A novel hybrid model for portfolio selection
Chorng-Shyong Ong
a, Jih-Jeng Huang
a,
Gwo-Hshiung Tzeng
b,c,*a
Department of Information Management, National Taiwan University, Taipei, Taiwan
b
Institute of Management of Technology, National Chiao Tung Universit, Hsinchu, Taiwan
c
College of Management, Kainan University, Taoyuan, Taiwan
Abstract
As we know, the performance of the mean–variance approach depends on the accu-rate forecast of the return accu-rate. However, the conventional method (e.g. arithmetic mean or regression-based method) usually cannot obtain a satisfied solution especially under the small sample situation. In this paper, the proposed method which incorporates the grey and possibilistic regression models formulates the novel portfolio selection model. In order to solve the multi-objective quadric programming problem, multi-objective evolution algorithms (MOEA) is employed. A numerical example is also illustrated to show the procedures of the proposed method. On the basis of the numerical results, we can conclude that the proposed method can provide the more flexible and accurate results.
Ó 2004 Elsevier Inc. All rights reserved.
Keywords: Mean–variance approach; Portfolio selection; Grey model; Possibilistic regression model; Multi-objective evolution algorithms (MOEA)
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.10.080
*
Corresponding author.
E-mail address:ghtzeng@cc.nctu.edu.tw(G.-H. Tzeng).
1. Introduction
The mean–variance approach was proposed by Markowitzto deal with the portfolio selection problem [1]. A decision-maker can determine the optimal investing rate to each security based on the sequent return rate. The formula-tion of the mean–variance method can be described as follows [1–3]:
minX n i¼1 Xn j¼1 rijxixj ð1Þ s:t:X n i¼1 lixiP E; Xn i¼1 xi¼ 1; xiP0 8i ¼ 1; . . . ; n:
where lidenotes the expected return rate of security 1, rijdenotes the
covari-ance coefficient between the ith security and the jth security, E denotes the acceptable least rate of the expected return.
Although the mean–variance model has been widely used in various portfo-lio selection problems, some issues should be highlighted to increase the accu-racy of this model. It is clear that the accuaccu-racy of the mean–variance approach depends on the accurate value of the expected return and the variance-covariance matrix. Several methods have been proposed to forecast the ade-quate excepted return and variance matrix such as arithmetic mean method
[1–3] and regression-based method [4]. Since these methods are based on the theory of large sample, they usually can not obtain a satisfied solution in the small sample situation[5].
In this paper, the grey prediction model is used to predict the further return rate. In addition, we divide the portfolio risk into the uncertainty risk and the relation risk. The uncertainty risk measures the possibilistic degree of the fu-ture return rate and the relation risk measures the trending degrees of the se-quences. These two risks can be calculated using the possibilistic regression model and the grey relation degree. Next, we can formulate the three-objective quadratic programming model (i.e. achieve the maximum return rate and the minimum uncertainty risk and relation risk simultaneously) to obtain the effi-cient frontier set using multi-objective evolutionary algorithms (MOEA). To summarize the above descriptions, we can depict the proposed method as shown inFig. 1.
A numerical example is also illustrated to show the proposed method. On this basis of the numerical results, we can conclude that the proposed method can provide the more flexible and accurate portfolio selection alter-natives.
The remainder of this paper is organized as follows. The grey and possibi-listic regression models are discussed in Section 2. Multi-objective evolutionary algorithms is proposed in Section 3. A numerical example is used to illustrate the proposed method in Section 4. The discussions of the numerical results are presented in Section 5 and the conclusions are presented in the last section.
2. Grey and possibilistic regression models
The grey prediction model is proposed to fit the sequence curve under the small sample[6–8]and this method has been recently used in various applica-tions such as stock price [9], and control system[10]. In this paper, the GM (1, 1) model, which is most commonly used, is employed to predict the future return rate.
Assume a sequence can be represented as x(0)= (x(0)(1), x(0)(2), . . . , x(0)(n)), then the corresponding first order accumulated generating operation (AGO) series and mean generating operation can be represented as x(1)= (x(1)(1), x(1)(2), . . . , x(1)(n)) and z(1)(k) = 0.5(x(1)(k) + x(1)(k 1)). Therefore, the grey differential equation of GM (1, 1) can be described as
xð0ÞðkÞ þ azð1ÞðkÞ ¼ b; 8k 2 f2; 3; . . . ; kg:
Using the ordinal least square (OLS) method, we can obtain the grey para-meter matrix Historical Date Grey Relation Possibilistic Regression Grey Prediction Relation Risk Uncertainty Risk Excepted Return MOEA Portfolio Selection
^ a¼ ðB0BÞ1 B0Yn where Yn¼ xð0Þð2Þ xð0Þð3Þ .. . xð0ÞðnÞ 2 6 6 6 6 4 3 7 7 7 7 5; B¼ zð1Þð2Þ 1 zð1Þð3Þ 1 .. . .. . zð1ÞðnÞ 1 2 6 6 6 6 4 3 7 7 7 7 5; ^a¼ a b : ð2Þ Last, the solution of the prediction value can be derived as
xð0ÞðkÞ ¼ 1 0:5 1þ 0:5 k2
b axð0Þð1Þ
1þ 0:5a : ð3Þ
Using the grey prediction model, we can predict the future return rate more accurately under the restriction of the small sample.
Next, we use the possibilistic regression[11]to obtain the uncertainty risk of the future return rate. The form of a possibilistic regression can be expressed as y¼ A0þ A1x1þ þ Anxn¼ A0x ð4Þ
where Aiis a symmetrical fuzzy number denoted as (ai, ci)L, and the form of the
membership function[12]of Eq.(4) can be obtained for x 5 0 as
lYðyÞ ¼ Lððy x0aÞ=c0jxjÞ ð5Þ
for x = 0 and y = 0, lY(y) = 1, and for x = 0 and y 5 0, lY(y) = 0. The h-level
set of y denoted as [y]hcan be obtained as following setting:
Lððy x0aÞ=c0jxjÞ ¼ h ð6Þ
Then, [y]hcan be obtained as
½yh¼ x0a jL1ðhÞjc0jxj
; x0aþ jL1ðhÞjc0jxj
¼ x½ ;xþ ð7Þ
On the basis of the above conditions, we can obtain the formulation of a pos-sibilistic regression model as follows:
min a;c J ¼ X j¼1;...;m hjc0jxjj ð8Þ s:t: yjP x0ja jL1ðh jÞjc0jxjj; yj6x0 jaþ jL 1ðh jÞjc0jxjj; j¼ 1; . . . ; m c P 0:
Solving the above mathematical programming model, we can calculate the uncertainty risk of the future return rate. Additionally, in order to obtain the relation risk of the security, the grey relational grade[6,7] is employed in this paper. Let two sequences xiand xjcan be represented as
and
xj¼ ðxjð1Þ; xjð2Þ; . . . ; xjðkÞ; . . . ; xjðnÞÞ:
Then, the grey relational coefficient can be obtained using the following formulation
cðxiðkÞ; xjðkÞÞ ¼
min
j mink jxiðkÞ xjðkÞj þ f minj mink jxiðkÞ xjðkÞj
jxiðkÞ xjðkÞj þ f min
j mink jxiðkÞ xjðkÞj
; ð9Þ
where f is the grey relation recognition coefficient with numerical value be-tween [0, 1]. The f can be adjusted for the requirement. In this paper, f is set at 0.1 to enlarge the scope of the grey relational coefficient. Finally, the grey relation grade can be expressed as follows
cðxi;xjÞ ¼ 1 n Xn k¼1 cðxiðkÞ; xjðkÞÞ: ð10Þ
After obtaining the results of the grey and possibilistic regression models, then, the proposed method can be formulated in the following mathematical pro-gramming equations maxX n i¼1 lixi ðExcepted ReturnÞ ð11Þ minX n i¼1 xþi xi xi ðUncertainty RiskÞ minX n i¼1 Xn j¼1 rijxixj ðRelation RiskÞ Xn i¼1 xi¼ 1 xiP0 8i ¼ 1; . . . ; n:
After solving the mathematical programming model, we can obtain the optimal portfolio selection alternative. However, it is clear that the above equations be-long to the three-objective quadratic programming problem and it is hard to obtain the optimal portfolio selection using the conventional methods. In addi-tion, the conventional method provides only one optimal portfolio selection rather than an efficient frontier set. Since the individual investor chooses the optimal portfolio selection based on his preference, the Pareto set should also be provided for various alternatives. In this paper, multi-objective evolutionary algorithms is employed to overcome the above problems.
3. Multi-objective evolutionary algorithms
Multi-objective evolutionary algorithms (MOEA) has been widely used since the 1990Õs to resolve the combinational problem in various domains such as scheduling[13], engineering[14]and finance[15]. The concept of MOEA is based on the method of genetic algorithms (GA). GA was pioneered in 1975 by Holland, and its concept is to mimic the natural evolution of a population by allowing solutions to reproduce, create new solutions, and compete for surviv-ing in the next iteration[16–20]. Then, the fitness is improved over generations and the best solution is finally achieved.
The procedures of MOEA are similar to GA. The initial population, P(0), is encoded randomly by strings. In each generation, t, the more fit elements are selected for the mating pool. Then, three basic genetic operators, reproduction, crossover, and mutation, are processed to generate new offspring. On the basis of the principle of survival of the fittest, the best chromosome of a candidate solution is obtained. The pseudo codes and the corresponding procedure graph of MOEA can be represented as shown inFigs. 2 and 3.
The power of evolution algorithms lies in its simultaneously searching a population of points in parallel, not a single point. Therefore, evolution algo-rithms can find the approximate optimum quickly without falling into a local optimum. In the conventional mathematical programming techniques, these methods generally assume small and enumerable search spaces[21]. However, MOEA can handle various function problems such as discontinuous or
cave form and scaling problems[21–23]. In addition, we can obtain the Pareto optimal set rather than a special solution using the method of MOEA.
Next, we describe the three basic genetic operators used in MOEA as follows:
Crossover. The goal of crossover is to exchange information between two parent chromosomes in order to produce two new offspring for the next pop-ulation. In this study, we use uniform crossover to generate the new offspring. The procedures of uniform crossover can be described as follows. Assume that two parents and a random template are selected by
Template¼ 0 1 0 0 1 1 0 1 Parent1¼ 1 1 0 1 0 0 1 1
Parent2¼ 0 0 1 0 1 1 0 0
then, two offspring will be generated as Initialize Population Fitness Evaluation Nondominated Solution Fitness Transformation Genetic Operators Satisfy? Loop Yes No
Offspring1¼ 0 1 1 0 0 0 0 1
Offspring2¼ 1 0 0 1 1 1 1 0
Mutation. Mutation is a random process where one genotype is replaced by another to generate a new chromosome. Each genotype has the probability of mutation, Pm, changing from 0 to 1, and vice versa.
Selection. The selection operator selects chromosomes from the mating pool using the ‘‘survival of the fittest’’ concept, as in natural genetic systems. Thus, the best chromosomes receive more copies, while the worst die off. The prob-ability of variable selection is proportional to its fitness value in the population, according to the formula given by
PðxiÞ ¼ fðxiÞ PN j¼1 fðxjÞ ð12Þ
where f(xi) represents the fitness value of the ith chromosome, and N is the
population size.
In addition, one of the crucial procedures of MOEA is to determine the fit-ness function. In this paper, the crowding distance[24,25] is used to sort the chromosomes and determine the Pareto set. In next section, we use a numerical example to illustrate the proposed method.
4. Numerical example
In this section, a numerical example is used to compare between the mean– variance approach and the proposed method. Let the sequent return rates of the six stocks from time t 6 to t can be represented as inTable 1. As men-tioned previously, in order to obtain the optimal portfolio selection, a deci-sion-maker should forecast the expected return in the t + 1 period as accurately as possible.
Table 1
The sequences of the six stocks
Period t 6 t 5 t 4 t 3 t 2 t 1 t Stock 1 0.07 0.06 0.10 0.08 0.09 0.12 0.14 Stock 2 0.03 0.05 0.11 0.05 0.13 0.14 0.09 Stock 3 0.07 0.11 0.07 0.07 0.05 0.10 0.09 Stock 4 0.06 0.12 0.16 0.08 0.05 0.10 0.12 Stock 5 0.06 0.10 0.09 0.06 0.15 0.07 0.13 Stock 6 0.04 0.01 0.07 0.10 0.11 0.07 0.12
Using the conventional arithmetic mean, we can obtain the further return rates and the variance-covariance matrix of the six stocks as shown inTables 2 and 3.
Then, we can use the weighted sum method and assume the weights are equal to resolve the mean–variance model to obtain the conventional optimal portfolio selection as shown inTable 4.
Now, we illustrate the proposed method as follows. First, according to the information inTable 1, we can use the grey prediction method, shown in Eqs.
(2) and (3), to calculate the future return rate of the six stocks in the t + 1 as shown inTable 5.
Next, we can obtain the possibilistic interval (PI) of each stock in the t + 1 period using the possibilistic regression model (i.e. Eq.(8)) and also derive the uncertainty risk as shown in Table 6. In order to obtain the relation risk, we can calculate the grey relation matrix using Eqs. (9) and (10) and the corre-sponding results can be shown as inTable 7.
Table 2
Arithmetic mean of the excepted return
Stock 1 2 3 4 5 6
Forecast value 0.09 0.09 0.08 0.10 0.09 0.07
Table 3
Variance-covariance matrix of the excepted return
Stock 1 Stock 2 Stock 3 Stock 4 Stock 5 Stock 6
Stock 1 0.00027 0.00045 0 0.00004 0.00036 0.00008 Stock 2 0.00179 0.00062 0.00028 0.00080 0.00002 Stock 3 0.00112 0.00008 0.00040 0.00032 Stock 4 0.000307 0.000270 0.00024 Stock 5 0.001707 0.00016 Stock 6 0.00076 Table 4
Optimal portfolio selection using the conventional method
Stock 1 2 3 4 5 6 Return Rate Portfoliorisk
Portfolio 0 0 0 1 0 0 0.10 0.0003
Table 5
The future return rate using the grey prediction model
Stock 1 2 3 4 5 6
Now, we can formulate the multi-objective mathematical programming based on the above information as the following equations:
max 0:16x1þ 0:13x2þ þ 0:14x6 min 0:8x1þ 0:14x2þ þ 0:1x6 min x2 1þ 0:289x1x2þ þ x26 x1þ x2þ x3þ x4þ x5þ x6¼ 1 xiP0 8i ¼ 1; . . . ; 6:
In order to deal with this three-objective quadratic programming problem, multi-objective evolutionary algorithms is employed in this paper and the cor-responding parameter value can also be shown as in Table 8.
Using MOEA, we can obtain the efficient frontier set and the 55 portfolio alternatives as shown inTable 9orAppendix A.
Table 7
The grey relation matrix
c(xi, xj) Stock 1 Stock 2 Stock 3 Stock 4 Stock 5 Stock 6
Stock 1 1 0.289 0.471 0.567 0.561 0.438 Stock 2 1 0.341 0.360 0.399 0.408 Stock 3 1 0.557 0.563 0.335 Stock 4 1 0.483 0.379 Stock 5 1 0.396 Stock 6 1 Table 6
The possibilistic interval and the uncertainty risk
Stock 1 2 3 4 5 6 PI (0.10, 0.18) (0.055, 0.195) (0.02, 0.14) (0.005, 0.205) (0.019, 0.263) (0.09, 0.19) Uncertainty risk 0.08 0.14 0.12 0.231 0.244 0.10 Table 8
Parameter setting in MOEA
Parameter Value
Chromosome Binary
Population size 100
Number of generations 2000
Selection strategy Tournament
Crossover type Uniform
Crossover probability 0.8
Table 9
Portfolio alternatives of the efficient frontier set
Stock 1 2 3 4 5 6 Return rate Uncertainty risk Relation risk
Alternative 1 0.279570 0.116325 0.077224 0.099707 0.203324 0.223851 0.1297 0.1429 0.3762 Alternative 2 0.310850 0.115347 0.077224 0.100684 0.203324 0.192571 0.1303 0.1424 0.3821 Alternative 3 0.371457 0.116325 0.030303 0.099707 0.187683 0.194526 0.1347 0.1379 0.4060 Alternative 4 0.371457 0.124145 0.022483 0.100684 0.187683 0.193548 0.1350 0.1382 0.4063 Alternative 5 0.373412 0.108504 0.053763 0.085044 0.125122 0.254154 0.1356 0.1271 0.4073 .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Alternative 52 0.621701 0.124145 0.022483 0.037146 0 0.194526 0.1476 0.0978 0.5537 Alternative 53 0.623656 0.124145 0.022483 0.037146 0 0.192571 0.1477 0.0978 0.5550 Alternative 54 0.624633 0.107527 0.022483 0.037146 0 0.208211 0.1478 0.0971 0.5595 Alternative 55 0.623656 0.108504 0.022483 0.022483 0 0.222874 0.1487 0.0953 0.5618 Ong et al. / Appl. Math. Comput. 169 (2005) 1195–1210 1205
On the basis ofTable 9, a decision-maker can determine the optimal port-folio alternative based on his preference. Next, we provide the discussion about our numerical example in next section.
5. Discussions
Mean–variance is widely used in the finance area to deal with the portfolio selection problem. However, the conventional method usually fails under the small sample situation. We can describe the shortcomings of the conventional method from its purpose and its theory, respectively, as follows.
The purpose of the mean–variance approach is to determine the t + 1 period optimal investing rate to each security based on the sequent return rate. The key is to forecast the t + 1 period return rate as accurately as possible. How-ever, it is clear that the arithmetic mean only reflects the average states of the past return rate instead of forecasting. Although many regression-based methods have been proposed to overcome the problem, these methods must obey the assumption of the large sample theory and cannot be used in the small sample situation theoretically. In this paper, we propose the grey and possibi-listic regression models to deal with the previously mentioned problem completely.
In order to highlight the shortcoming of the conventional method and to compare it to the proposed method, a numerical example is used. We can de-pict the sequence of the Stock 4 to describe the irrational results using the arith-metic mean as shown inFig. 4.
First, it is clear that the sequence shows the dramatically decreasing trend when the sequence rises to the peak. Second, the possibilistic interval is very large in Stock 4. This characteristic shows the large uncertainty risk in Stock
7 6 5 4 3 2 1 0.175 0.150 0.125 0.100 0.075 0.050 Period Return Rate
Portfolio alternatives using MOEA Stock 1 2 3 4 5 6 Return rate Uncertainty risk Relation risk Alternative 1 0.27957 0.116325 0.077224 0.099707 0.203324 0.223851 0.1297 0.1429 0.3762 Alternative 2 0.31085 0.115347 0.077224 0.100684 0.203324 0.192571 0.1303 0.1424 0.3821 Alternative 3 0.371457 0.116325 0.030303 0.099707 0.187683 0.194526 0.1347 0.1379 0.406 Alternative 4 0.371457 0.124145 0.022483 0.100684 0.187683 0.193548 0.135 0.1382 0.4063 Alternative 5 0.373412 0.108504 0.053763 0.085044 0.125122 0.254154 0.1356 0.1271 0.4073 Alternative 6 0.31085 0.233627 0.049853 0.085044 0.00391 0.316716 0.1357 0.1158 0.4086 Alternative 7 0.373412 0.108504 0.053763 0.069404 0.140762 0.254154 0.1362 0.1273 0.4088 Alternative 8 0.357771 0.233627 0.053763 0.085044 0 0.269795 0.1365 0.1144 0.4115 Alternative 9 0.365591 0.233627 0.053763 0.085044 0 0.261975 0.1366 0.1142 0.4126 Alternative 10 0.373412 0.233627 0.053763 0.085044 0 0.254154 0.1368 0.1141 0.414 Alternative 11 0.343109 0.108504 0.053763 0.052786 0.125122 0.316716 0.1369 0.1235 0.4123 Alternative 12 0.373412 0.249267 0.053763 0.069404 0 0.254154 0.1376 0.1127 0.4165 Alternative 13 0.342131 0.233627 0.053763 0.053763 0 0.316716 0.1381 0.1106 0.4201 Alternative 14 0.342131 0.233627 0.049853 0.053763 0.00391 0.316716 0.1382 0.1111 0.4201 Alternative 15 0.373412 0.077224 0.116325 0.022483 0 0.410557 0.1384 0.1009 0.4663 Alternative 16 0.342131 0.233627 0.038123 0.053763 0.01564 0.316716 0.1387 0.1126 0.4202 Alternative 17 0.342131 0.249267 0.049853 0.038123 0.00391 0.316716 0.139 0.1097 0.4236 Alternative 18 0.343109 0.107527 0.053763 0.053763 0 0.441838 0.1393 0.1056 0.4685 Alternative 19 0.374389 0.108504 0.022483 0.052786 0.125122 0.316716 0.1394 0.1222 0.4285 Alternative 20 0.342131 0.108504 0.049853 0.053763 0.00391 0.441838 0.1395 0.1061 0.468 Alternative 21 0.342131 0.233627 0.049853 0.022483 0.00391 0.347996 0.1401 0.107 0.4336 Alternative 22 0.357771 0.233627 0.053763 0.022483 0 0.332356 0.1402 0.1062 0.4336 Alternative 23 0.373412 0.233627 0.049853 0.022483 0.00391 0.316716 0.1407 0.1064 0.4344 Alternative 24 0.373412 0.077224 0.053763 0.022483 0.062561 0.410557 0.1409 0.1087 0.4653 Alternative 25 0.623656 0.108504 0.147605 0.022483 0 0.097752 0.1412 0.0978 0.5483 Alternative 26 0.373412 0.12219 0.022483 0.037146 0.062561 0.382209 0.1414 0.1117 0.4517 Alternative 27 0.373412 0.100684 0.053763 0.022483 0.00782 0.441838 0.1417 0.1017 0.4824 Alternative 28 0.373412 0.108504 0.053763 0.022483 0 0.441838 0.1418 0.1009 0.4828
(continued on next page)
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Table 10 (continued) Stock 1 2 3 4 5 6 Return rate Uncertainty risk Relation risk Alternative 29 0.373412 0.108504 0.018573 0.053763 0.00391 0.441838 0.142 0.1048 0.4848 Alternative 30 0.373412 0.077224 0.053763 0.022483 0 0.473118 0.1421 0.0996 0.5027 Alternative 31 0.373412 0.124145 0.022483 0.037146 0 0.442815 0.1426 0.1028 0.4855 Alternative 32 0.373412 0.12219 0.022483 0.037146 0 0.44477 0.1427 0.1027 0.4866 Alternative 33 0.373412 0.124145 0.016618 0.038123 0.005865 0.441838 0.1428 0.1037 0.4852 Alternative 34 0.55914 0.092864 0.085044 0.037146 0 0.225806 0.1429 0.0991 0.5092 Alternative 35 0.622678 0.053763 0.092864 0.037146 0 0.193548 0.1441 0.0964 0.5582 Alternative 36 0.498534 0.100684 0.049853 0.022483 0.01173 0.316716 0.1444 0.0997 0.4937 Alternative 37 0.621701 0.124145 0.069404 0.037146 0.01564 0.131965 0.1445 0.101 0.543 Alternative 38 0.561095 0.124145 0.022483 0.068426 0 0.223851 0.1445 0.1032 0.5121 Alternative 39 0.55914 0.116325 0.030303 0.037146 0.062561 0.194526 0.1447 0.108 0.5033 Alternative 40 0.621701 0.053763 0.077224 0.037146 0 0.210166 0.145 0.0961 0.5602 Alternative 41 0.623656 0.053763 0.077224 0.037146 0 0.208211 0.1451 0.0961 0.5615 Alternative 42 0.623656 0.092864 0.069404 0.037146 0 0.176931 0.1452 0.0975 0.5518 Alternative 43 0.621701 0.059629 0.069404 0.037146 0.00391 0.208211 0.1454 0.0968 0.5589 Alternative 44 0.623656 0.108504 0.053763 0.022483 0.062561 0.129032 0.1456 0.1049 0.5463 Alternative 45 0.621701 0.115347 0.030303 0.037146 0.065494 0.13001 0.1459 0.1071 0.5456 Alternative 46 0.622678 0.092864 0.053763 0.037146 0 0.193548 0.1461 0.0972 0.5541 Alternative 47 0.624633 0.100684 0.030303 0.037146 0.062561 0.144673 0.1462 0.106 0.5499 Alternative 48 0.621701 0.124145 0.022483 0.037146 0.064516 0.13001 0.1463 0.1071 0.5457 Alternative 49 0.623656 0.124145 0.022483 0.037146 0.062561 0.13001 0.1464 0.1068 0.5473 Alternative 50 0.623656 0.092864 0.022483 0.037146 0.062561 0.16129 0.1467 0.1056 0.5521 Alternative 51 0.621701 0.116325 0.030303 0.037146 0 0.194526 0.1472 0.0977 0.5533 Alternative 52 0.621701 0.124145 0.022483 0.037146 0 0.194526 0.1476 0.0978 0.5537 Alternative 53 0.623656 0.124145 0.022483 0.037146 0 0.192571 0.1477 0.0978 0.555 Alternative 54 0.624633 0.107527 0.022483 0.037146 0 0.208211 0.1478 0.0971 0.5595 Alternative 55 0.623656 0.108504 0.022483 0.022483 0 0.222874 0.1487 0.0953 0.5618 C.-S. Ong et al. / Appl. Math. Comput. 169 (2005) 1195–1210
4. To summarize the above finding, it is risky to invest too much money in Stock 4 over the next period. On the other hand, the proposed method can accurately reflect this characteristic of Stock 4. On the basis of Table 9 or
Appendix A, we can conclude that the portfolio selection of Stock 4 should not exceed 10 percent.
In addition, the proposed method can provide the more flexible portfolio alternatives. A decision-maker can select his optimal alternative based on the results of the Pareto set. For example, a risk averse may choose the alternative 1 to obtain the excepted return rate 0.1297. However, a risk lover may choose the alternative 55 to obtain the excepted return rate 0.1487 but a higher risk than a risk averse.
6. Conclusions
Portfolio selection problem has been a popular issue in the finance area since the 1950Õs. However, the conventional mean–variance method can not provide the satisfied solution under the small sample situation. In this paper, we pro-pose a hybrid method which incorporates the grey and possibisitic regression models to deal with this situation. In order to resolve the three-objective quad-ric programming, MOEA is employed here. In addition, a numequad-rical example is illustrated to show the procedures of the proposed method. On the basis of the numerical results, the proposed method can provide the more flexible and accu-rate results.
Appendix A. The full portfolio alternatives can be shown as in Table 10.
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