Proceedings

2. FUZZY MIXED SECURITIES PORTFOLIO SELECTION MODEL
We are going to describe the management method for the decision taker’s
expectations, by highlighting the interpretation of his rational behaviours (Albrecht,
2003; Bădescu et al, 2005). Thus, because investments are generally influenced by
changes in social and economical factors, an approach towards optimization is not
always the best solution. In many cases, a “satisfactory” approach from the investor’s
perspective is preferred, rather than an approach towards optimization. An investor
always has different levels of aspiration for the anticipated profit and risk. In the real
world of financial management, the experts’ knowledge and expertise are very
important with taking decisions.
Relying on the experts’ knowledge, an investor can decide his level of aspiration for
the portfolio’s anticipated profit and risk. Watada (2001) proposed a logistic function,
a sigmoid membership function, for expressing the levels of aspiration of an
individual, concerning the anticipated profit and risk. The sigmoid membership
function is
1
f ( x)
=
1 + exp( −αx)
.
We consider that it is more appropriate to use the logistic function in order to reveal a
vague/fuzzy level of the objective which an investor can consider (Bellman et al.,
1970; Konno et al., 1993). According to the maximization principle and using the
variance to measure the portfolio risk, Watada (2001) proposed a fuzzy portfolio
selection model which extends Markowitz’s Mean-variance model for the fuzzy case.
Regarding the suggested portfolio rebalancing model, the two objectives are taken
into consideration (profit and risk), as well as the limitations of the portfolio’s
liquidity. Since the anticipated profile, risk and liquidity are vague and uncertain, we
use the sigmoid membership function introduced by Watada (2001) to express the
aspiration level of the anticipated profile, risk and portfolio liquidity. Using the risk
function of the semi-absolute deviation to measure the portfolio risk, we suggest a
rebalancing model for the fuzzy portfolio, which is based on the Bellman-Zadeh
maximization principle (Bellman et al., 1970).
The membership functions for the objective for anticipated profile, risk and liquidities
are given as follows:
a) The membership function for the objective „the portfolio’s anticipated profit”:
1
μ r ( x)
=
1 + exp( −α
( E ( r(
x))
− r
where r is the inflexion point where the membership function takes the value 0.5 and
M
α can be given by the investor, according to its own degrees of satisfaction for the
r
anticipated profile. r M
~ 339 ~
r
represents the central aspiration level for the anticipated profile
returned by the portfolio. Figure 2 shows the membership function for the established
profit objective.
M
))