A possibilistic approach to selecting portfolios
with highest utility score ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi
Péter Majlender
peter.majlender@abo.fi
Abstract
The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz, has been one of the most important research
fields in modern finance. In this paper we will assume that (i) each investor
can assign a welfare, or utility, score to competing investment portfolios
based on the expected return and risk of the portfolios; and (ii) the rates
of return on securities are modelled by possibility distributions rather than
probablity distributions. We will present an algorithm of complexity o(n3 )
for finding an exact optimal solution (in the sense of utility scores) to the
n-asset portfolio selection problem under possibility distributions.
1
A utility function for ranking portfolios
The mean-variance methodology for the portfolio selection problem, originally
proposed by Markowitz [4], has been one of the most important research fields
in modern finance theory [7]. The key principle of the mean-variance model is
to use the expected return of a portfolio as the investment return and to use the
variance of the expected returns of the portfolio as the investment risk.
Following [1] we shall asssume that each investor can assign a welfare, or
utility, score to competing investment portfolios based on the expected return and
risk of those portfolios. The utility score may be viewed as a means of ranking
portfolios. Higher utility values are assigned to portfolios with more attractive riskreturn profiles. One reasonable function that is commonly employed by financial
theorists assigns a risky portfolio P with a risky rate of return rP , an expected rate
∗
The final version of this paper appeared in: C. Carlsson, R. Fullér and P. Majlender, A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems, 131(2002)
13-21. DOI: 10.1016/S0165-0114(01)00251-2
1
of return E(rP ) and a variance of the rate of return σ 2 (rP ) the following utility
score [1]:
U (P ) = E(rP ) − 0.005 × A × σ 2 (rP ),
(1)
where A is an index of the investor’s risk aversion (A ≈ 2.46 for an average investor in the U.S.A.). The factor of 0.005 is a scaling convention that allows us
to express the expected return and standard deviation in equation (1) as percentages rather than decimals. Equation (1) is consistent with the notion that utility is
enhanced by high expected returns and diminished by high risk.
Because we can compare utility values to the rate offered on risk-free investments when choosing between a risky portfolio and a safe one, we may interpret a
portfolio’s utility value as its certainty equivalent rate of return to an investor.
That is, the certainty equivalent rate of a portfolio is the rate that risk-free
investments would need to offer with certainty to be considered as equally attractive
as the risky portfolio. Now we can say that a portfolio is desirable only if its
certainty equivalent return exceeds that of the risk-free alternative. In the meanvariance context, an optimal portfolio selection can be formulated as the following
quadratic mathematical programming problem
U
X
n
i=1
ri xi
=E
X
n
ri xi
− 0.005 × A × σ
i=1
2
X
n
ri xi
→ max
(2)
i=1
subject to {x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n},
where n is the number of available securities, xi is the proportion invested in security (or asset) i, and ri denotes the risky rate of return on security i, i = 1, . . . , n.
Denoting the rate of return on the risk-free asset by rf , a portfolio is desirable for
the investor if and only if
X
n
U
ri xi > rf .
i=1
In this paper we will assume that the rates of return on securities are modelled
by possibility distributions rather than probablity distributions. That is, the rate
of return on the i-th security will be represented by a fuzzy number ri , and ri (t),
t ∈ R, will be interpreted as the degree of possibility of the statement that ’t will
be the rate of return on the i-th security’. In our method we will consider only
trapezoidal possiblity distributions, but our method can easily be generalized to the
case of possibility distributions of type LR.
In standard portfolio models uncertainty is equated with randomness, which
actually combines both objectively observable and testable random events with
subjective judgments of the decision maker into probability assessments. A purist
2
on theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to
probabilities.
The use of probabilities has another major drawback: the probabilities give
an image of precision which is unmerited - we have found cases where the assignment of probabilities is based on very rough, subjective estimates and then the
subsequent calculations are carried out with a precision of two decimal points. This
shows that the routine use of probabilities is not a good choice. The actual meaning
of the results of an analysis may be totally unclear - or results with serious errors
may be accepted at face value.
In standard portfolio theory the decision maker assigns utility values to consequences, which are the results of combinations of actions and random events. The
choice of utility theory, which builds on a decision maker’s relative preferences
for artificial lotteries, is a way to anchor portfolio choices in the von NeumannMorgenstern axiomatic utility theory. In practical applications the use of utility
theory has proved to be problematic (which should be more serious than having
axiomatic problems): (i) utility measures cannot be validated inter-subjectively,
(ii) the consistency of utility measures cannot be validated across events or contexts for the same subject, (iii) utility measures show discontinuities in empirical
tests (as shown by Tversky (cf. [5])), which should not happen with rational decision makers if the axiomatic foundation is correct, and (iv) utility measures are
artificial and thus hard to use on an intuitive basis.
As the combination of probability assessments with utility theory has these
well- known limitations we have explored the use of possibility theory as a substituting conceptual framework.
Let us introduce some definitions we shall need in the following section. A
fuzzy number A is called trapezoidal with tolerance interval [a, b], left width α and
right width β if its membership function has the following form

a−t



1−
if a − α ≤ t ≤ a,


α


 1
if a ≤ t ≤ b,
A(t) =
t−b



1
−
if a ≤ t ≤ b + β,


β



0
otherwise,
and we use the notation A = (a, b, α, β). It can easily be shown that
[A]γ = [a − (1 − γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1].
where [A]γ denotes the γ-level set of A.
3
Figure 1: Trapezoidal fuzzy number.
Let [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)] be fuzzy numbers and let
λ ∈ R be a real number. Using the extension principle we can verify the following
rules for addition and scalar muliplication of fuzzy numbers
[A + B]γ = [a1 (γ) + b1 (γ), a2 (γ) + b2 (γ)], [λA]γ = λ[A]γ .
Let A ∈ F be a fuzzy number with [A]γ = [a1 (γ), a2 (γ)], γ ∈ [0, 1]. In [2] we
introduced the (crisp) possibilistic mean (or expected) value and variance of A as
Z
1
γ(a1 (γ) + a2 (γ))dγ,
E(A) =
σ 2 (A) =
0
1
2
Z
1
2
γ a2 (γ) − a1 (γ) dγ.
0
It is easy to see that if A = (a, b, α, β) is a trapezoidal fuzzy number then
1
Z
γ[a − (1 − γ)α + b + (1 − γ)β]dγ =
E(A) =
0
a+b β−α
+
.
2
6
and
(b − a)2 (b − a)(α + β) (α + β)2
b − a α + β 2 (α + β)2
σ (A) =
+
+
=
+
+
.
4
6
24
2
6
72
2
2
A possibilistic approach to portfolio selection problem
Watada [6] proposed a fuzzy portfolio selection model where he used fuzzy numbers to represent the decision maker’s aspiration levels for the expected rate of
return and a certain degree of risk. Inuiguchi and Tanino [3] introduced a novel
possibilistic programming approach to the portfolio selection problem: their approach, which prefers a distributive investment solution, is based on the minimax
regret criterion (the regret which the decision maker is ready to undertake).
In many important cases it might be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability
distributions.
Consider now the portfolio selection problem with possibility distributions
U
X
n
i=1
ri xi
=E
X
n
ri xi
− 0.005 × A × σ 2
i=1
X
n
ri xi
→ max
i=1
subject to {x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n}.
4
(3)
where ri = (ai , bi , αi , βi ), i = 1, . . . , n are fuzzy numbers of trapezoidal form. It
is easy to compute that
E
X
n
ri xi
i=1
n
X
1
1
=
ai + bi + (βi − αi ) xi ,
2
3
i=1
and
X
X
n
2
2
n
n
1
1
1 X
2
σ
ri xi =
bi − ai + (αi + βi ) xi +
(αi + βi )xi .
2
3
72
i=1
i=1
i=1
Introducing the notations
√
1
1
1
0.005A
bi − ai + (αi + βi ) ,
ui = ai + bi + (βi − αi ) , vi =
2
3
2
3
√
0.005A
(αi + βi ),
wi = √
72
we shall represent the i-th asset by a triplet (vi , wi , ui ), where ui denotes its possibilistic expected value, and vi2 + wi2 is its possibilistic variance multiplied by the
constant 0.005 × A. We will also assume that there are at least three distinguishable assets, with the meaning that if two assets have the same expected value and
variance then they are considered indistinguishable (or identical in the framework
of mean-variance analysis). That is, we assume that ui 6= uj or vi2 + wi2 6= vj2 + wj2
for i 6= j.
Then we will state the possibilistic portfolio selection problem (3) as
hu, xi − hv, xi2 − hw, xi2 → max; s.t.{x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n}.
(4)
The convex hull of {(vi , wi , ui ) : i = 1, . . . , n}, denoted by T , and defined by
T = conv{(vi , wi , ui ) : i = 1, . . . , n}
X
X
n
n
n
n
X
X
xi = 1, xi ≥ 0, i = 1, . . . , n .
=
vi xi ,
wi x i ,
ui xi :
i=1
i=1
i=1
i=1
is a convex polytope in R3 . Then (4) turns into the following three-dimensional
nonlinear programming problem
−(v02 + w02 − u0 ) → max; subject to (v0 , w0 , u0 ) ∈ T,
5
or, equivalently,
f (v0 , w0 , u0 ) := v02 + w02 − u0 → min; subject to (v0 , w0 , u0 ) ∈ T,
(5)
where T is a compact and convex subset of R3 , and the implicit function
gc (v0 , w0 ) := v02 + w02 − c,
is strictly convex for any c ∈ R. This means that any optimal solution to (5) must
be on the boundary of T .
We will now present an algorithm for finding an optimal solution to (4) on
the boundary of T . Note that T is a compact and convex polyhedron of R3 and
that any optimal solution to (5) must be on the boundary of T , which imply that
any optimal solution can be obtained as a convex combination of at most 3 extreme
points of T . In the algorithm by lifting the non-negativity conditions for investment
proportions we shall calculate: (i) the (exact) solutions to all conceivable 3-asset
problems with non-colinear assets, (ii) the (exact) solutions to all conceivable 2assets problems with distinguishable assets, and (iii) the utility value of each asset.
Then we compare the utility values of all feasible solutions (i.e. solutions with
non-negative weights) and portfolios with the highest utility value will be chosen
as optimal solutions to portfolio selection problem (5). Our algorithm will require
O(n3 ) steps, where n is the number of available securities.
Consider three assets (vi , wi , ui ), i = 1, 2, 3, which are not colinear:
@(α1 , α2 , α3 ) ∈ R3 , (α1 , α2 , α3 ) 6= 0, such that






v1
v2
v3
α1  w1  + α2  w2  − (α1 + α2 )  w3  = 0.
u1
u2
u3
Then the 3-asset optimal portfolio selection problem with not-necessarily nonnegative weights reads
(v1 x1 + v2 x2 + v3 x3 )2 + (w1 x1 + w2 x2 + w3 x3 )2 − (u1 x1 + u2 x2 + u3 x3 ) → min
(6)
subject to x1 + x2 + x3 = 1.
Let us denote
L(x, λ) = (v1 x1 + v2 x2 + v3 x3 )2 + (w1 x1 + w2 x2 + w3 x3 )2
−(u1 x1 + u2 x2 + u3 x3 ) + λ(x1 + x2 + x3 − 1),
6
(7)
the Lagrange function of the constrained optimization problem (6). The KuhnTucker necessity conditions are
2v1 (v1 x1 + v2 x2 + v3 x3 ) + 2w1 (w1 x1 + w2 x2 + w3 x3 ) − u1 + λ = 0,
2v2 (v1 x1 + v2 x2 + v3 x3 ) + 2w2 (w1 x1 + w2 x2 + w3 x3 ) − u2 + λ = 0,
2v3 (v1 x1 + v2 x2 + v3 x3 ) + 2w3 (w1 x1 + w2 x2 + w3 x3 ) − u3 + λ = 0,
x1 + x2 + x3 = 1,
which lead us to the following linear equality system
2
q1 + r12
q1 q2 + r1 r2
x1
1/2(u1 − u3 ) − q1 v3 − r1 w3
=
, (8)
q1 q2 + r1 r2
q22 + r22
x2
1/2(u2 − u3 ) − q2 v3 − r2 w3
where we used the notations q1 = v1 − v3 , q2 = v2 − v3 , r1 = w1 − w3 and
r2 = w2 − w3 .
Now we prove that if (vi , wi , ui ), i = 1, 2, 3, are not colinear then equation (8)
has a unique solution. Suppose that the solution to equation (8) is not unique, i.e.
2
q1 + r12
q1 q2 + r1 r2
det
= 0.
q1 q2 + r1 r2
q22 + r22
That is,
2
q1 q2 + r1 r2
q1 + r12
= (q12 + r12 )(q22 + r22 ) − (q1 q2 + r1 r2 )2
det
q1 q2 + r1 r2
q22 + r22
2
q1 r1
2
= 0.
= (q1 r2 − q2 r1 ) = det
q2 r2
Thus, the rows of
q1 r1 q2 r2
are not linearly independent: ∃(α1 , α2 ) 6= 0 such that
α1 [q1 , r1 ] + α2 [q2 , r2 ] = 0 ⇐⇒ α1 [v1 − v3 , w1 − w3 ] + α2 [v2 − v3 , w2 − w3 ] = 0.
(9)
We find that equation (8) turns into
α22
−α1 α2
x1
1/2α1 (u1 − u3 ) + α2 (q2 v3 + r2 v3 )
2
2
(q2 +r2 )
= α1
.
−α1 α2
α12
x2
1/2α1 (u2 − u3 ) − α1 (q2 v3 + r2 v3 )
Multiplying both sides by [α1 , α2 ] we get that u1 , u2 and u3 have to satisfy the
equation
"
#
1
2 1
α1 α1 (u1 − u3 ) − α2 (u2 − u3 ) = 0.
2
2
7
If α1 6= 0, then we obtain α1 (u1 − u3 ) + α2 (u2 − u3 ) = 0, and from equation (9)
it follows that






v1
v2
v3
α1  w1  + α2  w2  − (α1 + α2 )  w3  = 0,
u1
u2
u3
i.e. (vi , wi , ui ), i = 1, 2, 3, were colinear.
If α1 = 0, then α2 6= 0, and from equation (9) it follows that q2 = r2 = 0.
Now we find that equation (8) turns into
2
q1 + r12 0
x1
1/2(u1 − u3 ) − q1 v3 − r1 v3 )
=
.
0
0
x2
1/2(u2 − u3 )
Multiplying both sides by [0, 1], we obtain
1
(u2 − u3 ) = 0.
2
We find that
v2 − v3 = w2 − w3 = u2 − u3 = 0,
which means that (vi , wi , ui ), i = 1, 2, 3, were colinear. Which ends the proof.
Using the general inversion formula
t1 t2
t3 t4
−1
1
t4 −t2
=
,
t1 t4 − t2 t3 −t3 t1
we find that the optimal solution to (8) is
x∗1
x∗2
1
−(q1 q2 + r1 r2 )
q22 + r22
=
q12 + r12
(q1 r2 − q2 r1 )2 −(q1 q2 + r1 r2 )
1/2(u1 − u3 ) − q1 v3 − r1 v3
×
.
1/2(u2 − u3 ) − q2 v3 − r2 v3
(10)
We will now show that
x∗ = (x∗1 , x∗2 , 1 − x∗1 − x∗2 )
satisfies the Kuhn-Tucker sufficiency condition, i.e. L00 (x, λ) is a positive definite
matrix at x = x∗ in the subset defined by
y = (y1 , y2 , y3 ) ∈ R3 : y1 + y2 + y3 = 0 .
8
Really, from (7) we get

2 + w2
v
v
v
+
w
w
v
v
+
w
w
1
2
1
2
1
3
1
3
1
1
1
v22 + w22
v 2 v 3 + w2 w3 
M := L00 (x∗ , λ) = v1 v2 + w1 w2
2
v 1 v 3 + w1 w3 v 2 v 3 + w2 w3
v32 + w32
   T    T
v1
v1
w1
w1
= v2  v2  + w2  w2  ,
v3
v3
w3
w3

and, therefore, the inequality
y T M y = (v1 y1 + v2 y2 + v3 y3 )2 + (w1 y1 + w2 y2 + w3 y3 )2 ≥ 0,
(11)
holds for any y ∈ R3 . So M is a positive semidefinite matrix. If y T M y = 0 for
some y = (y1 , y2 , y3 ) 6= 0, y1 + y2 + y3 = 0, then from (11) we find
v1 y1 + v2 y2 + v3 y3 = 0, w1 y1 + w2 y2 + w3 y3 = 0,
and we would get that


v1 v2 v3
q q
q r
det w1 w2 w3  = det 1 2 = det 1 1 = 0,
r1 r2
q2 r2
1
1
1
which would lead us to a contradiction with the non-colinearity condition. So L00 is
positive definite. Thus x∗ is the unique optimal solution to (6) and x∗ is an optimal
solution to (4) (with n = 3) if x∗1 > 0, x∗2 > 0 and x∗3 > 0 (the Kuhn-Tucker
regularity condition). The optimal value of (6) will be denoted by U∗ .
Consider now a 2-asset problem with two assets, say (v1 , w1 , u1 ) and (v2 , w2 , u2 ),
such that (v1 , w1 , u1 ) 6= (v2 , w2 , u2 ):
(v1 x1 +v2 x2 )2 +(w1 x1 +w2 x2 )2 −(u1 x1 +u2 x2 ) → min; s.t. x1 +x2 = 1. (12)
Let us denote
L(x, λ) = (v1 x1 +v2 x2 )2 +(w1 x1 +w2 x2 )2 −(u1 x1 +u2 x2 )+λ(x1 +x2 −1), (13)
the Lagrange function of the constrained optimization problem (6). The KuhnTucker necessity conditions are
2v1 (v1 x1 + v2 x2 ) + 2w1 (w1 x1 + w2 x2 ) − u1 + λ = 0,
2v2 (v1 x1 + v2 x2 ) + 2w2 (w1 x1 + w2 x2 ) − u2 + λ = 0,
x1 + x2 = 1,
9
which leads us to the following linear equation
1
2
2
(v1 − v2 ) + (w1 − w2 ) x1 = (u1 − u2 ) − (v1 − v2 )v2 − (w1 − w2 )w2 . (14)
2
If (v1 − v2 )2 + (w1 − w2 )2 6= 0 then we find that x∗ = (x∗1 , 1 − x∗1 ), where
x∗1
1
1
=
(u1 − u2 ) − (v1 − v2 )v2 − (w1 − w2 )w2 , (15)
(v1 − v2 )2 + (w1 − w2 )2 2
is the unique solution to equation (14). If v1 = v2 and w1 = w2 then from (14) we
find u1 = u2 , which would contradict the initial assumption that the two assets are
not identical. It can easily be seen that L00 (x∗ , λ) is a positive definite matrix in the
subset defined by
y = (y1 , y2 ) ∈ R2 : y1 + y2 = 0 .
So, x∗ is the unique optimal solution to (12), and if x∗ > 0 then x∗ is an optimal
solution to (4) with n = 2.
3
An algorithm
In this Section we provide an algorithm for finding an optimal solution to the nasset possibilistic portfolio selection problem (4). The algorithm will terminate in
o(n3 ) steps.
Step 1 Let c := +∞ and xc := [0, . . . , 0].
Step 2 Choose three points from the bag {(vi , wi , ui ) : i = 1, . . . , n} which have
not been considered yet. If there are no such points then go to Step 9, otherwise denote these three points by (vj , wj , uj ), (vk , wk , uk ) and (vl , wl , ul ).
Let (v1 , w1 , u1 ) := (vj , wj , uj ), (v2 , w2 , u2 ) := (vk , wk , uk ) and (v3 , w3 , u3 ) :=
(vl , wl , ul ).
Step 3 If
q1 r1
v 1 − v 3 w1 − w3
det
= det
= 0,
q2 r2
v 2 − v 3 w2 − w3
then go to Step 2, otherwise go to Step 4.
Step 4 Compute the first two component, [x∗1 , x∗2 ], of the optimal solution to (6)
using equation (10).
Step 5 If [x∗1 , x∗2 , 1 − x∗2 − x∗1 ] > 0 then go to Step 6, otherwise go to Step 2.
10
Step 6 If U∗ < c then go to Step 7, otherwise go to Step 2.
Step 7 Let c = U∗ , where U∗ is the optimal value of (6), and let
j-th
k-th
l-th
z}|{
z}|{
z}|{
∗
∗
xc = [0, . . . , x1 , 0, . . . , 0, x2 , 0, . . . , 0 x∗3 , 0, . . . , 0].
Step 8 Go to Step 2.
Step 9 Choose two points from the bag {(vi , wi , ui ) : i = 1, . . . , n} which have
not been considered yet. If there are no such points then go to Step 16,
otherwise denote these two points by (vj , wj , uj ) and (vk , wk , uk ). Let
(v1 , w1 , u1 ) := (vj , wj , uj ) and (v2 , w2 , u2 ) := (vk , wk , uk ).
Step 10 If (v1 − v2 )2 + (w1 − w2 )2 6= 0 then go to Step 9, otherwise go to Step
11.
Step 11 Compute the first component, x∗1 , of the optimal solution to (12) using
equation (15).
Step 12 If [x∗1 , x∗2 ] = [x∗1 , 1 − x∗1 ] > 0 then go to Step 13, otherwise go to Step 9.
Step 13 If U∗ < c then go to Step 14, otherwise go to Step 9.
Step 14 Let c = U∗ , where U∗ is the optimal value of (12), and let
j-th
k-th
z}|{
z}|{
∗
xc = [0, . . . , x1 , 0, . . . , 0, x∗2 , 0, . . . , 0].
Step 15 Go to Step 9.
Step 16 Choose a point from the bag {(vi , wi , ui ) : i = 1, . . . , n} which has not
been considered yet. If there is no such points then go to Step 20, otherwise
denote this point by (vi , wi , ui ).
Step 17 If vi2 + wi2 − ui < c then go to Step 18, otherwise go to Step 16.
Step 18 Let c = vi2 + wi2 − ui and let
i-th
z}|{
xc = [0, . . . , 0, 1 , 0, . . . , 0].
Step 19 Go to Step 16.
Step 20 xc is an optimal solution and −c is the optimal value of the original portfolio selection problem (4).
11
4
Example
We shall illustrate the proposed algorithm by a simple example. Consider a 3-asset
problem with A = 2.46 and with the following possibility distributions
r1 = (−10.5, 70.0, 4.0, 100.0),
r2 = (−8.1, 35.0, 4.4, 54.0),
r3 = (−5.0, 28.0, 11.0, 85.0)
and, therefore,
(v1 , w1 , u1 ) = (6.386, 1.359, 45.750),
(v2 , w2 , u2 ) = (3.469, 0.763, 21.717),
(v3 , w3 , u3 ) = (3.604, 1.255, 23.833).
It should be noted that the first asset may yield negative rates of return with
degree of possibility one. Usually, the support of fuzzy numbers representing the
possibility distributions of rates of return can not contain any return that is less than
-100%, because one can never lose more money than the original investment.
First consider the 3-asset problem with (v1 , w1 , u1 ), (v2 , w2 , u2 ) and (v3 , w3 , u3 ).
Since
q1 r1
2.782
0.105
det
= −1.352 6= 0,
= det
q2 r2
−0.135 −0.491
we get
x∗1
x∗2
=
1
0.124
0.800
0.259 0.427
,
=
×
0.373
0.044
−1.3522 0.427 7.751
and, since,
[x∗1 , x∗2 , x∗3 ] = [0.124, 0.373, 0.503] > 0.
we get (Step 7)
U∗ := −9.386 and x∗ := [0.124, 0.373, 0.503].
Thus [0.124, 0.373, 0.503] is a qualified candidate for an optimal solution to (3).
Let us consider all conceivable 2-asset problems (1, 2), (1, 3) and (2, 3), where
the numbers stand for the corresponding assets chosen from the bag
{(v1 , w1 , u1 ), (v2 , w2 , u2 ), (v3 , w3 , u3 )}.
Here we are searching for optimal solutions on the edges of the triangle generated
by the assets.
12
Select (1,2). Since
(v1 − v2 )2 + (w1 − w2 )2 = 8.864 6= 0,
we get
U∗ := −9.336 and [x∗1 , x∗2 ] = [0.163, 0.837] > 0.
Thus [0.163, 0.837, 0] is a qualified candidate for an optimal solution to (3).
Select (1,3). Since
(v1 − v3 )2 + (w1 − w3 )2 = 7.751 6= 0,
we get
U∗ := −9.352 and [x∗1 , x∗3 ] = [0.103, 0.897] > 0.
Thus [0.103, 0, 0.897] is a qualified candidate for an optimal solution to (3).
Select (2,3). Since
(v2 − v3 )2 + (w2 − w3 )2 = 0.259 6= 0,
we get
U∗ := −9.277 and [x∗2 , x∗3 ] = [0.171, 0.829] > 0.
Thus [0, 0.171, 0.829] is a qualified candidate for an optimal solution to (3).
Finally, we compute the utility values of all the three vertexes of the triangle
generated by the three assets:
v12 + w12 − u1 = −3.122,
and [1, 0, 0] is the corresponding feasible solution to (3).
v22 + w22 − u2 = −9.101,
and [0, 1, 0] is the corresponding feasible solution to (3).
v32 + w32 − u3 = −9.269,
and [0, 0, 1] is the corresponding feasible solution to (3).
Comparing the utility values of all feasible solutions we find that the only solution to the 3-asset problem is x∗ = [0.124, 0.373, 0.503] with a utility value of
9.386. The optimal risky portfolio will be preferred to the risk-free investment (by
an investor whose degree of risk-aversion is equal to 2.46) if rf < 9.386%.
13
5
Summary
In this paper we have considered portfolio selection problems under possibility
distributions and have presented an algorithm for finding an exact (i.e. not approximate) optimal solution to these problems. First we have proved that the boundary
of the set of feasible solutions (which is a convex polytope) must contain all optimal solutions to the problem. Then we have considered all possible sides, edges
and vertexes that could be generated from the given triplets and computed the optimal portfolios of (i) three assets that could generate a side, and (ii) two assets that
could generate an edge of the convex hull of all assets. Then we have compared
the utility values of all feasible solutions (i.e. solutions with non-negative weights)
and portfolios with highest utility value have been chosen as optimal solutions to
portfolio selection problem.
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[A12] Christer Carlsson, Robert Fullér and Péter Majlender, A possibilistic approach to selecting portfolios with highest utility score, FUZZY SETS AND
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With the introduction of fuzzy set theory by Zadeh [18] in 1965,
researchers began to realize that they could employ fuzzy set theory to manage portfolio in another type of uncertain environment
called fuzzy environment. Since then a lot of researchers began
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[A12], Inuiguchi and Tanino [5], Léon et al [6] and Tanaka and
Guo [15]. (page 34)
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Some fuzzy approaches to the portfolio selection problem have
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Carlsson [A12] introduced a possibilistic approach to selecting
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On the other hand, based on possibility theory [4] [18], a lot of researchers such as Carlsson, Fullér and Majlender [A12], Inuiguchi
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and Vercher [8] have devoted their efforts to the fuzzy portfolio
selection problem. (page 377)
A12-c4 Huang, X., Credibility based fuzzy portfolio selection, IEEE International
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18
Many scholars such as Watada [34], Tanaka and Guo [30], Tanaka,
Guo and Turksen [31], Parra et al [27] and Carlsson et al [A12]
have employed possibility measure to describe security returns
and extended Markowitzs mean-variance modelling idea in different ways. (page 159)
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