ON THE POSSIBILISTIC APPROACH
TO A PORTFOLIO SELECTION PROBLEM
SUPIAN SUDRADJAT
Predictions about investor portfolio holdings can provide powerful tests of asset
pricing theories. In the context of Markowitz portfolio selection problem, this
paper develops a possibilistic mean VaR model with multi assets. Furthermore,
through the introduction of a set of investor-specific characteristics, the methodology accommodates either homogeneous or heterogeneous anticipated rates of
return models. Thus, we consider a mathematical programming model with probabilistic constraints and solve it by transforming this problem into a multiple
objective linear programming problem. Also, we obtain our results in the framework of weighted possibilistic mean value approach.
AMS 2000 Subject Classification: Primary: 90B28, 90C15, 90C29, 90C48, 90C70;
Secondary: 46N10, 60E15, 91B06, 90B10.
Key words: portfolio selection, VaR possibilistic mean value, possibilistic theory,
weighted possibilistic mean, fuzzy theory.
1. INTRODUCTION
The process of selecting a portfolio can be divided into two stages. The
first stage starts with observation and experience and ends with beliefs about
the future performance of available securities. The second stage starts with
the relevant beliefs about future performances and ends with the choice of
portfolio [9]. This paper is concerned with the second stage.
The problem of standard portfolio selection is as follows. Assume (a) n
securities, (b) an initial sum of money to be invested, (c) the beginning of a
holding period, (d) the end of the holding period.
The portfolio selection model of Markowitz [8, 9] consists of two interrelated modules:
– a nonlinear programming problem where risk-averse investors solve a
utility maximization problem involving the risk and the expected rate of return
of any portfolio, subject to the constraint of an efficiency frontier. The latter
is defined pointwise, as a sequence of solutions to a quadratic programming
problem which minimizes the risk associated with each possible portfolio’s
MATH. REPORTS 9(59), 3 (2007), 305–317
306
Supian Sudradjat
2
expected rate of return subject to the constraint that the elements of the
portfolio be non-negative and sum to unity, and
– a parametric stochastic returns-generating process by which, in each
period, the investors determine the requisite vector of expectations and the
variance-covariance matrix of the investors’ anticipated rates of return on all
risky assets.
The managers are constantly faced with the dilemma of guessing the
direction of market moves in order to meet the return target for assets under management. Given the uncertainty inherent in financial markets, the
managers are very cautious in expressing their market views. The information content in such cautious views can be best described as being “fuzzy” or
vague, in terms of both the direction and size of market moves.
The rest of the paper is organized in the following manner. Section 2
proposes a formulation of the mean V aR portfolio selection model with multi
assets problem. Section 3 considers an overview of the possibility theory and
proposes a possibilistic mean V aR portfolio selection model. Some results
relatively to efficient portofolios are stated. Also, in Section 4 we use results
obtained for efficient portfolios in the frame of the weighted possibilistic mean
value approach. Thus, we are able to extend some recent results in this field
[4, 6, 7].
2. MEAN V aR PORTFOLIO SELECTION MULTIOBJECTIVE MODEL
WITH TRANSACTION COSTS
2.1. MEAN DOWNSIDE-RISK FRAMEWORK
In this section we extend [7, 8, 11] to the case of i ≥ 1 assets. In practice,
investors are concerned about the risk that their portfolio value falls below a
certain level. That is the reason why different measures of downside-risk are
considered in the multi asset allocation problem. Let υi , i = 1, . . . , k, be the
future portfolio value, i.e., the value of the portfolio by the end of the planning
period. Then the probability
P (υi < (V aR)i ),
i = 1, . . . , k,
that the portfolio value υi falls below the (V aR)i level, is called the shortfall
probability. The conditional mean value of the portfolio given that the portfolio
value has fallen below (V aR)i , called the expected shortfall, is defined as
E(υi |υi < (V aR)i ).
Other risk measures used in practice are the mean absolute deviation
E {(|υi − E(νi )|) |υi < E(υi ) } ,
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On the possibilistic approach to a portfolio selection problem
307
and the semi-variance
E((υi − E(νi ))2 |υi < E(υi )),
where we only consider the negative deviations from the mean.
Let xj , j = 1, . . . , n, be the proportion of the total amount of money
devoted to security j, and M1j and M2j the minimum and maximum proportions of the total amount of money devoted to security j. For j = 1, . . . , n,
i = 1, . . . , k let rji be a random variable which is the rate of the i return of
n
rji xj .
security j. We then have υi =
j=1
Assume that an investor wants to allocate his/her wealth among n risky
securities. If the risk profile of the investor is determined in terms of (V aR)i ,
i = 1, . . . , k, a mean-V aR efficient portfolio will be a solution of the multiobjective optimization problem
(2.1)
max [E(υ1 ), . . . , E(υk )]
x∈Rn
subject to
(2.2)
(2.3)
P {υi ≤ (V aR)i } ≤ βi ,
n
i = 1, . . . , k,
xj = 1,
j=1
(2.4)
M1j ≤ xj ≤ M2j ,
j = 1, . . . , n.
In this model, the investor tries to maximize the future value of his/her
portfolio, assuming the probability that the future value of his portfolio falls
below (V aR)i is not greater than βi , i = 1, . . . , k.
2.2. THE PROPORTIONAL TRANSACTION COST MODEL
The introduction of transaction costs adds considerable complexity to the
optimal portfolio selection problem. The problem is simplified if one assumes
that the transaction costs are proportional to the amount of the risky asset
traded, and there are no transaction costs on trades in the riskless asset.
Transaction cost is one of the main sources of concern to managers see [1, 16].
Assume the rate of transaction cost of security j, j = 1, . . . , n, and allocation of i, i = 1, . . . , k, assets is cji , thus the transaction cost of security j and
allocation of i assets is cji xj . The transaction cost of portfolio x = (x1 , . . . , xn )
n
cji xj , i = 1, . . . , k. Considering the proportional transaction cost and
is
j=1
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Supian Sudradjat
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the shortfall probability constraint, we propose the mean V aR portfolio selection model with transaction costs formulated as


n
n
cj1 xj , . . . , E(υk ) −
cjk xj 
(2.5)
maxn E(υ1 ) −
x∈R
j=1
j=1
subject to
(2.6)
(2.7)
P {υi ≤ (V aR)i } ≤ βi ,
n
i = 1, . . . , k,
xj = 1,
j=1
(2.8)
M1j ≤ xj ≤ M2j ,
j = 1, . . . , n.
3. POSSIBILISTIC MEAN V aR PORTFOLIO SELECTION MODEL
3.1. POSSIBILISTIC THEORY
We consider the possibilistic theory proposed by Zadeh [17]. Let a and
b be two fuzzy numbers with membership functions µa and µ , respectively.
b
The possibility operator (Pos) is defined (see [5]) as follows:


a ≤ b) = sup{min(µa (x), µb (y)) | x, y ∈ R, x ≤ y }

 Pos(
(3.1)
Pos(
a < b) = sup{min(µa (x), µb (y)) | x, y ∈ R, x < y }



Pos(
a = b) = sup{min(µa (x), µ (x)) | x ∈ R}
b
In particular, when b is a crisp number b, we have

a ≤ b) = sup{µa (x) | x ∈ R, x ≤ b }

 Pos(
Pos(
a < b) = sup{µa (x) | x ∈ R, x < b }
(3.2)


Pos(
a = b) = µa (b).
Let f : R× R → R be a binary operation over real numbers. Then it can
be extended to the set of fuzzy numbers. If for fuzzy numbers a, b we define
c = f (
a, b), then the membership function µc is obtained from the membership
functions µa and µb by the equation
(3.3)
µc (z) = sup{min(µa (x), µb (y)) | x, y ∈ R, z = f (x, y)}
for z ∈ R. That is, the possibility that the fuzzy number c = f (
a, b) achives
value z ∈ R is as great as the most possibility combination of the real numbers
x, y such that z = f (x, y), where the value of a and b are x and y, respectively.
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On the possibilistic approach to a portfolio selection problem
309
3.2. TRIANGULAR AND TRAPEZOIDAL FUZZY NUMBERS
Let the rate of return on security given by a trapezoidal fuzzy number be
r = (r1 , r2 , r3 , r4 ), where r1 < r2 ≤ r3 < r4 . Then the membership function of
the fuzzy number r is
 x−r
1

r1 ≤ x ≤ r2 ,



r
−
r
2
1



 1
r2 ≤ x ≤ r3 ,
(3.4)
µ(x) =
x
−
r

4

r3 ≤ x ≤ r4 ,



r
−
r
3
4



0
otherwise.
We recall that a trapezoidal fuzzy number is said to be triangular if
r2 = r3 . Let us consider two trapezoidal fuzzy numbers r = (r1 , r2 , r3 , r4 ) and
b = (b1 , b2 , b3 , b4 ), as shown in Figure 3.1.
mb~ ( x)
d
m~r ( x)
1
0 b1
b2 r1
b3
d x r2
r3
b4
r4
Fig. 3.1. Two trapezoidal fuzzy numbers r and b.
If r2 ≥ b3 then we have
Pos(
r ≤
b) = sup{min(µr (x), µb (y)) | x ≤ y} ≥
≥ min{(µr (r2 ), µb (b3 )} = min{1, 1} = 1,
which implies that Pos(
r ≤ b) = 1. If r2 ≥ b3 and r1 ≤ b4 , then the supremum
occurs at the abscissa δx of the intersection of the graphs of the functions
µr (x) and µb (x). A simple computation shows that
Pos{
r ≤ b} = δ =
b4 − r1
(b4 − b3 ) + (r2 − r1 )
and
δx = r1 + (r2 − r1) δ .
If r1 > b4 then for any x < y at least one of the equations
µr (x) = 0,
µb (y) = 0
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Supian Sudradjat
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holds. Thus we have Pos{
r ≤ b} = 0. Now, we summarize the above results as

 1 r2 ≤ b3 ,
δ r2 ≥ b3 , r1 ≤ b4 ,
(3.5)
Pos{
r ≤ b} =

0 r1 ≥ b4 .
Especially, when b is the crisp number 0 we have

 1 r2 ≤ 0,
δ r1 ≤ 0 ≤ r2 ,
(3.6)
Pos{
r ≤ 0} =

0 r1 ≥ 0,
where
δ=
(3.7)
r1
.
r1 − r2
We now state
Lemma 3.1 ([5]). Consider the trapezoidal fuzzy number r = (r1 , r2 , r3 , r4 ).
Then for any given confidence level α, 0 ≤ α ≤ 1, we have Pos(
r ≤ 0) ≥ α if
and only if (1 − α)r1 + αr2 ≤ 0.
The λ level set of a fuzzy number r = (r1 , r2 , r3 , r4 ) is a crisp subset of
R denoted by [
r]λ = {x | µ(x) ≥ λ, x ∈ R} . According to Carlsson et al. [4]
we have
[
r]λ = {x | µ(x) ≥ λ, x ∈ R} = [r1 + λ(r2 − r1 ), r4 − λ(r4 − r3 )] .
Given [
r ]λ = {a1 (λ), a2 (λ)], the crisp possibilistic mean value of r = (r1 , r2 ,
r3 , r4 ) is
1
λ(a1 (λ) + a2 (λ))dλ,
E(
r) =
0
stands for the fuzzy mean operator. We can see that if r = (r1 , r2 ,
where E
r3 , r4 ) is a trapezoidal fuzzy number, then
1
r2 + r3 r1 + r4
+
.
λ(r1 + λ(r2 − r1 ) + r4 − λ(r4 − r3 ))dλ =
(3.8) E(
r) =
3
6
0
3.3. EFFICIENT PORTFOLIOS
Let xj the the proportion of the total amount of money devoted to security j, M1j and M2j the minimum and maximum proportions of the total
amount of money devoted to security j. The trapezoidal fuzzy number of
rji is rji = (r(ji)1 , r(ji)2 , r(ji)3 , r(ji)4 ), where r(ji)1 < r(ji)2 ≤ r(ji)3 < r(ji)4 .
In addition, let us represent the (V aR)i level by the trapezoidal fuzzy number bi = (b1i , bi2 , bi3 , bi4 ), i = 1, . . . , k. So, we see that the model given by
(2.5)–(2.8) reduces to the form from the next result.
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On the possibilistic approach to a portfolio selection problem
311
Theorem 3.1. The possibilistic mean V aR portfolio selection model for
the vector mean V aR efficient portfolio model (2.5)–(2.8) is
n
n
n
n
rj1 xj −
cj1 xj − · · · − E
rjk xj −
cjk xj
(3.9) max E
x∈Rn
i=1
subject to
(3.10)
i=1
Pos
n
i=1
i=1
rjk xj < bi
≤ βi ,
i = 1, . . . , k,
i=1
n
(3.11)
xj = 1,
i=1
M1j ≤ xj ≤ M2j ,
(3.12)
j = 1, . . . , n.
In what follows, to obtain the efficient portfolios given by Theorem 3.1
we use White [15], namely,
Theorem 3.2. If λi > 0, i = 1, . . . , k, then an efficient portfolio for the
possibilistic model is an optimal solution of the problem
n
n
n
λi E
rji xj −
cji xj
(3.13)
max
x∈Rn
subject to
(3.14)
Pos
i=1
n
i=1
i=1
rjk xj < bi
≤ βi ,
i = 1, . . . , k,
i=1
n
(3.15)
xj = 1,
i=1
(3.16)
M1j ≤ xj ≤ M2j ,
j = 1, . . . , n.
We can now state
Theorem 3.3. Let the rate of return on security j, j = 1, . . . , n, be
represented by the trapezoidal fuzzy number rji = (r(ji)1 , r(ji)2 , r(ji)3 , r(ji)4 ),
bi = (b1i , bi2 , bi3 , bi4 ) be the trapezoidal
where r(ji)1 < r(ji)2 ≤ r(ji)3 < r(ji)4 . Let fuzzy number for V aR level and λi > 0, with i = 1, . . . , k. Then for the
312
Supian Sudradjat
8
possibilistic mean V aR portfolio selection model an efficient portfolio is an
optimal solution of the problem
(3.17)


n
n
n
n
r
x
+
r
x
r
x
+
r
x
n
n
(ji)1 j
(ji)4 j
 i=1 (ji)2 j i=1 (ji)3 j

i=1
i=1
+
−
λi 
cji xj 
maxn


x∈R
3
6
i=1
i=1
subject to
n
n
r(ji)1 xj − bi4 + βi
r(ji)2 xj − bi3 ≥ 0,
(3.18) (1 − βi )
i=1
i = 1, . . . , k,
j=1
n
(3.19)
xj = 1,
i=1
M1j ≤ xj ≤ M2j ,
(3.20)
j = 1, . . . , n.
Proof. From equation (3.8) we get
n
n
n
n
r
x
+
r
x
r
x
+
r(ji)4 xj
j
j
j
(ji)2
(ji)3
(ji)1
n
j=1
j=1
j=1
j=1
+
,
rji xj =
E
3
6
j=1
i = 1, . . . , k. By Lemma 3.1,
n
rjk xj < bi ≤ βi ,
Pos
i = 1, . . . , k,
i=1
is equivalent to
n
n
r(ji)1 xj − bi4 + βi
r(ji)2 xj − bi3 ≥ 0.
(1 − βi )
j=1
j=1
Therefore, Theorem 3.3 follows from Theorem 3.2.
Problem (3.17)–(3.20) is a standard multi-objective linear programming
problem. For finding the optimal solution, we can use several algorithms
[12, 14].
4. WEIGHTED POSSIBILISTIC MEAN VALUE APPROACH
In this section, after introducing a weighting function that measures the
importance of the λ-level sets of fuzzy numbers, we define the weighted lower
possibilistic and upper possibilistic mean values as well as the crisp possibilistic
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On the possibilistic approach to a portfolio selection problem
313
mean value of fuzzy numbers, which is consistent with the extension principle and with the well-known definition of expectation in probability theory.
We also show that the weighted interval-valued possibilistic mean is always a
subset (moreover, a proper subset excluding some special cases) of the intervalvalued probabilistic mean for any weighting function.
A trapezoidal fuzzy number r = (r1 , r2 , r3 , r4 ) is a fuzzy set of the real
line R with a normal, fuzzy convex and continuous membership function of
bounded support. The family of fuzzy numbers will be denoted by F. A λ-level
set of a fuzzy number r = (r1 , r2 , r3 , r4 ) is defined as [
r ]λ = {x | µ(x) ≥ λ, x ∈
R}, so that
[
r ]λ = {x | µ(x) ≥ λ, x ∈ R} = [r1 + λ(r2 − r1 ), r4 − λ(r1 − r4 )] ,
if λ > 0 and [
r ]λ = cl{x ∈ R | µ(x) ≥ 0 } (the closure of the support of r) if
λ = 0. It is well-known that if r is a fuzzy number, then [
r]λ is a compact
subset of R for all λ ∈ [0, 1].
Definition 4.1 ([6]). Let r ∈ F be a fuzzy number with [
r ]λ = [a1 (λ), a2 (λ)],
λ ∈ [0, 1]. A function w : [0, 1] → R is said to be a weighting function if it is
nonnegative, increasing, and satisfies the normalization condition
1
w(λ)dλ = 1.
(4.1)
0
We define the w-weighted possibilistic mean (or expected) value of a fuzzy
number r as
1
a1 (λ) + a2 (λ)
w(λ)dλ.
r) =
E w (
(4.2)
3
0
Note that if w(λ) = 2λ, λ ∈ [0, 1], then
1
r) =
[a1 (λ) + a2 (λ)]λdλ.
E w (
0
Thus, the w-weighted possibilistic mean value defined by (4.2) can be
considered as a generalization of possibilistic mean value in [6]. From the
definition of a weighting function, it can be seen that w(λ) might be zero for
certain (nonimportant) λ-level sets of r. So, by introducing different weighting
functions we can give different (case-dependent) importance to λ-levels sets of
fuzzy numbers.
Let r = (r1 , r2 , α, β) be a trapezoidal fuzzy number with peak [r1 , r2 ],
left-width α > 0 and right-width β > 0, and let
(4.3)
w(λ) = (2q − 1)[(1 − λ)−1/2q − 1],
314
Supian Sudradjat
10
where q ≥ 1. It is clear that w is a weighting function with w(0) = 0 and
lim w(λ) = ∞. Then the w-weighted lower and upper possibilistic mean
q→1−0
values of r are given by
1
α(2q − 1)
−
r) =
[r1 − (1 − λ)α](2q − 1)[(1 − λ)−1/2q − 1]dλ = r1 −
Ew (
2(4q − 1)
0
and
+ (
r) =
E
w r )(
Therefore,
(4.4)
0
1
[r2 + (1 − λ)β](2q − 1)[(1 − λ)−1/2q − 1]dλ = r2 +
β(2q − 1)
.
2(4q − 1)
α(2q − 1)
β(2q − 1)
, r2 +
,
r ) = r1 −
Ew (
2(4q − 1)
2(4q − 1)
r1 + r2 (2q − 1)(β − α)
w (
+
.
r) =
E
2
4(4q − 1)
This remark along with Theorem 3.1 leads to the following result.
Theorem 4.1. The mean V aR efficient portfolio model is
n
k
k
w
λi E
rji xj −
cji xj
(4.5)
maxn
x∈R
i=1
j=1
i=1
subject to
(4.6)
n
rji xj < bi ≤ βi ,
Pos
i = 1, . . . , k,
j=1
(4.7)
n
xj = 1,
i=1
(4.8)
M1j ≤ xj ≤ M2j ,
j = 1, . . . , n.
The next result extends Theorem 3.3 to the case of weighted possibilistic
mean value approach, with the special weighting function (4.3).
1
Theorem 4.2. Let w(λ) = (2q −1)[(1−λ)− 2q −1], q ≥ 1, and let the rate
of return on security j, j = 1, . . . , n, be represented by the trapezoidal number
rji = (r(ji)1 , r(ji)2 , r(ji)3 , r(ji)4 ), where r(ji)1 < r(ji)2 ≤ r(ji)3 < r(ji)4 . Let
bi = (b1i , bi2 , bi3 , bi4 ) be the trapezoidal fuzzy number for (V aR)i , i = 1, . . . , k.
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On the possibilistic approach to a portfolio selection problem
315
Then the possibilistic mean V aR portfolio selection model is
(4.9)
 n

n
n
n
r(ji)4 xj − r(ji)3 xj
k
n
 r(ji)1 xj + r(ji)2 xj (2q−1)

j=1
j=1
j=1
 j=1

+
− cji xj
maxn λi
x∈R


2
4(4q−1)
i=1
j=1
subject to
n
n
(4.10) (1 − βi )
r(ji)1 xj − bi4 + βi
r(ji)2 xj − bi3 ≥ 0,
j=1
i = 1, . . . , k,
j=1
n
(4.11)
xj = 1,
j=1
M1j ≤ xj ≤ M2j ,
(4.12)
j = 1, . . . , n.
Proof. From equation (4.2) we get
E
n
rji xj =
n
r(ji)1 xj +
j=1
2
j=1
n
n
n
(2q−1)
r(ji)4 xj − r(ji)3 xj
r(ji)2 xj
j=1
j=1
+
j=1
,
4(4q − 1)
i = 1, . . . , k, q ≥ 1. By Lemma 3.1,
n
rjk xj < bi ≤ βi ,
Pos
i = 1, . . . , k,
i=1
is equivalent to
n
n
r(ji)1 xj − bi4 + βi
r(ji)2 xj − bi3 ≥ 0.
(1 − βi )
j=1
j=1
Therefore, Theorem 4.2 follows from Theorem 3.2.
Problem (4.9)–(4.12) is a standard multi-objective linear programming
problem. For finding its optimal solution, we can use several algorithms
[12, 15].
w (
r ) = r1 +r2 + β−α . Thus,
Letting q → ∞ in (4.4), we obtain lim E
we get
q→∞
2
8
316
Supian Sudradjat
12
Corollary 4.1. For q → ∞, the weighetd possibilistic mean V aR efficient portfolio selection model can be reduced to the linear programming problem
 n

n
n
n
r(ji)2 xj
r(ji)4 xj −
r(ji)3 xj
k
n
 r(ji)1 xj +

j=1
j=1
j=1
 j=1

cji xj 
+
−
maxn λi 
x∈R
2
8


i=1
subject to
j=1
n
n
r(ji)1 xj − bi4 + βi
r(ji)2 xj − bi3 ≥ 0,
(1 − βi )
j=1
j=1
n
xj = 1,
j=1
M1j ≤ xj ≤ M2j ,
j = 1, . . . , n.
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Received 8 February 2007
Padjanjaran University
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Jl. Raya Bandung-Sumedang Jatinangor 40600
Bandung, Indonesia
[email protected]
and
University of Bucharest
Faculty of Mathematics and Computer Science
Str. Academiei 14
010014 Bucharest, Romania