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Portfoliooptimizationbased Onreturn, Riskandliquiditywith Archive of SID Portfolio optimization based on return, risk and liquidity with the approach of Goal programming Ali Akbar Abedi Sharabiani [email protected] Mahsa Ghandehari [email protected] Department of Management, Faculty of Administrative Sciences and Economics, University of Isfahan, Iran. Abstract Purpose - The main purpose of this article is providing a generalized model for portfolio's optimization. In this paper, we have attempted to develop Markowitz mean- variance model and as well enter liquidity criterion into multi-criteria decision- making model which leads to the optimization with using of goal programming. Design/methodology/approach – The methodology that has been used in this study consists of three stages; first, the six-objective criterion has been obtained with entering liquidity criterion (including: variations,12-Months Performance, Return of Asset(ROA), bid ask spread, trade value and turnover ratio) into the model. Secondly, the multi-objective was studied with using of the goal Programming and modeling. Finally, the portfolio has been optimized with using of the collected information about criterion which were gathered from data of Tehran Stock Exchange (TSE) and with creating goals related to investors with using of LINGO software analyzed data and specified investments share of investors in any industry. Findings – Liquidity criteria could be considered as important variables for investors because of investor’s portfolios, optimization of portfolio and reduced risk. Originality/Value – Since Liquidity is an important criterion for investors in reducing portfolio risk, and the MCDM approach has been constructed with entering Liquidity in the Markowitz model that the MCDM approach could be optimized with using of Goal programming, finally. Therefore, innovation of this article is due to constructing the MCDM approach that focused on liquidity criterion and also optimizing it with goal programming. Keywords: Portfolio Optimization, Liquidity, Goal programming, Multi-Objective www.SID.ir Archive of SID 1. Introduction In his portfolio selection, Markowitz assumes that all investors have their options based on the two opposite criteria of return and risk (Markowitz, 1959). However it is also possible that investors add some other criteria or measures into their portfolio model and pursue different goals in addition to risk and return. Multi-objective optimization methods has been studied with different techniques such as genetic algorithm, goal programming and adaptive scheduling with random constraints.(Abdelaziz et al.,Yun et al.,2001) Scenario Planning is a new method for designing portfolio under uncertainty which can make investment decisions easier for investment companies that do portfolio planning in uncertainty situations (Hanafizadeh et al.,2011). Fuzzy approach has been used to select the portfolio and future probable returns under uncertainty, ( Hasuike et al., 2009). Smimou and Thulasiram (2010) applied a simple parallel algorithm for portfolio large-scale problem solving which cause to efficiency in portfolio selection process. Ehrgott and his co-workers(2004) developed Markowitz mean- variance model and applied 5 secondary goals instead of mean-variance. Ralph and Yue (2005) introduce multi-objective model for portfolio selection in which they propose other criteria such as dividend, liquidity, social responsibility and other criteria instead of returns or risk. Liquidity is one of the criteria that have been considered by investors in different aspects and also research results depict that the expected return of stocks is in inverse relation with liquidity (Marshall, 2006; Datar and Naik, 1998; Jacoby et al., 2000; Amihud, 2002). Liquidity can be defined as "The ability of a high volume transaction quickly with low cost and low impact of prices» (Weimin, 2005). Andrew and Wierzbicki (2003) used mean variance and liquidity for portfolio optimization in three dimensions of liquidity filtering, liquidity constraints, and mean objective function. Their studies also show that liquidity is effective in reducing portfolio risk. Given the multidimensional nature of liquidity, it can be measured with different measures, in this article we intend study Ehrgott 5 objective model by changing some variables and adding criterion called liquidity to this model. We also will use criteria that Andrew has used for liquidity used to develop MCDM approach so the model based on these criteria can be effective in the optimal portfolio Also, it can with MCDM modeling help to get better responses on the efficient frontier because we expect the liquidity criteria be effective in reducing portfolio risk. In next we attempt to examine Markowitz model and multi-objective criteria in the model and then do it’s modeling with using goal planning technique and optimize it, and finally we do conclusions with an example of available data from Tehran Stock Exchange. 2. Markowitz model In 1952 Harry Markowitz introduces the basic model of portfolio that provided the basis for modern portfolio theories. Before Markowitz, investors were familiar with concepts of return and www.SID.ir Archive of SID risk but could not measure them. They knew that diversity is good and should not be "put all their eggs in a same basket" However Markowitz was the first one who expressed Portfolio theory scientifically(Jones,1996). Markowitz mean variance model is formulated as follows (Ehrgott et al., 2004). xi xixj , M Here M is a number of available assets, represents the share of investment in the ith asset and represents the share of investment in j asset .. i ϵ {1, ..., M} and are expected return related to the i asset and is the covariance between i and j assets. Markowitz then posed the concept of efficient frontier. Efficient portfolio is the optimal combination securities as portfolio risk for a given rate of return is minimum. Curve shown in Figure 1 shows the efficient set of the portfolio and a point on the curve is selected as the best portfolio that has more return in terms of same risk or has less risk in terms of specific return. Fig1. The Risk-Return efficient frontier www.SID.ir Archive of SID 3. Multi-criteria model and hierarchy of objectives Portfolio optimization has been welcomed by various researchers. For example, Ehrgott used multi -objective model instead of Markowitz average returns. They had applied 5 secondary objectives in their model. We optimize the model with changing in it and adding liquidity variable in this model: Fig 2. Example for an objective hierarchy based on the Markowitz model. 6 available secondary objectives in the above model can be formulated as follows: xi⩾0 , i=1,…,M. Here M is a number of available assets, Xi represent investment share in ith asset. 3.1. 12-Months Performance 12-Months Performance is showed with: where i ϵ {1, ..., M} and ri calculates asset price changes from beginning to end of year (month to month), and with assuming that there isn’t deviated data between data, yearly performance is calculated from monthly average returns The first criteria of model will be as follow( Ehrgott et al.,2004): = Note that is the price of ith asset in the next month and is the price of ith asset in the earlier month, also in this function and the subsequent functions represents shares invested in i asset, and if we show it as f(x), this will be as follow: F(x1)= www.SID.ir Archive of SID 3.2. Volatility The risk of a particular investment is the standard deviation of changes in asset price in past. Price volatility related to price changes can be measured with different methods. Here volatility for 12 month is calculated. Thus the second criteria of the model are as follow (Ehrgott et al., 2004): F(x2) = Such in the Markowitz model, σij is covariance between returns of i and j asset and j,iϵ , M is a number of available assets, xi represents the share of investment in i assets and xj represents the share of investment in j assets. 3.3. Return of Asset Return of asset ratio is a measure that shows how much incomes the company have earned from in possession asset, and makes it possible to find out how much resources are consumed and the management to which extent has used optimally from limited resource, when this ratio is much higher, company performance is better and it will be showed with following relationship: ROA = Note that NI is Company’s(i) net income in year t and TA is total assets of Company(i) in year t, and xi represents the investment share in the company(i), and if we show that as a function, it will be as follow: F(x3) = 3.4. Bid Ask Spread The difference between the lowest proposal sales price and highest proposal purchase price is shown with BASi , t . If the distance between these two values is low or close to zero, liquidity capability will be more .Bid Ask Spread is obtained from following relation: (Ryan ,1996; Stoll, 1989) BASi,t = In this relation, is the proposal sell price for each stock at time t , is the proposal purchase price for i stock at time t, and if we show that as a function, will be as follow: F(x4) = www.SID.ir Archive of SID 3.5. Turn over volume This criteria is obtained by dividing the number of shares traded on the number of diffused stocks of company, higher measure of this ratio indicates high liquidity of ith company’s stock. This criteria is obtained from following relation (Andrew and Wierzbick, 2003): TOV= In this relation is the number of each company’s traded stocks at time t and the number of diffused stocks of each company at time t, and if we show that as a function, will be as follow: F(x5)= 3.6. Dollar Volume This is an old criterion for measuring liquidity and is obtained from multiplying the number of share volume (SV) in stock price (P), higher value of this criteria indicating high liquidity of shares and is obtained from following relationship: DV=(Share Volume × Price) In this relationship the daily last price is used to calculate the price, o and if we show that as a function, will be as follow: F(x6)= 4.
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