of a portfolio return being less than or equal to a target valueis less than one, the possibility value of the opposite event, i.e. the portfolio return being greater than the target value,has the maximum value of one, as well. These results are quite awkward and confuse the decision maker (Huang,2009).

In order to define a self-dual measure, Liu and Liu (2002) have introduced the concept of credibility measurewhich is more appropriate than the possibility measure in portfolio selection problems. Let be a fuzzy variable withmembership function μ, and u and r are real numbers. The credibility of a fuzzy event characterized by , is thendefined as in Eq. (1).

Now consider a triangular fuzzy variable which is fully determined by the triplet of crisp numbers with and themembership function given by:

The credibility of is then defined as follows.

Liu and Liu (2002) have also provided a more general definition of the expected value of a fuzzy variable basedon the credibility measure given in Eq. (4).

where at least one of the two integrals is finite.According to Liu and Liu (2002), if the fuzzy variable has a finite expected value, its variance is then defined by:

Now, let be a fuzzy variable with finite expected value. The skewness of is defined as follows (Li et al., 2010).

3. The mean-variance-skewness models of portfolio selection

Let be a fuzzy variable representing the return of the th security and be the proportion of total

capital invested in security . In general, is given as where is the closing price

of the th security at present, is the estimated closing price in the next year, and is the estimateddividends during the coming year.

The following notations are used in developing the models:

E: The operator of expected return

V: The operator of risk

S: The operator of skewness

Minimal expected return which the investors can accept

i ixi i ( ' ) /i i i ip d p p ip

i 'ip id

:


Maximal risk which the investors can bear

Minimal skewness which is desired by the investors

Now suppose that minimal expected return and maximal risk are given, the investors interested in theuse of skewness prefer a portfolio with large skewness. Therefore, we have the following mean-variance-skewness model:

1 1

1 1

1 1

1

max .....

: .....

..... 7..... 1

0, 1,2,..,

n n

n n

n n

n

i

S x x

subject to V x x

E x xx x

x i n

The second model is as follows:

1 1

1 1

1 1

1

max .....

: .....

..... 8..... 1

0, 1,2,..,

n n

n n

n n

n

i

E x x

subject to V x x

S x xx x

x i n

The purpose of the model presented in (11) is to select a portfolio with maximum return where theminimal variance as well as the maximal skewness is given.

The third model is as follows:

1 1

1 1

1 1

1

min .....

: .....

..... 9..... 10, 1,2,..,

n n

n n

n n

n

i

V x x

subject to S x x

E x xx xx i n

The purpose of this model is to choose a portfolio with minimum risk where the maximal skewness andexpected return are given.

Theorem 2: Assume that be independent triangular fuzzy variables forTherefore, the above models can convert to deterministic programming. For instance, the model (8) in thiscase is as follows (Li et al. 2010):

:

:

, ,i i i ia b c 1, 2,..., .i n


1

1 13 2 2

1 1 1 1 1 1

1

2max

4:

. 2 32 10

33 21 11

384

ni i i

ii

n n

i i i i i i ii i

n n n n n n

i i i i i i i i i i i i i i i i i ii i i i i i

n

i i ii

a b cx

subject to

c a x c a b x

c b x c b x b a x c b x b a x b a x

c b x

3

1

1 2

384

... 10, 1, 2,...,

n

i i ii

n

i

c b x

x x xx i n

Proof: This deterministic programming for mean-variance-skewness model has been proved in Li et al.(2010). The presented model is a high-dimensional nonlinear programming. In next step, a GWO algorithmis proposed for solving the model.

4. Gray Wolf Optimization (GWO)

The GWO algorithm mimics the leadership hierarchy and hunting mechanism of gray wolves in nature proposedby (Mirjalili et al. in 2014). Four types of grey wolves such as alpha, beta, delta, and omega are employed forsimulating the leadership hierarchy. In addition, three main steps of hunting, searching for prey, encircling prey, andattacking prey, are implemented to perform optimization. In addition to the social hierarchy of wolves, group huntingis another interesting social behavior of grey wolves. According to (Muro et al. 2011), the main phases of gray wolfhunting are as follows:

Tracking, chasing, and approaching the prey

Pursuing, encircling, and harassing the prey until it stops moving

Attack towards the prey

1.1. Mathematical model

The hunting technique and the social hierarchy of grey wolves are mathematically modeled in order to designGWO and perform optimization. The proposed mathematical models of the social hierarchy, tracking, encircling,and attacking prey are as follows:

1.1.1. Gray wolf social hierarchy

In order to mathematically model the social hierarchy of wolves when designing GWO, we consider the fittestsolution as the alpha (α). Consequently, the second and third best solutions are named beta (β) and delta (δ)respectively. The rest of the candidate solutions are assumed to be omega (ω). In the GWO algorithm the hunting(optimization) is guided by α, β, and δ. The ω wolves follow these three wolves.

1.1.2. Encircling prey

As mentioned above, grey wolves encircle prey during the hunt. In order to mathematically model encirclingbehavior the following equations are proposed:

| . ( ) ( ) | 12pD C X t X t

( 1) ( ) . 13pX t X t A D


where indicates the current iteration, A

and C

are coefficient vectors , pX

is the position vector of the prey,

and X

indicates the position vector of a grey wolf.

The vectors A

and C

are calculated as follows:

12 . 14A a r a

22. 15C r

Where components of a are linearly decreased from 2 to 0 over the course of iterations and 1r , 2r are randomvectors in [0,1].

With the above equations, a grey wolf in the position of (X,Y) can update its position according to the position ofthe prey (X*,Y*). Different places around the best agent can be reached with respect to the current position byadjusting the value of A

and C

vectors. For instance, (X*-X,Y*) can be reached by setting (1,0)a and

(1,1)C

. Note that the random vectors 1r , 2r allow wolves to reach any position between the two particularpoints. So a grey wolf can update its position inside the space around the prey in any random location by theabove-mentioned equations.

The same concept can be extended to a search space with n dimensions, and the grey wolves will move in hyper-cubes (or hyper-spheres) around the best solution obtained so far.

1.1.3. Hunting

Grey wolves have the ability to recognize the location of prey and encircle them. The hunt is usually guided bythe alpha. The beta and delta might also participate in hunting occasionally. However, in an abstract search spacewe have no idea about the location of the optimum (prey). In order to mathematically simulate the huntingbehavior of grey wolves, we suppose that the alpha (best candidate solution) beta, and delta have betterknowledge about the potential location of prey. Therefore, we save the first three best solutions obtained so farand oblige the other search agents (including the omegas) to update their positions according to the position of thebest search agent. The following formulas are proposed in this regard.

1| . | 16D C X X

2| . | 17D C X X

3| . | 19D C X X

1 1.( ) 20X X a D

2 2.( ) 21X X a D

3 3.( ) 22X X a D

1 2 3( 1) 233

X X XX t

With these equations, a search agent updates its position according to alpha, beta, and delta in a n dimensionalsearch space. In addition, the final position would be in a random place within a circle which is defined by the


positions of alpha, beta, and delta in the search space. In other words alpha, beta, and delta estimate the positionof the prey, and other wolves updates their positions randomly around the prey.

1.1.4. Attacking prey (exploitation)

As mentioned above the grey wolves finish the hunt by attacking the prey when it stops moving. In order tomathematically model approaching the prey we decrease the value of a . Note that the fluctuation range of A

is

also decreased by a . In other words A

is a random value in the interval [-2a,2a] where a is decreased from 2 to 0over the course of iterations. When random values of A

are in [-1,1], the next position of a search agent can be in

any position between its current position and the position of the prey.

With the operators proposed so far, the GWO algorithm allows its search agents to update their position based onthe location of the alpha, beta, and delta; and attack towards the prey. However, the GWO algorithm is prone tostagnation in local solutions with these operators. It is true that the encircling mechanism proposed showsexploration to some extent, but GWO needs more operators to emphasize exploration.

1.1.5. Search for prey (exploration)

Grey wolves mostly search according to the position of the alpha, beta, and delta. They diverge from each other tosearch for prey and converge to attack prey. In order to mathematically model divergence, we utilize A

with

random values greater than 1 or less than -1 to oblige the search agent to diverge from the prey. This emphasizesexploration and allows the GWO algorithm to search globally. |A|>1 forces the grey wolves to diverge from the

prey to hopefully find a fitter prey. Another component of GWO that favors exploration isC

, which containsrandom values in [0, 2]. This component provides random weights for prey in order to stochastically emphasize(C>1) or de-emphasize (C<1) the effect of prey in defining the distance in Equation (3.1). This assists GWO toshow a more random behavior throughout optimization, favoring exploration and local optima avoidance. It isworth mentioning here that C is not linearly decreased in contrast to A. We deliberately require C to providerandom values at all times in order to emphasize exploration not only during initial iterations but also finaliterations. This component is very helpful in case of local optima stagnation, especially in the final iterations.

The C vector can be also considered as the effect of obstacles to approaching prey in nature. Generally speaking,the obstacles in nature appear in the hunting paths of wolves and in fact prevent them from quickly andconveniently approaching prey. This is exactly what the vector C does. Depending on the position of a wolf, it canrandomly give the prey a weight and make it harder and farther to reach for wolves, or vice versa.

To sum up, the search process starts with creating a random population of grey wolves (candidate solutions) in theGWO algorithm. Over the course of iterations, alpha, beta, and delta wolves estimate the probable position of theprey. Each candidate solution updates its distance from the prey. The parameter a is decreased from 2 to 0 in orderto emphasize exploration and exploitation, respectively. Candidate solutions tend to diverge from the prey when

| | 1A

and converge towards the prey when | | 1A

. Finally, the GWO algorithm is terminated by thesatisfaction of an end criterion.

1.2. Constraints handling

During the algorithm running, the position of Gray Wolves may be out of feasible solution. Therefore, we employan approach for constraints handling. Here, we try to convert constraint problem to unconstraint problem. There aremany specific methods for constraints handling has been proposed with Evolutionary Algorithms, of which the mostpopular are the penalty function methods for their simplicity and their easy application. Here, a penalty method isconsidered for constrained optimization with the proposed algorithm.

Let the following constrained problem:


min.

0 241, 2,..., 1, 2

k i

i i

f xstg xL x U i n k

Among various approaches to handle the constraints in such an optimization problem, the most popular is the penalty

function method for its simplicity and easy application. This method is applied to inequality constraints as follows

(Yang et al., 2008).

1min

: 25

0 01,

exp 0

k

k k kk

kk k k

k k k

f x g x

w here

if g xg x

e g x if g x

where k=1,2,… is the number of inequality constraints.

5. Numerical experiments

In this section, some numerical examples are provided to evaluate performance of the proposed GWO. Thealgorithm is coded using Matlab software and tested on a Pentium Dual 2.6 GHz with 256 GB memory. The proposedalgorithm is compared with model results in LINGO which can generate global optima for small problems, butbecause of LINGO’s limitation, this comparison couldn’t be performed in large scale data. Herein, the mean-variance-skewness model is applied to the data adopted from Huang (2008) which is given in Table 1. The parameters of theproposed GWO used to find the optimal solution for the portfolio selection problem are also given in Table 2. Inaddition to, we assume that in all examples upper bounds on assets are

Table 1: Fuzzy returns of 10 securities

Security i Return Security i Return

1 (-0.4,2.7,3.4) 6 (-0.1,2.5,3.6)

2 (-0.1,1.9,2.6) 7 (-0.3,2.4,3.5)

3 (-0.2,3.0,4.0) 8 (-0.1,3.3,4.5)

4 (-0.5,2.0,2.9) 9 (-0.7,1.1,2.7)

5 (-0.6,2.2,3.3) 10 (-0.2,2.1,3.8)

Table 2: Parameters of GWO algorithm approach

Parameters Value

The number of gray wolves 30

Iterations 200

0.5,0.3,0.25,0.45,0.5,0.25,0.35,0.25,0.15,0.45 .


Example 1: Assume that an investor wants to select his portfolio from 10 securities given in Table 1. Let the bearablemaximum risk is 1.2 and the skewness of his portfolio is at least -1. Based on the model presented in Eq. (8), we havethe following model:

1 1

1 1

1 1

1

max .....

: ..... 1.2

..... 1..... 1

, 1,2,..,10

n n

n n

n n

n

i i i

E x x

subject to V x x

S x xx x

l x u i

According to the results of the GWO algorithm, the investor should allocate his money as shown in Table 3. Thecorresponding maximum expected value in this example is 2.355.

Table 3: Asset allocation of the mean-variance-skewness model in Example 1

Security i 1 2 3 4 5 6 7 8 9 10 Expectedvalue Time(min)

LINGO 0 0 0.25 0 0.5 0 0.25 0 0 0 2.356 18

GWO 0.0001 0 0.249 0 0.499 0.0006 0.247 0.002 0.0003 0. 002 2.355 1

Example 2: Assume that an investor wishes to choose his portfolio from 10 securities given in Table 1. Let theminimum expected return that the investor can accept is 1.8, and the skewness of his portfolio is at least -1. Based onthe model presented in Eq. (9), we have the following model:

1 1

1 1

1 1

1

min .....

: ..... 1

..... 1.8..... 1

, 1,2,..,10

n n

n n

n n

n

i i i

V x x

subject to S x x

E x xx xl x u i

Results of the GWO algorithm show that the investor should allocate his money as given in Table 4.

Table 4: Asset allocation of the mean-variance-skewness model in Example 2

Security i 1 2 3 4 5 6 7 8 9 10 Variance value Time(min)

LINGO 0.5 0 0 0.2 0 0 0.15 0 0.15 0 0.1620 546

GWO 0.49 0.0003 0 0.188 0.0107 0.023 0.139 0 0.149 0 0.1623 1

The corresponding minimum variance in this example is 0.1623.


6. Summary and Conclusions

As an extension of the fuzzy portfolio selection model, a fuzzy mean-variance-skewness model was presented in thisresearch. Since the variables are triangular fuzzy variables, the models were converted to deterministic programming.In order to solve these models a GWO was developed to find the final optimal portfolio solutions. The proposedalgorithm was tested against LINGO software in some numerical examples. The results clearly showed that that theproposed algorithm is robust and effective.

There still remain some further directions for future research. The new other algorithms such as Krill herd (KH),Gravitational search (GS), Roach Infestation Optimization (RIO), and ect can be employed instead of GWO andcompared with the results obtained in current paper. In addition to, other realistic constraints, e.g. minimumtransaction lots, sector capitalization, and cardinality constraint with various measures of risk such as semi-variance,value at-risk, entropy, and so on can be added to develop more complex portfolio optimization models.

References

[1] Arditti, F. D. (1967). Risk and the required return on equity. Journal of Finance, 22(1), 19–36.

[1] Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrainedportfolio optimization. Computers and Operations Research, 27, 1271–1302.

[2] Crama, Y., and Schyns, M. (2003). “Simulated Annealing for Complex Portfolio Selection Problems,”European Journal of Operational Research, 150(3), 546–571.

[3] Díaz, A., González, M., Navarro, E., Skinner, F.S. (2009). An evaluation of contingent immunization.Journal of Banking and Finance, 33, 1874–1883

[4] Golmakani, H. R. Fazel, M. (2011). Constrained portfolio optimization using Particle Swarm Optimization.Expert Systems with Applications, 38, 8327–8335.

[5] Huang, X. (2008). Mean-Entropy Models for Fuzzy Portfolio Selection. IEEE Transactions on FuzzySystems, 16 (4), 1096–1101

[6] Konno, H., Suzuki, K. (1995). A mean-variance-skewness optimization model. Journal of the OperationsResearch Society of Japan, 38:137–187

[7] Lai, T. (1991). Portfolio optimization with skewness: a multiple–objective approach. Review of theQuantitative Finance and Accounting, 1, 293–305.

[8] Li, X. Qin, Z. Kar, S. (2010). Mean-variance-skewness model for portfolio optimization with fuzzy returns.European Journal of Operational Research, 202, 239–247

[9] Liu, B., Liu, Y-K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEETransactions on Fuzzy Systems, 10, 445–450

[10] Maringer, D. (2001). “Optimizing Portfolios with Ant Systems,” in International ICSC Congress onComputational Intelligence: Methods and Applications (CIMA 2001), 288–294. ICSC.

[11] Maringer, D., and P.Winker (2003). “Portfolio Optimization under Different Risk Constraints with ModifiedMemetic Algorithms,” Discussion Paper No. 2003-005E, Faculty of Economics, Law and Social Sciences,University of Erfurt.

[2] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77–91.

[12] Rolland, E. (1997). A tabu search method for constrained real-number search: Applications to portfolioselection. Technical Report, Department of Accounting & Management Information Systems. Ohio StateUniversity, Columbus.

[3] Samuelson, P. (1970). The fundamental approximation theorem of portfolio analysis in terms of means.variances, and higher moments, Review of Economic Studies, 37, 537–542


[13] Soleimani, H. Golmakani, H.R. Salimi, M.H. (2009). Markowitz- based portfolio optimization with minimumtransaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. ExpertSystems with Applications, 36, 5058–5063.

[14] Yang, X.S. (2008). Nature-Inspired Metaheuristic Algorithms, Luniver Press.

[15] Zadeh, LA. (1965). Fuzzy sets. Information and Control, 8, 338-353

[16] Zadeh, LA. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3- 28

[17] Xu, F., Chen, W., & Yang, L. (2007). Improved Particle Swarm Optimization for realistic portfolio selection.InEighth ACIS international conference on software engineering, artificial intelligence, networking, andparallel/distributed computing (pp. 185–190). IEEE Computer Society.

[18] Zakamouline, V., Koekebakker, S. (2009). Portfolio performance evaluation with generalized Sharpe ratios:Beyond the mean and variance. Journal of Banking and Finance, 33, 1242–1254.

[19] Mirjalili, S., Mirjalili, S. M., and Lewis, A. (2014). "Grey Wolf Optimizer," Advances in EngineeringSoftware, vol. 69, pp. 46-61.

[20] Muro, C, Escobedo, R, Spector, L, Coppinger, R. (2011). Wolf-pack (Canis lupus) hunting strategies emergefrom simple rules in computational simulations. Behav Process; 88:192–7.

an application of grey wolf optimization algorithm for fuzzy portfolio selection problem

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