Optimization of the Technology Sector Stock Portfolio during the Covid-19 Pandemic Using the Markowitz Model

Stock investment in Indonesia has grown quite rapidly in recent years and has attracted the attention of many people . When investing, analysis is needed to get future profits by minimizing stock risk. One of the ways that can be used to minimize risk is to form an optimal portfolio using the Markowitz model. The technology sector is one of the stock sectors that has progressed because of the high public interest in technology-related matters during the Covid-19 pandemic. In this study , the optimal portfolio is a portfolio containing EMTK with a weight of 63% and TFAS with a weight of 37% which produces an expected return value of 0.007488 ( 0.7488%) and a risk value of 0.035000 (3.5%).

Stock investment in Indonesia has grown quite rapidly in recent years and has attracted the attention of many people . When investing, analysis is needed to get future profits by minimizing stock risk. One of the ways that can be used to minimize risk is to form an optimal portfolio using the Markowitz model. The technology sector is one of the stock sectors that has progressed because of the high public interest in technology-related matters during the Covid-19 pandemic. In this study , the optimal portfolio is a portfolio containing EMTK with a weight of 63% and TFAS with a weight of 37% which produces an expected return value of 0.007488 ( 0.7488%) and a risk value of 0.035000 (3.5%).

INTRODUCTION
Stock investment in Indonesia in recent years has developed quite rapidly and has attracted the attention of many parties. Based on data released by PT Kustodian Sentral Efek Indonesia (Sadono et al., 2021) , although the Covid-19 pandemic is still ongoing, the performance of the Jakarta Composite Index (JCI) on December 29, 2021 has reached the level of 6,600.68, the number increased by 10 .4% of the position in December 2020, this shows that many people continue to invest amidst the uncertainty of the Covid-19 pandemic.
The Covid-19 pandemic was first discovered in Indonesia in early March 2020. This condition has changed many things, such as how to study, work, worship, socialize, and other things. One of the areas that has experienced a major impact due to the Covid-19 pandemic is the economic sector, this also affects investment activities so that investors need to adapt and rearrange optimal portfolios in investing because in the midst of the Covid-19 pandemic, the state of the stock market is changing and investors need to be careful (Nugroho et al., 2021) . There are several sectors in the stock, including the property industry sector and the technology industry sector. Based on previous research conducted by Saraswati (Saraswati, 2020) it was found that the share of the property industry sector experienced a high decline due to low public interest in buying property during the Covid-19 pandemic, in contrast to the technology industry sector which experienced rapid progress due to the high interest in the needs of the public. related to technology during the pandemic due to restrictions on activities.
In investing, analysis is needed to obtain future profits, because there are risks that cannot be eliminated but can be minimized. The risk in investing can be minimized by forming a portfolio that maximizes profits and minimizes risk or in other words provides optimal results (State et al., 2020) . Determining the optimal portfolio can use various models, one of which is the Markowitz model which emphasizes efforts to maximize the expected return and minimize risk to develop an optimal portfolio. The Markowitz model is also referred to as the mean-variance model, which means that when forming an optimal portfolio, a quantitative approach is used that relates the risk measured using standard deviation or variance with the expected return (NP Hartono et al., 2021) .
This study will examine the Markowitz model to optimize the stock portfolio of the technology sector on the Indonesia Stock Exchange (IDX) during the Covid-19 pandemic using data from the first year of the Covid-19 pandemic entering Indonesia, which is in the period March 2020 -February 2021. This research was conducted to help investors form an optimal portfolio so that they get maximum profit even though they invest during the Covid-19 pandemic.

Data
The data used in this study is data on closing stock prices of the technology sector listed on the Indonesia Stock Exchange from March 2020 -February 2021. The source of company data is obtained from the website www.idx.co.id which is the official website of the Indonesia Stock Exchange and price data daily closings can be accessed on the website www.finance.yahoo.com which is part of the Yahoo! which provides official stock data.

Step Analysis
The analysis in this study was carried out using the python programming language on Google Colab. Some general steps used to determine the optimal portfolio using the Markowitz model are as follows: 1. Collect closing stock price data Closing stock price data collected in the form of daily data for each technology sector stock listed on the Indonesia Stock Exchange and can be accessed through the www.finance.yahoo.com page for the period March 2020 -February 2021.

Calculating the return of each stock
Stock return is one of the things that motivates investors to invest because from the return investors can see the results of their investments. The formula used to calculate stock returns is (Assof et al., 2022) : where,

Calculating the expected return of each stock
Expected Return is the level of return or profit expected by investors on each company's stock in the future. The calculation of expected return is useful for making investment decisions. The expected return value is obtained by the following formula (Muis, 2008) :

Calculating the risk of each stock
Stock risk is a measure of spread that aims to determine the possibility of deviations between returns and expected returns. Risk is calculated for each stock that has a positive expected return value using the standard deviation (  ) (Lestari and Erdiana, 2021) . The formula used to calculate the risk of each stock is as follows (Country et al., 2020) : Determining the stocks that are candidates for the portfolio Determination of candidate portfolios is done by taking into account the expected return value. If the expected return is positive and not equal to zero, then the stock is included in the candidate portfolio. 6. Determine the number of portfolios that can be formed Determination of the number of portfolios that can be formed using the following combination calculations (A, 2017) :

Calculating the covariance between stocks in a portfolio
Covariance is a measurement to show the direction of movement of two variables. A negative covariance value of a portfolio means that an increase in the return of one stock is influenced by a decrease in the other stock, so this shows that both stocks can be diversified. The formula used to calculate the covariance value is as follows (Assof et al., 2022) :

Calculating the correlation coefficient between stocks
Correlation is useful to show the relationship of a stock with other stocks in the portfolio. If two stocks have returns with a correlation coefficient value of +1 (perfectly positive), then all risks cannot be diversified, whereas if two stocks have returns with a correlation coefficient value of -1 (perfect negative), then all risks can be diversified. To calculate the correlation coefficient, the following formula is used (Muthohiroh et al., 2021) :

Define an efficient portfolio
An efficient portfolio is determined based on the calculation of covariance and correlation coefficient, if the value of covariance and correlation coefficient of a portfolio is negative, it means that both stocks can be diversified and included in an efficient portfolio. 10. Determine the proportion of funds per share Determination of the proportion of funds to be given to each stock in the selected efficient portfolio is calculated using the Lagrange method with the following steps (Afriana et al., 2017) : The Lagrange function L(W)is derived from the first derivative L ′ (W) = 0,so that the value wcan be determined by the following formula: In this study, each portfolio contains two stocks, then the portfolio variance based on the matrix (10) can be expressed as follows, From the formula obtained, the portfolio risk can be calculated by finding the standard deviation of the portfolio by rooting the variance of the portfolio. So the formula to get the risk value of a portfolio of two stocks is as follows, where, = p  portfolio risk/standard deviation.

Determining the optimal portfolio
The optimal portfolio is determined based on the relationship between the expected return and risk of each portfolio by calculating the value of the

Draw a conclusion
The conclusion is drawn by presenting the selected optimal portfolio data in the form of the proportion of funds per share in the portfolio as well as the return and portfolio risk values obtained from several efficient portfolios that have been formed.

Research data
The data used in this study is based on data on closing stock prices of the

Calculating the Return and Expected Return of Each Share
After the closing price of the shares is obtained, the return value of each share can be calculated using equation (1). Furthermore, if the return value has been obtained, then the expected return for each stock can be calculated based on equation (2). The results of the calculation of the expected return of each stock during the observation period are as follows:  Table 2. it is shown that the highest stock return is TFAS stock with a value of 0.008277, while the stock with the lowest return is KREN stock. with a value of -0.005924.

Calculating the Risk of Each Share
In calculating the risk or standard deviation of each stock, (σ i ) it is necessary to first know the return value and expected return of each stock. Calculation of risk is based on equation (3). The results obtained from the calculation of the risk value of each share are as follows: The results of the preparation of the 78 portfolios formed are:

Calculating Covariance Between Stocks in Portfolio
The next step is to calculate the covariance value between stocks in the formed portfolio. The calculation of covariance is based on equation (5), which aims to show the direction of movement of the two stocks in the portfolio. Some of the results of the covariance between stocks can be seen as follows: The covariance value can be in the form of positive or negative numbers, however, the expected condition is that the covariance is negative, because if it is negative then the two stocks in the portfolio move in opposite directions and can be diversified or combined.

Calculating the Correlation Coefficient Between Stocks
Correlation is useful for showing the relationship of a stock with other stocks in a portfolio. The expected condition in the correlation is the same as the covariance, i.e. the result is negative, because if the correlation coefficient is negative, then all the risks can be diversified. Some of the results of the calculation of the correlation coefficient of each stock in the portfolio based on equation (6) are as follows: Determination of an efficient portfolio is based on the results of the calculation of covariance and correlation coefficient of each portfolio that has been formed. Portfolios that have a covariance value and a negative correlation coefficient are included in the efficient portfolio. Based on the value of covariance and correlation coefficient, there are 28 portfolios that include efficient portfolios, namely: Determination of the proportion of funds for each stock in an efficient portfolio is carried out to produce an optimal portfolio. The proportion of funds for each share in the portfolio is determined based on the Langrange method in equation (7). The results of the calculation of the proportion of funds per share in each portfolio are: ). While the calculation of portfolio risk (σ p )is based on the Markowitz model in equation (8) which aims to minimize portfolio variance. the expected return and risk values of each efficient portfolio formed are as follows: Table 9. Expected Return and Portfolio Risk

Determining the Optimal Portfolio
The optimal portfolio is determined based on the relationship between expected return and risk for each portfolio formed by determining the value of the Coefficient of Variation (CV) according to equation (9). The portfolio with the smallest Coefficient of Variation (CV) is the best portfolio. The results obtained for the Coefficient of Variation (CV) portfolio are: Table 10. Coefficient of Variation (CV) portfolio The relationship between the expected return and risk of each portfolio adjusted for the smallest portfolio Coefficient of Variation (CV) value can be expressed in the form of a graph made using python, which produces the following graph:  Based on Figure 1. it can be seen that the optimal portfolio is portfolio 15 because it produces the smallest portfolio Coefficient of Variation (CV) value, which is 4.674268. Portfolio 15 contains 63% EMTK (Elang Mahkota Teknologi Tbk.) shares and 37% TFAS (Telefast Indonesia Tbk.) shares with the highest expected return value compared to other portfolios, namely 0.007488 (0.7488%) and the risk value is not the smallest value compared to other portfolios. the other portfolio is 0.035000 (3.5%).

CONCLUSIONS AND RECOMMENDATIONS
The results showed that of the 28 available efficient portfolios, it was found that the optimal portfolio is a portfolio containing EMTK and TFAS shares with the proportion of funds for each share is 63% : 37%. In other words, investors must invest in EMTK (Elang Mahkota Teknologi Tbk.) companies with a share of 63% and in TFAS companies (Telefast Indonesia Tbk.) with a share of 37%. The expected return and risk resulting from the optimal portfolio formed by the Markowitz model are 0.007488 (0.7488%) and 0.035000 (3.5%). The risk value generated from the portfolio has a lower value than the risk value of individual stocks, where EMTK shares are worth 0.044701 (4.4701%) and TFAS shares are worth 0.059440 (5.9440%), so that with the diversification between EMTK and TFAS stocks, the risk value has been minimized. .