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Portfolio rebalancing model with transaction costs based on fuzzy decision theory

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Abstract

The fuzzy set is one of the powerful tools used to describe an uncertain environment. As well as quantifying any potential return and risk, portfolio liquidity is taken into account and a linear programming model for portfolio rebalancing with transaction costs is proposed. The level of return that an investor might aspire to, the risk and the liquidity of portfolio are vague in an uncertain financial environment. Considering them as fuzzy numbers, we propose a portfolio rebalancing model with transaction costs based on fuzzy decision theory. An example is given to illustrate the behavior of the proposed model using real data from the Shanghai Stock Exchange.

Introduction

In 1952, Markowitz [14] published his pioneering work which laid the foundation of modern portfolio analysis. Markowitz’s model has served as a basis for the development of modern financial theory over the past five decades. However, contrary to its theoretical reputation, it is not used extensively to construct large-scale portfolios. One of the most important reasons for this is the computational difficulty associated with solving a large-scale quadratic programming problem with a dense covariance matrix. Konno and Yamazaki [9] used the absolute deviation risk function to replace the risk function in Markowitz’s model and formulated a mean absolute deviation portfolio optimization model. It turns out that the mean absolute deviation model maintains the favorable properties of Markowitz’s model and removes most of the principal difficulties in solving Markowitz’s model. Simaan [21] provided a thorough comparison of the mean variance model and the mean absolute deviation model. Furthermore, Speranza [22] used the semi-absolute deviation to measure the risk and formulated a portfolio selection model.

Transaction cost is one of the main concerns for portfolio managers. Arnott and Wagner [2] found that ignoring transaction costs would result in an inefficient portfolio. Yoshimoto’s empirical analysis [24] also drew the same conclusion. Mao [13], Jacob [8], Patel and Subhmanyam [18] and Morton and Pliska [15] studied portfolio optimization with fixed transaction costs. Pogue [19], Chen et al. [5], and Yoshimoto [24] studied portfolio optimization with changeable transaction costs. Mulvey and Vladimirou [16] and Dantzig and Infanger [6] incorporated transaction costs into the multi-period portfolio selection model. Li et al. [11] gave a linear programming algorithm to solve a general mean variance model for portfolio selection with transaction costs. Due to changes of situation in financial markets and investors’ preferences towards risk, most of the applications of portfolio optimization involve a revision of an existing portfolio, i.e., portfolio rebalancing.

Usually, expected return and risk are two fundamental factors which investors consider. In some cases, investors may consider other factors such as liquidity. Liquidity has been measured as the degree of probability involved in the conversion of an investment into cash without any significant loss in value. Arenas et al. [1] took into account three criteria (return, risk and liquidity) and used a fuzzy goal programming approach to solve the portfolio selection problem.

In 1970, Bellman and Zadeh [3] proposed fuzzy decision theory. Ramaswamy [20] presented a bond portfolio selection model using fuzzy decision theory. A similar approach for portfolio selection using fuzzy decision theory was proposed by León et al. [10]. Using the fuzzy decision principle, Östermark [17] proposed a dynamic portfolio management model by fuzzifying the objective and the constraints. Watada [23] presented another type of portfolio selection model based on the fuzzy decision principle. The model is directly related to the mean–variance model, where the goal rate (or the satisfaction degree) for an expected return and the corresponding risk are described by logistic membership functions.

In this study, considering liquidity, we propose a linear programming model for portfolio rebalancing with transaction costs. Furthermore, based on fuzzy decision theory, a portfolio rebalancing model with transaction costs is proposed.

This paper is organized as follows. In Section 2, we introduce briefly fuzzy decision theory and the maximization decision. In Section 3, as well as quantifying any potential return and risk, liquidity of a portfolio is taken into account and a linear programming model for portfolio rebalancing with transaction costs is proposed. In Section 4, the investor’s vague aspiration levels are considered as fuzzy numbers. Based on the fuzzy decision theory, a portfolio rebalancing model with transaction costs is proposed. In Section 5, an example is given to illustrate the behavior of the proposed model using real data from the Shanghai Stock Exchange. A few concluding remarks are given in Section 6.

Section snippets

Fuzzy decision and maximization decision

Due to incomplete knowledge and information, it is not enough to use precise mathematics to model a complex system. In order to represent vagueness in everyday life, Zadeh [25] introduced the concept of fuzzy sets in 1965. Based on this concept, Bellman and Zadeh [3] presented fuzzy decision theory. They defined decision-making in a fuzzy environment with a decision set which unifies a fuzzy objective and a fuzzy constraint.

Suppose that fuzzy sets are defined on a set of alternatives, X. Let

Linear programming model for portfolio rebalancing with transaction costs

Suppose an investor allocates his/her wealth among n securities offering random rates of return. The investor starts with an existing portfolio and decides how to reallocate assets.

The expected rate of return ri of security i without transaction costs is given byri=1Tt=1Trit,i=1,2,,n,where rit can be determined by historical or forecast data.

Let x+=(x1+,x2+,,xn+) and x-=(x1-,x2-,,xn-), where xi+ is the proportion of the security i, i = 1, 2,  , n bought by the investor, xi- is the proportion of

Portfolio rebalancing model based on fuzzy decision theory

Since investment is generally influenced by disturbances to social and economical circumstances, an optimization approach is not always the best. In some cases, a satisfaction approach is much better than an optimization one. An investor always has aspiration levels for expected return and risk. In the real world of financial management, the expert’s knowledge and experience are very important in decision-making. Based on an experts’ knowledge, the investor may decide his/her aspiration levels

An example

In this section, we give a numerical example to illustrate the proposed portfolio rebalancing model. Assume that an investor chooses 30 different stocks from the Shanghai Stock Exchange for his/her investment. The exchange codes of the 30 stocks are given in Table 1.

The rate of transaction costs for stocks is 0.0055 in the two securities markets on the Chinese mainland. Assume that the investor already owns an existing portfolio and he/she will not invest additional capital during the portfolio

Conclusion

In addition to the more usual factors of expected return and risk, portfolio liquidity is considered in the portfolio rebalancing process. The turnover rates of securities are used to measure their liquidity. Considering all three factors, a linear programming model for portfolio rebalancing with transaction costs is proposed. An investor’s aspiration levels for the expected return: risk and liquidity are vague in an uncertain environment. The vague aspiration levels are considered to be fuzzy

Acknowledgements

Supported by the National Natural Science Foundation of China under Grant No. 70221001 and City University of Hong Kong under Grant No. 7100289. The authors are grateful to the two referees for their valuable comments and suggestions.

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