ArticleMoments and Semi-Moments for fuzzy portfolios selection

7/29/2019 ArticleMoments and Semi-Moments for fuzzy portfolios selection

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Moments and Semi-Moments for fuzzy

portfolios selection

Louis Aime FonoDepartement de Mathematiques et Informatique, Faculte des Sciences,

Universite de Douala, B.P. 24157 Douala, Cameroun.

Jules Sadefo Kamdem

LAMETA CNRS UMR 5474 et Faculte dEconomieUniversite de Montpellier 1, France.

Christian Deffo TassakLaboratoire de Mathematiques Appliquees aux Sciences Sociales

Departement de Mathematiques - Faculte des SciencesUniversity of Yaounde I, Cameroon

February 18, 2011

Abstract

The aim of this paper is to consider the moments and the semi-moments (i.e semi-kurtosis) for portfolio selection with fuzzy risk fac-tors (i.e. trapezoidal risk factors). In order to measure the lep-tokurtocity of fuzzy portfolio return, notions of moments (i.e. Kur-tosis) kurtosis and semi-moments(i.e. Semi-kurtosis) for fuzzy port-

folios are originally introduced in this paper, and their mathematicalproperties are studied. As an extension of the mean-semivariance-skewness model for fuzzy portfolio, the mean-semivariance-skewness-semikurtosis is presented and its four corresponding variants are alsoconsidered. We briefly designed the genetic algorithm integratingfuzzy simulation for our optimization models.

Corresponding author: Faculte Economie, Avenue Raymond DUGRAND C.S. 7960634960 Montpellier cedex 2 Email: [email protected]

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1 INTRODUCTION

In portfolio analysis, an asset return is usually characterized as a randomvariable on a probability space (see Bachelier [1] and Markowitz [14]). Animportant area of finance research is portfolio selection which is to select acombination of assets under the constraints of the investor objectives. Sincereturns are uncertain in nature, allocation capital in different risky assetsto minimize risk and to maximize the return is the main concern of portfo-lio selection. Modern portfolio selection theory has been introduced by theseminal work of [14] which consider trade-off between return and risk. Asin Markowitz[15], variance has been widely accepted as a risk measure by

numerous portfolio selection models. However variance as a risk measure hassome shortcomings and limitations (see [15]).One important shortcoming is that analysis based on variance considers

high returns as equally undesirable low returns (i.e. it does not take intoaccount the asymmetry of the probability distribution). Then there is acontroversy over the issue of whether higher moments should be consideredin portfolio selection models. Some authors such as Samuelson [18], Krausset al. [11], Konno et al.[9]-[10], Briec et al.[4] have argued that it is importantto take into account higher moments than the first and second ones (see alsoFor instance Samuelson [18] showed that investors would prefer a portfoliowith a larger third order moment if first and second moments are same.

The above literature assumed that the securities returns are random vari-ables with fixed expected returns and variances values. However, since in-vestors receive efficient or inefficient information from the real world, am-biguous factors usually exist in it. Consequently, we need to consider notonly random conditions but also ambiguous and subjective conditions forportfolio selection problems. A recent litterature has recognized the fuzinessand the uncertainty of portfolios returns. As discussed in [6], investors canmake use of fuzzy set to reflect the vagueness and ambiguity of securities(i.e. incompleteness of information due to the lack of data). Therefore, the

probability theory becomes difficult to used. For example, some authorssuch as Tanaka and Guo [19] quantified mean and variance of a portfoliothrough fuzzy probability and possibility distributions, Carlsson et al.[2]-[3]used their own definitions of mean and variance of fuzzy numbers. In par-ticular, Huang [7] quantified portfolio return and risk by the expected valueand variance based on credibility measure. Recently, Huang [7] has proposedthe mean-semivariance model for portfolio selection and Li et al.[5], Kar etal.[8] introduced mean-variance-skewness for portfolio selection with fuzzyreturns.

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Different from Huang [7] and Li et al.[5], after recalling the definition

of mean, variance, semi-variance and skewness, this paper consider the k-moments (i.e. Kurtosis for k = 4) and semi-moments (i.e. semi-Kurtosis fork = 4) for portfolio selection with fuzzy risk factors (i.e. returns). Several em-pirical studies show that portfolio returns have fat tails. Generally investorswould prefer a portfolio return with smaller kurtosis which indicates the lep-tokurtosis (fat-tails or thin-tails) when the mean value, the variance and theasymmetry are the same. In order to measure the leptokurtocity of fuzzyportfolio return, notions of moments and the semi-moments of fuzzy portfo-lio are originally introduced in this paper, and their mathematical propertiesare studied. As an extension of the mean-semivariance-skewness model for

fuzzy portfolio, the mean-semivariance-skewness-semikurtosis is presentedand the corresponding variants (the mean-variance-skewness-kurtosis, themean-variance-skewness-semikurtosis and the mean-variance-skewness-semikurtosismodels) are also considered. We briefly designed the genetic algorithm inte-grating fuzzy simulation for our optimization models.

The paper is organized as follows. In Section 2, we review some prelimi-nary knowledge on fuzzy variable and credibility measure. In Section 3, werecall the notions of mean, variance, and skewness of a fuzzy variable. Weintroduce kurtosis for fuzzy variables, study some of its properties and deter-mine, for an integer k > 1, the k-moment of a symmetric fuzzy trapezoidal

fuzzy variable. We compute variance, skewness and kurtosis of trapezoidalnumbers and triangular numbers. In Section 4, we introduce the notion ofsemi-moment of order n=2p (p N) of a fuzzy variable. We justify that theparticular cases of the semi-moment are the known notion of semi varianceand the new notion of semi-Kurtosis for p = 1 and p = 2 respectively. Wecompute the semi-variance and the semi-kurtosis of a trapezoidal fuzzy vari-able. We establish some links between moment and semi-moment of fuzzyvariable. After a brief introduction of fuzzy-simulation-based genetic algo-rithm, Section 5 suggests some determinist optimization programs with afamily of independent triangular fuzzy numbers and, proposes a genetic al-

gorithm to compute Kurtosis and semi-kurtosis of a fuzzy variable. Section6 contains some concluding remarks and the proofs are in Section 7.

2 Fuzzy variable and credibility

Let be a fuzzy variable with membership function . For any x R, (x)represents the possibility that takes value x. For any set B, Liu definedthe credibility measure as the average of possibility measure and necessity

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measure as follows:

Cr({ B}) = 12

supxB

(x) supxBc

(x) + 1

. (1)

It is easy to show that credibility measure is self-dual. That is,

Cr({ B}) + Cr({ Bc}) = 1.Remark 1. Note that for taking values in B, Zadeh has defined the pos-sibility measure of B by

P os({

B}

) = supxB

(x)

and the necessity measure of by

N ec({ B}) = 1 supxBc

(x).

But neither, of these measures are self-dual. That reason also justified theintroduction of the credibility measure by Liu [12].

Example 1. 1. Let = (a,b,c,d) be a trapezoidal fuzzy number (witha b c d). For any r R, Cr( r) is defined as follows:

Cr({ r}) =

0 if r < a12

( raba) if a r < b

12

if b r < c1 1

2( rdcd) if c r < d

1 if d rand

Cr({ r}) =

1 if r < a1 1

2(raba ) if a r < b

12

if b r < c12(

rd

cd ) if c r < d0 if d r

.

2. For all r R, the credibility of a triangular fuzzy variable = (a,b,c)(with a b c) is given by:

Cr({ r}) =

0 if r < a12

( raba ) if a r < b

1 12

( rcbc) if b r < c

1 if c r.

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and

Cr({ r}) =

1 if r < a1 1

2(raba ) if a r < b

12

(rcbc ) if b r < c

0 if c r.

Let us end this Section by giving some notations useful throughout thispaper.

For a trapezoidal fuzzy variable = (a,b,c,d) such that a = b andc = d, supp() = [a, d] its support, cor() = [b, c] its core, ls the lengthof supp() and lc the length of cor(). We set:

= b a, = d b, ls() = d a and lc() = c b. For a triangular fuzzy variable = (a,b,c) such that b = a and c = a,

we set:

1 = max{b a, c b} and = min{b a, c b}.

= (a,b,c,d) is symmetric (that is t R, r R, (t r) = (t + r))if = , and = (a,b,c) is symmetric if 1 = ,

3 Moments of the trapezoidal of fuzzy vari-ables

3.1 Expected Value, Variance and Skewness of fuzzyrandom variables

The definitions of the expected value, variance and skewness of fuzzy variablesare obtained from Li et al. [5].

Definition 1. Let be a fuzzy variable. Then its expected value is definedas

E[] = e =

+0

Cr{ r} dr 0

Cr{ r} dr (2)

provided that at least one of the above integrals is finite.

Remark 2. Note that, expected value is one of the most important conceptsof fuzzy variable, which gives the center of its distribution.

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Example 2. The expected value of a trapezoidal fuzzy variable denoted =

(a,b,c,d) is given by E[] = a+b+c+d4 and the expected value of a triangularfuzzy variable denoted = (a,b,c) is given by E[] = a+2b+c

4.

Definition 2. Let be a fuzzy variable with finite expected value e. Then itsvariance is defined as

V[] = E[( e)2]. (3)Let us determine the variance of a trapezoidal fuzzy variable and that of

a triangular fuzzy variable.

Example 3. 1. Let = (a,b,c,d) be a fuzzy trapezoidal variable withexpected value E[] = a+b+c+d4

= e. The variance V[] of is given by:

V[] = [ 14

(ls() + lc())]3(

| |3

) + max(( ||

4 1

2lc())

3

6 , 0)+

| |2

[1

2ls()( + )

4][

1

4(ls()+lc())]

2+( ||

4+ 1

2ls())

3

6 ( ||

4+ 1

2lc())

3

6 .

2. We can easily check that if is symmetric ( = ), V[] simply becomes

V[] =3[lc() + ]

2 + 2

24.

3. Let = (a,b,c) be a triangular fuzzy variable such thatE[] = a+2b+c4

=e.The varianceV[] of can be deduced from the variance of a trapezoidalone by this way :

V[] =

3331 + 2121+ 111

2

3

3841 .

4. More precisely:- The variances of the following three trapezoidal fuzzy variables are:V[(1, 2, 3, 4)] = 41

24, V[(1, 2, 3, 4)] = 13

24and V[(1, 0, 1, 4)] = 41

24.

- The variances of the following three triangular fuzzy variables are:V[(1, 0, 4)] = 2491

1536and V[(1, 1, 2)] = V[(1, 2, 4)] = 123

256.

Let us end this Subsection with some useful preliminaries on the Skewnessof a fuzzy variable.

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Definition 3. Let be a fuzzy variable with finite expected value e. Then its

skewness is defined asSk() = E[( e)3]. (4)

Remark 3. If has a symmetric membership, then Sk [] = 0, see [5].

In the following example, we determine the skewness of trapezoidal andtriangular fuzzy variable respectively.

Example 4. 1. The skewness of a trapezoidal fuzzy variable = (a,b,c,d)is given by

Sk [] =1

8(b a)[(

b e4

)4

(

a e4

)4] +1

8(c d)[(

c e4

)4

(

d e4

)4].

2. The skewness of a triangular fuzzy variable = (a,b,c) is given by

Sk [] =1

8(b a) [(b e

4)4 (a e

4)4] +

1

8(b c) [(b e

4)4 (c e

4)4].

that is,

Sk [] =(c a)2

32(c + a 2b).

In the following Subsection, we determine, for an integer k > 1, the k-

moment of a symmetric trapezoidal fuzzy variable.

3.2 k-moment of a symmetric trapezoidal fuzzy vari-able

Proposition 1. Let = (a,b,c,d) be a symmetric trapezoidal fuzzy variablewith expected value E[] = e. For an integer k > 1, the k-moment mk =E[( e)k] is given by:

mk = 0 if k is oddk

2i=0

C2i+1k+1

[(c

b)+]k2i

2k+1(k+1) if k is even

Corollary 1. Let = (a,b,c) be a symmetric triangular fuzzy variable withexpected valueE[] = e. For an integerp 1, thek-momentmk = E[(e)k]is given by:

If k = 2p + 1, thenmk = m2p+1 = 0 (5)

If k = 2p, then mk = m2p = k4p+2 .

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3.3 Kurtosis: definitions, first properties and some

particular cases

In this section, we introduce the kurtosis of a fuzzy variable. We study itsproperties and give some examples.

Definition 4. Let be a fuzzy variable such that E[] = e < . The kurtosis of , denoted K[], is given by:

K[] = E[( e)4].

The normalized kurtosis of , denoted K1[], is given by:

K1[] =E[( e)4]

([])4.

We can rewrite K[] and K1[] by means of a credibility measure. Forsuch, we have:

Let be a fuzzy variable such that E[] = e < . The kurtosis K[] is given by:

K[] = +

0

Cr{

(

e)4

r}

dr. (6)

The normalized kurtosis K1[] is given by:

K1[] =

+0

Cr{( e)4 r}dr[+0

Cr{( e)2 r}dr]2 . (7)

The following result determines the Kurtosis by means of a credibilitymeasure. It also establishes some properties on the linearity of the Kurtosis.

Proposition 2. Let be a fuzzy variable such that E[] = e.

1. The kurtosis of is defined by

K[] =

+0

Cr{ e 4r} Cr{ e 4r}dr. (8)

2. The normalized kurtosis of is defined by

K1[] =

+0

Cr{ e 4r} Cr{ e 4r}dr[+0

Cr{ e 2r} Cr{ e 2r}dr]2 . (9)

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3.

a, b

R, K[a+ b] = a4K[].

4. a, b R, K1[a+ b] = K1[].When becomes a symmetric fuzzy variable, then the previous formulas

become

Corollary 2. If is a symmetric fuzzy variable, then (8) becomes

K[] =

+0

Cr{ e 4r}dr. (10)

Then (9) becomes

K1[] =

+0

Cr{ e 4r}dr[+0

Cr{ e 2r}dr]2 . (11)

Let us end this Section with the following Proposition which determinethe kurtosis of trapezoidal and triangular fuzzy variable.

Proposition 3. 1. Let = (a,b,c,d) a fuzzy trapezoidal variable withexpected value E[] = e. The kurtosis K[] of is given by:

K[] =

[1

4(ls()+lc())]

5(| |

5)+max(

( ||4

12

lc())5

10 , 0)+

( ||4

+ 12

ls())5

10 | |

2[1

2ls() ( + )

4][

1

4(ls() + lc())]

4 (||

4+ 1

2lc())

5

10 2. If = (a,b,c,d) is symmetric, the previous expression of K[] becomes:

K[] =5[lc() + ]

4 + 102[lc() + ]2 + 4

160.

3. Let = (a,b,c) be a triangular fuzzy variable such thatE[] = a+2b+c4

=

e.The kurtosis K[] of can be deduced from the kurtosis of a trapezoidalone by this way :

K[] =25351 + 395

41+ 171

4 + 290312 + 7021

3 510.2401

.

4. Let = (a,b,c,d) a fuzzy trapezoidal variable with expected valueE[] =e.

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The normalized kurtosis of is given by:

K1[] =K[]

V2[]

where K[] and V[] have been already given before.

For = the new expression of K1[] is:

K1[] =5[lc() + ]

4 + 102[lc() + ]2 + 4

160[ 3[lc()+]2+2

24]2

We deduce from the previous formulae that:The normalized Kurtosis of some examples of trapezoidal fuzzy variables are:K1[(1, 2, 3, 4)] = 27414

8405, K1[(1, 2, 3, 4)] = 2178

845, K1[(2, 1, 3, 4)] = 3798

1805and

K1[(1, 2, 2, 4)] = 9092825215

.We notice that: for = (a,b,c) a triangular fuzzy number, we have:

- if b=a, then K[] = 25310.240

4 with E[] = 3b+c4

.- if b=c, then K[] = 253

10.2404 with E[] = a+3b

4.

3.4 Moments of portfolio

Example 5. Let (i = (ai, bi, ci, di))i=1,2,...,n be a family of n independenttrapezoidal fuzzy variables and x = (x1, . . . , xn) a family of n positive reals.The portfolio =

ni=1 i defined by

(x) =n

i=1

xii = (n

i=1

xiai,n

i=1

xibi,n

i=1

xici,n

i=1

xidi)

is a fuzzy variable and its expectation is:

e(x) = E[(x)] =1

4

n

i=1

(ai + bi + ci + di) xi. (12)

Proposition 4. The variance of is

V[] = 1192

nk=1

nl=1 xkxlkl

[n

k=1

xk(ls(k)+lc(k))]3|

nk=1

xk(kk)|+

(1

32n

k=1

nl=1 xkxlkl

[n

k=1

xk(ls(k) + lc(k))]2|

nk=1

xk(k k)|)

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2. The variance of the portfolio return is:

V[(x)] =3331 + 21

21+ 111

2 33841

.

3. The Skewness of the portfolio return is:

SK[(x)] = (n

i=1

xi(ci ai))2.n

i=1

xi(ci 2bi + ai)

4. The Kurtosis of the portfolio return is:

K[] = 2535

1 + 3954

1+ 1714

+ 2903

12

+ 702

13

5

10.2401.

4 Semi-Moment of fuzzy variable

Let be a fuzzy variable with finite expected value e. We define the variable( e) as follows:

( e) =

e if e0 if > e

. (13)

4.1 Definitions

Definition 5. 1. The semi-Moment of order n = 2p with p N is

MS2p[] = MSn [] = E[[(e)]2p] =

+0

Cr{[(e)]2p r}dr. (14)

2. The normalized semi-moment of is defined by:

MS2p[] =MS2p[]

(MS2 [])p

=MS2p[]

(VS[])p.

In the case where p = 1, we obtain the well-known semivariance of described as follows.

Definition 6. Let be a fuzzy variable with expected value e.

1. The semivariance of is defined as

VS[] = E[[( e)]2] =+0

Cr{[( e)]2 r}dr. (15)

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Remark 4. The variance ofis used to measure the spread of its distribution

about e = E[]. Note that, variance concerns not only the part is lessthan e, but also the part is greater than e. If we are only interestedwith the first part, then we should use the concept of semi-variance.

For the example of semivariance of triangular and trapezoidal fuzzy vari-ables, we have the following example:

Example 6. 1. The semivariance of a trapezoidal fuzzy number = (a,b,c,d)(where a,b,c,d R such that a = b and c = d) with expected valuee = a+b+c+d

4is given by:

VS

[] =

1

6(b a)[(e

a

4 )3

+min(0, (

b

e

4 )3

)]+

1

6(d c) max(0, (e

c

4 )3

)

2. The semivariance of a triangular fuzzy number = (a,b,c) with ex-pected value e = a+2b+c

4is deduced from the semivariance of a trape-

zoidal one by this way:

VS[] =1

6(b a) [(e a

4)3 +

1

(b c) (b e

4)3 min(0, (b e))].

4.2 Semi-kurtosis: Definitions and examples

In this Subsection, we focus on the semi-Kurtosis (i.e. p=2 in (14))Definition 7. Let be a fuzzy variable with finite expected value e. Thenthe semikurtosis of is defined

KS[] = E[[( e)]4] =+0

Cr{[( e)]4 r}dr. (16)

Let us give the semikurtosis of a trapezoidal fuzzy number and a trian-gular fuzzy number.

Example 7. 1. The semikurtosis of a trapezoidal fuzzy variable = (a,b,c,d)with expected value e = a+b+c+d

4

is given by:

KS[] =1

10(b a)[(e a

4)5+min(0, (

b e4

)5)]+1

10(d c) max(0, (e c

4)5).

2. The semikurtosis of a triangular fuzzy number = (a,b,c) with expectedvalue e = a+2b+c

4is deduced from the semikurtosis of a trapezoidal one

by this way:

KS[] =1

10(b a) [(e a

4)5 +

1

(b c) (b e

4)5 min(0, (b e))]

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Definition 8. Let a fuzzy variable with expected value e.

The normalized semi-kurtosis of is defined by:

KS1 [] =KS[]

(VS[])2.

Example 8. 1. The normalized semikurtosis of a trapezodal fuzzy vari-able = (a,b,c,d) with expected value e is defined as follows:

KS1 [] =

110(ba) [(

ea4

)5 + min(0, ( be4

)5)] + 110(dc) max(0, (

ec4

)5)

[ 16(ba) [(

ea4

)3 + min(0, ( be4

)3)] + 16(dc) max(0, (

ec4

)3)]]2

2. The normalized semikurtosis of a triangular fuzzy variable = (a,b,c)with expected value e is defined as follows:

KS1 [] =

110(ba) [(

ea4

)5 + 1(bc)(

be4

)5 min(0, (b e))][ 16(ba) [(

ea4

)3 + 1(bc)(

be4

)3 min(0, (b e))]]2

Proposition 5. Let(k)k=1,...,n be a family of independent trapezoidal fuzzyvariables with finite expected values (ek)k=1,...,n, (xk)k=1,...,nbe a family of npositive reals and =

n

k=1 xkk be a portfolio. Then

The semivariance of is

VS[] =1

6n

k=1 xk(bk ak)[(

nk=1 xk(ek ak)

4)3+min(0, (

nk=1 xk(bk ek)

4)3)]+

1

6n

k=1 xk(dk ck)max(0, (

nk=1 xk(ek ck)

4)3)

The semikurtosis of is

KS

[] =

1

10

nk=1 xk(bk ak) [(

nk=1 xk(ek ak)4 )5+min(0, (n

k=1 xk(bk

ek)

4 )5

)]+

1

10n

k=1 xk(dk ck)max(0, (

nk=1 xk(ek ck)

4)5).

We end this section by establishing a link between Moment and Semi-Moment.

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4.3 Links between Moments and Semi-Moments

Proposition 6. Let be a fuzzy variable with finite expected value e, MS2p[]and M2p[] the semi-kurtosis and kurtosis of respectively. Then

0 MS2p[] M2p[]. (17)

Proposition 7. Let be a fuzzy variable with finite expected value e. Then

M2p[] = 0 if and only if Cr{ = e} = 1. (18)

Proposition 8. Let be a fuzzy variable with finite expected value e. Then

MS2p[] = 0 if and only if Cr{ = e} = 1,i.e.,M2p[] = 0. (19)

Proposition 9. Let be a symmetric fuzzy variable with finite expected valuee. Then

MS2p[] = M2p[]. (20)

Remark 5. The previous results generalize those established by Huang [7]when we consider moment and semi-moment as variance and semi-variance.

Furthermore, we can deduce the links between Kurtosis and semi-Kurtosisof a fuzzy variable.

Corollary 4. Let be a fuzzy variable with finite expected value e, KS[]and K[] the semi-kurtosis and kurtosis of respectively. Then

1.0 KS[] K[]. (21)

2.K[] = 0 if and only if Cr{ = e} = 1. (22)

3.

KS[] = 0 if and only if Cr{ = e} = 1,i.e.,K[] = 0. (23)4.

KS[] = K[]. (24)

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5 An application in finance

5.1 Review, model, and a determinist program with afamily of triangular fuzzy numbers

Let i be a fuzzy variable representing the return of the ith security, and letxi be the proportion of the total capital invested in security i. In general, i is

given as(pi+dipi)

pi, where pi is the closing price of the ith security at present,

pi is the estimated closing price in the next year, and di is the estimateddividends during the coming year.It is clear that pi and di are unknown at present. If they are estimated

as fuzzy variables, then i is also a fuzzy variable. Thereby, the portfolios1,...,n and the total return = 1x1 + 2x2 + ... + nxn are also fuzzyvariables.

When minimal expected return, minimal skewness and maximal risk aregiven, the investors prefer an asymmetric portfolio with small kurtosis .Therefore, we propose the following mean-semivariance-skewness-semikurtosismodel:

minimize KS

[x11 + x22 + ... + xnn]subject toE[x11 + x22 + ... + xnn] s1VS[x11 + x22 + ... + xnn] s2S[x11 + x22 + ... + xnn] s3x1 + x2 + ... + xn = 1xi 0, i = 1, 2,...,n.

. (25)

The first constraint of this model ensures the expected return is no less thansome target value s1, the second one assures that risk does not exceed somegiven level s2 the investor can bear, the third one assures that the skewness

is no less than some target value s3. The last two constraints imply that allthe capital will be invested to n securities and short-selling is not allowed.The other variants of this model can be deduced from this one by changingthe objective function either by mean or semi-variance or skewness.

Theorem 1. Let(i = (ai, bi, ci))i=1,2,...,n be a family of n independent trian-gular fuzzy variables.Then the model (25) becomes the following determinist programm:

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min 110n

i=1 xi(biai) [(

n

i=1 xi(eiai)4 )5 + 1ni=1 xi(bidi)(

n

i=1 xi(biei)4 )5 min(0,

ni=1 xi(bi ei))]subject ton

i=1 xi(ai + 2bi + ci) 4s11

5n

i=1 xi(biai)[(n

i=1 xi(eiai)4

)3 + 1ni=1 xi(bidi)

(n

i=1 xi(biei)4

)3 min(0,n

i=1 xi(bi ei))] s2(n

i=1 xi(ci ai))2n

i=1 xi(ci 2bi + ai) 32s3x1 + x2 + ... + xn = 1xi 0, i = 1, 2,...,n

Remark 6. It is important to notice that, similarly to above, one can writefour variants of the previous model and deterministic program. These vari-

ants are described as follows.

1. This model minimizes risk (semi-variance) when expected return andskewness are both no less than some given target values s1 and s3 re-spectively and the kurtosis is no more than the given target value s4. Ifthe second and the third constraints (skewness and Kurtosis) do not ex-ist, then the above model degenerates to mean-variance model proposedearlier by Huang [7].

2. This model maximizes the expected return. Similarly, if the first andthe third constraints (semi-variance and kurtosis) do not exist, then

the above model degenerates to mean-variance model proposed earlierby Huang [7].

3. The third variant of the first model is a model which maximizes theskewness. If we cancel the third constraint (kurtosis), then the abovemodel degenerates to mean-variance-skewness model proposed by Li [5].

4. The fourth and last variant of the first model is the multi-objectivenonlinear programming. The aim of this model is to minimize the risk(semi-variance) and the kurtosis, to maximize the expected value andthe skewness when the different target values are unknown.

5.2 Random fuzzy simulation and Genetic algorithm

Genetic algorithm (GA) has been successfully used to solve many industrialoptimization problems, and has been well discussed in Goldberg[?] and re-cently in [12]. In [12], the author designed the hybrid intelligent algorithmintegrating random fuzzy fuzzy simulation and GA is designed to solve theproposed models. Roughly speaking, in the proposed algorithm, randomsimulation and fuzzy simulation are employed to computed the credibility of

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a fuzzy investment return X = 1x1 + 2x2 + ... + nxn, the Kurtosis and

semi-Kurtosis of X.

In the following, we provide a method to compute the kurtosis and semi-kurtosis of general fuzzy variables describing security returns. Fuzzy simula-tion was first introduced by Liu and Iwamura [13], and then was successfullyapplied to solving fuzzy optimisation problems by Liu [12]. For the compu-tation of other parameters as mean, variance, semivariance and skewness, wecan refer to [5] and [7].Let j be a fuzzy variable with membership function j, and decision variablexj , for all 1 j n. It is obvious to see that the computation of the kurtosisand semikurtosis of the variable 1x1+2x2+...+nxn depends on the compu-tation ofCr{1x1+2x2+...+nxn r} where r is a nonnegative real number.

Computation of the credibility measureRandomly generate real numbers ji such that j(ji) , j = 1, 2,...,n,i =1, 2,...,Nrespectively, where is a sufficiently small number, and N is asufficiently large integer. Then, the value ofCr{1x1+2x2+...+nxn r} can be estimated by the formula

1

2( max1iN{ min1jn j(ji)/n

j=1jixj r}+1 max1iN{ min1jn j(ji)/

nj=1

jixj r}).

In the same way, we can deduce the computation of the expressionCr{1x1 + 2x2 + ... + nxn r}.

Computation of the kurtosisNote that we write = E[1x1 + 2x2 + ... + nxn] which may becalculated by fuzzy simulation [12].The following algorithm is used to compute K[1x1 + 2x2 + ... + nxn].step 1. Set e = 0

step 2. Randomly generate jk such that j(jk) , j = 1, 2,...,n,k =1, 2,...,N where is a sufficiently small number.step 3. Set two numbers

a = min1kN

(1kx1 + 2kx2 + ... + nkxn )4,

b = max1kN

(1kx1 + 2kx2 + ... + nkxn )4.

step 4. Randomly generate a number r [a, b].

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step 5. Set e

e + Cr

{1x1 + 2x2 + ... + nxn

r

}.

step 6. Repeat the fourth to fifth steps for N times.step 7. Return a 0 + b 0 + e(ba)

Nas the target value.

Computation of the semikurtosisNote that we write = E[1x1 + 2x2 + ... + nxn] which may becalculated by fuzzy simulation [12].the following algorithm is used to compute KS[1x1 + 2x2 + ... + nxn].step 1. Set e = 0step 2. Randomly generate jk such that j(jk) , j = 1, 2,...,n,k =1, 2,...,N, where is a sufficiently small number.

Step 3. If1kx1 + 2kx2 + ... + nkxn 0, go to step 4 else go tostep 2.Step 4. Set two numbers

a = min1kN

(1kx1 + 2kx2 + ... + nkxn )4,

b = max1kN

(1kx1 + 2kx2 + ... + nkxn )4.

step 5. Randomly generate a number r [a, b].step 6. Set e e + cr{1x1 + 2x2 + ... + nxn r}.step 7. Repeat the fifth to sixth steps for N times.step 8. Return a 0 + b 0 + e(ba)

Nas the target value.

In the following, we recall general principle of GA.

Liu[12] has successfully applied GA to solve many optimization prob-lems with fuzzy parameters. In in our work, following [5], a solution x =(x1, . . . , xn) is encoded by a chromosome c = (c1, . . . , cn) where the genes cifor i = 1, . . . , n are non-negative numbers. We can then find the decodingprocesses by the relation xi = ci/

n

j=1 cj. The preceded relation ensuresthat nj=1 xj = 1 always holds.

In GA, we employ the rank-based-evaluation(RBE) function to measurethe likelihood of reproduction for each chromosome. The RBE function isdefined by

Eval(ci) = (1 )i1 ; i = 1, . . . , p o p size, (26)

where (0, 1). The procedure of the genetic algorithm is summarized asfollows:

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Step 1. Initialize pop

size feasible chromosomes, in which fuzzy sim-

ulation is use d to check the feasibility of the chromosomes (i.e. xi 0,xi = ci/

ni=1 ci and therefore

nj=1 xj = 1).

Step 2. Employ random fuzzy simulation to compute the objectives ofall chromosomes, and then gives an order of the chromosomes based ofthe objectives values.

Step 3. Find the evaluation function of each chromosome according tothe RBE function. Then calculate the fitness of each chromosome bythe evaluation function.

Step 4. Select the chromosomes according to spinning roulettes wheel. Step 5. Update the chromosomes by crossover operation and muta-

tion operation where random fuzzy simulation is utilized to check thefeasibility of each child.

Step 6. Repeat Steps 2-5 for a given number of generations. Step 7. Report the best chromosome, and then decoded into the opti-

mal soplution.

6 Concluding remarks

Different from Huang [7] and Li et al.[5], after recalling the definition of mean,variance, semi-variance and skewness, this paper consider the k-moments(i.e. Kurtosis for k = 4) and semi-moments (i.e. semi-Kurtosis for k = 4)for portfolio selection with fuzzy risk factors (i.e. returns). In order to mea-sure the leptokurtocity of fuzzy portfolio return, notions of moments andthe semi-moments of fuzzy portfolio are originally introduced in this paper,and their mathematical properties are studied. As an extension of the mean-semivariance-skewness model for fuzzy portfolio, the mean-semivariance-skewness-

semikurtosis is presented and the corresponding variants (the mean-variance-skewness-kurtosis, the mean-variance-skewness-semikurtosis and the mean-variance-skewness-semikurtosis models) are also considered. We briefly de-signed the genetic algorithm integrating fuzzy simulation for our optimizationmodels. The next step of our research will be an application to real financialportfolios data.

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7 Proof of the results

Throughout this Section, is a fuzzy variable with E[] = e.

Proof of Proposition 1: For a symmetric trapezoidal fuzzy variable = (a,b,c,d), we can easily show the following result:Cr{( e)k r} = Cr{ e kr} Cr{ e kr}.

Cr{( e)k r} =

12

, if 0 r ( cb2

)k

kr

2+ cb

4+ 1

2, if ( cb

2)k r ( cb

2+ )k

0, if r ( cb2

+ )k

where = d c = b a.So, we can conclude that:mk[] =

( cb2

+)k

0Cr{(e)k r} =

ki=0

kij=0C

jki

(2)j(cb)kj2k+1(k+1)

=k

j=0Cj+1k+1

(2)j(cb)kj2k+1(k+1)

=k

2i=0C

2i+1k+1

[(cb)+]k2i2k+1(k+1)

. The proof is complete.

Proof of Corollary 1: We show that, for a symmetric fuzzy variable, mk[] is nil when k is an odd number.By definition, we have:mk[] = E[(E[])k] =

+0

Cr{(E[])k r}dr0 Cr{(E[])k r}dr, k N.In [5], X. Li has already proved that for a symmetric fuzzy variable , E[] = eand Cr{ e r} = Cr{ e r}, where e is a real number such that(e r) = (e + r), r R and is the membership function of .Furthermore, we have:mk[] =

+0

Cr{(e)k r}dr0 Cr{(e)k r}dr = +0 krk1Cr{e r}dr 0 krk1Cr{ e r}dr = +0 krk1Cr{ e r}dr +0

krk1Cr{ e r}dr = 0.Now, we assume that k is an even integer.For a symmetric triangular fuzzy variable = (a,b,c), we can easily showthe following result:

Since Cr{( e)k

r} = Cr{ e k

r} Cr{ e k

r}, we have:Cr{( e)k r} =

kr2

, if 0 r k0, if r k

where = c b = b a.So, we can conclude that: mk[] =

k0

kr2

dr = 12k+2

k.

Proof of Proposition 2: 1) It is easy to show that: Cr{(e)4 r} =

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Cr

{

e

4

r

} Cr

{

e

4

r

}. Hence we have the following egality:

K[] =

+0

Cr{(e)4 r}dr =+0

Cr{e 4r}Cr{e 4r}dr.

2) We deduce the second result from the definition ofK1[] and by using thefact that:

V[] =

+0

Cr{(e)2 r}dr =+0

Cr{e 2r}Cr{e 2r}dr.

3) i) Let a, b R. We have K[a + b] = E[(a + b E[a + b])4]. SinceE[a+ b] = aE[] + b, we deduce that K[a+ b] = E[(a+ b aE[] b)4] =E[(a

aE[])4] = a4E[(

E[])4] = a4K[].

ii) Since V[a+ b] = a2V[], we deduce K1[a+ b] = K1[].

Proof of Corollary 2: When is a symmetric fuzzy variable, we have:Cr{(e)4 r}dr = Cr{e 4r} and Cr{(e)2 r}dr = Cr{e 2

r} and the proof is complete.

Proof of Proposition 3: 1) Let = (a,b,c,d) be a trapezoidal fuzzyvariable such that E[] = e, = b a, = d c.By using the fact that Cr{(e)4 r} = Cr{e 4r}Cr{e 4r},we can easily obtain the following results:

i)When > , then e < c. We can so distinguish the two following cases asfollows:1stcase: e < b

Cr{( e)4 r} =

1 4r+ea2

, if 0 r (b e)412

, if (b e)4 r (c e)4 4

r+ed2

, if (c e)4 r (e a+b2

)4

4r+ea2

, if (e a+b2

)4 r (e a)40, if r (e a)4.

and finally we get:

K[] =+0 Cr{(e)4 r}dr = ( (ea)+(eb)2 )5.(5 )+( (ea)+(eb)2 )4.((de)+(ea)2 )+

(ea)510

+ (be)5

10 (ce)5

10.

2ndcase: e > b

Cr{( e)4 r} =

12

, if 0 r (c e)4 4

r+ed2

, if (c e)4 r (e a+b2

)4

4r+ea2

, if (e a+b2

)4 r (e a)40, if r (e a)4.

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and finally we get:

K[] =+0 Cr{(e)4 r}dr = ( (ea)+(eb)2 )5.(5 )+( (ea)+(eb)2 )4.((de)+(ea)2 )+

(ea)510

(ce)510

.

ii) When < , we use a similar way to calculate K[].iii)When = , we have:

Cr{( e)4 r} =

12

, if 0 r ( cb2

)4

4r

2+ cb

4+ 1

2, if ( cb

2)4 r ( cb

2+ )4

0, if r ( cb2

+ )4

= d c = b aand this result implies that:

K[] =

+0

Cr{(e)4 r}dr = 5[(c b) + ]4 + 102[(c b) + ]2 + 4

160.

2) Let = (a,b,c) be a triangular fuzzy variable such that E[] = e, =b a, = c b. By using the fact that Cr{( e)4 r} = Cr{ e 4

r} Cr{ e 4r}, we can easily obtain the following results:i)When > , then e < b and

Cr{( e)4 r} =

1 4

r+ea2 , if 0 r (b e)4 4

r+ec2

, if (b e)4 r (+4

)4

4r+ea2

, if (+4

)4 r (e a)40, if r (e a)4

and finally we get:

K[] =

+0

Cr{(e)4 r}dr = 2535 + 3954+ 174 + 29032 + 7023 5

10.240.

ii) When < , we use a similar way to calculate K[].iii)When = , we have:

Cr{( e)4 r} =

k42

, if 0 r k0, if r 4.

where = c b = b a and this result implies that:

K[] =

+0

Cr{( e)4 r}dr = 4

10.

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Proof of Corollary 3: We deduce these results from Proposition 3.

Proof of Proposition 6: Let and r R. With (13), we have:[(e)]2p =

( e)2p si e0 si > e

. Thus we distinguish two cases as follows:

i) If () e, then [(() e)]2p = (() e)2p. And [(() e)]2p r (() e)2p r.ii) If () > e, then [(() e)]2p = 0 and (() e)2p [(() e)]2p.Thus the inequality [(() e)]2p r implies (() e)2p r. We de-duce that ,r, {/[(() e)]2

p

r} {/(() e)2p

r}. Since Cr ismonotone, we have: r,Cr{[( e)]2p r} Cr{( e)2p r}. HenceK[] =

+0

Cr{( e)2p r}dr +0

Cr{[( e)]2p r}dr = KS[]. For p = 2, we show (21).

Proof of Proposition 7: Assume that is symmetric anf let p N .() : Assume that Cr{ = e} = 1. Thus we have: Cr{ e = 0} = 1 iffCr{( e)2p = 0} = 1. With the self-duality of Cr, we have Cr{( e)2p =0} = 0.Let r > 0. We have: Cr{(e)2p r} Cr{(e)2p > 0} Cr{(e)2p =0} = 0. That means r > 0, Cr{( e)

2p

r} = 0. And we deduceK[] = +0

Cr{( e)2p r}dr = 0.(:) Assume that K[] = 0. Since Cr takes values in [0; 1], this equal-ity means Cr{( e)2p r} = 0, r > 0. Since Cr is self-dual, we haveCr{( e)2p = 0} = 1 and we deduce that Cr{ e = 0} = 1, that is,Cr{ = e} = 1.

Assume that is symmetric and replace p = 2 in the precede proof toobtain (22).

Proof of Proposition 8: Let pN. Assume that M2p[] = 0. With

Proposition 6, we have MS2p[] = 0.Assume that MS2p[] = 0. that is, E[[( e)]2p] = 0. Since E[[( e)]2p] =+0

Cr{[( e)]2p r}dr, and the credibility measure Cr takes its valuein [0; 1], then Cr{[( e)]2p r} = 0, r > 0. By the self-duality of Cr, wehave Cr{[( e)]2p = 0} = 1 and, deduce that

Cr{( e) = 0} = 1. (27)Since e = ( e) + ( e)+, then 27 implies e = ( e)+.

And E[( e)] = E[( e)+] =+0

Cr{( e)+ r}dr = 0. This equality

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implies that Cr

{(

e)+

r

}= 0,

r > 0. Since Cr is self dual, we obtain

Cr{( e)+ = 0} = 1.With Cr{( e) = 0} = 1 and Cr{( e)+ = 0} = 1, we deduceCr{( e) = 0} = 1, that is, Cr{ = e} = 1. With Proposition 7, wehave M2p[] = 0. When p = 2 and is symmetric we obtain (23).

Proof of Proposition 9: p N. Assume that is symmetric and letus show (24).Since M2p[] =

+0

Cr{(e)2p r}dr and MS2p[] =+0

Cr{[(e)]2p r}dr, it suffices to show that: Cr{( e)2p r} = Cr{[( e)]2p r}. Forthat we distinguish two cases:- Ifr < 0, then we have Cr{(e)2p r} = Cr{[(e)]2p r} = Cr{} =1.- If r 0, then (with r = r2p) and assume that r > 0. We have ( e)2p r ( e) ] ; r] [r; +[, and [( e)]2p r ( e) ]; r][r; +[. Therefore, we obtain Cr{(e)2p r} = 1Cr{r < e < r}, Cr{[( e)]2p r} = 1 Cr{r < ( e) < r}.It rests to show that Cr{r < e < r} = Cr{r < ( e) < r}.Let be the membership function ofe and be the membership functionof (

e). Let us recall that = if < e

0 otherwise

.

We have:Cr{r < e < r} = 1

2[1+supx]r;r[ (x)max(supx];r[ (x), supx]r;+;[ (x))] =

12

[1 + supx]r;0[ supx];r[ (x)].We also have Cr{r < ( e) < r} = Cr{r < ( e) 0} since( e) 0. Therefore

Cr{r < ( e) < r} = 12

1 + sup

x]r;0[(x) max( sup

x];r[(x), sup

x]0;+;[(x))

=

1

2

1 + supx]r;0[ (x) supx];r[ (x) . (28)Since (x) = 0, x ]0;+[, hence

Cr{r < e < r} = Cr{r < ( e) < r}.

Assume that is symmetric. In the the precede proof, when p = 2 we obtain(24).

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References

[1] L. Bachelier (1900): Theorie de la speculation: Thesis: Annales Scien-tifiques de lEcole Normale Superieure , (III 17), 2186.

[2] C. Carlsson, R. Fuller (2001): On possibilistic mean value and varianceof fuzzy numbers, Fuzzy Sets and Systems ,(122), 239247.

[3] C. Carlsson, R. Fuller and P. Majlender (2002): A approach to selectingportfolios with highest score, Fuzzy Sets and Systems ,(131), 131.

[4] W. Briec, K. Kerstens and O. Junkung (2007): Mean-variance-skewness

portfolio performance gauging: A general shortage function and dual ap-proach, Management Science ,(53), 135149.

[5] X. Li, Z. Qin, S. Kar (2010): Mean-variance-skewness model for portfolioselection with fuzzy returns, European Journal of Operational Research,vol. 202; (1), 239247.

[6] T. Hasuike, H. Katagiri, H. Ishii (2009): Portfolio selection problemswith random fuzzy variable returns Fuzzy Sets and Systems, Vol. 160,(18), Pages 25792596.

[7] X. Huang (2008): Mean-semivariance models for fuzzy portfolio selection,Journal of Computational and Applied Mathematics, Vol. 217, (1), Pages18.

[8] S. Kar, R. Bhattacharyya and D. D. Majumbar (2011): Fuzzy-mean-skewness portfolio selection models by interval analysis , Computers andMathematics with applications, (61), Pages 126137.

[9] H. Konno, K. Suzuki (1995), A mean-variance-skewness optimizationmodel, Journal of the Operations Research Society of Japan, (38), Pages137187.

[10] H. Konno, H. Shirakawa, H. Yamazaki (1993): A mean-absolutedeviation-skewness portfolio optimization model, Annals of Operations Re-search, (45), Pages 205220.

[11] A. Kraus, R. Litzenberger (1976): Skewness preference and the valuationof risky assets, Journal of Finance, (21), Pages 1085-1094.

[12] B.Liu (2002), Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg.

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2011


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[13] B.Liu, K.Iwamura (1998): Chance constrained programming with fuzzy

parameters, Fuzzy Sets and Systems, 94, Pages 227237.

[14] H. Markowitz (1952): Portfolio selection, Journal of Finance (7), 1952,Pages 7791.

[15] H. Markowitz (1959): Portfolio selection: Efficient diversification of in-vestments, Wiley, New York.

[16] N.M. Pindoriya, S.N. Singh, S.K. Singh (2010): Multi-objective meanvarianceskewness model for generation portfolio allocation in electricitymarkets ,Electric Power Systems Research, vol. 80, (10), Pages 13141321.

[17] A. Saeidifar, E. Pasha (2009): The possibilistic moments of fuzzy num-bers and their applications, Journal of Computational and Applied Math-ematics, Vol. 223, Issue 2, Pages 10281042.

[18] P. Samuelson (1970): The fundamental approximation theorem of port-folio analysis in terms of means, variances and higher moments, ,Reviewsof Economics Studies, (37), Pages 537542.

[19] H. Tanaka, P. Guo (1999): Portfolio selection based based on upperand lower exponential possibility distributions, ,European Journal of op-

erational research, (114), Pages 115126.

[20] Z. Wang , F. Tian (2010): A Note of the expected value and variance ofFuzzy variables, International Journal of Nonlinear Science, Vol.9, No.4,pages 486492.

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ArticleMoments and Semi-Moments for fuzzy portfolios selection