چند دوره بهینه سازی سبد سرمایه گذاری تحت اقدامات احتمالی

A single-period portfolio selection theory provides optimal tradeoff between the mean and the variance of the portfolio return for a future period. However, in a real investment process, the investment horizon is usually multi-period and the investor needs to rebalance his position from time to time. Hence it is natural to extend the single-period fuzzy portfolio selection to the multi-period case based on the possibility theory. In this paper, we propose the possibilistic expected value and variance for the terminal wealth with fuzzy forms after T periods by using the central value operator. Classes of multi-period possibilistic mean-variance models are formulated originally under the assumption that the proceeds of risky assets are fuzzy variables. Besides, we apply a particle swarm optimization algorithm to solve the proposed multi-period fuzzy portfolio selection models. A numerical example is given to illustrate the performance of the proposed models and algorithm.

Portfolio selection is seeking the best allocation of wealth among different assets. Numerous studies on portfolio selection are based on the probabilistic mean-variance methodology first proposed by Markovitz (1952), such as, Perold (1984), Pang, (1980), Best (2010) and Maringer and Kellerer (2003). It has also gained widespread acceptance as a practical tool for portfolio optimization. However, it is a single period model which makes a one-off decision at the beginning of the period and holds on until the end of the period, while it was natural to extend Markowitz's work to multi-period portfolio selections, such as Smith (1967), Mossin (1968), Merton (1969), Samuelson (1969), Fama (1970), Hakansson (1971), Elton and Gruber (1974), Francis and Kirzner (1991), Dumas and Luciano (1991), Östermark (1991), Grauer and Hakansson (1993), Pliska (1997), Li and Ng (2000) and Chen (2005). The literatures mentioned often used the probability distribution of asset returns. However, in the real world, the financial market behavior is affected by several non-probabilistic factors such as vagueness and ambiguity (see (Lacagnina and Pecorella, 2006)). The returns of assets are usually affected by many factors including economic, social, political and people's psychological factors as proposed by Huang (2011). Decision-makers are usually provided with information which is characterized by vague linguistic descriptions such as high risk, low profit, and high interest rate. In these cases, it is impossible for us to get the precise probability distribution we need. Furthermore, even if we know all the historical and current data, it is difficult that we predict the future return as a fixed value. Hence we need to consider that the future return has ambiguousness. There are several approaches dealing with ambiguous situations. On the one hand, some authors characterize uncertain distributions by defining a confidence region of their first two moments, so that the portfolio is robust against such uncertainty, see Pflug and Wozabal (2007) and Wozabal (2012). On the other hand, fuzzy set theory and Possibility theory, proposed by Zadeh (1978) and advanced by Dubois et al. (1988), may help to solve problems in uncertain and imprecise environments. In particular, in the field of portfolio selection, investors are faced with forecasting the performance of the assets they manage. Given the uncertainty inherent in financial markets, analysts are very cautious in expressing their guesses. Hence, there exist a lot of published works in the field of finances, which incorporate the approach of fuzzy set theory. Watada (1997) and León et al. (2002) discussed portfolio selection by using fuzzy decision theory. Inuiguchi and Tanino (2000) introduced a possibilistic programming approach to the portfolio selection problem under the minimax regret criterion. Carlsson et al. (2002) and Zhang et al. (2009a) introduced a possibilistic approach for selecting portfolios with the highest utility value under the assumption that the returns of assets are trapezoidal fuzzy numbers. Zhang et al. (2009b) discussed portfolio selection problem under possibilistic mean-variance utility and presented a SMO algorithm for finding the optimal solution. Zhang et al. (2010) proposed a risk tolerance model for portfolio adjusting problem with transaction costs based on possibilistic moments. Gupta et al. (2008) applied multi criteria decision making via fuzzy mathematical programming to develop comprehensive models of asset portfolio optimization for the investors pursuing either of the aggressive or conservative strategies. In addition, Bilbao-Terol et al. (2006), Zhang and Wang (2008), Li and Xu (2009), Huang (2010) and Zhang et al. (2011) discussed the portfolio selection problems in a fuzzy uncertain environment by following the ideas of probabilistic mean-variance model. However, the researches mentioned above are on the single-period portfolio selection problems in fuzzy environment. There have been few literatures on multi-period fuzzy portfolio selection based on possibility theory. Zhang et al. (2012) presented a mean-semivariance-entropy model for multi-period portfolio selection based on possibility theory, in which the risk level is characterized by the sum of the lower possibilistic semivariance of portfolio return in each period. The aim of this paper is to develop a multi-period mean-variance portfolio selection model with fuzzy returns based on possibility theory. By using the central value operator introduced by Fullér and Majlender (2004) and Fullér et al., 2010a, Fullér et al., 2010b and Fullér et al., 2011, we formulate the possibilistic expected value and possibilistic variance for the terminal wealth after T periods. A class of multi-period possibilistic mean-variance models is formulated originally. Moreover, an efficient solution is achieved for this class of fuzzy multi-period portfolio selection formulation, which makes the derived investment strategy an easy implementation task. The organization of this paper is as follows. Section 2 introduces some basics of possibility distributions. In Section 3, we develop a class of multi-period portfolio selection models based on possibility theory. The optimization models are converted into crisp forms when the return of risky assets is taken as symmetrical triangular fuzzy variables. An efficient solution is derived in Section 4 to generate the optimal portfolio policy. In Section 5, an example is given to illustrate the behavior of the proposed models and algorithm. This paper concludes in Section 6 with some suggestions for future study.

Financial markets are usually very sensitive and the influences on the returns of assets include general economic, industry and the performances of concerned company, which are affected by human's subjective intention. Fuzzy models are useful to handle the vagueness and ambiguity of the predictions. The possibilistic mean-variance approach has been extended in this paper to multi-period fuzzy portfolio selection problems. With the central value operator and its properties, we have derived the possibilistic expected value and possibilistic variance value for the terminal wealth after some periods. Under the assumption that the proceeds of risky assets are symmetrical triangular fuzzy variables, a class of multi-period possibilistic mean-variance models and their crisp forms are formulated originally. Moreover, we formulate a particle swarm optimization algorithm for this class of multi-period fuzzy portfolio selection problems, which makes the derived investment strategy an easy implementation task. An example is given to show the whole idea of the proposed models and algorithm. Although the assumption of non-interactivity is unduly restrictive, the determination of multi-period portfolio selection in the presence of possibilistic distributions gives insight into the nature of dynamic portfolio optimization in uncertain environment. And it is also pointed out that the covariance between its marginal distributions becomes zero for any weighting function if all γ-level sets of a fuzzy set are symmetrical. Moreover, the theory of interactive fuzzy numbers is still under development. Some of the future research subjects are to investigate the multi-period portfolio optimization when the returns within and/or between periods are interactive, and to investigate the constrained multi-period portfolio problems with transaction costs based on the possibility theory and their efficient solution methodology.