Fuzzy portfolio selection using fuzzy analytic hierarchy process
Introduction
In multiple-criteria (or attribute) decision-making problems, a decision maker (DM) or (DMs) often needs to select or rank alternatives that are associated with intangible or conflicting attributes. These problems arise in many real-world situations. For example, in financial problems, extrinsic attributes, such as economic, political, social and technological factors; intrinsic attributes, such as profitability and size; and investor’s objectives, such as profit and security are considered in the selection of a satisfactory portfolio.
The portfolio selection problem has been one of the most important issues in modern finance since the 1950s. In 1952, Markowitz proposed his pioneering work, the mean–variance method for the portfolio selection problem. Several papers dealt with the problem using both quantitative and qualitative analysis methods. One of the hot research topics in this area is the use of fuzzy set theory. Fuzzy set theory is a powerful tool used to describe an uncertain environment with vagueness, ambiguity or some other type of fuzziness, which appears in many aspects of financial markets, such as the unpredictable behavior of financial managers [41].
This paper addresses the huge task of ranking and choosing from among a large set of stocks in a fuzzy environment. It thus demonstrates the usefulness of fuzzy methodology in financial problems. To do this, we first deal with two constrained fuzzy AHP methods given by Enea and Piazza [9]. We revise the first of these methods in order to correct some of its fallacies and obtain a new version, which we denote, the revised constrained fuzzy AHP method (RCFAHP). Then we apply the RCFAHP and the second method of Enea and Piazza [9] to the task of picking stocks on the Istanbul Stock Exchange (ISE). The methodology used for the hierarchy construction is based on the paper of Saaty et al. [33].
In portfolio selection through the conventional AHP [33], the basic approach is to set up a model that incorporates the firm’s behavior (high, medium, or low risk), the risk class of the investor (high, medium or low), the investor’s objectives, and extrinsic and intrinsic factors. The portfolio manager, as the decision maker, has a large set of criteria for selecting stocks. These criteria depend on various interdependent primary and secondary factors. The problem is to compare the various factors and determine their priorities. The decision maker will then be able to compare each list of stocks with this criteria list and determine the portfolio. To make this comparison, he/she will need a scale by which to compare the factors pairwise. The scale (1–9) is used in AHP and reflects the decision maker’s belief as to which of two factors or criteria is more important (and to what degree), relative to some quality that both of them share.
AHP is widely used for multi-criteria decision making and has successfully been applied to many practical problems [32]. In spite of its popularity, this method is often criticized for its inability to adequately handle the inherent uncertainty and imprecision associated with the mapping of the DM’s perceptions to exact numbers. Traditional AHP requires exact or crisp judgements (numbers). However, due to the complexity and uncertainty involved in real world decision problems, decision makers might be more reluctant to provide crisp judgements than fuzzy ones. Furthermore, even when people use the same words, individual judgments of events are invariably subjective, and the interpretations that they attach to the same words may differ. Moreover, even if the meaning of a word is well-defined (e.g., the linguistic comparison labels in the standard AHP questionnaire responses), the boundary criterion that determines whether an object does or does not belong to the set defined by that word is often fuzzy or vague. This is why fuzzy numbers and fuzzy sets have been introduced to characterize linguistic variables. A linguistic variable is a variable whose values are not numbers but words or sentences from a natural or artificial language. Linguistic variables are used to represent the imprecise nature of human cognition when we try to translate people’s opinions into spatial data. The preferences in AHP are essentially human judgments based on human perceptions (this is especially true for intangibles), so fuzzy approaches allow for a more accurate description of the decision-making process [5]. A number of methods have been developed to handle fuzzy AHP.
The rest of the paper is organized as follows. Section 2 provides a literature review for portfolio selection and the fuzzy AHP methodology. Section 3 summarizes two constrained fuzzy AHP methods given by Enea and Piazza [9] and presents our modifications of their first Constrained Fuzzy AHP method. Section 4 applies the revised first method (RCFAHP) and the second method of Enea and Piazza [9] to the problem of stock selection at the ISE and explains the hierarchy constructed in this paper, which is adapted from [33]. Finally, Section 5 draws some general conclusions.
Section snippets
Literature review for portfolio selection
In the literature, there are several approaches to constructing a portfolio. Historically, the mean–variance model is the first example of a portfolio optimization problem, and it is credited to Markowitz, who presented his ideas in 1952 [23]. This model is important because mean–variance analysis provides a basis for the derivation of the equilibrium model known variously as the Capital Asset Pricing Model (CAPM), Sharpe–Lintner model, black model and two-factor model [34]. Furthermore, by
Constrained fuzzy AHP methods
In this section we will present some modifications to the first constrained fuzzy AHP method of Enea and Piazza [9], which we call the revised constrained fuzzy AHP method (RCFAHP). For this purpose, first we need to review some basic fuzzy concepts and summarize the two constrained fuzzy AHP methods given by Enea and Piazza [9].
An application to ISE (Istanbul Stock Exchange)
In this section, we will show an application of RCFAHP and Enea and Piazza’s second method to the selection of stocks on the ISE. First of all, we will explain how to construct the hierarchy, which is adapted from [33].
Results and comparisons
In this study, our main objective was to rank a set of stocks in a fuzzy environment and thus to demonstrate the usefulness of fuzzy methodology in financial problems. The ranking or preference numbers serve as a guide to how to allocate the available monetary resources among the given stocks. Here this is accomplished by modifying the first method of Enea and Piazza and then by showing how the Constrained Fuzzy AHP methods—both the RCFAHP and the second method of Enea and Piazza—can be used
Acknowledgements
Both authors are very indebted to the Editor-in-Chief, Professor Witold Pedrycz, and the anonymous referees for their valuable comments and suggestions that helped us for improving our work.
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