Fuzzy Multiple Objective Decision Making

161

13

Novel Algorithm

for Uncertain

Portfolio Selection

The mean-variance approach was proposed by Markowitz (1952) to deal with the port-

folio selection problem. A decision maker can determine the optimal investing ratio for

each security based on the historical return rate. The formulation of the mean-variance

method can be described as follows (Markowitz, 1952, 1959, and 1987):

…

∑∑

∑

∑

σ

µ≥

=

≥∀=

==

=

=

xx

st xE

x

xi

n

min

.. ,

1,

01,,

.

ij ij

j

n

i

n

ii

i

n

i

i

n

i

11

1

1

(13.1)

where μ

i

denotes the expected return rate of the ith security, σ

ij

denotes the covari-

ance coefcient between the ith security and the jth security, and E denotes the

acceptable least rate of the expected return.

It is clear that the accuracy of the mean-variance approach depends on the accu-

rate values of the expected return rate and the covariance matrix. Several methods

have been proposed to forecast the appropriate acceptable return rate and variance

matrix such as the arithmetic mean method (Markowitz, 1952, 1959, and 1987) and

the regression-based method (Elton and Gruber, 1995). However, these methods

derive only the precise expected return rate and covariance matrix and do not con-

sider the problem of uncertainty.

Since the decision maker wants to determine the optimal portfolio strategy to

gain maximum prots, how can we ignore future uncertainty? We should note that

the possible area of the return rate and the covariance matrix should be derived to

allow the decision maker to determine the future optimal portfolio selection strategy.

In addition, these methods are based on the large sample theory and cannot provide

satisfactory solutions in small sample situations (Elton et al., 1978).

In this chapter, the possible area of the return rate and the covariance matrix are

derived using asymmetrical possibilistic regression. Then, the Mellin transformation is

162 Fuzzy Multiple Objective Decision Making

employed to calculate the uncertain return rate and the variance with specic distribu-

tion. Finally, the optimal portfolio selection model can be reformulated based on these

concepts. In addition, a numerical example is used to illustrate the proposed method and

compared with the conventional mean-variance method. On the basis of the simulated

results, we can conclude that the proposed method can provide a better portfolio selec-

tion strategy than the conventional mean-variance method by considering uncertainty.

13.1 POSSIBILISTIC REGRESSION

The possibilistic regression model was rst proposed by Tanaka and Guo (2001) to

reect the fuzzy relationship between the dependent and the independent variables.

The upper and the lower regression boundaries are used in possibilistic regression

to reect the possibilistic distribution of the output values. By solving the linear

programming (LP) problem, the coefcients of the possibilistic regression can be

obtained easily.

Next, we describe the use of the possibilistic regression model (Tanaka and Guo,

2001) to obtain the uncertain return rate and the variance. To obtain accurate results,

we extend the symmetrical fuzzy numbers to the asymmetrical fuzzy numbers. The

general form of a possibilistic regression can be expressed as:



=+ ++ =

′

yA

xAAxAx

nn011

(13.2)

where A

i

is an asymmetrical possibilistic regression coefcient denoted as

−+acaa c(,,)

iiLi

ii

R

. To achieve minimum uncertainty, the tness function of the

possibilistic regression can be dened as:

…

∑

=

′

+

′

=

cx cx

ac

LR

Jmin (|

||

|)

jj

jm

,

1, ,

(13.3)

In addition, the dependent variable should be restricted to satisfy the following

two equations:

≥

′

−

′

j

ax

cx

L

y

||

,

jj

(13.4)

≤

′

+

′

ax cx

R

y

||

.

jj j

(13.5)

On the basis of the concepts above, we can obtain the formulation of a possibilistic

regression model:

…

…

∑

=

′

+

′

≥

′

−

′

≤

′

+

′

=

≥

=

cx cx

ax cx

ax cx

cc

LR

L

R

L

J

st y

yj

m

min (||||)

.. ||,

||

,1

,,

,.

ac

jj

jm

jj j

jj j

R

,

1, ,

0

(13.6)