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 method- ology accommodates either homogeneous or heterogeneous anticipated rates of return models. Thus, we consider a mathematical programming model with prob- abilistic constraints and solve it by transforming this problem into a multiple objective linear programming problem. Also, we obtain our results in the frame- work 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 interre- lated 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. REPORTS9(59),3 (2007), 305–317
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 un- der management. Given the uncertainty inherent in financial markets, the managers are very cautious in expressing their market views. The informa- tion 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 meanV 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. MEANV aRPORTFOLIO SELECTION MULTIOBJECTIVE MODEL WITH TRANSACTION COSTS
2.1. MEAN DOWNSIDE-RISK FRAMEWORK
In this section we extend [7, 8, 11] to the case ofi≥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. Letυi, 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 valueυi 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)},
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 securityj, andM1j and M2j the minimum and maximum propor- tions 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 security j. We then haveυi = n
j=1rjixj.
Assume that an investor wants to allocate his/her wealth amongnrisky 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 multiob- jective optimization problem
(2.1) max
x∈Rn[E(υ1), . . . , E(υk)]
subject to
(2.2) P{υi ≤(V aR)i} ≤βi, i= 1, . . . , k, (2.3)
n j=1
xj = 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 thanβi,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 allo- cation ofi, i= 1, . . . , k, assets iscji, thus the transaction cost of securityjand allocation ofiassets iscjixj. The transaction cost of portfoliox= (x1, . . . , xn) is n
j=1cjixj, i= 1, . . . , k. Considering the proportional transaction cost and
the shortfall probability constraint, we propose the meanV aRportfolio selec- tion model with transaction costs formulated as
(2.5) max
x∈Rn
E(υ1)−n
j=1
cj1xj, . . . , E(υk)−n
j=1
cjkxj
subject to
(2.6) P{υi ≤(V aR)i} ≤βi, i= 1, . . . , k,
(2.7) n
j=1
xj = 1,
(2.8) M1j ≤xj ≤M2j, j= 1, . . . , n.
3. POSSIBILISTIC MEANV 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 µb, respectively.
The possibility operator (Pos) is defined (see [5]) as follows:
(3.1)
Pos(a≤b) = sup{min(µa(x), µb(y))|x, y∈R, x≤y} Pos(a <b) = sup{min(µa(x), µb(y))|x, y∈R, x < y} Pos(a=b) = sup{min(µa(x), µb(x))|x ∈R}
In particular, whenbis a crisp numberb, we have (3.2)
Pos(a≤b) = sup{µa(x)|x∈R, x≤b} Pos(a < b) = sup{µa(x)|x∈R, x < b} Pos(a=b) =µa(b).
Letf :R×R→Rbe a binary operation over real numbers. Then it can be extended to the set of fuzzy numbers. If for fuzzy numbersa,b we define c=f(a,b), then the membership functionµc is obtained from the membership functionsµa and µb by the equation
(3.3) µc(z) = sup{min(µa(x), µb(y))|x, y ∈R, z =f(x, y)}
forz ∈R. That is, the possibility that the fuzzy number c=f(a,b) achives valuez∈Ris as great as the most possibility combination of the real numbers x, ysuch thatz=f(x, y), where the value ofaandbarexandy, respectively.
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), wherer1< r2 ≤r3 < r4. Then the membership function of the fuzzy numberr is
(3.4) µ(x) =
x−r1
r2−r1 r1 ≤x≤r2, 1 r2 ≤x≤r3,
x−r4
r3−r4 r3 ≤x≤r4,
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.
d 1
) (
~ x
mb m~r(x)
0 b1 b2 r1 b3 dx r2 r3 b4 r4
Fig. 3.1. Two trapezoidal fuzzy numbersrandb.
Ifr2 ≥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.Ifr2≥b3 andr1≤b4, then the supremum occurs at the abscissa δx of the intersection of the graphs of the functions µr(x) andµb(x). A simple computation shows that
Pos{r ≤b}=δ= b4−r1 (b4−b3) + (r2−r1) and
δx=r1+ (r2−r1)δ .
Ifr1 > b4 then for anyx < y at least one of the equations µr(x) = 0, µb(y) = 0
holds. Thus we have Pos{r ≤b}= 0. Now, we summarize the above results as (3.5) Pos{r≤b}=
1 r2 ≤b3,
δ r2 ≥b3, r1 ≤b4, 0 r1 ≥b4.
Especially, whenbis the crisp number 0 we have (3.6) Pos{r≤0}=
1 r2 ≤0, δ r1 ≤0≤r2, 0 r1 ≥0, where
(3.7) δ = r1
r1−r2. We now state
Lemma3.1 ([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 λlevel 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
E(r) = 1
0 λ(a1(λ) +a2(λ))dλ,
whereE stands for the fuzzy mean operator. We can see that ifr= (r1, r2, r3, r4) is a trapezoidal fuzzy number, then
(3.8) E(r) = 1
0 λ(r1+λ(r2−r1) +r4−λ(r4−r3))dλ= r2+r3
3 +r1+r4 6 .
3.3. EFFICIENT PORTFOLIOS
Let xj the the proportion of the total amount of money devoted to se- curity 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 num- ber 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.
Theorem3.1. The possibilistic mean V aR portfolio selection model for the vector mean V aR efficient portfolio model (2.5)–(2.8) is
(3.9) max
x∈Rn
E
n
i=1
rj1xj
−n
i=1
cj1xj− · · · −E n
i=1
rjkxj
−n
i=1
cjkxj
subject to
(3.10) Pos
n
i=1
rjkxj <bi
≤βi, i= 1, . . . , k,
(3.11) n
i=1
xj = 1,
(3.12) M1j ≤xj ≤M2j, j= 1, . . . , n.
In what follows, to obtain the efficient portfolios given by Theorem 3.1 we use White [15], namely,
Theorem3.2. Ifλi >0, i= 1, . . . , k, then an efficient portfolio for the possibilistic model is an optimal solution of the problem
(3.13) max
x∈Rn
n i=1
λi
E n
i=1
rjixj
−n
i=1
cjixj
subject to
(3.14) Pos
n
i=1
rjkxj <bi
≤βi, i= 1, . . . , k,
(3.15)
n i=1
xj = 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), wherer(ji)1 < r(ji)2 ≤r(ji)3< r(ji)4. Letbi = (b1i, bi2, bi3, bi4)be the trapezoidal fuzzy number for V aR level and λi > 0, with i = 1, . . . , k. Then for the
possibilistic mean V aR portfolio selection model an efficient portfolio is an optimal solution of the problem
(3.17)
x∈Rmaxn n i=1
λi
n
i=1r(ji)2xj+n
i=1r(ji)3xj
3 +
n
i=1r(ji)1xj +n
i=1r(ji)4xj
6 −n
i=1
cjixj
subject to (3.18) (1−βi)
n
i=1
r(ji)1xj−bi4
+βi n
j=1
r(ji)2xj−bi3
≥0, i= 1, . . . , k,
(3.19)
n i=1
xj = 1,
(3.20) M1j ≤xj ≤M2j, j= 1, . . . , n.
Proof. From equation (3.8) we get
E n
j=1
rjixj
= n
j=1r(ji)2xj+ n
j=1r(ji)3xj
3 +
n
j=1r(ji)1xj+ n
j=1r(ji)4xj
6 ,
i= 1, . . . , k. By Lemma 3.1, Pos
n
i=1
rjkxj <bi
≤βi, i= 1, . . . , k, is equivalent to
(1−βi) n
j=1
r(ji)1xj−bi4
+βi n
j=1
r(ji)2xj−bi3
≥0.
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 theλ-level sets of fuzzy numbers, we define the weightedlower possibilistic and upper possibilistic mean values as well as the crisp possibilistic
mean value of fuzzy numbers, which is consistent with the extension princi- ple 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 interval- valued 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 byF. Aλ-level set of a fuzzy numberr= (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)], ifλ > 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 ofR for all λ∈[0,1].
Definition4.1 ([6]). Letr∈ 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
(4.1)
1
0 w(λ)dλ= 1.
We define the w-weighted possibilistic mean (or expected) value of a fuzzy numberr as
(4.2) Ew(r) = 1
0
a1(λ) +a2(λ)
3 w(λ)dλ.
Note that if w(λ) = 2λ,λ∈[0,1], then Ew(r) =
1
0 [a1(λ) +a2(λ)]λdλ.
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 thatw(λ) might be zero for certain (nonimportant)λ-level sets ofr. So, by introducing different weighting functions we can give different (case-dependent) importance toλ-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-widthβ >0, and let
(4.3) w(λ) = (2q−1)[(1−λ)−1/2q−1],
where q ≥ 1. It is clear that w is a weighting function with w(0) = 0 and
q→1−0lim w(λ) = ∞. Then the w-weighted lower and upper possibilistic mean values ofr are given by
Ew−(r) = 1
0 [r1−(1−λ)α](2q−1)[(1−λ)−1/2q−1]dλ=r1−α(2q−1) 2(4q−1) and
E+w(r)(r) = 1
0 [r2+ (1−λ)β](2q−1)[(1−λ)−1/2q−1]dλ=r2+β(2q−1) 2(4q−1). Therefore,
(4.4)
Ew(r) =
r1−α(2q−1)
2(4q−1), r2+β(2q−1) 2(4q−1)
, Ew(r) = r1+r2
2 +(2q−1)(β−α) 4(4q−1) .
This remark along with Theorem 3.1 leads to the following result.
Theorem4.1. The meanV aR efficient portfolio model is
(4.5) max
x∈Rn
k i=1
λi
Ew
n j=1
rjixj
− k
i=1
cjixj
subject to
(4.6) Pos
n
j=1
rjixj <bi
≤βi, i= 1, . . . , k,
(4.7)
n i=1
xj = 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).
Theorem4.2. Let w(λ) = (2q−1)[(1−λ)−2q1 −1],q≥1,and let the rate of return on securityj,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.
Then the possibilistic meanV aR portfolio selection model is (4.9)
x∈Rmaxn k i=1
λi
n
j=1r(ji)1xj+n
j=1r(ji)2xj
2 +
(2q−1) n
j=1r(ji)4xj−n
j=1r(ji)3xj
4(4q−1) −n
j=1
cjixj
subject to (4.10) (1−βi)
n
j=1
r(ji)1xj−bi4
+βi n
j=1
r(ji)2xj−bi3
≥0, i= 1, . . . , k,
(4.11)
n j=1
xj = 1,
(4.12) M1j ≤xj ≤M2j, j= 1, . . . , n.
Proof. From equation (4.2) we get
E n
j=1
rjixj
= n
j=1r(ji)1xj+n
j=1r(ji)2xj
2 +
(2q−1) n
j=1r(ji)4xj−n
j=1r(ji)3xj
4(4q−1) ,
i= 1, . . . , k,q≥1. By Lemma 3.1, Pos
n
i=1
rjkxj <bi
≤βi, i= 1, . . . , k,
is equivalent to (1−βi)
n
j=1
r(ji)1xj −bi4
+βi n
j=1
r(ji)2xj−bi3
≥0. 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].
Letting q → ∞ in (4.4), we obtain lim
q→∞Ew(r) = r1+r2 2 + β−α8 . Thus, we get
Corollary 4.1. For q → ∞,the weighetd possibilistic mean V aR effi- cient portfolio selection model can be reduced to the linear programming prob- lem
x∈Rmaxn k
i=1
λi
n
j=1r(ji)1xj+ n
j=1r(ji)2xj
2 +
n
j=1r(ji)4xj−n
j=1r(ji)3xj
8 −n
j=1
cjixj
subject to
(1−βi) n
j=1
r(ji)1xj −bi4
+βi n
j=1
r(ji)2xj−bi3
≥0, n
j=1
xj = 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
adjat03@yahoo.com and
University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania
