Optimal Portfolio Using Single Index Model (SIM) For Health Sector Stocks

Investment is one of the fund management activities with the aim of obtaining future profits. In addition to profits, investors also need to consider the risks that will be faced by diversifying. Diversification is done by forming an optimal portfolio. This research aims to determine the proportion of stocks in the optimal portfolio and calculate the expected return and risk value of the optimal portfolio. The object used to form the optimal portfolio is health sector stock group for the period January 2020 - December 2022. The method used to form the optimal portfolio is Single Index Model (SIM). The results showed that there were 6 combinations of health sector stock in the optimal portfolio, such as IRRA, PRDA, SAME, SILO, MERK, and HEAL stocks of 8.94%, 9.24%, 9.34%, 11.92%, 27.15%, and 33.41% respectively with expected return of 2.68% and a risk value of 1.85%.


Introduction
Indonesia's economy after the Covid-19 pandemic has increased.One of the factors driving economic growth in Indonesia is investment.Investment is the commitment of a number of funds or other resources invested at this time to obtain a number of benefits in the future (Tandelilin, 2010).Stock is one of the most popular investments for investors.The Indonesia Stock Exchange (IDX) is one of the places where stock transactions in Indonesia.There are many sectoral indices available on the IDX.The performance of sectoral stocks for the 2018-2022 period based on stocks closing prices is shown in Table 1.
Based on Table 1, there is positive and negative growth from each sectoral stock on the IDX.Health sector is one of the sectors that has good index growth.In 2020, when other sector stock prices decreased due to the Covid-19 pandemic, health sector stocks increased compared to other stocks by 17.8%, indicating that companies in the health sector are always needed by the public in various circumstances.So that health sector stocks can be an option for investors in investing in stocks.
An investor who buys stocks has the main goal to obtain returns in the future.Investors must take into account not only the potential gains but also the potential risks that they may encounter.Return and risk have a strong relationship, the greater the expected return, the greater the risk faced (Hidayat et al., 2022).An investor can minimize the risk by diversifying.Diversification is the formation of a portfolio that contains a combination of several stocks.Diversification is done to minimize risk without reducing returns by forming an optimal portfolio.This research aims to form an optimal portfolio of health sector stocks using single index model (SIM).The closing price of stocks of each month from January 2020 to December 2022 serves as the historical data employed in the process of portfolio formation.After that, calculate the expected return and risk value of the portfolio that has been formed.

Portfolio
A portfolio is formed by combining assets that are anticipated to yield a return, but also entail multiple risks (Alkindi et al., 2023).However, these risks can be mitigated by distributing them among different assets.Portfolios are formed to reduce risk without reducing returns by diversifying.With diversification, an investor can form an optimal portfolio.
The first step to forming an optimal portfolio is form an efficient portfolio first.An efficient portfolio is characterized by having the highest anticipated yield relative to portfolios of equal risk, or alternatively, by maintaining a portfolio with low risk compared to portfolios that possess the same return.(Hidayat et al., 2022).From the group of efficient portfolios, the best one can be selected to become the optimal portfolio.

Stock
Stock return is the profit obtained in investing in stocks.Furthermore, the stock expected return is the value of profit expected by investors in the future.The return and expected return value are written as follows (Qin, 2023): (1) : Stock return, : Stock price of period , : Stock price of period ( ), ( ) : Stock expected return.
Standard deviation is a value to measure the level of risk that will be obtained when making an investment.The stock risk value are written as follows (Zhang, 2022): : Stock risk.

Kolmogorov-Smirnov Normality Test
Stock return data is assumed to distribute normally with hypothesis (Awan and Wang, 2022): : return data follows a normal distribution, : return data is not follows normal distribution, with significance level: ( ) : Portfolio expected return, : Portfolio risk.

Materials
The object used for this research is the closing price of the health sector group stocks (IDXHEALTH) listed on the Indonesia Stock Exchange (IDX).The data used is secondary data and time series data.There are 18 stocks listed in the health sector group stocks during the period January 2020 -December 2022).

Stock Selection based on Expected Return
The first step to form an optimal portfolio is select stocks that have a positive expected return value.The expected return of stocks is calculated using equation (2).

Kolmogorov-Smirnov Normality Test on Stock Return Data
After the normality test of Kolmogorov-Smirnov, normally distributed stocks are added to the next calculation.

Determining Stocks for The Optimal Portfolio Using Single Index Model (SIM)
Next, calculate the using equation ( 7) to determine the optimal portfolio stocks.After that, calculate the proportion and weight of stocks in the optimal portfolio using equations ( 9) and ( 10).

Calculation of Expected Return and Risk Value of The Optimal Portfolio
Last, calculate the expected return and risk value of the optimal portfolio using equations ( 12) and ( 13).

Stock Selection based on Expected Return
The first step to form an optimal portfolio using Single Index Model (SIM) is calculated the expected return of stocks using equation (2).Stocks added to the next calculation are stocks that have positive expected return values.Stocks that have negative expected return are assumed not to be profitable for the portfolio.Based on the previous calculation, there are 14 stocks for the Kolmogorov-Smirnov normality test.Kolmogorov-Smirnov test value is calculated using equation ( 4) and the value is compared to the value in the Kolmogorov-Smirnov table with a significance level of 5%.Next, determine the candidate portfolio stocks from the 10 normally distributed stocks.
value is used to determine the relationship between return and risk in an investment.ERB value is calculated using equation (7).4, 9 stocks are selected as optimal portfolio candidates.Only 6 stocks were selected as the optimal portfolio, which are HEAL, IRRA, MERK, PRDA, SAME, and SILO.Proportion and weight of each stock in the optimal portfolio using the Single Index Model is calculated using equations ( 9) and (10).5, the composition for the optimal portfolio are HEAL, IRRA, MERK, PRDA, SAME, and SILO with stock weights of 33.41%, 8.94%, 27.15%, 9.24%, 9.34%, and 11.92% respectively.

Calculation of Expected Return and Risk Value of The Optimal Portfolio
After forming the optimal portfolio, calculate the expected return and risk value of optimal portfolio using Single Index Model using equations ( 12) and ( 13).6, expected return value is 2.68% and risk value is 1.85% of optimal portfolio.

Conclussion
In this research, optimal portfolio in the health sector stock group formed with the Single Index Model (SIM) contains 6 stock combinations with the composition of IRRA, PRDA, SAME, SILO, MERK, and HEAL stocks respectively of 8.94%, 9.24%, 9.34%, 11.92%, 27.15%, and 33.41%.Expected return value of the optimal portfolio that has been formed with the Single Index Model (SIM) is 2.68% with a risk value of 1.85%.

Table 1 :
Stock Price Index Growth Data

Table 2 :
Expected Return of Stocks Value

Table 4 :
Determination of Candidate Portfolio Stocks

Table 5 :
Proportions and Weights of Optimal Portfolio Stock

Table 6 :
Expected Return and Risk Value of Optimal Portfolio