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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, .

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 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

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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 . 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

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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

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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)=

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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) =

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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. Multi-objective approach and goal programming There are several methods for multi-objective problem solving. One of these methods is goal programming which for first time were presented by Charnz Kooper in 1961. Goal programming provides the way to move toward several objectives (even conflicting objectives). Although Markowitz is the most reliable model for securities portfolio selection, but technical problems and computational and also not taking the demands of investors in this model making techniques have caused that experts have suggested multi-criteria techniques. (Lee and Lerro, 1973) General form of goal programming is as follow: Minimize Z=

Subject to:

In this model are deviated variables, and goal is minimization or maximization of these variables, so negative deviation in goals of profit types, and positive deviation in goals of cost types must be minimized. is as weights associated with each variable and is used if a goal be more

important than other. Xj are decision variables and k is the number of desired goals and k are

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coefficients of decision variables or desired criteria. is the number of goals specified from investor and we seek an answer to reach the specified goals (Hiller and Lieberman, 1980). Here we tried to achieve a model that has 6 objectives and model it with goal programming technique, with assuming that the problem constraints are linear, the model will be the 6 objective modeling will be as follow:

subject to:

2.

7.

, f={1,…,6}, M={1,…,18}

Objective (limitation) number 1: is maximizing the annual performance (expected return). Objective (limitation) number 2: is concerned about asset volatility existed in portfolio that is Max in market optimistic condition and Min in market pessimistic condition. Objective (limitation) number 3: This limitation is related to the return of assets, and objective is maximizing the return of assets. Objective (limitation) number 4: This objective represents Bid Ask Spread and if market is effective, this objective is Max, and in condition of asymmetry of information this is Min Objective (limitation) number 5: represents the turn over volume and in optimistic conditions is Max and in pessimistic conditions is Min. Objective (limitation) number 6: This shows the dollar volume and in optimistic conditions is Max and in pessimistic conditions is Min. Objective (limitation) number 7: This limitation represents the diversification and shows the maximum acceptability for and investment in the each share, the amount of investment (wi) in each

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asset can be determined through the interview method or weighting methods such as entropy or combination of them. Objective (limitation) number 8: This objective represent the investment ratio in each share that is equal to 1. 5. Data and Results Suppose a person or an Investment Fund wants to have a basket of different company’s shares to select the optimal portfolio and by considering three criteria of returns, liquidity and risk we determine the optimal portfolio: Statistical population of research are accepted companies in Tehran Securities Exchange (TSE) and our selected sample are TSE top 50 companies which was selected from the end of 2010, to select the final sample we impose restrictions that are: 1. Financial year of Companies to end the last month of each year 2. Trading Symbol of company don’t close more than one month 3. Required data is available 4. Companies activity, isn’t financial or investment 5. Companies are selected from years between 2006-2009. With these restrictions our final sample consisted of 18 companies in 4 years so the size of final sample is 72, their information collected through website of the Tehran Securities Stock, and calculations performed by Excel software. Table 1 Estimate 6 Objective criteria for selected companies Company 12-Months Volatility Return Bid Turn Dollar Performance of Ask over Volume Asset Spread Volume (×107Rials)

Farsit Doroud 13 11 21 2.9 110 1.71 Fars and Khuzestan Cement 23 9 14 2.1 10 1.06 Iranian Copper Industries 14 12 38 3.1 100 40.17 Tehran Cement 12 11 13 2.5 20 1.20 16 11 13 2.5 40 0.23 Saipa Azin 14 9 15 2.6 10 0.90 Northern Cement 15 13 12 2.2 12 0.50 Minerals and Industrial Chadormalu 17 12 44 2.2 12 13.98

Electric Khodro Sharg 12 12 12 2.2 20 0.22 Gol-Gohar Iron Ore 14 10 39 2.3 10 21.58 Sobhan Pharmaceutical 15 10 11 2.3 30 1.37 Jaber Ebne Hayyan Pharmaceuticals 15 12 21 1.9 20 0.69

Saipa 17 12 14 1.3 68 7.91 Diesel 14 9 11 3.2 20 6.08 Mehrkam pars 13 10 15 2.4 40 0.33 Mashhad Wheel 11 9 13 1.9 20 0.04 Loghman Pharmaceutical 13 9 15 1.9 7 0.27 18 8 21 2.3 30 0.27

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According to the 6 objective model which used in previous section, we have show data about each objective in Table 1, in this table, columns are related to 18 companies in the sample and rows are related to 6 objectives existed in portfolio development and have been calculated using the formulas described in the previous section. In order to modeling data from the table, at first step we obtained to weights (Wj) of objective function thorough entropy method, also we allocated each of these 18 companies in 6 industries, to acquire investor goals and to determine a person or an investment fund have invested what percentage of its capital in these 18 companies, then we obtained the percentage of investment for each industry, using interview technique from active investors in market and with minor adjustments in the associated weights of each industry. Therefore, the modeling related to goal programming is as follow: Minimize Z=

Subject to:

X2+X4+ X5+X7 - = 0.13 (8)

X8 = 0.16 (9)

X18 (10)

X11+X12+X17 0.11 (11)

X6+X9+X13+X14+X15+X16 (12)

, f={6,…,12}

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Data analysis was performed using the LINGO software and we could obtain investment percentage in these 6 industries which is related to the 18 companies, and is specified in

Table 2 Determine the percentage of investment in each industry Company Industry Type of Percentage of investment Farsit Doroud, Iranian Non-metallic minerals, Metals 30 Copper Industries, Gol- Gohar Iron Ore 13 Fars and Khuzestan Cement and Chemical Cement, Tehran Cement, Northern Cement Tolypers

Minerals and Industrial Mine 16 Chadormalu

Pars Oil products 10

Sobhan Medical 11 Pharmaceutical,Jaber Ebne Hayyan Pharmaceuticals, Loghman Pharmaceutical

Saipa Azin, Electric Iran Khodro (machine) 20 Khodro Sharg, Saipa, , Mehrkam pars, Mashhad Wheel

In Table 2,The First column represents the number of companies existed in the sample. The second column represents the industry for each company and the third column shows the percentage invested in each share. LINGO output shows that if invest investors or investment funds to invest 30% of their capital in the Non-metallic minerals, Metals, 13% in the Cement and Chemical industry, 16% in the Mine industry, 10% in the Petroleum products industry, 11% in the Medical industry and 20% in the Iran Khodro(machine) industry, Then Investment risk will reduce Considering liquidity criteria. 6. Conclusions and recommendations Although Markowitz model is the most reliable model for securities portfolio but because of its technical and computational problems and neglecting the investor demands, In this paper we paid attention to liquidity criteria in addition to return and risk. By entering this criteria and combine them with the average variance we develop MCDM approach that had 6 objectives. This six goals

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are: 12-Months Performance, Volatility, Return of Asset, Bid Ask Spread, Turn over Volume, Dollar Volume. we used goal programming technique to optimize the 6 objectives model. In this programming, we presented goals for the investor and MCDM modeling approach we covered the investor goals and our claim was that individual investors or investment funds with developing similar portfolios and with attention to multiple criteria reduce the risk of complex. The results showed that in selected sample is Most of the investment in Non-metallic minerals & Metals And least amount of investment in Petroleum products. This study can be useful for investors and Companies so that for optimizing portfolio, consider different criteria to reduce the risk of portfolio and to have the highest satisfaction of their investments. for Future researchers is recommended to consider other liquidity criteria to reduce portfolio risk. Limitations However this study was not free limitations. In this study selected sample has been elected from 50 top companies in Tehran stock exchange and selected sample was small and the other hand number of selected companies were manufacturing companies and investment companies not been considered. References Abdelaziz F., Aouni B., Fayedh R. (2005), “Multi-Objective Stochastic Programming for Portfolio Selection”,(2005), European Journal of Operational Research, Vol. 1, No. 1, pp. 1-13. Amihud, Y. (2002),”Illiquidity and stock returns: cross-section and timeseries effects”. Journal of Financial Markets, Vol. 2,No. 5,pp. 31–56. Andrew W., Constantin p. and Wierzbicki, M.( 2003),” it is 11 pm- do you know your liquidity is? the mean-variance liquidity frontier”, Journal of investment management,Vol. 1,No. 1,pp. 55-93. Datar, V.T., Naik, N.Y., Radcliffe, R. (1998), “Liquidity and stock returns: an alternative test”. Journal of Financial Markets , Vol. 1,No. 1,pp. 203–219. Hanafizadeh, P.,Kazazi,A.and Jalili Bolhasani,A. (2011), “ Portfolio design for investment companies through scenario planning”, Management Decision, Vol. 49 Iss: 4 pp. 513 – 532. Hasuike, T., Katagiri, H., & Ishii, H. (2010), “Portfolio selection problems with random fuzzy variable returns”, Fuzzy Sets and Systems 160, PP.2579–2596. Hillier, Frederick S. and Lieberman, Gerald J. (1980), Introduction to Operations Research, 3d. ed., Holden-Day, Inc. Jacoby, G., Fowler, D.J., Gottesman, A.A. (2000), “The capital asset pricing model and the liquidity effect: a theoretical approach”. Financial Markets , Vol. 3,No. 3,pp. 69–81. Jones, charles P. (1996), Investments Analysis and Management , 5 d. ed.,John Wiley & Sons, New York. Lee, Sang M., and Lerro, A J.)1973(, "Optimizing the Portfolio Selection for Mutual Funds"., The Journal of Finance, Vol. 28, Issue 5, pp. 1087–1101. Markowitz, H. (1959), ”Portfolio Selection: Efficient Diversification of

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Investments”. John Wiley & Sons, New York. Marshal, B. R. (2006), “Liquidity and stock returns: Evidence from a pure order-driven market using a new liquidity proxy”; International Review of Financial Analysis ,Vol. 5,No. 15,pp. 21– 38. Ehrgott, M., Klamroth, K. and Schwehm, C. (2004), “An MCDM approach to portfolio optimization”. European Journal of Operational Research , Vol. 1,No. 155,pp. 752–770. Ralph, E.S., Yue Qi & Hirschberger, M. (2005), “Suitable-Portfolio Investors, Frontier Sensitivity, and the Effect of Multiple Objectives of GeorgiaAthens, on Standard Portfolio Selection”, Terry College of Business, University Georgia 30602-6253 USA. Ryan, H. (1996), "The Use of Financial Ratios as Measures of Determinants of Risk in The Determination of The Bide-Ask Spread", Journal of Financial and Strategic Decisions. Smimou, K., Thulasiram, R.K. (2010), “ A simple parallel algorithm for large-scale portfolio problems”, The Journal of Risk Finance, Vol. 11 Iss: 5,pp. 481 – 495. Stoll, H. (1989), " Inferring The Components of the Bid Ask Spread: Theory and Emprical tests", The Journal of Finance ,Vol. 44, No. 1: 115-134. Weimin, Liu. ( 2005), “A liquidity-augmented capital asset pricing model”, Journal of Financial Economics, Vol. 1,No. 1,pp. 1-41.

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