Decision Making of a Portfolio Selection ModelBased on Fuzzy Statistic Test

Pei-Chun LinGraduate School of Information,

Production and SystemWaseda UniversityKitakyushu, Japan

Email: peichunpclin@gmail.com

Junzo WatadaGraduate School of Information,

Production and SystemWaseda UniversityKitakyushu, Japan

Email: watada@waseda.jp

Berlin WuDepartment of Mathematical Sciences

National Chengchi UniversityTaipei, Taiwan

Email: berlin@nccu.edu.tw

Abstract—The objective of our research is to build a statisticaltest that can evaluate the sensitivity of a portfolio selectionmodel with fuzzy data. The central point and radius are used todetermine the portfolio selection model and we make a decisionfor the best return by a fuzzy statistical test. Empirical studiesare presented to illustrate the risk of the portfolio selection modelwith interval values. We conclude that the evaluation by the fuzzystatistical test enables us to obtain a stable expected return andlow risk investment with different choices based on the risk levelk, which is taken for the risk level.Index Terms—Portfolio selection; Optimization; Fuzzy proba-

bility distributions; Fuzzy statistics and data analysis

I. INTRODUCTION

The portfolio selection model has been well developed onthe basis of a mean-variance approach. It was first proposedby Markowitz [14],[15],[16], who combined probability andoptimization theories to analyze the performance of economicagents. The key principle of the mean-variance model is touse the expected return of a portfolio as investment returnand the variance of the expected returns of the portfolioas investment risk. Most of the existing portfolio selectionmodels are based on probability theory. The mean-varianceportfolio selection problem has been studied by Sharpe [23],Merton [17], Perold [21], Pang [19], Voros [27], and Best[2], [3]. To analyze uncertain phenomena in the real world,multivariate data analysis has been applied to investigateportfolio selection problems. For example, Zhang et al. [40]discussed the portfolio selection problem when the returnsof assets are fuzzy numbers. Hasuike et al. [7] discussedtwo portfolio selection problems including probabilistic futureand ambiguous expected returns. Moreover, Hasuike and Ishii[6] discussed a portfolio selection problem with type-2 fuzzyreturns involving interval numbers and considering investor’ssubjectivity.Some other research works discussed how to solve fuzzy

portfolio selection models, as Peng et al. [20] addressedthe portfolio selection problems in fuzzy environments bya credibility programming approach based on a credibilitymeasure (see [13]). Wang et al. [29] proposed a new realoptions analysis approach by combining the binomial lattice-based model with a fuzzy random variable. Zhang et al. [36]

proposed a model to convert the forecasted uncertain valuesinto some normal fuzzy numbers. Tanaka and Guo [24],[25]proposed two types of portfolio selection model based onfuzzy probabilities and exponential possibility distributions,respectively. Wang and Zhu [30], and Lai et al. [8] constructedinterval programming models of portfolio selection. Zhangand Wang [39] and Zhang et al. [38] discussed the portfolioselection problem based on the (crisp) possibilistic mean andvariance when short sales are not allowed at all risky assets.Watada [28], Ramaswamy [22], and Leon et al. [9] discussedportfolio selection using fuzzy decision theory.Most studies did not consider any kind of probability

distribution function with fuzzy random variables. Moreover,no statistical test was applied to examine the results of theportfolio selection model with fuzzy data. In view of thisweakness, the objective of this paper is to develop a statisticaltest to test the results of the portfolio selection model withfuzzy data. First, we deal with the problem of defining thestatistic test with fuzzy data.In the real world, occasionally, vague information is given

when describing data in natural language (see [18]). Whenwe want to work with fuzzy data, the underlying probabilitydistribution of the fuzzy data is not known. It is not easy todescribe such information in terms of statistics. Therefore, wemust establish some techniques to handle such informationand knowledge. Following Zadeh [34] [35], we will usefuzzy set theory and take the concept of fuzzy statisticsinto consideration. Fundamental statistical measurements suchas mean, median and mode are useful for illustrating somecharacteristics of a sample distribution. More research worksshould focus on fuzzy statistical analysis and its applicationsin engineering, medical and social science. Wu and Cheng [31]identified a model structure through qualitative simulation;Casalino et al. [4], Esogbue and Song [5], and Wu and Sun[32] demonstrated the concepts of fuzzy statistics and appliedthem to social surveys. Wu and Tseng [33] used the fuzzyregression method of coefficient estimation to analyze theTaiwan monitoring index of economics. All of the above-mentioned studies dealt with problems by using the centralpoint values. Lin et al. [10] proposed a new weight functionwith fuzzy numbers defined by the central point and radius.

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Moreover, Lin et al. [11] proposed a method to recognizethe underlying distribution function using its central point andradius, which gives us more information about the originalfuzzy data.The objective of this paper is to build a statistical test of

fuzzy data and apply it to a portfolio selection model withinterval values and statistically evaluate decision for the bestreturn. In the first step, we need to find out the probabilitydistribution function and each parameter in the probabilitydistribution functions. When we know the distribution functionof each parameter, we can easily calculate the expected returnand variance. Those values can enable us to define a portfolioselection model with interval values. We also give a decisionby a fuzzy statistical test, which explicitly tells us whether ornot we statistically accept the risk in investment.The rest of the paper consists of the following. Section 2

gives a brief review of the related studies. The main methodis described in Section 3. Section 4 illustrates the empiricalstudies with interval values and how to use a fuzzy statisticaltest on the portfolio selection model. Finally, the concludingremarks and the topics of further studies are presented inSection 5.

II. NOTATIONS AND PRELIMINARY DEFINITIONSA. Fuzzy NumbersFuzzy set theory was proposed by Zadeh [34] to deal with

the vagueness in data. The following definitions will be usedin this context.Definition 1 Fuzzy Sample Mean (data with interval values)[18]Let U be the universe set and {Fi = [ai, bi], ai, bi ∈

�, i = 1, · · · , n} be a sequence of a random fuzzy sampleon U . Then, the fuzzy sample mean value is defined asF̄ = [ 1

n

∑n

i=1 ai,1n

∑n

i=1 bi].Definition 2 [10] An interval value is denoted as F = [a, b]with a central point o = a+b

2and radius l = b−a

2. We give the

notation as F ≡ (o, l).When we have fuzzy numbers, we need statistical methods

to deal with the data. We call that a fuzzy statistical testmeans a statistical test which can deal with fuzzy data. First,we introduce a traditional statistical test in the followingsubsection.

B. Statistical AnalysisLet X1, · · · , Xn be a sequence of random variables (not

necessarily normally distributed). We say that the Xi are in-dependently identically distributed (i.i.d) if theXi are indepen-dent and have the same distribution. We writeWn =

∑n

i=1 Xi

and X̄n = Wn

nto denote the total and average, respectively,

of the nX ′is. We introduce the most important theorem in

statistics as follows.Theorem 1 Central Limit Theorem [1]Let X1, · · · , Xn be i.i.d. random variables with mean m

and variance σ2. Let

Zn =n

1

2 (X̄n −m)

σ=

Wn − nm

n1

2 σ.

Then, Zn converges in distribution to Z as n→∞. We denoteZn → Z ∼ N(0, 1) as n→∞,

where Z distributes according to a standard normal distributionfunction N(0, 1).Note that m denotes mean and μ expresses a membership

function.In statistics, functions of observations are often important.

Therefore, we introduce the following statistical test.Theorem 2 T -test [26]For samples X1, X2, · · · , Xn of a normally distributed

stochastic quantity X ∼ N(m,σ2), a statistic for the hypoth-esis H0 : m = m0 is

T =X̄n −m0

Sn√n

, (3)

and the acceptance region A for T under probability α for anerror of the first type is

A = {t ∈ �| | t |=| X̄n −m0

Sn√n

|≤ tn−1;1−α2}, (4)

where X̄n and Sn are the sample mean and the samplestandard deviation of the data, m0 is a specified value, nthe sample size and tn−1;1−α

2the (1 − α

2)-fractile of the t-

distribution with degrees of freedom n− 1, respectively.When the above information is given, we have provided all

prior knowledge for dealing with portfolio selection problem.Now, we give a brief description of the portfolio selectionmodel in the following subsection.

C. Markowitz’s Portfolio Selection ModelMarkowitz’s mean-variance model is based on a probability

distribution in which uncertainty is equated with randomness[14] [15] [16]. That is, the return on the ith asset, ri, will beregarded as a random variable.Consider a market with n risky assets. An investor’s po-

sition in this market is described by a portfolio vector x =[x1, x2, · · · , xn]

′, where the ith component xi represents theproportion invested in asset i. The return vector on portfoliovector x is denoted r = [r1, r2, · · · , rn]′, where ri representsthe return rate of asset i. In the conventional mean-variancemethodology for portfolio selection, ri is regarded as a randomvariable, ∀i = 1, 2, · · · , n. Let r̄ = [r̄1, r̄2, · · · , r̄n]′ andV = [σij ]n×n be the expected return vector and covariancematrix, respectively. The return R on the portfolio x is givenby R =

∑n

i=1 rixi. Set I = [1, 1, · · · , 1]′. The objective ofthe investor is to choose a portfolio that maximizes the returnon the investment subject to some constraints on the risk ofthe investment.A mean-variance model for portfolio selection can be for-

mulated mathematically as

max r̄x

s.t.√x′Vx ≤ ε

I′x ≤ 1

xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎬⎪⎪⎭

, (5)

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where ε(ε ≥ 0) represents the risk level, x′Vx =∑n

i=1 σ2iix

2i +

∑n

j=1

∑n

i=1,i�=j σijxixj , σij = cov(ai, aj) isthe covariance, and ai and aj are random variables, ∀i, j =1, 2, · · · , n.It means that we need to solve the following programmingproblem:

max∑n

i=1 E(ri)xi

s.t.√x′Vx ≤ ε∑n

i=1 xi ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎬⎪⎪⎭

. (6)

We will introduce the fuzzy statistical test on portfolio selec-tion model in the following section.

III. FUZZY STATISTICAL TEST FOR THE PORTFOLIOSELECTION MODEL

Assume that there are n distinct tradable assets in themarket. The terminal rate of return for asset i, denoted as ri,∀i = 1, 2, · · · , n, is assumed to be a fuzzy random variable.By the widely accepted definition, the expectation of a fuzzyrandom variable is a fuzzy variable. We give some definitionsto express the total fuzzy return R on a portfolio vector x,where x = [x1, x2, · · · , xn]

′ is a n row vector.

A. Portfolio Selection Model with Interval Values

Let Fi = [ai, bi] ≡ (oi, li), ∀i = 1, 2, · · · , n, be intervalvalues on the probability space Ω, where oi is a randomvariable with central point of Fi, and li is a random variablewith the radius of Fi, ∀i = 1, 2, · · · , n.For the proportion invested in asset i, ∀i = 1, 2, · · · , n, we

have x = [x1, x2, · · · , xn]′. The fuzzy expected return of Fi,

i.e. Fi ≡ (oi, li), ∀i = 1, 2, · · · , n, was defined as follows.Definition 3 Fuzzy Expected Return (data with interval values)

E[R(x)] ≡n∑

i=1

E(Fi)xi = (n∑

i=1

E(oi)xi,

n∑i=1

E(li)xi).

Note that R[x] ≡∑n

i=1 Fixi, ∀i = 1, 2, · · · , n.Moreover, the fuzzy portfolio variance of Fi, ∀i =

1, 2, · · · , n, was defined as follows.Definition 4 Fuzzy Portfolio Variance (data with intervalvalues)

var[R(x)] ≡n∑

i=1

var(Fi)xi = (var(

n∑i=1

oixi), var(

n∑i=1

lixi)),

wherevar(

∑n

i=1 oixi) =∑n

i=1 σ2oii

x2i +

∑n

j=1

∑n

i=1,i�=j σoijxixj ,σoij = cov(oi, oj), and var(

∑n

i=1 lixi) =∑n

i=1 σ2liix2i +∑n

j=1

∑n

i=1,i�=j σlijxixj , σlij = cov(li, lj), ∀i, j =1, 2, · · · , n.Model 1 Portfolio Selection Model with Interval ValuesLet Fi ≡ (oi, li), ∀i = 1, 2, · · · , n, be interval values. The

portfolio selection model with interval values is described as

follows:max

∑n

i=1 E(oi)xi

min∑n

i=1 E(li)xi

s.t. x′Vox ≤ ε2

x′Vlx ≤ ε2

n∑i=1

xi ≤ 1

xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (7)

where ε (ε ≥ 0) represents the risk level, x′Vox =∑n

i=1 σ2oii

x2i +

∑n

j=1

∑n

i=1,i�=j σoijxixj , σoij = cov(oi, oj),and x

′Vlx =

∑n

i=1 σ2liix2i +

∑n

j=1

∑n

i=1,i�=j σlijxixj , σij=

cov(li, lj), ∀i, j = 1, 2, · · · , n.To reduce the calculation, we rewrite formula (7) clearly in

the following formula:max

∑n

i=1 E(oi)xi

min∑n

i=1 E(li)xi

s.t.∑n

i=1

√var(oi)xi = k∑n

i=1

√var(li)xi ≤ k∑n

i=1 xi ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(8)

Note that becausex′Vox =

∑n

i=1 σ2oii

x2i +

∑n

j=1

∑n

i=1,i�=j σoijxixj ≤(∑n

i=1 σoiixi)2 ≤ ε2, we set (

∑n

i=1 σoiixi)2 = k1

2, i.e.,∑n

i=1 σoiixi = k1, where σoii =√var(oi). We can obtain

another restriction in the same way, i.e.,∑n

i=1 σliixi = k2,where σlii =

√var(li). We can choose an k such that∑n

i=1

√var(oi)xi = k1 = k and

∑n

i=1

√var(li)xi = k2 ≤

k.Therefore, we can obtain the optimal solution of the model

1 in (9) by choosing different values for k (k ≥ 0), whichindicates the acceptable risk level. We obtain the optimalsolution vector x∗ = [x∗1, x

∗2, · · · , x∗n]′.

Moreover, when we obtain the optimal solution vector x∗ =[x∗1, x

∗2, · · · , x∗n]′, we can calculate the fuzzy expected return

E(R(x∗)):

E[R(x∗)] ≡ (

n∑i=1

E(oi)x∗i ,

n∑i=1

E(li)x∗i ).

B. Decision Making of Portfolio Selection Model by FuzzyStatistical Test

Definition 5 T -test with interval valuesFor interval values Fi = [ai, bi] ≡ (oi, li), samples

o1, o2, · · · , on can be approximated as a normally distributedstochastic quantity O ∼ N(mo, σ

2o) by Theorem 1, and the

samples l1, l2, · · · , ln can be approximated as a normallydistributed stochastic quantity L ∼ N(ml, σ

2l ) by Theorem 1.

Hence, the bivariate normal distribution of the 2-dimensionalrandom vector F can be written in the notation F ∼ N(m,Σ),wherem ≡ (mo,ml) is a mean vector andΣ = [cov(Xo, Xl)]is a 2× 2 covariance matrix.In fact, Σ =

[σ2o 00 σ2

l

]because o and l are independent.

Statistics for each hypothesis are as follows.

H0 : m = m0, for statistic vector T ≡ (To, Tl), (9)

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where m0 ≡ (mo0 ,ml0) is a specified vector, To =X̄on −mo0

Son√n

and Tl =X̄ln −ml0

Sln√n

.

Note that, here mo0 =∑n

i=1 E(oi)x∗ and ml0 =∑n

i=1 E(li)x∗. We set m0 = E[R(x∗)] in this paper.

The acceptance region A for T under probability α for anerror of the first type is defined as

A = Ao ×Al. (10)

Note that

Ao = {to ∈ �| | to |=| X̄on −mo0

Son√n

|≤ tn−1;1−α2}, and

(11)

Al = {tl ∈ �| | tl |=| X̄ln −ml0

Sln√n

|≤ tn−1;1−α2}, (12)

where X̄on is the sample mean of o, Son is the sample standarddeviation of o, X̄ln is the sample mean of l, Sln is the samplestandard deviation of l, n is the sample size and tn−1;1−α

2is

the (1− α2)-fractile of the t-distribution with n− 1 degrees of

freedom.Definition 6 Fuzzy Statistical Test on the Portfolio SelectionModel with Interval ValuesAccording to Definition 6 and Formula (8) in Model 1, we

say that if m0 ∈A, then we do not reject the null hypothesisH0. In this situation, the decision of K is

K = {k|k ≥ 0 such that m0 ≡ (mo0 ,ml0) ∈A,

we do not reject H0}.Note that m0 ∈A means that mo0 ∈ Ao and ml0 ∈ Al.Hence, we obtain the solution of Model 1 with different k

and get a set K .

C. Procedure of Solving Portfolio Selection Model with Inter-val Values by Fuzzy Statistic TestThe procedure can be written in the following to solve

portfolio selection model with interval values by a fuzzystatistical test:Step 1. Collect the fuzzy data and calculate the return of eachexchange currency with interval values.Step 2. Compute oi and li, ∀i = 1, 2, · · · , n.Step 3. Identify the underlying distribution by simulating oiand li. [12]Step 4. Calculate the parameters for the expected value andvariance in model (8). [12]Step 5. Solve the optimization model (8) with different valuesof k.Step 6. Calculate K by Definition 6.Step 7. Obtain the optimal vector solution x∗ with different kin K .Step 8. Compute the possibility distribution of the fuzzyexpected return R(x∗) with different k in K .To illustrate our proposed effective meanings and ap-

proaches to obtain efficient portfolios, we exemplify a realportfolio selection problem in the following section.

IV. EMPIRICAL STUDIESExample 1 We selected five exchange currencies (USD,

EUR, AUD, GBP and CHF) from the Bank of Tokyo-Mitsubishi. The original data came from the closing prices ofevery day from July 2010 to December 2010. There were 124interval values in this period [a, b], where a is the minimumprice, and b is the maximum price in one day. We use the datain Reference [12] and present the simulation results in Table1.

TABLE IPARAMETERS OF PROBABILITY DISTRIBUTION FUNCTIONS FOR

INTERVAL VALUES

O L

USD LOG(−4.23, 1.21) W (0.45, 3.26)EUR LOG(4.44, 1.33) Γ(7.69, 0.08)AUD LOG(6.03, 1.56) Γ(7.51, 0.07)GBP N(0.80, 2.412) Γ(9.87, 0.06)CHF W (3.01, 1.58) Γ(9.99, 0.05)

Note that the abbreviation LOG denotes logistic distri-bution, W , N , and Γ denote Weibull distribution, normaldistribution and gamma distribution, respectively.When we knew the distribution function, we could find out

the expected values and variances by using those parameters.Table 2 shows the results.

TABLE IIEXPECTED VALUES AND VARIANCES FOR INTERVAL VALUES

O1 O2 O3 O4 O5

Expected value -4.23 4.44 6.03 0.80 2.70Variance 4.79 5.78 8.03 2.412 3.06

L1 L2 L3 L4 L5

Expected value 0.40 0.62 0.53 0.59 0.50Variance 0.02 0.05 0.04 0.04 0.02

Now, we have all the data that we need in our portfolioselection model with interval values. We put these data inModel (8) and present the results as follows:max −4.23x1 + 4.44x2 + 6.03x3 + 0.80x4 + 2.70x5min 0.40x1 + 0.62x2 + 0.53x3 + 0.59x4 + 0.50x5

s.t.√4.79x1 +

√5.78x2 +

√8.03x3 +

√5.81x4

+√3.06x5 = k√

0.02x1 +√0.05x2 +

√0.04x3 +

√0.04x4

+√0.02x5 ≤ k

x1 + x2 + x3 + x4 + x5 ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

.

(13)We rewrite the model with estimated parameters as follows:max −4.23x1 + 4.44x2 + 6.03x3 + 0.80x4 + 2.70x5min 0.40x1 + 0.62x2 + 0.53x3 + 0.59x4 + 0.50x5

s.t. 2.19x1 + 2.40x2 + 2.83x3 + 2.41x4 + 1.75x5 = k

0.14x1 + 0.22x2 + 0.20x3 + 0.20x4 + 0.14x5 ≤ k

x1 + x2 + x3 + x4 + x5 ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

.

(14)

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We solved Model (14) by using GP-IGP (Linear and IntegerGoal Programming). The result depends on the selection withdifferent values of k. We present the result in Table 3.

Explanation of decision with T -test

In Table 3, for example, when k = 2.5, we have x∗ =

[0.52, 0, 0.48, 0, 0]′. We calculated the return

R =

n∑i=1

rixi = r1x1 + r2x2 + r3x3 + r4x4 + r5x5

= 0.52 ∗ x1 + 0.48 ∗ x3.

Therefore, we obtained 124 new data points.We calculated the expected valuem and standard deviation

Sn by Minitab15. The results arem ≡ (0.68, 0.45) and Sn ≡(1.05, 0.13). The hypothesis was H0 : m = (0.74, 0.46), forstatistic vector T ≡ (To, Tl), where To = |0.68− 0.74

1.05√124

| =

0.55 and Tl = |0.45− 0.460.13√124

| = 0.17. The 95 percent confi-

dence interval was [0.49, 0.87]and [0.43, 0.48], respectively.Becasuse (0.74, 0.46) ∈ [0.49, 0.87] × [0.43, 0.48], we

accepted the hypothesis. Note that we say (0.74, 0.46) ∈[0.49, 0.87]× [0.43, 0.48]; this means that 0.74 ∈ [0.49, 0.87]and 0.46 ∈ [0.43, 0.48].Hence, we obtained the fuzzy expected return

E[R(x∗)] ≡ (

n∑i=1

E(oi)x∗i ,

n∑i=1

E(li)x∗i )

= (0.74, 0.46). (15)

The interval value of the fuzzy expected return was[0.28, 1.20].Using the same method for the other values of k, we can also

obtain the other interval value of the fuzzy expected return.We present the results in Table 4.In Table 4, we can see that we accepted the hypothesis

for k ≤ 2.83 and that we had a stable return for k ≥ 2.81because we had the same expected return of radius. Moreover,we cannot solve model (14) for k > 2.83. We obtained anegative return for k ≤ 2 and the maximum return for k =2.83.In this example, we conclude that the maximum fuzzy

expected return was [5.50, 6.56].

Example 2 In [37], Zhang implemented the concept ofthe γ-level to deal with the optimization model. He alsoadopted an additional condition of the upper and lowerbounds in the model proposed by Markowitz. We applied hismodel to solve our problem but deleted the upper bound andlower bound of his example.First, we calculated the expected value of 124 inter-

val values by Definition 1. We obtained 5 interval num-bers as follows: USD=r1 = [−4.56,−3.73], EUR=r2 =

[3.59, 4.89], AUD=r3 = [5.41, 6.43], GBP=r4 = [0.17, 1.44],and CHF=r5 = [2.05, 3.06].Hence, the lower possibilistic mean-standard deviation

model wasmax −4.56x1 + 3.59x2 + 5.41x3 + 0.17x4 + 2.05x5

s.t. x1 + x2 + x3 + x4 + x5 ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎬⎭ ,

(16)and the upper possibilistic mean-standard deviation model was

max −3.73x1 + 4.89x2 + 6.43x3 + 1.44x4 + 3.06x5

s.t. x1 + x2 + x3 + x4 + x5 ≤ 1xi ≥ 0 ∀i = 1, 2, · · · , n

⎫⎬⎭ .

(17)Table 5 shows the results, where L∗ denotes the optimalsolution in model (16), and U∗ denotes as the optimal solutionin model (17).From Table 5, we can see that when we choose

∑5

i=1 xi = 1,the optimal solution of model (16) and (17) are the same asx∗ = [0, 0, 1, 0, 0]′.Hence, we obtained the fuzzy expected return

E[R(x∗)] ≡ (

n∑i=1

E(oi)x∗i ,

n∑i=1

E(li)x∗i )

= (5.92, 0.51). (18)

The interval value of the fuzzy expected return was[5.41, 6.43].

V. DISCUSSIONSIn this paper, we proposed a method to defuzzify fuzzy

data. Therefore, we can estimate the probability distributionfunction with fuzzy data. When we found the probabilitydistribution function with fuzzy data, we could calculate theexpected value, variance and covariance by using the estimatedparameters of the underlying distribution function. Then, wecould solve the portfolio selection model and evaluate it byfuzzy statistical test.An empirical study of a portfolio selection model was

conducted based on a fuzzy statistical test in example 1.Example 1 showed that our method is suitable for short-terminvestments because the largest k decides the largest and stablereturn. Moreover, when being smaller than 2, k might giveus a negative profit. Although in Example 1, the decisionsare acceptable for all k, we need to consider situations inthe real world. We do not want to buy the negative returnwhen k ≤ 2. Example 1 also showed that our method of thefuzzy statistical test can indicate whether or not the underlyingprobability distribution function and the estimated parametersare suitable. Moreover, the statistical test can also check thevalidity of the model in (14).Example 2, indicated to simply buy the single exchange

currency with the highest interval value. The authors took theleft and right points of the interval values to build a separatemodel, thus obtaining the maximum return. The results showedthat Zhang obtain the maximum return with one exchange

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TABLE IIIFUZZY STATISTICAL TEST FOR THE RESULTS OF MODEL (14) AND (15) WITH DIFFERENT CONDITIONS k

k 0.5 1 1.5 2 2.5 2.6∑

5

i=1xi 0.23 0.46 0.68 0.91 1 1

x∗ [.23, 0, 0, 0, 0]′ [.46, 0, 0, 0, 0]′ [.68, 0, 0, 0, 0]′ [.91, 0, 0, 0, 0]′ [.52, 0, .48, 0, 0]′ [.36, 0, .64, 0, 0]′

m0 (-0.97, 0.09) (-1.93, 0.18) (-2.90, 0.27) (-3.86, 0.37) (0.74, 0.46) (2.34, 0.48)m (-0.95, 0.09) (-1.90, 0.18) (-2.81, 0.28) (-3.77, 0.37) (0.68, 0.45) (2.29, 0.47)Sn (0.47, 0.03) (0.95, 0.07) (1.41, 0.11) (1.89, 0.15) (1.05, 0.13) (1.36, 0.14)A [-1.03, -0.86] [-2.07, -1.73] [-3.06, -2.56] [-4.10, -3.43] [0.49, 0.87] [2.05, 2.54]

×[0.08, 0.10] ×[0.17, 0.20] ×[0.25, 0.30] ×[0.34, 0.40] ×[0.43, 0.48] ×[0.44, 0.49]Decision m0 ∈A m0 ∈A m0 ∈ A m0 ∈A m0 ∈A m0 ∈ A

k 2.7 2.8 2.81 2.82 2.83∑

5

i=1xi 1 1 1 1 1

x∗ [.2, 0, .8, 0, 0]′ [.05, 0, .95, 0, 0]′ [.03, 0, .97, 0, 0]′ [.02, 0, .98, 0, 0]′ [0, 0, 1, 0, 0]′

m0 (3.95, 0.50) (5.55, 0.52) (5.71, 0.53) (5.87, 0.53) (6.03, 0.53)m (3.90, 0.48) (5.41, 0.50) (5.61, 0.50) (5.71, 0.50) (5.92, 0.50)Sn (1.87, 0.15) (2.42, 0.17) (2.49, 0.18) (2.53, 0.18) (2.61, 0.18)A [3.57, 4.24] [4.98, 5.84] [5.17, 6.06] [5.26, 6.17] [5.45, 6.38]

×[0.46, 0.51] ×[0.47, 0.53] ×[0.47, 0.53] ×[0.47, 0.53] ×[0.47, 0.54]Decision m0 ∈A m0 ∈A m0 ∈ A m0 ∈A m0 ∈A

TABLE IVTHE FUZZY EXPECTED RETURN WITH DIFFERENT CONDITIONS k

k 0.5 1 1.5 2 2.5 2.6x∗ [.23, 0, 0, 0, 0]′ [.46, 0, 0, 0, 0]′ [.68, 0, 0, 0, 0]′ [.91, 0, 0, 0, 0]′ [.52, 0, .48, 0, 0]′ [.36, 0, .64, 0, 0]′

E[R(x∗)] (-0.97, 0.09) (-1.93, 0.18) (-2.90, 0.27) (-3.86, 0.37) (0.74, 0.46) (2.34, 0.48)Interval Value [-1.06, -0.88] [-2.11, -1.75] [-3.17, -2.63] [-4.23, -3.49] [0.28, 1.20] [1.86, 2.82]

k 2.7 2.8 2.81 2.82 2.83x∗ [.2, 0, .8, 0, 0]′ [.05, 0, .95, 0, 0]′ [.03, 0, .97, 0, 0]′ [.02, 0, .98, 0, 0]′ [0, 0, 1, 0, 0]′

E[R(x∗)] (3.95, 0.50) (5.55, 0.52) (5.71, 0.53) (5.87, 0.53) (6.03, 0.53)Interval Value [3.45, 4.45] [5.03, 6.07] [5.18, 5.70] [5.34, 6.40] [5.50, 6.56]

TABLE VFUZZY STATISTICAL TEST FOR THE RESULTS OF MODEL (17) AND (18) WITH DIFFERENT CONDITIONS

∑5

i=1xi 0.23 0.46 0.68 0.91 1

x∗ [0, 0, 0.23, 0, 0]′ [0, 0, 0.46, 0, 0]′ [0, 0, 0.68, 0, 0]′ [0, 0, 0.91, 0, 0]′ [0, 0, 1, 0, 0]′

[L∗, U∗] [1.24, 1.48] [2.49, 2.96] [3.68, 4.37] [4.92, 5.85] [5.41, 6.43]m0 (1.36, 0.12) (2.725, 0.235) (4.025, 0.345) (5.385, 0.465) (5.92, 0.51)m (1.36, 0.12) (2.72, 0.23) (4.02, 0.35) (5.39, 0.46) (5.92, 0.51)Sn (0.60, 0.04) (1.20, 0.09) (1.78, 0.13) (2.38, 0.17) (2.61, 0.19)A [1.26, 1.47] [2.51, 2.94] [3.71, 4.34] [4.97, 5.81] [5.46, 6.38]

×[0.11, 0.12] ×[0.22, 0.25] ×[0.32, 0.37] ×[0.43, 0.49] ×[0.47, 0.54]Decision m0 ∈A m0 ∈ A m0 ∈ A m0 ∈A m0 ∈ A

currency. Although we make decision by a fuzzy statistical testin this example and accepted all the solution [L∗, U∗] underdifferent conditions, we concluded that whereas there was noproblem in calculating the interval data, the model has someproblems for portfolio selection because one simply buys thehighest exchange currency.

In both this two examples, we can see that our model cangive a return larger than Zhang’s model and also can give apositive risk for people to consider whether or not they have tobuy this asset. Moreover, the fuzzy statistic test also providesa reasonable results in our model.

VI. CONCLUSIONS

The empirical studies showed that the estimated methodis more explicit and meaningful to use for the probabilitydistribution function and the fuzzy statistical test can statis-tically evaluate decisions for the best return. We can thereforeestimate the maximum return by considering more informationbased on vagueness, which comes from fuzzy data. We havemany choices depending on k to decide which expected returnis better and to obtain a maximum return and a lower risklevel in our model. This method can also provide not only onechoice of portfolio selection, but also two or more choices for

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decision makers.We have further points to improve in the future as follows:1) In this paper, we used the estimated method to find theunderlying probability distribution function. It is difficultto find the underlying probability distribution functionbecause we cannot always find out the underlying dis-tribution function based on the known distribution. Weneed to find a new distribution function. Hence, we candetermine the data that we need in the model.

2) We solve the portfolio selection model and statisticallyevaluate decision of the best return by a fuzzy statisticaltest. We can know that whether or not the estimatedparameters and the results in our model are correct.However, we cannot buy the asset based solely on thefuzzy statistical test. We need to consider the literaturedata, situations in the real world, psychology, politics,among other factors.

3) Moreover, if we can obtain the triangular fuzzy numbersor trapezoid fuzzy numbers, it would be useful for usto examine the portfolio selection model by a fuzzystatistical test with these numbers, which we proposeto be a more explicit and easier method for decisionmakers to deal with fuzzy data.

REFERENCES[1] S. F. Arnold. Mathematical Statistics. Prentice-Hall, Englewood Cliffs,

NJ, 1990.[2] M.J. Best and R. R. Grauer. The efficient set mathematics when mean-

variance problems are subject to general linear constrains. Journal ofEconomics and Business, Vol.42, No.2, pp.105-120, 1990.

[3] M.J. Best and J. Hlouskova. The efficient frontier for bounded assets.Mathematical Methods of Operations Research, Vol.52, No.2, pp.195-212, 2000.

[4] G. Casalino, F. Memola and C. Minutolo. A model for evaluation oflaser welding efficiency and quality using an artificial neural networkand fuzzy logic. Proceedings of the Institution of Mechanical Engineers,Vol.218, No.6, pp.641-646, 2004.

[5] A. O. Esogbue and Q. Song. On optimal defuzzification and learningalgorithms: theory and applications. Fuzzy Optimization and DecisionMaking, Vol.2, No.4, pp.283-296, 2003.

[6] T. Hasuike and H. Ishii, A Portfolio Selection Problem with Type-2Fuzzy Return Based on Possibility Measure and Interval Programming,Proceedings of FUZZ-IEEE International conference on Fuzzy Systems,pp. 267-272, 2009.

[7] T. Hasuike, H. Katagiri and H. Ishii, Portfolio Selection Problems withRandom Fuzzy Variable Returns, Fuzzy Sets and systems, Vol. 160, No.18, pp. 2579-2596, 2009.

[8] K.K. Lai, S.Y. Wang, J.P. Xu, S.S. Zhu and Y. Fang, A class of linearinterval programming problems and its application to portfolio selection,IEEE Transactions on Fuzzy Systems, Vol.10, No.6, pp.698-704, 2002.

[9] T. Leon, V. Liem and E. Vercher. Viability of infeasible portfolioselection problems: A fuzzy approach. European Journal of OperationalResearch, Vol.139, No.1, pp.178-189, 2002.

[10] P.-C. Lin, B. Wu and J. Watada. Kolmogorov-Smirnov Two Sample Testwith Continuous Fuzzy Data. Integrated Uncertainty Management andApplications, Vol. 68, pp. 175-186, 2010.

[11] P.-C. Lin, J. Watada and B. Wu. A Database For A New Fuzzy Prob-ability Distribution Function and Its Application. International Journalof Innovative Management, Information and Production, Vol.2, No.2,pp.1-7, 2011.

[12] P.-C. Lin, J. Watada and B. Wu. Fuzzy Portfolio Selection Model withInterval Values Base on Probability Distribution Functions. InternationalJournal of Innovative Computing, Information and Control, Vol.8, No.8,August 2012.

[13] B. Liu, A Survey of Credibility Theory, Fuzzy Optimization and Deci-sion Making, Vol.5, No.4, pp.387-408, 2006.

[14] H. Markowitz. Portfolio selection. Journal of Finance, Vol.7, No.1,pp.77-91, 1952.

[15] H. Markowitz. Portfolio selection: Efficient Diversification of Invest-ments, Wiley, New York, 1952.

[16] H. Markowitz. Analysis in Portfolio Choice and Capital Markets, BasilBlackwell, Oxford, 1987.

[17] R.C. Merton. An analytic derivation of the efficient frontier. Journal ofFinance and Quantitative Anaysis, Vol.7, No.4, pp.1851-1872, 1972.

[18] H. Nguyen and B. Wu. Fundamentals of Statistics with Fuzzy Data,Springer, Netherlands, 2006.

[19] J.S. Pang. A new efficient Algorithm for a class of portfolio selectionproblems. Operational Research, Vol.28, No.3, pp.754-767, 1980.

[20] J. Peng, H.M.K. Mok and Wai-Man Tse, Credibility Programming Ap-proach to Fuzzy Portfolio Selection Problems, Proceedings of IEEE In-ternational Conference on Machine Learning and Cybernetics, pp.2523-2528, 2005.

[21] A.F. Perold. Large-scale portfolio optimization. Management Science,Vol.30, No.10, pp.1143-1160, 1984.

[22] S. Ramaswamy, Portfolio Selection Using Fuzzy Decision Theory, Bankfor International Settlements Monetary and Economic Dept., Basle,Switzerland, 1998.

[23] W. F. Sharpe. Portfolio Theory and Capital Markets, McGraw-Hill, NewYork, 1970.

[24] H. Tanaka and P. Guo. Portfolio selection based on upper and lowerexponential possibility distributions. European Journal of OperationalResearch, Vol.114, No.1, pp.115-126, 1999.

[25] H. Tanaka, P. Guo and I. B. Turksen. Portfolio selection based onfuzzy probabilities and possibility distributions. Fuzzy Sets and Systems,Vol.111, No.3, pp.387-397, 2000.

[26] R. Vertl. Statistical Methods for Non-Precise DATA, CRC, NewYork,1995.

[27] J. Voros. Portfolio analysis-An analytic derivation of the efficient port-folio frontier. European Journal of Operational Research, Vol.23, No.3,pp.294-300, 1986.

[28] J. Watada. Fuzzy Portfolio Selection and Its Applications to DecisionMaking. Tatra Mountains Mathematical Publication, Vol. 13, pp.219-248, 1997.

[29] B. Wang, S. Wang and J. Watada, Fuzzy portfolio selection based onvalue-at-risk, IEEE Transactions on Fuzzy Systems, Vol.19, No.4, pp.758 -769, 2011.

[30] S.Y. Wang and S. S. Zhu. On fuzzy portfolio selection problems. FuzzyOptimization and Decision Making, Vol.1, No.4, pp.361-377, 2002.

[31] B. Wu and M. Chen. Use of fuzzy statistical technique in changeperiods detection of nonlinear time series. Applied Mathematics andComputation, Vol.99, No.2-3, pp.241-254, 1999.

[32] B. Wu and C. Sun. Interval-valued statistics, fuzzy logic, and their usein computational semantics. Journal of Intelligent and Fuzzy Systems,Vol.11, No.1-2, pp.1-7, 2001.

[33] B. Wu and N. Tseng. A new approach to fuzzy regression models withapplication to business cycle analysis. Fuzzy Sets and System, Vol.130,No.1, pp.33-42, 2002.

[34] L.A. Zadeh. Fuzzy Sets. Information and Control, Vol.8, no.3, pp.338-353, 1965.

[35] L.A. Zadeh. Outline of a new approach to the analysis of complexsystems and decision processes. IEEE Trans. Syst. Man Cybern, SMC-3,no.1, pp.28-44, 1973.

[36] H. Zhang, P.-C. Lin, B. Wu, and J. Watada. Fuzzy Valuation in RealOptions Analysis. The 25th Fuzzy System Symposium (FSS2009) En-glish Section: Decision Making under Uncertain Environment, Tsukuba,Japan, July 14-16, 2009.

[37] W.G. Zhang. Possibilistic mean-standard deviation models to portfolioselection for bounded assets. Applied Mathematics and Computation,Vol.189, No.1, pp.1614-1623, 2007.

[38] W.G. Zhang, W. A. Liu, Y. L. Wang. A class of possibilistic portfolioselection models and algorithms. Lecture Notes in Computer Science,Vol.3828, pp.464-472, 2005.

[39] W.G. Zhang, Y. L. Wang. Using fuzzy possibilitic mean and variancein portfolio selection model. Lecture Notes in Artificial Intelligence,Vol.3801, pp.291-296, 2005.

[40] W.-G. Zhang, Q.-M. Zhang and Z.-K. Nie, A Class of Fuzzy PortfolioSelection Problems, Proceedings of IEEE International Conference onMachine Learning and Cybernetics, pp. 2654 - 2658, 2003.

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