Optimal Portfolio Using Roy’s Safety-First Method on Primary Consumer Goods Sector Stocks

Before carrying out investment activities, investors need to form an optimal investment portfolio. This study aims to form an optimal portfolio in primary consumer goods sector stocks that sell the basic needs of the community so that stocks in the sector tend to be stable. The method used in forming the optimal portfolio is Roy's Safety-first method. The portfolio formed produces 6 combinations of stocks consisting of WIIM, DSNG, MRAT, CAMP, SIMP, and MBTO stocks respectively with a proportion of funds of 44.05%, 16.38%, 18.61%, 15.06%, 4.32%, and 1.59% with an expected return portfolio of 3.10% and a portfolio risk of 1.65%.


Introduction
Investment is the allocation of a certain amount of money or resources in the present with the expectation of obtaining returns in the future.Investments can be made in various types of assets, either in the form of physical assets such as land, gold, property, or financial assets through capital markets such as stocks, bonds, or mutual funds (Vraneš et al., 1996).
Nowadays, investment has begun to be favoured and practised by the public to protect the value of money from inflation.One of the popular investments is investment through the capital market, such as stocks.Stocks provide high returns with high risks as well.Stocks are proof of capital or fund ownership in a company (Castrén, 2019).
The goal of investors investing in stocks is to make a profit.However, in addition to providing profits, stock investment also has risks so investors need to understand the characteristics of assets to maximize profits and minimize risks.To reduce investment risk, investors can diversify, namely allocating capital to different assets through the formation of a stock portfolio (Martínez-Nieto et al., 2021).
A stock portfolio is a collection of stocks invested by an investor with the aim of maximizing return and minimizing risk.Investors need to determine the weight or proportion of funds to be allocated to each stock in the portfolio.To achieve maximum profit, the formation of an optimal portfolio is required.The optimal portfolio is a portfolio that provides the highest rate of return for the lowest level of risk (Zhang et al., 2020).
The optimal portfolio can be formed with Roy's Safety-first method.This method prioritises safety and focuses on downside risk Downside risk is the financial risk associated with losing or earning lower returns than expected.
Investors can invest in various types of sectors listed on the Indonesia Stock Exchange (IDX).One of the sectors listed on the IDX is the primary consumer goods sector which sells the basic needs of the community so that under any circumstances these products are always needed.Therefore, stocks in the primary consumer goods sector tend to be stable (Herwany et al., 2021).

Investment
Investment is the allocation of a certain amount of money or resources in the present with the expectation of obtaining returns in the future.Things that need to be considered in making an investment are return and risk.Return is the profit obtained through stock ownership and risk is the difference between the actual return and the expected return (Hidayatullah and Manda, 2021).

Stock
Stocks are proof of capital ownership in a company.Stocks are classified into several sectors, one of which is the primary consumer goods sector.Primary consumer goods sector stocks are sectors that provide primary needs, such as food, beverages, and household products.Population density in Indonesia continues to increase every year.This shows that the need for primary goods will continue to grow and be sustainable so that the shares of this sector tend to be stable (Ferdinanda, & Gantino, 2021).

Stock Return
Investors invest in stocks to get profits or investment returns.Stock return consists of realized return, which is the return that has occurred, and expected return, which is the return that investors expect to get in the future (Sholehah et al., 2020).Realized return and expected return of stocks are calculated using equations ( 1) and ( 2). (1) where, : return of stock at time , : price of stock at time , : price of stock at time , : expected return.

Stock Risk
Stock risk is the difference between actual return and expected return.Stock risk can be calculated by standard deviation using equation (3).

Portfolio
An investment portfolio is a collection of two or more investment instruments to achieve maximum return with a minimum level of risk.The best portfolio is the optimal portfolio.In forming an optimal portfolio, diversification is carried out to reduce investment risk.Diversification is allocating different assets.If a stock experiences a decline in performance, then other stocks that are in stable conditions can help reduce the risk of loss (Anastasia and Chrestella, 2021).

Portfolio Expected Return and Portfolio Risk
Portfolio expected return and portfolio risk can be calculated by equations ( 4) and (5).In forming the optimal portfolio, it is necessary to calculate the portfolio expected return and portfolio risk.Portfolio expected return and portfolio risk can be calculated by equations ( 4) and ( 5).

Roy's Safety-first Method
The method used to form the optimal portfolio is the Safety-first method or the method that prioritizes safety.This method is used by investors who tend to avoid risk or risk averse because this method prioritizes capital safety over potential profits.Roy's safety-first method is one of the methods in the Safety-first method which aims to minimize risk so that the expected return is not smaller than the minimum return limit (Manurung et al., 2024).Roy's safety-first method is expressed in equation ( 6).
(6) where, : expected return, : minimum return limit.Equation ( 6) has not considered the risk factor.Therefore, it is necessary to add risk factors in portfolio formation by assuming normally distributed returns so that is expressed as an average value and risk is expressed as a standard deviation.Then, normalisation is performed on the variables and using the normalization technique.Therefore, equation ( 6) can be expressed as equation ( 7). ( 7) where, : value of .The objective of Roy's Safety-first method is to minimize the risk that the expected return of the stock is smaller than the minimum return limit so that the value of must be negative.For negative values of , we can use equation ( 8). ( 8) Equation ( 8) is equivalent to equation ( 9). ( 9) The result of the comparison or ratio between the difference between and against can be written with the symbol as in equation ( 10).
(10) where, : roy ratio.Stocks selected into the portfolio are those that have a positive value because investors do not want a negative return or loss.After the portfolio is formed, then determine the stock weight using equation (11).

Jarque-Bera Normality Test
In determining stock weights with Roy's Safety-first method, stock returns must be normally distributed.Therefore, the normality test of stock returns is carried out using the Jarque-Bera test.The Jarque-Bera test is expressed by equation ( 12).
where, : x value to be tested, ̅ : the mean to be tested, Return is normally distributed if: where, : Jarque-Bera probability, : significance level which is 5%.

Materials
The object used in this study is the closing price of monthly shares in the primary consumer goods sector (IDXNONCYC) listed on the Indonesia Stock Exchange (IDX) for the period January 2020 to December 2023.

Results and Discussion
The first step in forming an optimal portfolio is to calculate the return value and expected return of each stock using equations ( 1) and ( 2).The results of the calculation of the expected return can be seen in Table 1.In forming a stock portfolio using Roy's Safety-first method, stock return data must be normally distributed.Therefore, a normality test is carried out on stock return data using the Jarque-Bera test From the Jarque-bera test conducted, the Jarque-Bera probability value is obtained in Table 2.  3).The results of the calculation of the risk of each stock can be seen in Table 3.In Table 3, it can be seen that the biggest stock risk is WIIM stock, which is 0.24954.The greater the risk, the greater the potential profit or loss received by investors.
The optimal portfolio formation uses Roy's Safety-first method.This method is one of the criteria in the Safetyfirst method.To determine the stocks selected into the portfolio can use equation (10).The selected stocks are stocks that have a positive value.The value or the minimum return limit value used in the calculation is the risk-free rate of return obtained from the BI 7 Days Repo Rate for the period January 2020 to December 2023.The value used is 0.00366.Optimal portfolio formation using Roy's Safety-first method can be seen in Table 4.Stocks selected into the portfolio are stocks that have a positive value because investors do not expect negative returns or losses.In Table 4, it can be seen that stocks that have a positive value and are selected into the optimal portfolio are CAMP, DSNG, MBTO, MRAT, SIMP, and WIIM stocks.
After the portfolio is formed, the weight of each stock is calculated using equation ( 11).The weight of each stock can be seen in Table 5.In Table 5, the weight of each stock is obtained, for WIIM stocks of 44.05%, DSNG of 16.38%, MRAT of 18.61%, CAMP of 15.06%, SIMP of 4.32%, and MBTO of 1.59%.
The optimal portfolio has been formed using Roy's Safety-first method.Next, the calculation of portfolio expected return and portfolio risk is carried out.Portfolio expected return and return portfolio can be seen in Table 6.In Table 6, the portfolio expected return is 3.10% and the portfolio risk is 1.65%.

Conclussion
Based on the results of the research that has been done, it is concluded that the optimal stock portfolio in primary consumer goods sector stocks formed using Roy's Safety-first method produces 6 stock combinations consisting of WIIM shares with a share weight of 44.05%, DSNG 16.38%, MRAT 18.61%, CAMP 15.06%, SIMP 4.32%, and MBTO 1.59% with an portfolio expected return value of 3.10% and a portfolio risk of 1.65%.

Table 1 :
Expected return of stocks value Stocks that have negative expected return values are not included in further calculations because these stocks do not provide benefits to investors.

Table 2 :
Jarque-bera normality test Based on Table 2, there are 9 stocks that are normally distributed, namely ANJT, CAMP, DSNG, ICBP, JPFA, MBTO, MRAT, SIMP, and WIIM.Stocks that are not normally distributed are not included in further calculations.Next, calculate the stock risk using equation (

Table 3 :
Stock risk

Table 4 :
Optimal portfolio formation using Roy's Safety-first method

Table 5 :
Weight of each stock

Table 6 :
Portfolio expected return and return portfolio