International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878 Volume 1 Issue 4, December 2013 Portfolio Rebalancing Model Using Fuzzy Optimization Manoj Jha 1 , Namita Srivastava 2 1,2 Department of Mathematics, Maulana Azad National Institute of Technology Bhopal, India Abstract: Uncertainty in stock market is a major issue for all the investor. Researchers have proposed portfolio selection as one of the measures to overcome uncertainty of stock market. The financial market is highly volatile and investor needs to change his/her investment strategy from time to time. Investor can construct portfolio using his/her own aspiration level with respect to return, dividend, risk, liquidity and transaction cost. In this paper portfolio is constructed and rebalanced using fuzzy optimization technique with user different aspiration level. Keywords: Portfolio selection, semi absolute deviation, fuzzy number, transaction cost, S-shape membership functions. 1. Introduction Uncertainty in stock market is a prolonged issue and there is no exact solution to solve the uncertainty problem. The stock market is an uncertain, complex, dynamic and noisy system. Researchers have proposed portfolio selection as one of the measures to overcome complication and uncertainty of stock market. Portfolio selection is a challenging problem as stocks do not follow a predefined or steady pattern. Investors consider return and risk as the two fundamental factors. The financial market is highly volatile and investor needs to change his/her investment strategy from time to time. In this paper portfolio is rebalanced in the presence of dividend, transaction cost and fuzzy turnover rate. The portfolio is rebalanced after every year because in Indian Stock Market the profit earned after a year is exempted from tax liability. The equity market with 25 Stocks is considered for creating and rebalancing the portfolio. 2. Related Work Harry Markowitz [1] in 1952 gave most significant theory for portfolio selection. The main principle of mean variance model was to use expected return of portfolio as investment return & variance as investment risk. Markowitz’s mean variance model has lead to development of number of models. Konno and Yamazaki [2] proposed mean absolute model as an alternative to mean variance model keeping all positive features of mean variance model. Markowitz [1] developed Mean Variance optimization model which became a very common quantitative model in finance today by for constructing an optimal portfolio. The MV model allocates each asset in the portfolio a proportion of the investment amount by taking into consideration each asset’s returns, risk and the correlations between the assets. Konno et.al [2] developed a linear programming model using Mean Absolute Deviation as risk function, thus replacing variance in Markowitz’s MV model. The LP model was however equivalent to Markowitz’s model when it possess a multivariate normal distribution of the asset returns. Markowitz [3] transformed the general mean-semi variance portfolio optimization problem into a general mean-variance optimization problem. Konno et.al [4] used mean-variance objective function and extended it to include skewness in portfolio optimization problem. Konno et.al [5] utilized a variant of the MV model by imposing fixed transaction cost and cardinality constraints on problems with up to 54 assets. Speranza[6] proposed semi-absolute deviation to evaluate risk in portfolio selection model. S.C.Liu et.al [7] proposed mean-variance-skewness model for portfolio selection with transaction costs assuming that the transaction cost is a V- shaped function of the difference between the existing portfolio and a new one. Chen et. al. [8] proposed Portfolio optimization of equity mutual funds with fuzzy return rates and risks. Liu [9] has solved portfolio optimization problem using fuzzy technique. He represented asset returns by fuzzy data. Chen et.al [10] proposed mean–variance–skewness model for optimal portfolio selection in intuitionistic fuzzy environment. Firstly, membership and non-membership functions of object and constrain functions were defined. Secondly, intuitionistic fuzzy programming model was presented based on intuitionistic fuzzy “min-max” operator. Pankaj Gupta et. al[11]., morphed mean–variance optimization portfolio model into semi-absolute deviation model, They 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. Yong Fang et. al. [12], proposed a linear programming model for portfolio rebalancing with transaction cost, they illustrated the behavior of the proposed model using data from the Shanghai Stock Exchange. 3. Problem Formulation It is assumed that an investor allocates his/her wealth among n assets. The investor starts with an existing portfolio and rebalances the portfolio while reallocating assets but does not invest additional capital during the rebalancing. It is assumed that i x is the proportion of total funds invested in the th i asset, i δ ε {0,1} indicates the absence or presence Paper ID: J201378 59 of 70