Optimizing Risk of Trading Stocks Using Markowitz Portfolio (A Case Study of Several Stocks in Indonesia)

Optimizing Risk of Trading Stocks Using Markowitz Portfolio (A Case Study of Several Stocks In Indonesia)

Ansari Saleh Ahmar1 & Andi Nurani Mangkawani Arifin2

1 Department of Statistics, Universitas Negeri Makassar, Makassar, Indonesia [email protected]

2 AHMAR Institute, Makassar, Indonesia [email protected] ______ABSTRACT— Stock trading in Indonesia and in the world which is sometimes up and sometimes down to make an investor should think hard to be able to obtain maximum profit and minimal risk. Investors can reduce risk by diversifying investment. One of diversify investment is the portfolio. Portfolio theory formed with the assumption that investors can appropriately in choosing an asset in a portfolio with the objective of maximising expected return on the particular level of risk. In this paper, the author tries to present how to determine optimal portfolio using Markowitz Portfolio Model. Markowitz portfolio theory emphasis on maximising return expectation (mean) and minimise risk (variance) to select and develop the optimal portfolio. In the case study conducted in 4 stocks, namely that Astra Agro Lestari Tbk. (ASTRA), Bank Mandiri (Persero) Tbk. (MANDIRI), Bank Negara Indonesia (Persero) (BNI), Bank Central Asia Tbk (BCA). Result of this study is obtained the optimal portfolio is the portfolio on α= 0,9 i.e. (19,52% for ASTRA; 8,97% for MANDIRI; 54,48% for BCA; and 17,03% for BNI).

Keywords— Markowitz, Portfolio, Risk, Return. ______

1. INTRODUCTION Stock trading in Indonesia and in the world which is sometimes up and sometimes down to make an investor should think hard to be able to obtain maximum profit and minimal risk. Composite stock price index (CSPI) in Indonesia Stock Exchange (IDX) on Thursday closed down by 34.46 points, or 0.66 percent, to 5150.48, along with foreign market players who returned to action off the stock [1]. Reza Priyambada as Head of Research NH Korindo Securities Indonesia6 says that the condition of the majority of global stock markets fell into one of the factors behind the foreign market players returned to action off the stock. And the rate of Composite Stock Price Index (CSPI) continued strength in stock trading [2]. CSPI still appreciate following a capital inflow is expected to continue. Jakarta Composite Index rose 27.78 points or 0.54 percent at 5198.735. LQ45 stock index rose 0.66 percent to 896.725. The whole green benchmark stock index. Based on this information, the investor in Indonesia requires a method to get maximum profit and minimum risk. One way that can use is portfolio Method. Portfolio theory introduced by Harry Markowitz (1952). Markowitz [3] suggest that an efficient portfolio is needed to use in investment. Markowitz's Portfolio was given portfolio value with the smallest risk for an expected return or in other words that investing with the lowest risk and provide maximum benefit for each investment. The concept called investment diversification or make investments that are not concentrated in one of investment but rather other of investment. Portfolio Markowitz have studied by many researchers for examples : Deng, Lin, & Lo (2012) discussed about Cardinality Constraints Markowitz Portfolio Optimization problem (CCMPO problem) [4]; Evstigneev, Hens, & Schenk- Hoppé (2015) talked about the minimum variance portfolio, the return-generating self-financing portfolio involved in the solution to the Markowitz optimization problem [5]; Markowitz (2014) discussed about mean-variance approximations to expected utility [6]; Dhrymes (2017) mentioned about CAPM extending the work of Markowitz in portfolio selection, and the role played by idiosyncratic risk, and the optimal composition of efficient portfolios [7]; Leung, Ng, & Wong (2012) discussed about derives explicit formulas for the estimator of the optimal portfolio return [8]; and Park & Shin (2013) talked about stock trading model using portfolio optimization on stocks which listed in KOSPI200 from January 2007 to August 2008 [9]. 2. LITERATURE REVIEW Subheadings may divide this section. It should provide a concise and precise description of the experimental results, their interpretation as well as the preliminary conclusions that can draw.

Assumptions to Markowitz Portfolio Theory [10] : 1. Investors consider each investment alternative as being represented by a probability distribution of expected returns over some holding period. 2. Investors maximize one-period expected utility, and their utility curves demonstrate the diminishing marginal utility of wealth. 3. Investors estimate risk by variability of expected returns. 4. Investors base decisions solely on expected return and risk. 5. Investors prefer higher returns to lower risk and lower risk for the same level of return. Markowitz Portfolio Theory emphasis on maximizing the return expectation (mean) and minimize the risk (variance) to select and develop the optimal portfolio [11].

Let n asset with mean return ri and covariansiσij. In mathematically, portfolio model Markowitz can write as: 1 n Minimize risk :  wi w j ij 2 j1 n n Boundary : wi ri  r ,wi  1 i1 i1 Solution Markowitz model using Multiplier Lagrange λ dan µ. And be formed Lagrange function as : n n n 1     L   wi w j ij   wi ri  r     wi 1 2 j1  i1   i1  The Lagrange function of the differential in each weight wi and be equal to zero. To illustrate the case with the three stock portfolios, we can easily generalize in n stocks. Lagrange function for three stocks : 1 w2 2  w w   w w   w w   w2 2  w w   L   1 1 1 2 12 1 3 13 2 1 21 2 2 2 3 23  2  2 2  w3w1 31  w3w2 32  w3  3    3   3    wi ri  r     wi 1  i1   i1  L function is differentiable in each weight wi: L 1  2 2 w  w  w  w  w  r   w 2 1 1 12 2 13 3 21 1 31 3 1 1 

L 1 2   13 w1  23 w2  31w1  32 w2  2 3 w3  r3   w3 2    It was known that ij ji and equation above equal to zero, is obtained: 2  1 w1   12w2   13w3  r1    0 2  21w1   2 w2   23w3  r1    0 2  31w1   32w2   3 w3  r3    0 From above, there are three equations, and two constraints, so total there are five equations. It equations can solve for five unknown variables such us, w1, w2, w3, λ and µ. The equation above can form n variable. n ijw j  r i   0; for i  1,2,..., n j1 n  wii r r i1 n  wi 1 i1 One of approximation to solute the general form equation Markowitz Model using two-fund theorem method. Two-fund theorem method can illustrate : From the Solving System Of Equation (Lagrange) obtain : n  ij w j  ri    0 , i= 1,2,...,n j1 Step by step using two-fund theorem method :

1. Determine the value of the parameter  =0 and  =1, obtained : 1 11  12  1n v1 1 1     v 1 21 22 2n 2     1  n1 nn vn 1 1 1 v1 11  12  1n 1 1  v    1 2  21 22 2n    1  vn n1 nn 1 2. Determine the value of the parameter =1 and =0, obtained :

2 11  12  1n v1 r 1   2      v r 21 22 2n 2   2       2   n1 nn vn r n  2 1 v1 11  12  1n  r 1  2     v    r 2  21 22 2n   2      2     vn n1 nn  r n  3. From the result of the variance values, returns, and covariance calculated from the data, obtain v1, v2. Furthermore, the value of v1dan v2normalized to obtain the same amount of equal weight one, with the formula : 1 2 v j v j 1 and 2 . w j  n w j  n 1 2  v j v j j1 j1 4. w values will use as portfolio weight. All linear combinations of the two weights( w1and w2), i.e. (αw1 + (1 – α)w2) is the efficient portfolio, with 0 ≤ α ≤ 1. The optimal portfolio can select from a collection of efficient portfolios following the hope of profit and risk of each. 3. RESEARCH METHOD In this case, I will be tested 4 (fours) stocks in the Indonesian Stock which are listed in LQ45, i.e., Astra Agro Lestari Tbk., Bank Mandiri (Persero) Tbk., Bank Negara Indonesia (Persero), Bank Central Asia Tbk. Data in the study obtained from Yahoo Finance in the period 1 January 2014 – 11 August 2014 (160 days). From the stock price data, obtained the average return, average variance, and variance-covariance matrix as follows: ASTRA MANDIRI BCA BNI Mean -0,0002 -0,00178 -0,00116 -0,00162 Variance 0,000419 0,000333 0,000189 0,000305

Variance-Covariance Matrix ASTRA MANDIRI BCA BNI ASTRA 0,000419 0,000076 0,000065 0,000075 MANDIRI 0,000076 0,000333 0,000141 0,000222 BCA 0,000065 0,000141 0,000189 0,000129 BNI 0,000075 0,000222 0,000129 0,000305

Variance CovarianceInvers Matriks ASTRA MANDIRI BCA BNI ASTRA 2555,695 -160,436 -571,8645 -270,953 MANDIRI -160,436 6474,886 -2213,049 -3737,73 BCA -571,8645 -2213,049 8334,953 -1769,04

BNI -270,9531 -3737,726 -1769,037 6809,852

From data above, obtained value of v1, v2, w1, and w2 :

v1 v2 w1 w2 ASTRA 1552,442 0,873057 0,2307 -0,12463 MANDIRI 363,675 -2,87875 0,054044 0,410946 BCA 3781,002 -2,73737 0,561875 0,390763 BNI 1032,136 -2,26212 0,15338 0,322921

1 2 From the value of w , w obtained will form an efficient portfolio weight (weff) with using the formula: 1 2 weff = αw +(1 – α )w . Here is presented value of weffwith various value of α :

w1 0,2307 0,054044 0,561875 0,15338 w2 -0,12463 0,410946 0,390763 0,322921 alpha ASTRA MANDIRI BCA BNI VAR 0,1 -0,0891 0,375256 0,407875 0,305967 0,000267 0,2 -0,05356 0,339565 0,424986 0,289013 0,00025 0,3 -0,01803 0,303875 0,442097 0,272059 0,000236 0,4 0,017502 0,268185 0,459208 0,255105 0,000223 0,5 0,053035 0,232495 0,476319 0,238151 0,000211 0,6 0,088568 0,196805 0,493431 0,221197 0,000202 0,7 0,124101 0,161114 0,510542 0,204243 0,000194 0,8 0,159634 0,125424 0,527653 0,187289 0,000187 0,9 0,195167 0,089734 0,544764 0,170335 0,000182

By holding the principle of minimizing the risk (variance), the optimal portfolio is the portfolio on α =0,9 i.e. (19,52%; 8,97%; 54,48%; 17,03%). So, if an investor have initial capital = Rp10.000.000,-, portfolio can be chosen as follows :

SEKURITAS Portfolio Sum ASTRA 19,52 Rp1.952.000 MANDIRI 8,97 Rp897.000 BCA 54,48 Rp5.448.000 BNI 17,03 Rp1.703.000 Total Rp10.000.000

4. CONCLUSION Markowitz Portfolio Theory emphasis on maximizing the return expectation (mean) and minimize the risk (variance) to select and develop the optimal portfolio. In the case study conducted in 4 securities, namely that Astra Agro Lestari Tbk., Bank Mandiri (Persero) Tbk., Bank Negara Indonesia (Persero), Bank Central Asia Tbk. obtained the optimal portfolio is the portfolio on α= 0,9 i.e. (19,52%; 8,97%; 54,48%; 17,03%).

5. REFERENCES

[1] Aditia Noviansyah, “IHSG di Bursa Efek Indonesia Turun Sebesar 34,46 poin,” 2015. [Online]. Available: https://bisnis.tempo.co/read/news/2015/05/07/088664451/ihsg-di-bursa-efek-indonesia-turun-sebesar-34-46-poin. [Accessed: 02-Dec-2016].

[2] Arthur Gideon, “IHSG Dibuka Menguat, Sempat Sentuh Level 5.201,59,” 2016. [Online]. Available: http://bisnis.liputan6.com/read/2556763/ihsg-dibuka-menguat-sempat-sentuh-level-520159. [Accessed: 02-Dec- 2016]. [3] H. Markowitz, “Portfolio Selection,” J. Finance, vol. 7, no. 1, pp. 77–91, 1952. [4] G.-F. Deng, W.-T. Lin, and C.-C. Lo, “Markowitz-based portfolio selection with cardinality constraints using improved particle swarm optimization,” Expert Syst. Appl., vol. 39, no. 4, pp. 4558–4566, 2012. [5] I. V. Evstigneev, T. Hens, and K. R. Schenk-Hoppé, “Solution to the Markowitz Optimization Problem,” in Mathematical Financial Economics, Springer International Publishing, 2015, pp. 19–25. [6] H. Markowitz, “Mean–variance approximations to expected utility,” Eur. J. Oper. Res., vol. 234, no. 2, pp. 346– 355, 2014. [7] P. J. Dhrymes, “Portfolio Theory: Origins, Markowitz and CAPM Based Selection,” in Portfolio Construction, Measurement, and Efficiency, Cham: Springer International Publishing, 2017, pp. 39–48. [8] P.-L. Leung, H.-Y. Ng, and W.-K. Wong, “An improved estimation to make Markowitz’s portfolio optimization theory users friendly and estimation accurate with application on the US stock market investment,” Eur. J. Oper. Res., vol. 222, no. 1, pp. 85–95, 2012. [9] K. Park and H. Shin, “Stock Trading Model using Portfolio Optimization and Forecasting Stock Price Movement,” J. Korean Inst. Ind. Eng., vol. 39, no. 6, pp. 535–545, Dec. 2013. [10] A. McKay, “5 Assumptions of the Markowitz Portfolio Theory,” 2012. [Online]. Available: https://cfacuecards.wordpress.com/2012/06/10/5-assumptions-to-markowitz-portfolio-theory/. [Accessed: 02- Dec-2016]. [11] Abdurakhman, Manajemen Investasi. Yogyakarta: Universitas Gadjah Mada, 2011.