The 3rd International Fuzzy Systems Symposium” (FUZZYSS'13), October 24-25, 2013, in Istanbul, Turkey.
A Novel Fuzzy Goal Programming Approach for Optimal Investment Decisions Ozan Kocadağlı and Rıdvan Keskin
Abstract— In the investment process, investors mostly encounter conflicting objectives. The multi-objective programming methods allow investors to handle these objectives simultaneously. Fuzzy goal programming is one of the most efficient methods used in the investment decision making since it assigns the fuzzy goals to all objectives in addition to priority among each others. In this paper; portfolio risk, return and Beta coefficient are defined as fuzzy goals, and then their fuzzy membership functions are constituted for behaviors of different investors in accordance with Market trends. By means of fuzzy goal programming approach that take into accounts priority levels among Goals and behaviors of different investors, a novel portfolio selection model is developed. In the application section, the optimal portfolios are constituted to determine the invest ratios of the stocks traded in ISE30 Index. Finally, the proposed portfolio selection models are analyzed in terms of the performances over test periods.
I. INTRODUCTION
I
N finance theory, the principal phenomenon is risk. For this reason, the evaluation of risky assets is one of the major research tasks in finance for years. In the development of portfolio theory, Markowitz [1] defined risk in terms of a well-known statistical measure known as the variance. Specifically, Markowitz quantified risk as the variance about the expected return of an asset. Although the total risk of an asset can be measured by its variance, this risk measure can be divided into two general types of risk: systematic risk and unsystematic risk. William Sharpe [2] defined systematic risk as the portion of an asset’s variability that can be attributed to a common factor, and the portion of an asset’s variability that can be diversified away as unsystematic risk [3]. In context of measuring the systematic risk, Capital Asset Pricing Model (CAPM), developed by William Sharpe [2] and John Lintner [4], afterward extended by Fischer Black [5], widely used in applications. According to this model as known the SharpeLintner Black mean-variance CAPM, the typical measure of asset riskiness is the beta coefficient that is known as systematic risk measure compares the variability of an asset’s historical returns to the market as a whole. That is, beta measures the expected change of an asset for every percentage change in the benchmark index [6]. While making investment decisions, investors are concerned only with the systematic risk, because the unsystematic risk is diversified Ozan Kocadağlı is with Department of Statistics, Mimar Sinan Fine Arts University, Istanbul, TURKEY (corresponding author to provide phone: 090212-246-0011 (5511); fax: 090-212-261-1121; e-mail: ozankocadagli@ msgsu.edu.tr). Rıdvan Keskin with Faculty of Economics and Administrative Sciences, Department of Management, Hitit University, Çorum, TURKEY (e-mail:
[email protected]).
away by a well-balanced portfolio. For this reason, Beta is only concern that investors have when they value securities [7]. The attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about how to measure risk and the relation between expected return and risk [8]. Although the well-known classic portfolio selection models such as mean-variance model of Markowitz [1] and meanabsolute deviation model of Konno and Yamazaki [9] minimize the risks defined in the their objective function at given return levels, there are prominent criteria that have to be taken into account like the liquidity, the currency exchange risk, the movement of financial market. When the liquidity and the currency exchange risk are overlooked in the investment process, the movement of financial market can be considered with Beta coefficient with respect to investment strategies. For this reason, Beta should be used as a goal or a restriction in any portfolio selection model [10]. However, decision makers encounter the natural uncertainties when they evaluate risk, return and Beta. These uncertainties arise being described and measured these quantities. In order to overcome this kind of uncertainty, the fuzzy membership functions are useful tools. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables Zadeh [11]. The other issue faced in decision making process is how to handle risk, return and beta simultaneously because they conflict with each others. Therefore, they can be considered conflict objectives. In this case, it is possible to assign the fuzzy goals to these criteria. Thus, the related problem can be transformed into Fuzzy goal programming. Goal programming, developed by Charnes and Cooper [12], is a mathematical programming technique that allows handling the multiple objectives simultaneously. Bellman and Zadeh [13] improved a basic framework for decision making in a fuzzy environment. Zimmermann [14] extended fuzzy linear programming approach to a conventional multi-objective linear programming problem. Afterwards, Narasimhan [15] and Hannan [16] extended the fuzzy set theory to the field of goal programming. The other references of Fuzzy Goal Programming are Rubin and Narasimhan [17], Tiwari, Dharmar, and Rao [18], Wang and Fu [19], Chen and Tsai [20], Yaghoobi and Tamiz [21], Hu et al. [22]. In context of constructing portfolio selection models; fuzzy linear and nonlinear programming, fuzzy goal programming methods play an important role due to handling a couples of criteria simultaneously. In the fuzzy programming
The 3rd International Fuzzy Systems Symposium” (FUZZYSS'13), October 24-25, 2013, in Istanbul, Turkey. applications to portfolio selection problems; Parra et al. [23], Watada [24], Fang et al. [25], Kocadağlı [10], [26], [27], Zarandi and Yazdi [28], Gupta et al. [29] improved multiobjective models for the multi-criteria cases. The purpose of this study is to constitute a portfolio selection model in which all criteria is taken into accounts like fuzzy goals by means of Fuzzy Goal Programming. In order to set this model, initially the membership function of risk is defined by solving the mean- variance model of Markowitz [1] and then ones of return and Beta are constructed with respect to market return and trend. By using these membership functions into the fuzzy goal programming model introduced by Wang and Fu [19], a novel portfolio selection model is improved. In the application section, increasing and decreasing periods in ISE30 Index are handled, and then the portfolio selection models are set by using membership functions of fuzzy goals established for different investor behaviors according to market trends in the related periods. After this, the optimal portfolios are determined by solving these models for different investment strategies. Finally, the proposed optimal portfolios are analyzed in terms of the performances over test periods. II. FUZZY GOAL PROGRAMMING MODEL According to the decision theory, the characters of decision makers can be divided into three categories of risk-averse, risk-seeking and risk-neutral as shown in Fig 1. U i ( Ax
U i ( Ax
MembershipFunction
1 u21
Risk -averse
u51
u22
Risk -neutral
u52
u23
Risk -seeking
u53
0
x1 x 2
x3
x4
x5
x6
In Fig. 1, the satisfaction level of the decision maker related to the fuzzy goals is denoted with ( x) . From Fig. 1, it can be inferred that if decision makers achieve their goals at least (at most) to a certain level, a unit less than (more than) that level will cause a lower degree of satisfaction for a riskseekers than that for a risk-averse. According to Wang and Fu [19], the general fuzzy goal programming structure can be written as follows: Model 1:
Gik1 : giK1 (x1 , x 2 ,..., x n ) Ax i Bik1
i1 I1 , k I 4
(1)
Gik2 : giK2 (x1 , x 2 ,..., x n ) Ax i Bik2
i2 I2 , k I4
(2)
G : g (x1 , x 2 ,..., x n ) Ax i B
i 3 I3 , k I 4
(3)
1
k
2
k i3
k
K i3
3
Pk G i j I j , j 1, 2, 3, k ij
k i3
k I4
(4)
(5)
I1 1, 2, ..., l1 , I2 1, 2, ..., l2 , I3 1, 2, ..., m, I 4 1, 2, ..., K
where G ikj : ith goal of problem j (j = 1, 2 3),
g iKj : Linear or nonlinear function, Bikj : The target value is desired to reach by gi (x) at kth priority level. Pk : The set of goals with kth level of priority. Here, the symbols “ ”, “ ” and “ ” denote the fuzzified aspiration levels with respect to the linguistic terms of "at most", "at least", and "around" referred to as diffrent types of problems in Eq. (1), (2) and (3) respectively [18]. In this frame, there is a priority among fuzzy goals. That is, it is not possible that the lower levels of priority are considered unless the higher levels of priority are satisfied. In other words, if P1>P2>… >PK and 1>λ1>λ2>….>λK>λ>0, then λ must satisfy the two following conditions: If the first (K-1) levels are satisfied, then λ= λK and λk = 1, k {1, 2 ..... K- 1}. If the first ( l -1) levels ( l K-2) are satisfied, once
l < l, then there are no feasible λ and λk, k { l +1, 2, ... K}. To meet preference priority of a decision maker, there is a condition that the satisfaction level of goals in priority Pi has to be larger than the satisfaction level of goals in priority Pi+1. This restriction causes a punishment cost for each one of priority level. If the preference priority among fuzzy goals is P1>P2>… >PK, then the punishment costs corresponding this priority becomes like 0 < M1 < M2 < … < MK. Wang and Fu [19] suggests to a punishment cost related to kth priority as Mk = (10K)k-1. From here, if Mk is the penalty cost of kth level of goals with 0
