Fuzzy Data Decision Support in Portfolio Selection: a Possibilistic Safety-first Model

Vast pools of historical financial information are available on economies, industry, and individual companies that affect investors' selection of appropriate portfolios. Fuzzy data provides a good tool to reflect investors' opinions based on this information. A possibilistic mean variance safety-first portfolio selection model is developed to support investors' decision making, to take into consideration this fuzzy information. The possibilistic-programming problem can be transformed into a linear optimal problem with an additional quadratic constraint using possibilistic theory. We propose a cutting plane algorithm to solve the programming problem. A numerical example is given to illustrate our approach.

principle can be combined. Watada(2001) presented another type of portfolio selection model based on the fuzzy decision principle. This model is directly related to the mean-variance model, where the goal rate for an expected return and the corresponding risk are described by logistic membership functions. Tanaka et al(2000) formulate fuzzy decision problems based on probability events. Carlsson et al(2002) studied a portfolio selection model in which the rate of return of securities follows the possibility distribution. Enriqueta et al(2007) presented a fuzzy downside risk approach for managing portfolio problems in the framework of risk-return trade-off using interval-valued expectations. Lacagnina and Pecorella (2006) proposed a multistage stochastic soft constraints fuzzy model to solve a portfolio management problem. Liu and Wang (1998) constructed a pair of two-level mathematical programming models, based on which the upper bound and lower bound of the objective values were obtained. Recent advances of portfolio selection model considered integration of various muti-criteria decision making models such as fuzzy AHP in Tiryaki and Ahlatcioglu (2009) and expert systems such as Smimou et al(2008). Xiaoxia(2007) gave a new defnition of risk for random fuzzy portfolio selection, Weiguo(2007) proposed two kinds of portfolio selection models based on lower and upper possibilistic means and possibilistic variances. Very limited research has been attempted on possibilistic mean-variance safety-first portfolio models.
In this paper, we assume the securities which has fuzzy rate of return and develop a possibilistic mean-variance safety-first portfolio model. Vague input data can be specified for these quantitative risk factors using historical data based on historical data quantile described in existing work such as Zmeskal(2001) and Wu et al(2009). Using the possibilistic means and variances, the possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problem there are no special standard algorithms, we propose a cutting plane algorithm to solve them.
Using our proposed approach, fuzzy variance and covariance are derived directly from fuzzy numbers, which are different from the probability theory where variance and covariance are derived from a great deal of historical data such as Markowitz's mean variance framework and the safety-first portfolio model. This will, on the one hand, reduce the computation complexity and on the other hand, overcome the hurdle of semi-positive covariance matrix as required in a great deal of portfolio models based on the probability theory.
The rest of the paper is organized as follows. In Section 2, we briefly introduce possibilistic mean variance approach and possibilistic mean safety-first approach and present a possibilistic mean variance portfolio selection model with safety-first. In Section 3 we present the proposed possibilistic mean variance safety-first portfolio selection model and the solution approach. In Section 4, a example is given to illustrate the proposed model. The last section concludes the paper.

Mean variance portfolio selection model with safety-first
The expected losses, conditional on the states where there are large losses, may be higher sometimes. The mean-variance approach encourages risk diversification, but the mean safety-first approach discourages risk diversification sometimes. The mean-variance approach not only controls the downside risk of security return, but also bounds the possible upside gains. In contrast, the mean safety-first approach only controls the downside risk of security return. Another limitation of both approaches is that the underlying distribution of the rate of return is not well understood, and there are no higher degree information is utilized except means, covariances (variances), so we propose the following mean variance safety-first portfolio selection model:

Possibility theory
Possibility theory was proposed by Zadeh(1978) and advanced by Dubois and Prade (1998) where fuzzy variables are associated with possibility distributions in a similar way that random variables are associated with probability distributions in probability theory. The possibility distribution function of a fuzzy variable is usually defined by the membership function of the corresponding fuzzy set. We call a fuzzy number a  of any fuzzy subset R with membership function , respectively. Based on the concepts and techniques of possibility theory founded by Zadeh(1978), we consider in this paper the trapezoidal fuzzy numbers which are fully determined by quadruples 1 2 3 4 ( , , , ) r r r r r   of crisp numbers such that 1 2 3 4 r r r r    . Their membership functions can be denoted by: We note that the trapezoidal fuzzy number is a triangular fuzzy number if 2 3 r r  .

Fuzzy numbers and operation.
The sum of two trapezoidal fuzzy numbers is also a trapezoidal fuzzy number, and the product of a trapezoidal fuzzy number and a scalar number is also a trapezoidal fuzzy number. The sum of   , then the possibility value of the first trapezoidal fuzzy number being no larger than the second is defined as (Liu, Iwamura,1998): The following lemma holds: Theorem 1. Assume the trapezoidal fuzzy number 1 2 3 4 ( , , , ) r r r r r   , then for any given confidence level  (2001) It is easy to see that if ) , , , Then the crisp possibilistic covariance value of

Model formulation
In standard portfolio models uncertainty is handled in the form of randomness using probability theory. One of the main difference between the possibility and probability measures is that probability is additive whereas possibility is subadditive, which means for the possibility measure that the possibility of an event being partitioned into smaller events, is less than or equal to the sum of the possibilities of the smaller events. The subadditive property of the possibility measure fits the requirements of risk metrics in financial theory. Moreover, in probability, estimation of prior probability distributions of parameters such as mean and variance are usually obtained from judgment. Determination of the probability distribution of the parameters is difficult (Leon,Liern, Vercher,2002). Using probability theory can hardly account for the uncertainty in the probability distribution of the uncertain variables. In contrast, measurement of the uncertainty in a possibilistic model can be done by the sum of the possibilities of an event and its complement minus one.
So we will assume that the rates of return on assets are modeled by possibility distributions rather than probability distributions. Applying possibilistic distribution may have two-fold advantages (Inuiguchi,1992): first, the knowledge of the expert can easily be introduced to the estimation of the return rates; Second, the reduced problem is more tractable than the result of the stochastic programming approach. The rate of return on www.ccsenet.org/cis Computer and Information Science Vol. 3, No. 4;November 2010ISSN 1913-8989 E-ISSN 1913 120 the j th asset will be represented by a fuzzy number j r in our method, and we will consider only trapezoidal possibility distributions for simplicity. In addition, we denote the disaster level by the trapezoidal fuzzy number Theorem 1 provides a simplified deterministic form of Model (FMVSF). Theorem 2: Solving (FMVSF) is equivalent to solving the following problem: , which completes the proof.
Remark: An investor yields his optimal portfolio by giving the value of V  ,  , w and solving the resulting model (FMVSF).

Cutting plane algorithm
Problem (FMVSF) is a linear optimal problem with an additional quadratic constraint. For such problems there are no special standard algorithms. Of course, one could treat this problem with general methods of nonlinear optimization, but this would lead to local solutions. In this paper, we propose to solve Problem (FMVSF) using a cutting plane algorithm, which was first introduced by Kelley (1960) and Cheney and Goldstein(1959) Further assume that G the feasible set of (FMVSF), is nonempty and contained in T. let contains a subsequence that converges to an optimal solution of (FMVSF).
Proof: First we observe from

Numerical example
In this section,we present a numerical examples to demonstrate our proposed approach: for a 3-security practical problem which allows us to show a step-by-step computation using the proposed approach.
We first consider a market risk manager's decision of choosing 3-securities: IBM, GE and MSFT. The manager structuring an equity portfolio only has vague views regarding equity return scenarios described as "bullish", "bearish" or "neutral". The manager forms such views as a result of the subjective or intuitive opinion of the decision-maker on the basis of information available at a given point in time. It is recognized that a fuzzy set can be used to characterize the range of acceptable solutions to the portfolio selection problem under this circumstance.
The manager may specify the following possibility distribution for expected rates: The above trapezoidal fuzzy data can also be yielded by fuzzifying historical stochastic data. The approach for stating vague input data using historical data is similar to an interesting and practically applicable method based on historical data quantile employed in Zmeskal(2001) and Wu et al(2009 Here, variance and co-variance are derived directly from fuzzy numbers, which is different from the probability theory where variance and co-variance are derived from a great deal of historical data such as Markowitz's mean variance framework and the safety-first portfolio model. Therefore, comparing to existing research based on probability theory, computation complexity is reduced. Moreover, the problem of semi-positive co-variance matrix is handled.
We continue to our computation and let 0.5, 0.01,