Fuzzy probability distribution with VaR constraint for portfolio selection

16th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT)

Fuzzy probability distribution with VaR constraint for portfolio selection M. Rocha1 L. Lima

2

H. Santos3 B. Bedregal4

1

2

Graduate program in Mathematics and statistics, UFPA, Belem, Brasil Graduate program in Electrical Engineering and Computing,UFRN, Natal, Br 3 Department of Informatics and Applied Mathematics, UFRN, Natal, Br 4 Department of Informatics and Applied Mathematics, UFRN, Natal, Br

Abstract

the decision makers should not consider parameters (goals and constraints) using numbers or unique distribuition functions, but instead they should use fuzzy numbers or fuzzy probability distribution functions (see, for example, Zdenek Zmeskal [16]). Recently, researchers investigated many fuzzy portfolio selection problems (see, e.g., Watada [13], Ramaswamy [9], Leon et al. [5], Wang and Zhu [12], Tanaka et al. [11] and Zhang et al. [17]). Carlsson and Fuller [2] introduced the notations ââof upper and lower possibilistic mean values, and introduced the notation of crisp possibilistic mean values and crisp possibilistic variance of continuous distributions. Zhang and Nie [18] extended the concepts of possibilistic mean and possibilistic variance proposed by Carlsson and Fuller [2], and introduced the concepts of upper and lower possibilistic variances and covariances of fuzzy numbers. We propose to make a comparison using their model, but instead we apply fuzzy Laplace. We also demonstrate the theorems which are necessary for the inclusion of these distributions to the model proposed by Li et al. So, we evaluate the behavior of this model when these distributions functions are changed and we also vary the VaR (Value at Risk). This paper is organized as follows. In section 2, we introduce the basic concepts of the possibilistic mean and variance of a fuzzy number. In section 3, we proposed a possibilistic portfolio model under constraints of VaR and risk-free investment. In section 4, we present a fuzzy normal distribution demonstrated in Li [6]. In section 5, we present fuzzy Laplce distribution. In Section 6, numerical examples are given to illustrate our effective proposed approaches. And finally, Section 7 presents our conclusions.

This work aims at comparing two models of fuzzy distribution: Normal and Laplace, whenever they are inside the context of possibilistic mean-variance model described by Li et al. in [6], where fuzzy Normal distribution is used. We propose to make a comparison using their model, but instead we apply fuzzy Laplace distribution. We also demonstrate the theorems which are necessary for the inclusion of these distributions to the model proposed by Li et al. So, we evaluate the behavior of this model when these distribution functions are changed and we also vary the VaR (Value at Risk). For financial analysts it is very important having other distributions as parameters, regarding the volatility of the stock market due to the behavior of financial market. Keywords: Number Fuzzy, VaR, Portfolio selection, Risk 1. Introduction The least complex and most natural way to represent the problem of optimal portfolio selection is a constrained optimization problem. The aim is to maximize or minimize an objective function (usually maximize returns or risk minimization) subject to constraints. However, the objective function and constraints are usually not simple functions. They often rely on more than one characteristic of each asset, and these characteristics are usually combinations of functions that are much more complex than a linear or quadratic function. So, finding a solution to this optimization problem requires more complex techniques. For these and other reasons, many researchers seek models that can measure all of these variables. In this sense, Markowitz [7],[8] proposed a model for the mean-variance portfolio selection and probability theory associated to optimization techniques to model the performance of investment under uncertainty. However, due to the complexity of financial systems, there are several situations where the input data are not precise but only fuzzy. Therefore, © 2015. The authors - Published by Atlantis Press

2. Possibilistic variance and mean value In this section, we introduce some concepts that will be used in the next sections. A fuzzy number A˜ is a fuzzy set of the real line R with a normal, convex and continuous membership function of bounded support. The family of fuzzy number will be denoted by F. A γ-level set of fuzzy A˜ is ˜ γ = {t ∈ R|A(t) ˜ defined by [A] ≥ γ} if γ > 0 and γ ˜ ˜ [A] = cl{t ∈ R|A(t) > 0} (the closure of the sup1479

˜ if γ = 0. It is well known that if A˜ is port of A) ˜ γ is a compact subset of R a fuzzy number then [A] ˜ for all γ ∈ [0, 1] and [A]γ = [a1 (γ), a2 (γ)]. Carlsson and Fuller [2] defined the upper and ˜ First, lower possibilistic means of a fuzzy number A. ¯ ˜ we can define M (A) as Z 1 ¯ (A) ˜ = M

γ(a1 (γ) + a2 (γ))dγ.

upper and lower possibilistc variances and covariances fuzzy numbers. The upper and lower possibilistic variances of a fuzzy set A˜ is defined as: Z

2 ˜ σU (A)

=

1

2

˜ − a2 (γ))2 dγ, γ(MU (A)

(10)

˜ − a1 (γ))2 dγ. γ(ML (A)

(11)

0

(1)

and

0

¯ (A) ˜ as We can rewrite M

Z

2 ˜ σL (A)

=

2

1

0

R1

¯ (A) ˜ = 1 M 2

0

γ(a1 (γ))dγ 1 2

R1 0

+

γ(a2 (γ))dγ

! ,

1 2

respectively. The upper and lower possibilistic covariances between fuzzy numbers, are defined, respectively, as:

(2)

or R1

¯ (A) ˜ = 1 M 2

0

γ(a1 (γ))dγ

R1 0

R1 +

0

γdγ

γ(a2 (γ))dγ

R1 0

! .

˜ B) e =2 CovU (A,

(3)

Z

1

˜ − a2 (γ))(MU (B) e − b2 (γ))dγ, (12) γ(MU (A)

0

γdγ

and Therefore, R1 ¯ (A) ˜ = M

0

γ(a1 (γ) + a2 (γ))/2dγ

R1 0

˜ B) e =2 CovL (A, .

R1 0

(4)

γdγ

˜ ≤ a1 (γ)](a1 (γ))dγ P os[A

R1 0

˜ ≤ a1 (γ)]dγ P os[A

,

sup

A(u) = γ.

˜ = σ 2 (A)

e B) e = Cov(A,

(5)

(6)

˜ ≥ a2 (γ)](a2 (γ))dγ P os[A

R1 0

·

(14)

e B) e + CovL (A, e B) e CovU (A, · 2

(15)

e + λ2 B) e = λ1 M (A) e + λ2 M (B)· e M (λ1 A

˜ as Similarly, we can denote ML (A) 0

2

e and B e be fuzzy numbers. Lemma 2.1 [2] Let A Then for each λ1 , λ2 ∈ R,

u≤a1 (γ)

˜ = ML (A)

2 (A) ˜ + σ 2 (A) ˜ σU L

The possibilistic covariance between the fuzzy e and B e is defined as: numbers A

|{z}

R1

˜ − a1 (γ))(ML (B) e − b1 (γ))dγ. (13) γ(ML (A)

The possibilistic variance of fuzzy number A˜ is defined as:

where Pos denotes possibility, i.e. ˜ ≤ a1 (γ)] = P os[A

1

0

¯ (A) ˜ is nothing but the levelIt follows that M weighted average of the arithmetic means of all γlevel sets, that is, the weight of the arithmetic mean of a1 (γ) and a2 (γ) is just γ. Lets look at the right-hand side of equation (4). ˜ can be reforThe first quantity, denoted by MU (A) mulated as ˜ = MU (A)

Z

˜ ≥ a2 (γ)]dγ P os[A

e and B e be fuzzy numbers. Lemma 2.2 [2] Let A Then for each λ1 , λ2 ∈ R, (7)

e + λ2 B) e = λ21 σ 2 (A) e + λ22 σ 2 (B) e σ 2 (λ1 A e φ(λ2 )B), e +2|λ1 λ2 |Cov(φ(λ1 )A,

where Pos denotes possibility, i.e. where φ(x) is a sign function of x ∈ R. ˜ ≥ a2 (γ)] = P os[A

sup

A(u) = γ

(8)

3. Portfolio Model under constraints of VaR and risk free investment

|{z}

u≥a2 (γ)

The possibilistic mean value of A˜ is the arithmetic mean of its lower and upper possibilistic mean values as follows: ˜ ˜ ˜ = MU (A) + ML (A) M (A) 2

In order to define the model, it is necessary to make the following considerations. First, there are n risk assets and one risk-free asset for investment and the asset return rate ϕei is a fuzzy number, i = 1, 2, . . . , n. xi represents the proportion invested in assets i, and rf is the risk-free asset return. From this, we can then define the return rep as

(9)

Similarly as shown for the upper and lower possibilistic means, Zhang and Nie [17] introduced the 1480

rep =

n X

1−

xi ϕei + rf

n X

i=1

! xi

,

[ξei ]γ = [µi − σi

(16)

M (rep ) =

MU (ϕei ) + ML (ϕei ) xi + rf 2

1−

n X

xi

n X

· (17)

According to Lemma 2.2, it is known that the possibilistic variance of rep is given by xi ϕei

=

n X

x2i σ 2 + 2 ϕi

e

i=1

i=1

" xi ϕei ∼ F N

n X

xi xj Cov(ϕei , ϕej ). (18)

i>j=1

˜ ≤ V aR)] = sup A(t) ≤ 1 − β. P os[A

(19)

    

s.t.

P

x2i σ 2 + 2

e

e

P

xi σ i

, (21)

i=1

i=1

1 2

σ2 =

min

−

π 8

Suppose that the return rate of asset i is a Laplace distribution fuzzy variable expressed as ϕ ei ∼ F L(µi , σi ), and its membership function is

P

e

!2 #

5. Fuzzy Laplace Distribution

n x x Cov(ϕi , ϕj ) i>j=1 i j ϕi n n M (ϕ )+M (ϕ ) x U i 2 L i + rf 1 − x ≥r i=1 i i=1 i pos(ϕi xi ≤ V aR) ≤ 1 − β, n x ≤ 1, i=1 i 0 ≤ li ≤ xi ≤ ui , i = 1, 2, ..., n, i=1

n  X

0 ≤ li ≤ xi ≤ ui , i = 1, 2, ..., n.

Finally, we can define the possibilistic meanvariance model regarding the possibilistic mean as the portfolio return and the possibilistic variance as the portfolio risk. Under this structure, the possibilistic portfolio model under constraint of VaR and risk-free investment can be formulated as: Pn

π 1 − 2 8


 Pn 2 2 Pn x σ + 2 1>j=1 xi xj σi σj i=1 i 2 Pn s.t. x (µ − rf ) + rf ≥ e r i=1 i i
Pn
2 Pn 2 ≤ ln(1 − β) 1 − π −(V aR − x µ ) xσ i i  2 8 i=1 i=1 i i P  n   x ≤1  i=1 i

t≥V aR

σ2 =

xi µ i ,



Proof Refer to [6]. Moreover, Li [6] defines the possibilistic portfolio model under constraints of VaR and risk-free investments, whereas the variables are fuzzy with fuzzy Normal distribution. Thus:      

|{z}

min

n X

where xi ≥ 0, i = 1, 2, ..., n.

A value at risk VaR constraint is imposed on our portfolio model, as follows. Replacing equation (6) and considering a1 (γ) = V aR ≤ 1 − β and β as confidence level, we obtain

     

(20)

!

i=1

σ2

ln γ − 1],

Theorem 4.1 Assume that the return rates of assets are fuzzy variables with fuzzy normal distribution expressed as ϕei ∼ F N (µi , σi ), i=1,2,...,n, then

i=1

!

p

i=1

i=1

n X

ln γ − 1, µi + σi

γ ∈ (0, 1), i = 1, 2, .., n.

since ϕei is a fuzzy membership rep is also a fuzzy number. Consequently, the possibilistic mean of the portfolio return rep is n X

p

e e


e

A

ϕi

(t/µi , b)

=

1 |t − µi | exp − 2b b

=

1 2b

e

eP





exp exp



−µi +t b −t+µi b



if t < µi

if t ≥ µi ,

√

where pos denotes the measure of possibilistic, re is the underestimated expected rate of return, li and ui represent the lower bound and upper bound on investment in asset i, respectively, and VaR is defined as the value at risk by the β-confidence level. The model shows that risk-averse investors wish not only to reach the expected rate of returns in their actual investment, but also to ensure that the maximum of their possible risk is lower than an expected loss.

where σi2 = 2b2 ⇒ b = 22 σi . The γ-level set of ϕei is defined as  √ √ √ 2 2 [ϕei ] = µi − σi ln 2σi γ, µi + σi ln 2σi γ . 2 2 

√

γ

(22)

Theorem 5.1 We assume that the rates of return of the assets are Laplace fuzzy distributions variables expressed as ϕei ∼ F L(µi , σi ),, i = 1, 2, ..., n. Then

4. Fuzzy Normal Distribution

n X

According to Li [6] the rate of return on asset i is a fuzzy variable with fuzzy normal distribution expressed by ξei ∼ F N (µi , σi ), and its membership function is

i=1

xi ϕei ∼ F L

n X

1 xi µi , 8

i=1

n X

!2 ! xi σi

,

(23)

i=1

where xi ≥ 0, i = 1, 2, ..., n.

Aξe (t) = exp{−[(t − µi )/σi ]2 }.

Proof According Pnto Lemma 2.1, the possibilistic mean value of i=1 xi ϕei can be calculated by

i

The γ-level set of ξe is 1481

M

n X

! xi ϕei

=

n X

i=1

xi M (ϕei ) =

n X

Cov(ϕei , ϕej )

xi µi .

=

i=1

i=1

From equations (5), (7) and (21), it can be deducted that 1

Z MU (ϕei )

=

√

 γ

2 0

√

µi +

=

Z

γ ln

2σi

√

2 σP n

n X

=

i=1

xi ϕi

e

2σi γdγ

√ 2 1 σi ln 2σi − 2 2

µi +

dγ



x2i σ 2 + 2 ϕi

e

i=1

0

√ =

1

According to Lemma 2.2 and (xi P ≥ 0), we can n compute the possibilistic variance of i=1 xi ϕei



√ 2 σi ln 2σi γ 2

µi +

CovU (ϕei , ϕej ) + CovL (ϕei , ϕej ) 2 1 σi σj . (27) 8

=

.

8

(24)

xi xj Cov(ϕei , ϕej )

i=1

n X 1

=



n X

x2i σi2 + 2

n X 1

8

xi xj σi σj

i=1

i=1

n X

1 = 8

and

!2 ·

xi σi

i=1 1

Z ML (ϕei )

=

 γ

2 0

√ 2 µi − σi ln 2σi γ 2

√

√ 2 1 σi ln 2σi − 2 2

µi −

=

√





So the proof of the theorem is complete.

dγ





.

(25)

According Pn to Theorem 5.1 the membership function of i=1 xi ϕei is defined by:

In the same way, the following results can be obtained: 2 σU

1

Z =

A(t) = 2



γ MU (ϕei ) − a2 (γ)

2 0

dγ

where

1 2 σ , 8 i

=

Pn   |t − ( i=1 xi µi )| 1 exp − , 2b b

 √ √ 2 2 1 σ ⇒ b= b= 2 2 8

and 2 σL

1

Z =

2



γ MU (ϕei ) − a2 (γ)

2 0

dγ

n X

pos

2 + σ2 σU L

2

ϕi

e

=

1 2 σ · 8 i

sup

(26)

t≤V aR

   

1

2b   

Furthermore, according to equations (12) and (13), the upper and lower possibilistic covariances are given by Z CovU (ϕei , ϕej ) = 2

=

1



0



MU (ϕei ) − b2 (γ) dγ =

1 σi σj , 8

CovL (ϕei , ϕej ) = 2

0

1



                  



γ ML (ϕei ) − a1 (γ)

1 ML (ϕei ) − b1 (γ) dγ = σi σj , 8



exp

! ϕi xi ≤ V aR



   

=

−

   n  P t −  x µ i i   i=1

b

  

  

|V aR−(

1 2b exp

n P

xi µi )|

i=1 b

−

 

= Z

1 2b exp



γ MU (ϕei ) − a2 (γ)



i=1

(28)

i=1

Thus, the possibilistic variance can be written as σ2 =

!2  xi σi  .

So

1 2 σ . 8 i

=

n X

  

.

   P  n  xi µi −V AR    − i=1

b

    

   n P V aR− x i µi  

 

 

1 2b exp

−

,

    

i=1 b

if,V aR