Fuzzy Portfolio Optimization

In this chapter, we present a brief overview of portfolio optimization. First, we discuss the classical mean-variance model of portfolio optimization developed by Markowitz. We then discuss the various extensions of the Markowitz’s model by considering alternative measures of risk, namely, semivariance, absolute deviation and semi-absolute deviation.

In this chapter, we discuss portfolio optimization models with interval coefficients, where the expected return, risk and liquidity of assets are treated as interval numbers. In addition, some realistic constraints such as number of assets held in the portfolio and the maximal and minimal fractions of the capital allocated to the various assets are considered. We present optimization models for portfolio selection in respect of three types of investment strategies, namely, conservative strategy, aggressive strategy and combination strategy.

In this chapter, we discuss a bi-objective fuzzy portfolio selection model that maximizes the portfolio return and minimizes the portfolio risk. We use an fuzzy interactive approach to solve the model so that the desired aspiration levels of the decision maker with regard to return and risk objectives are achieved as closely as possible.

In this chapter, we describe possibilistic programming approaches to portfolio optimization. First we briefly introduce the foundations of possibility theory. Then we describe the portfolio selection problem with fuzzy coefficients. The problem is a fuzzy counterpart of Markowitz model. The classical possibilistic programming approaches are described. They are the fractile optimization approach, the modality optimization approach and the spread minimization approach. We show that the reduced problems become simple linear programming problems and that the solutions obtained by those approaches suggest concentrated investments or semi-concentrated investments when fuzzy coefficients are non-interactive. To obtain diversified investment solutions, regret-based possibilistic programming approach is proposed. It is shown that the reduced problem is also an linear programming problem and that the solution can be a diversified investment solution. As the other way to obtain a diversified investment solution, three models to treat the interaction among fuzzy coefficients are described. The necessity fractile optimization approach and usual minimax regret approach are applied to portfolio selection problems with interactive fuzzy coefficients. It is shown that the reduced problems are solved by linear programming techniques and that more diversified investment solutions are obtained due to the interaction among fuzzy coefficients.

In this chapter, we present a hybrid bi-objective credibility-based fuzzy mathematical programming model for portfolio selection under fuzzy environment. The expected value and chance constrained programming techniques are used to formulate the mathematical model in which return, risk and liquidity are considered for measuring performance of an asset. The model seeks to maximize the portfolio return while minimizing the portfolio risk. The portfolio liquidity is considered as a constraint. To solve the fuzzy optimization model, a two-phase approach is discussed

In this chapter, we describe fuzzy portfolio selection models using five criteria: short term return, long term return, dividend, risk and liquidity. For portfolio return, we consider short term return (average performance of the asset during a 12-month period), long term return (average performance of the asset during a 36-month period) and annual dividend. This is done in order to capture subjective preferences of the investors for portfolio return. For a given expected return, the negative semi-absolute deviation is penalized which quantifies portfolio risk. Further, we categorize all individual investor attitudes towards bearing portfolio risk into one of the following two distinct classes: aggressive (weak risk aversion attitude) and conservative (strong risk aversion attitude). The nonlinear S-shape membership functions are employed to express vague aspiration levels of the investor regarding the multiple criteria used for portfolio selection.

In this chapter, we present fuzzy framework of portfolio selection by simultaneous consideration of suitability and optimality. Suitability is a behavioral concept that refers to the propriety of the match between investor preferences and portfolio characteristics. The approach described in this chapter for portfolio selection is based on multiple methodologies. We evolve a typology of investors using the inputs from a primary survey of investor preferences. A cluster analysis is done on the basis of three evaluation indices to categorize the chosen sample of financial assets into different clusters. Further, using analytical hierarchy process (AHP), we determine weights of the various assets within a cluster from the point of view of the investor preferences. The optimal asset allocation based on a mix of suitability and optimality is obtained using fuzzy portfolio selection models. The criteria used for portfolio selection are short term return, long term return, risk, liquidity and AHP weighted score of suitability.

In this chapter, we present an approach based on AHP and fuzzy multiobjective programming (FMOP) to attain the convergence of suitability and optimality in portfolio selection. We use a typology of investors with a view to discriminate among investors types and asset clusters categorized on the basis of three evaluation indices. The local weights (performance scores) of each asset within a cluster with respect to the four key criteria, namely, return, risk, liquidity and suitability are calculated using AHP. These weights are used as coefficients of the objective functions corresponding to the four criteria in the multiobjective programming model. The multiobjective programming model is transformed into a weighted additive model using the weights (relative importance) of the four key criteria that directly influence the asset allocation decision. These criteria weights are also calculated using AHP. To improve portfolio performance on individual objective(s) as per investor preferences, we use an interactive fuzzy programming approach.

Of late, the investors have shown great interest in socially responsible investment, also called ethical investment. Ideally, the investors may have a portfolio that is based not only on financial considerations but also incorporates a set of ethical values. The ethical investment movement that began from the USA in 1960s has gained tremendous momentum the world over recently. The growing instances of corporate scams and scandals have made it incumbent upon the investors to consider the quality of governance of corporations and ethicality of their conduct. Indeed, there has been a spate of reforms relating to corporate laws and capital markets all over the world. Also, the investors are becoming conscious of the desirability of ethical evaluation of the assets. The growing influence of institutional investors has reinforced this consciousness. The focus of this chapter is to present a comprehensive three-stage multiple criteria decision making framework for portfolio selection based upon financial and ethical criteria simultaneously. Fuzzy analytical hierarchy process (Fuzzy-AHP) technique is used to obtain the ethical performance score of each asset based upon investor preferences. A fuzzy multiple criteria decision making (Fuzzy-MCDM) method is used to obtain the financial quality score of each asset based upon investor-ratings on the financial criteria. Two hybrid portfolio optimization models are presented to obtain well diversified financially and ethically viable portfolios.

Given that not all the assets available in the market are appropriate for a given investor, it is desirable to stratify these assets into different classes on the basis of some predefined characteristics. Furthermore, using investor preferences, one needs to select some good quality assets from a given class to build an optimal portfolio. The focus of this chapter is to present a hybrid approach to portfolio selection using investor preferences in terms of selection of assets from a particular class that suits the given investor-type. The support vector machine (SVM) with radial basis function kernel is used to classify the assets into three classes. The optimal portfolio selection is achieved using a model that is based on four financial criteria: short term return, long term return, risk, and liquidity. A real coded genetic algorithm (RCGA) is designed to solve the portfolio selection model.