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\title{Linear Programming on Portfolio Optimization: Empirical Evidence from Bist Mining Industry Index}
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             \author[1]{Fatih  Konak}

             \author[2]{Bugra  Bagci}

             \affil[1]{  Hitit University}

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\date{\small \em Received: 6 December 2015 Accepted: 5 January 2016 Published: 15 January 2016}

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\begin{abstract}
        


A lot of methods are improved for the portfolio optimization within classical approach. Quadratic programming, one of these methods, has many disadvantages, so alternative methods are studied to improve. MAD Method, an improved new method, is converted portfolio optimization problem into a linear programming problem. MAD Method is demonstrated and a case study is done by using stock certificate which belongs to BIST Mining Sector.

\end{abstract}


\keywords{BIST mining sector, linear programming, portfolio optimization.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
any models have been developed in the financial literature under the heading of portfolio optimization and these models were named as traditional portfolio and modern portfolio optimization. While attempting to minimize the risk of the portfolio through diversification of securities too much, not minding interrelationship between them, in traditional portfolio management approach; portfolio optimization has been made through mean-variance model  {\ref (Markowitz, 1952}:77-91) in modern portfolio management approach.\par
Harry Markowitz is called as the founder of the theory of modern investment with his study, by the name of "Portfolio Selection", that he presented as phd dissertation in the year 1952. In this study, Markowitz targeted the selection of the lowest-risk portfolio corresponding to a certain return on the basis of mean variances.\par
Various scientists attempted to develop portfolio selection model on the basis of mean -variance model. Tobin  {\ref (1958)},  {\ref Sharpe (1964)} ve Lintner (1965) adapted real-life constraints to the model, such as investor's decision on percentage of portfolio consisting of risky assets, borrowing-lending situation, short-term sales, transaction costs and taxes.  {\ref Brennan (1971)} investigated the subject of borrowing and lending; Turnbull (1977) investigated the subjects of personal taxation, uncertain inflation, nonmarket assets.  {\ref Levy (1983)} and Schnabel (1984) dealt with short-term sales problem.\par
The difficulties caused by the increase in the number of securities for expected return of optimum portfolio and determination of variance were overcome by single index model developed by  {\ref Sharpe (1963)} and multiple index models developed by  {\ref Perold (1984)}.The studies conducted on mean-variance model revealed the Capital Assets Pricing Model (CAPM), which is both mathematical and logical extension of the meanvariance model  {\ref (Harrington, 1983)}.  {\ref Sharpe (1964)},  {\ref Lintner (1965)} and Mossin (1966) added moving riskfree financial asset to the model on the basis of Markowitz's studies. \hyperref[b3]{Konno and Yamazaki (1991)} proposed the mean absolute deviation (MAD) model, which is also a portfolio optimization model, alternatively to the meanvariance portfolio optimization model of Markowitz. MAD Model has used mean absolute deviation instead of variance intended to be minimized in the objective function of mean-variance model. Thus, portfolio selection problem was degraded from a quadratic program to a linear program \hyperref[b10]{(Simaan, 1997} {\ref (Simaan, : 1437))}.\par
In this study, it is intended to inform about MAD model proposed by \hyperref[b3]{Konno and Yamazaki (1991)} for also solution of large-scale portfolio optimization problems that can't be solved with Markowitz's classical mean -variance model and investigate its some properties. In the practice section, portfolio optimization was performed through MAD model for trading securities at BIST. 
\section[{II.}]{II.} 
\section[{Mean Absolute Deviation (mad) Model}]{Mean Absolute Deviation (mad) Model}\par
MAD model is an alternative method simplifying Markowitz's classical formulation by using absolute deviation as a risk scale. When these two mathematically equivalent formulas have been considered in terms of calculation, significant differences are noticed between them. As well as approach of risk measurement through variance converts the problem to quadratic programming problem, absolute deviation approach degrades the problem to linear programming problem  {\ref (Konno ve Koshizuka, 2005: 893)}.\par
Konno and Yamazaki revealed that mean absolute deviation of normal distribution was proportional with standard deviation of that. Consequently, MAD model and Markowitz model show the same activity under the multi variability of return of assets. Provided that these returns of (R\textunderscore 1,R\textunderscore 2,?,R\textunderscore n ) suggest multi variability normal distribution, then these two measures are the same. In other words, when returns of (R\textunderscore 1, R\textunderscore 2,?,R\textunderscore n ) suggest multi variability normal distribution, it means that minimizing the function ??(??) is minimizing the function ? (x) at the same time \hyperref[b10]{(Simaan, 1997} {\ref (Simaan, : 1437))}. Furthermore, Rudolf, Wolter and Zimmermann (1999) revealed that minimizing mean deviation was equivalent to maximizing expected utility in case of avoiding risk \hyperref[b8]{(Rudolf et al. 1999}: 85 103). 
\section[{a) Mathematical Model}]{a) Mathematical Model}\par
?? ?? , (?? = 1,2, ? ??), ??. represents the random variable implying return of asset and ?? ?? , (?? = 1,2, ? , ??) ??. representing the ratio to be invested to asset, total return of portfolio consisting of assets is calculated as follows:??(??) = ? ?? ?? ?? ?? ?? ?? =1\par
Here, ??. represents return of asset, ?? ?? ; asset's price at the end of the period, ?? ???1 ; asset's price at the beginning of the period?? ?? = (?? ?? ??? ???1 ) ?? ???1\par
, is calculated with this formula.\par
Standard deviation used as the scale of variance and risk in standard portfolio analysis is calculated as follows:??(??) = ??[(??(??) ? ??[??(??)]) 2 ] ??(??) = ???(??)\par
Mean absolute deviation used as a measure of the risk in MAD model is defined as follows  {\ref (Konno ve Yamazaki, 1991: 523 524)}.??(??) = ?? ??? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? =1 ? ?? ?? =1 
\section[{??}]{??}\par
This function is also the objective function that will be minimized.????????(??) = ?? ??? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? =1 ? ?? ?? =1 ?? ? ????? ?? ??? ?? ? ?? ?? ?? =1 ? ?? ?? = 1 ?? ?? =1 0 ? ?? ?? ? 1 , ?? = 1,2, ? , ??\par
??: mimimum return desired by investor ?? ???? : ?? is the acquired return for time period ?? = (1,2, ? , ??) and it is assumed that this return could be acquired from historical data or predictions for future; besides, expected value of random variable would converge these data resulted mean?????? ?? ?? = ????? ?? ? = ? ?? ???? ?? ?? ??=1\par
??(??) converges in the way given below :?? ??? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? =1 ? ?? ?? =1 ?? = 1 ?? ? ????? ???? ? ?? ?? ??? ?? ?? ?? =1 ? ?? ??=1 ?????? ?? ???? = ?? ???? ? ?? ?? ; ?? = 1,2, ? , ??; ?? = 1,2, ? , ??\par
In this case, the problem is converted to the minimizasyon problem given below:????????(??) = ? ?? ?? ???? ?? ?? ?? ?? =1 ? ?? ??=1 ?? ? ?? ?? ?? ?? ? ?? ?? ?? =1 ? ?? ?? = 1 ?? ?? =1 0 ? ?? ?? ? 1 , ?? = 1,2, ? , ??\par
By means of equations mentioned above, objective function became linear. This model will be equivalent to such a linear programming problem given below.????????(??) = ? ?? ?? ?? ??=1 ?? ?? ?? + ? ?? ???? ?? ?? ? 0 , ?? = 1,2, ? , ?? ?? ?? =1 ?? ?? ? ? ?? ???? ?? ?? ? 0 , ?? = 1,2, ? , ?? ?? ?? =1 ? ?? ?? ?? ?? ? ?? ?? ?? =1 0 ? ?? ?? ? 1 , ?? = 1,2, ? , ?? ? ?? ?? = 1 ?? ?? =1\par
III. 
\section[{Aplication}]{Aplication}\par
In this section, portfolio optimization was carried out through that MAD model was applied to the actual data obtained from BIST. In the application part of this study, returns of mining sector concerned shares included in the SIST index between the dates 04.01.2010 to 4.12.2014 were calculated on the basis of daily closing prices and MAD model was applied to these data. It was assumed in the model that investors would create their portfolios with fully risky investments and risk-free investment and short selling wouldn't be allowed. MAD model was applied to the data by being written in an econometric package program. The shares used in the study belong to ?hlas Mining (IHMIN), Ipek Natural Energy (IPEKE), Koza Mining (KOZAA), Koza Gold (KOZGO), Park Electric and Mining (PRKM).The statistics concerning these 5 shares have been given in Table \hyperref[tab_0]{1}. For an investor targeting different returns and will make an investment on the basis of MAD model; in which ratio from which shares he should invest to his portfolio have been shown in Table  {\ref 2}. 
\section[{Table 2 : Minimum Risk Ratio of Shares by a different Target Return}]{Table 2 : Minimum Risk Ratio of Shares by a different Target Return} 
\section[{Table 2 (Con't) : Minimum Risk Ratio of Shares by a different Target Return}]{Table 2 (Con't) : Minimum Risk Ratio of Shares by a different Target Return}\par
The expected return and variances of these portfolios acquired for different returns using MAD model have been shown in Table \hyperref[tab_1]{3}.  When examined the tables, it has been seen that the shares that will make contribution to portfolio in terms of profit that will be created for investor's desire and natural expectation are IHMIN, KOZGO and PRKM yielding positive return. Accordingly, it is obvious that other two shares will not make any contribution for profit growth. Two mainly recommended shares in the portfolios created by MAD model are KOZGO and PRKM. The reason why IHMIN securities haven't been included in the optimum portfolios is that there is so much risk due to excessive fluctuations between beginning of period and end of period related market closing prices of the years selected. Remembering that through MAD model it is intended to minimize the equation that is objective function, yielding mean absolute deviation; naturally, the shares to be selected are supposed to minimize the risk as well as increase profit. Therefore, not only the shares with positive returns but also the ones with minimum risk were selected in optimum portfolios created by MAD model. When these considerations taken into account, it has been noticed that MAD model has yielded positive results and can be used in daily life.\par
On the other hand, comparing the variation coefficients (coefficient of change), another criterion in the selection of shares; it can be decided that which shares should be included in the portfolio that will be created and which ones shouldn't. Variation coefficient is defined as follows. 
\section[{?????????????????? ?????????????????????? = ???????? ????????????}]{?????????????????? ?????????????????????? = ???????? ????????????}\par
Since standard deviation has been used as a risk scale in portfolio optimization, ?????????????????? ?????????????????????? = ???????????????? ?????????????????? ???????????? Table \hyperref[tab_4]{6} includes standard deviation and value of returns of the shares belonging to mine sector in question and the results of variation coefficient calculated on the basis of these values. It was found in the evaluations mentioned above that IPEKE and KOZGO securities' returns were negative, accordingly they shouldn't be included in the portfolios created. On the basis of that, when other three shares compared, the risks of the ones that must be included in portfolio, are supposed to be small as much as possible. When their variance coefficients were compared, the order from high to low value would be respectively IHMIN, KOZAA, and PRKM. On the basis of that securities with small variation coefficient should be included in the optimum portfolio for portfolio optimization, since IHMIN return's variation coefficient is so high, KOZAA and PRKM securities are supposed to be included in the optimum portfolio to be created as much as possible. When all of these taken into account, it is seen that portfolio optimization carried out with MAD model comply with daily life and not conflict with other portfolio selection criteria or methods in finance sector. 
\section[{IV.}]{IV.} 
\section[{Conclusion}]{Conclusion}\par
Mean-Variance Model creating major changes in Markowitz's portfolio selection understanding is a currently used quadratic programming model revealing interrelationships between assets through risk-return variation, accordingly, taking into account diversification and the evaluation of entire portfolio. MAD model proposed by Konno and Yamazaki is one of the models proposed in time to overcome several problems encountered in the selection of portfolio. In MAD model, which is a linear programming model, risk is expressed with mean absolute deviation, not with variance.\par
In this study, MAD model was theoretically introduced. In the study performed with actual data, daily values of returns of the securities between January 2010-December 2014 of mine sector being included in SIST's index were used and portfolios were acquired on the basis of different target returns through application of the model.\par
Model was tested firstly comparing the fluctuations between values of returns of actual data and market closing prices; secondly, variation coefficient comparison method, which is another criterion used for selection of share in portfolio optimization, was used. It was seen according to both these two considerations that there wasn't any conflict with the consequences of the portfolios created through MAD model.\par
MAD model brought a new perspective to the classic portfolio optimization problem and degraded the problem to linear programming problem by defining the risk on the basis of mean absolute deviation. Thus, model has brought along the advantages such as transaction easiness, not requiring distribution assumption, ability to be reformulated for various constraints. The only disadvantage of MAD model encountered in the literature is that it can lead to prediction error due to not taking covariance matrix into account. When theoretical benefits and application performance of the model have been considered together, it has showed itself as a preferable portfolio optimization model.\begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.16666666666666666\textwidth}P{0.16666666666666666\textwidth}P{0.18333333333333335\textwidth}P{0.1611111111111111\textwidth}P{0.17222222222222222\textwidth}}
Shares\tabcellsep Mean\tabcellsep Std.\tabcellsep Skw.\tabcellsep Kur.\\
\tabcellsep \tabcellsep Dev.\tabcellsep \tabcellsep \\
IHMAD\tabcellsep 0,038\tabcellsep 0,050\tabcellsep 1,325\tabcellsep -5,341\\
IPEKE\tabcellsep -0,045\tabcellsep 0,031\tabcellsep 1,291\tabcellsep 0,274\\
KOZAA\tabcellsep 0,049\tabcellsep 0,090\tabcellsep 5,594\tabcellsep -2,584\\
KOZAL\tabcellsep 0,089\tabcellsep 0,046\tabcellsep 3,863\tabcellsep 1,209\\
PRKM\tabcellsep 0,058\tabcellsep 0,025\tabcellsep 1,580\tabcellsep 1,158\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.85\textwidth}}
of MAD Model; in order to test the reliability of this\\
model, Tablo4 and Tablo5 will be examined.\\
Having constituted optimum portfolios with\\
different minimum returns acquired through application\end{longtable} \par
 
\caption{\label{tab_1}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_2}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.20198019801980197\textwidth}P{0.2103960396039604\textwidth}P{0.17673267326732672\textwidth}P{0.2608910891089109\textwidth}}
\tabcellsep Min.\tabcellsep Max.\tabcellsep Difference\\
IHMAD\tabcellsep 0.64\tabcellsep 9.8\tabcellsep 9.16\\
IPEKE\tabcellsep 1.45\tabcellsep 6.18\tabcellsep 4.73\\
KOZAA\tabcellsep 1.48\tabcellsep 2.32\tabcellsep 0.84\\
KOZAL\tabcellsep 12.15\tabcellsep 48\tabcellsep 35.85\\
PRKM\tabcellsep 2.34\tabcellsep 7.28\tabcellsep 4.94\end{longtable} \par
 
\caption{\label{tab_3}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{P{0.1457142857142857\textwidth}P{0.20642857142857143\textwidth}P{0.24285714285714283\textwidth}P{0.255\textwidth}}
\tabcellsep Std. Dev.\tabcellsep Return\tabcellsep Variation Coeff.\\
IHMAD\tabcellsep 0,050\tabcellsep 0,00388\tabcellsep 129\\
IPEKE\tabcellsep 0,031\tabcellsep -4,5E-05\tabcellsep -704,72\\
KOZAA\tabcellsep 0,090\tabcellsep 0,0049\tabcellsep 18,31\\
KOZAL\tabcellsep 0,046\tabcellsep 0,00089\tabcellsep 52,22\\
PRKM\tabcellsep 0,025\tabcellsep 0,0058\tabcellsep 43,725\end{longtable} \par
 
\caption{\label{tab_4}Table 6 :}\end{figure}
 			\footnote{© 2016 Global Journals Inc. (US) 1} 			\footnote{© 2016 Global Journals Inc. (US)} 			\footnote{BLinear Programming on Portfolio Optimization: Empirical Evidence from Bist Mining Industry Index} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}